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# IEEE Transactions on Computers

## Filter Results

Displaying Results 1 - 25 of 27
• ### [Front cover]

Publication Year: 1973, Page(s): c1
| PDF (462 KB)
• ### IEEE Computer Society

Publication Year: 1973, Page(s): nil1
| PDF (230 KB)
• ### [Breaker page]

Publication Year: 1973, Page(s): nil1
| PDF (230 KB)
• ### The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer

Publication Year: 1973, Page(s):805 - 812
Cited by:  Papers (332)  |  Patents (3)
| | PDF (3690 KB)

Constrained least squares estimation is a technique for solution of integral equations of the first kind. The problem of image restoration requires the solution of an integral equation of the first kind. However, application of constrained least squares estimation to image restoration requires the solution of extremely large linear systems of equations. In this paper we demonstrate that, for convo... View full abstract»

• ### Combinational and Sequential Cellular Structures

Publication Year: 1973, Page(s):813 - 823
Cited by:  Papers (11)  |  Patents (6)
| | PDF (2259 KB)

Cellular arrays composed of identical cells with uniform interconnections are presented. The basic cell is a switching device with two inputs, two outputs, and two control variables x and T. The cell structure is either purely combinational or with unit delay depending upon the control variable T. The control variable x sets up either a crossing mode'' or a bending mode'' in the cell. Thus a t... View full abstract»

• ### Logical Network Cost and Entropy

Publication Year: 1973, Page(s):823 - 826
Cited by:  Papers (21)
| | PDF (1337 KB)

A measure of the minimum cost of a logical network is important in the evaluation of such networks. A hypothesis is investigated which states that the average minimum cost depends on both the number of input variables and the entropy (entropy'' in this sense is based on the probability of a 1'' in the functions' truth table) of the function realized. Some experimental results are presented to ... View full abstract»

• ### On Minimization of Fuzzy Functions

Publication Year: 1973, Page(s):826 - 832
Cited by:  Papers (47)
| | PDF (1865 KB)

The study of fuzziness in combinational switching systems by means of a suitable fuzzy algebra is discussed. The insufficiency of the methods for simplification of fuzzy functions as presented by Lee and Chang [9] and by Siy and Chen [10] is discussed. A new technique for minimization of fuzzy functions is developed. Special properties of fuzzy functions are discussed and their relationships to tw... View full abstract»

• ### Remarks on Base Extension for Modular Arithmetic

Publication Year: 1973, Page(s):833 - 835
Cited by:  Papers (9)
| | PDF (1177 KB)

It is shown that in a residue system with only the two moduli m-1 and m+1, base extension to modulus m is possible with savings in time and sometimes in hardware over conventional techniques. It is also shown that the underlying theory generalizes to provide comparable savings when extending any system with two moduli nm-1 and nm+1 (where n may be any integer) to include the third modulus m. For s... View full abstract»

• ### Universal Test Sets for Logic Networks

Publication Year: 1973, Page(s):835 - 839
Cited by:  Papers (45)
| | PDF (2278 KB)

This paper examines the problem of finding a single universal test set that will test any of a variety of different implementations of a given switching function. It is shown that, for AND/OR networks, universal test sets may be found that detect not only all single faults but all multiple faults as well. The minimality and size of these sets are examined and their derivation for incomplete functi... View full abstract»

• ### On Performance Analysis of Retrieval Functions

Publication Year: 1973, Page(s):840 - 844
Cited by:  Papers (1)
| | PDF (1431 KB)

Threshold-type retrieval functions have been suggested by many people and used widely in clustering algorithms. In this paper, a discriminating capacity is defined for these retrieval functions. Explicit methods for computation are presented. The discriminating capacity is useful as a quantitative measure for the retrieval power of each function and can serve as a guide for selection. View full abstract»

• ### On Control Memory Minimization in Microprogrammed Digital Computers

Publication Year: 1973, Page(s):845 - 848
Cited by:  Papers (34)
| | PDF (1437 KB)

The problem of minimizing the bit dimension of control memories in microprogrammed digital computers is considered in this paper. We start essentially with the same basic formulation as that of Grasselli and Montanari [2]. However, in order to minimize the computational requirements, we start directly with the set of maximum compatibility classes of microcommands whose number is usually small, and... View full abstract»

• ### Parallel Processing in Boolean Algebra

Publication Year: 1973, Page(s):848 - 851
Cited by:  Papers (2)
| | PDF (2137 KB)

A processor called Boolean analyzer has been presented at the 1968 IFIP Congress to introduce parallel processing of Boolean expressions [1]. The present paper shows how to increase its speed many times by making its processing more parallel. The applications of the Boolean analyzer are limited to few but important problems. A typical problem of that kind: listing of all implicants of a function o... View full abstract»

• ### A Hybrid Digital/Optical Computer System

Publication Year: 1973, Page(s):852 - 858
Cited by:  Papers (18)
| | PDF (4335 KB)

A brief review of the principles of optical data processing (ODP) with its high throughput data rate and parallel processing potential is given, followed by a description of a viable on-line electron-beam addressed electrical-to-optical input transducer. A proposed on-line hybrid digital/optical two-dimensional processing system is then described. An output plane optical-to-digital interface and s... View full abstract»

• ### Pattern Recognition by Piecewise Linear Discriminant Functions

Publication Year: 1973, Page(s):859 - 862
Cited by:  Papers (8)
| | PDF (1405 KB)

A piecewise linear function is represented in terms of a set of linear functions through the use of the maximum and minimum functions. A procedure for finding piecewise linear discriminant functions for pattern recognition is described. The procedure iteratively uses the accelerated relaxation method to find every linear function in a piecewise linear function. The procedure was implemented by a F... View full abstract»

• ### Comments on "Identification of Totally Symmetric Boolean Functions

Publication Year: 1973, Page(s):863 - 864
| | PDF (403 KB)

A modification in the algorithm for the detection of totally symmetric functions as expounded by the author in an earlier note1 is presented here. The modified algorithm takes care of a limited number of functions that escape detection by the previous method. View full abstract»

• ### Comments on "The Relationship Between Multivalued Switching Algebra and Boolean Algebra Under Different Definitions of Complement

Publication Year: 1973, Page(s): 864
Cited by:  Papers (4)
| | PDF (230 KB)

This correspondence identifies the definitions of the above paper in mathematical terms. Both algebraic and logical terms are given. View full abstract»

• ### On Efficient Computation of Matrix Chain Products

Publication Year: 1973, Page(s):864 - 866
Cited by:  Papers (21)
| | PDF (608 KB)

It is pointed out that the number of scalar multiplications (additions) required to evaluate a matrix chain product depends on the sequence in which the associative law of matrix multiplication is applied. An algorithm is developed to find the optimum sequence that minimizes the number of scalar multiplications. A program is written for use on the CDC 6600 computer to implement this algorithm and ... View full abstract»

• ### A Simply Solvable Class of Simultaneous Linear Equations

Publication Year: 1973, Page(s):866 - 867
Cited by:  Papers (4)
| | PDF (414 KB)

A solution is presented for a class of a very large system of linear algebraic equations. The given set of simultaneous equations is initially transformed to yield an equivalent system of equations in triangular form whose solution is straightforward. With the present technique, the number of operations on the computer is minimized while the convergence to a unique solution is guaranteed. View full abstract»

• ### Boundary Conditions for Iteratively Realized Sequential Circuits

Publication Year: 1973, Page(s):867 - 868
| | PDF (421 KB)

A more compact and structured way of obtaining the boundary conditions for iteratively realized sequential circuits is presented. The notion of a predecessor tree'' is used. View full abstract»

• ### Some Binary Output Sequence Properties of Deterministic Autonomous Finite-State Machines with Probabilistic Initialization

Publication Year: 1973, Page(s):868 - 869
Cited by:  Papers (1)
| | PDF (429 KB)

An improved bound on the number of low-weight sequences that may be generated by L-step observable autonomous finite-state machines is derived. An upper and lower bound on the range of L for this same class of machines is also developed. View full abstract»

• ### A Nonlinear Transformation for Digital Picture Processing

Publication Year: 1973, Page(s):869 - 873
Cited by:  Papers (8)
| | PDF (1000 KB)

A nonlinear picture-to-picture transformation is introduced and compared with a straightforward linear transformation. The criterion for comparison is based on the desire to smooth noisy or textured regions while retaining edge definition. Selected examples from a large number of simulations are presented. The transformation presented is believed to be superior to many techniques presently being e... View full abstract»

• ### Remarks on the SHR-Optimal State Assignment Procedure

Publication Year: 1973, Page(s):873 - 875
Cited by:  Papers (2)
| | PDF (648 KB)

The optimal state assignment procedure by Story, Harrison, and Reinhard [1] depends computationally on the number of distinct state assignment columsn Cdi, i = 1, 2, ..., C(R). C(R) can be computed exactly by see the [Eqa] in pdf file but a bound for C(R) is shown to be C(R) = 2R-1 - 1 for R - 1 =2n. Furthermore, for this value of R a distinct state assignment set ... View full abstract»

• ### Angle Detection on Digital Curves

Publication Year: 1973, Page(s):875 - 878
Cited by:  Papers (305)
| | PDF (602 KB)

A simple parallel procedure for selecting significant curvature maxima and minima on a digital curve is described. View full abstract»

Publication Year: 1973, Page(s):878 - 879
| | PDF (323 KB)

The load on any input wire of a decoder is calculated and shown to be independent of the state of the other inputs. View full abstract»

• ### Book Reviews

Publication Year: 1973, Page(s):879 - 884
| PDF (1410 KB)

## Aims & Scope

The IEEE Transactions on Computers is a monthly publication with a wide distribution to researchers, developers, technical managers, and educators in the computer field.

Full Aims & Scope

## Meet Our Editors

Editor-in-Chief
Paolo Montuschi
Politecnico di Torino
Dipartimento di Automatica e Informatica
Corso Duca degli Abruzzi 24
10129 Torino - Italy
e-mail: pmo@computer.org


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# Lower bound on volume of minimal hypersurface contained in a unit ball with curvature bounds

I was just wondering, if I have a geodesic ball of radius one in a manifold M whose sectional curvature lies between -epsilon and epsilon for epsilon small, and the injectivity radius of my manifold is two say, and I have a SINGULAR minimal hypersurface in M passing through the centre of my unit ball, can I find a lower bound, away from zero, on the volume of the singular minimal hypersurface contained in the ball, in terms of only the dimension of the ambient manifold M? Thanks!

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I know nothing about this stuff. Is such a bound already known if M is flat? –  Deane Yang Oct 26 '09 at 14:10

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# How can I solve this infinite sum?

I calculated (with the help of Maple) that the following infinite sum is equal to the fraction on the right side.

$$\sum_{i=1}^\infty \frac{i}{\vartheta^{i}}=\frac{\vartheta}{(\vartheta-1)^2}$$

However I don't understand how to derive it correctly. I've tried numerous approaches but none of them have worked out so far. Could someone please give me a hint on how to evaluate the infinite sum above and understand the derivation?

Thanks. :)

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I suggest you to consider the geometric series $1 + x + x^2 + x^3 + \cdots = (1 - x)^{-1}$. Differentiating both sides will give you a result that leads to the given identity. –  sos440 Jun 6 '11 at 8:21
Terminological aside to OP: one doesn't "solve" infinite sums, one $\it evaluates$ them. –  Gerry Myerson Jun 6 '11 at 13:28
Thanks to all you guys :) @Gerry Myerson Thanks :P My command of English isn't that bad but math terms are seriously giving me a hard time! –  Julian Jun 6 '11 at 19:08

let $$S=\sum_{i=1}^\infty\frac{i}{\theta^i}, (\theta>1),$$ then \begin{align} S-\frac{1}{\theta}S&=\sum_{i=1}^\infty\frac{i}{\theta^i}-\sum_{i=1}^\infty\frac{i}{\theta^{i+1}}\\ &=\sum_{i=1}^\infty\frac{i}{\theta^i}-\sum_{i=2}^\infty\frac{i-1}{\theta^{i}}\\ &=\frac{1}{\theta}+\sum_{i=2}^{\infty}\frac{1}{\theta^i}\\ &=\frac{1}{\theta}+\frac{1}{\theta^2-\theta}, \end{align} which yields $$S=\frac{\theta}{(\theta-1)^2}$$

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Several good methods have been suggested. Here's one more. \eqalign{\sum{i\over\theta^i}&={1\over\theta}+{2\over\theta^2}+{3\over\theta^3}+{4\over\theta^4}+\cdots\cr&={1\over\theta}+{1\over\theta^2}+{1\over\theta^3}+{1\over\theta^4}+\cdots\cr&\qquad+{1\over\theta^2}+{1\over\theta^3}+{1\over\theta^4}+\cdots\cr&\qquad\qquad+{1\over\theta^3}+{1\over\theta^4}+\cdots\cr&\qquad\qquad\qquad+{1\over\theta^4}+\cdots\cr&={1/\theta\over1-(1/\theta)}+{1/\theta^2\over1-(1/\theta)}+{1/\theta^3\over1-(1/\theta)}+{1/\theta^4\over1-(1/\theta)}+\cdots\cr} which is a geometric series which you can sum to get the answer.

-

Hint: Consider the expectation (first moment) of the geometric distribution. Specifically, letting $1 - p = 1/\vartheta$ in the derivation of the formula ${\rm E}(Y)=(1-p)/p$ (in that link) gives exactly what you are looking for.

Elaborating. It is shown in the Wikipedia link how to derive the equality $$\sum\limits_{k = 0}^\infty {(1 - p)^k pk} = \frac{{1 - p}}{p}, \;\; 0 < p \leq 1.$$ Letting $1-p = 1/\vartheta$, so that $p=(\vartheta - 1)/\vartheta$, this gives $$\sum\limits_{k = 1}^\infty {k\frac{1}{{\vartheta ^k }}} = \frac{{1 - p}}{{p^2 }} = \frac{1}{\vartheta }\frac{{\vartheta ^2 }}{{(\vartheta - 1)^2 }} = \frac{\vartheta }{{(\vartheta - 1)^2 }}.$$ That is, $$\sum\limits_{i = 1}^\infty {\frac{i}{{\vartheta ^i }}} = \frac{\vartheta }{{(\vartheta - 1)^2 }}.$$

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# Thread: function is even, odd, or neither?

1. ## function is even, odd, or neither?

How would I go about doing that?

Let's say f(x) = 17x^322 for example....

2. Hello,
Originally Posted by realintegerz
How would I go about doing that?

Let's say f(x) = 17x^322 for example....
To say so, find $f(-x)$

Here, $f(-x)=17*(-x)^{322}=17*((-1)*x)^{322}$

Using this rule : $(ab)^c=a^c b^c$, we get

$f(-x)=17*x^{322}*(-1)^{322}=f(x)*(-1)^{322}$

Using the rule of exponents : $a^{bc}=(a^b)^c$, we can say that $(-1)^{322}=[(-1)^2]^{161}=1^{161}=1$

Thus $f(-x)=f(x)$

Got it ?

3. so if f(-x) = f(x) it is even, and if it doesnt it is odd?

how do i know when its neither...

4. Originally Posted by realintegerz
so if f(-x) = f(x) it is even, and if it doesnt it is odd?

how do i know when its neither...
NO!

Even: f(-x) = f(x).

Odd: f(-x) = - f(x).

Otherwise neither even nor odd.


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# Index

## kerygma Index

index of speeches and parables in the Gospels
(referrals to Marc are in green, Matthew in Lila, John in blue, Luke in orange)

.

# Conclusions

After my work with Luke

Like Mark, John uses the word Kingdom of God instead of Matthews Kingdom of Heaven.

Jesus talked about the Kingdom of God together with Nicodemus saying “unless one is born of water and spirit, he can’t enter into the Kingdom of God“. This seemingly contraddicts Jesus  parable desciption of the Kingdom of Heaven in Matthew where the Earth may be the Kingdom of Heaven.   If not the Kingdom of Heaven is to be intended as a Kingdom after the Resurrection day.

I consider more and more miracles that Jesus talked about. The apostles would probably not put words in Jesus that are not his. Maybe the words may not be 100% accurate as Talmidins may not remember every detail.

I will continue …

to work with these Gospels, updating the kerygma index and writing other post like posts about Jesus parables.

“if you are not astonished by the words of Jesus,
then you have not understood Jesus.”

## A pluralist agnostic seeker

Insert math as
$${}$$


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# Degeneracy and orthogonality

1. Nov 5, 2008

### quantumfireball

1. The problem statement, all variables and given/known data

Are all electronis states orthonormal?
I mean the degenerate states ie [n,l,m>states corresponding to same energy
for example can one write
[2,0,0>=a[2,1,-1>+b[2,1,0>+c[2,1,+1>

2. Relevant equations

3. The attempt at a solution

for example can one write
[2,0,0>=a[2,1,-1>+b[2,1,0>+c[2,1,+1>???

2. Nov 6, 2008

### borgwal

$$<n',l',m'|n,l,m>=\delta_{n'n}\delta_{l'l}\delta_{m'm}$$

So you cannot write |2,0,0> as a superposition of the three l=1 states.

3. Nov 6, 2008

### quantumfireball

Fine but how do you go about the proof???


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# What types of Diophantine equations are unsolvable?

### What types of Diophantine equations are unsolvable?

Question: What types of Diophantine equations are unsolvable over the set of positive rational numbers?

Guest

### Re: What types of Diophantine equations are unsolvable?

FYI: 'From the result of Hilbert's tenth problem, does it follow that there exist infinitely numbers of equation...',

https://math.stackexchange.com/questions/118662/from-the-result-of-hilberts-tenth-problem-does-it-follow-that-there-exist-infi?rq=1.
Guest

### Re: What types of Diophantine equations are unsolvable?

Good Reference Book: 'An Introduction to Diophantine Equations' by Prof. T. Andreescu et al.

https://www.springer.com/gp/book/9780817645489.
Guest

### Re: What types of Diophantine equations are unsolvable?

Guest wrote:Question: What types of Diophantine equations are unsolvable over the set of positive rational numbers?

I had a very bad experience with https://math.stackexchange.com, and I strongly do not recommend it to anyone.

Dave.
Guest

### Re: What types of Diophantine equations are unsolvable?

A General Example:

Can we generally determine the solvability or the unsolvability for the following Diophantine equation,

$$x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} = c \in \mathbb{Z_{+}}$$?

Remark: $$x, y \in \mathbb{Z}$$.

Hmm. The value of c is important!

A Specific Example:

Can we solve the following Diophantine equation,

$$x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123$$?
Guest

### Re: What types of Diophantine equations are unsolvable?

Guest wrote:A General Example:

Can we generally determine the solvability or the unsolvability for the following Diophantine equation,

$$x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} = c \in \mathbb{Z_{+}}$$?

Remark: $$x, y \in \mathbb{Z}$$.

Hmm. The value of c is important!

A Specific Example:

Can we solve the following Diophantine equation,

$$x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123$$?

If we are unhappy with $$x, y \in \mathbb{Z}$$, then we could choose $$x, y \in \mathbb{Q}$$.

It is still a fair game.
Attachments
Silhouette of Sherlock Holmes.jpg (5.13 KiB) Viewed 217 times
Guest

### Re: What types of Diophantine equations are unsolvable?

Remark: We need to develop a theory that works well. Good luck!
Guest

### Re: What types of Diophantine equations are unsolvable?

Remark: The simplest case occurs when $$x = y$$. Right?

We should assume $$x \ne y$$.
Guest

### Re: What types of Diophantine equations are unsolvable?

Some Food for Thought: 'NP-Complete decision problems for binary quadratics', by Profs., K. L. Manders and L. Adleman,

https://www.sciencedirect.com/science/article/pii/0022000078900442?via%3Dihub.
Guest

### Re: What types of Diophantine equations are unsolvable?

Wow! A single number has so much meaning/richness attached to it.

And the study of numbers tells us or reveals to us the truth about the nature of existence/things (energy, etc.)

Why? (A Hint: https://www.math10.com/forum/viewtopic.php?f=63&t=8579).

But we must be energetic, bold, resourceful, and imaginative to know their hidden/subtle/powerful secrets.

The great Gauss was right to say that number theory is the queen of the queens (the sciences including its greatest science, mathematics). Amen!
Guest

### Re: What types of Diophantine equations are unsolvable?

"Simple seeks simplest (best) solution."

A Specific Example:

Can we solve the following Diophantine equation,

$$x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123$$ for some $$x \ne y \in \mathbb{Z}$$?

Let s = $$\sum_{i=1}^{7} \beta^{i} + 1$$ for some $$\beta \in \mathbb{Q}$$.

Question 1: Does $$s | 234,567,890,123$$ where $$\beta < 0$$ or $$0 <\beta < 1$$ or $$\beta >1$$?

Question 2: Does $$y = (\frac{234,567,890,123}{s})^{ \frac{1}{7} } \in \mathbb{Z}$$?

Question 3: Does $$x = \beta y \in \mathbb{Z}$$?

If our questions, one, two, and three, have affirmative answers, then we can solve our equation, otherwise, we cannot solve it. Right?
Guest

### Re: What types of Diophantine equations are unsolvable?

An Update:

Guest wrote:"Simple seeks simplest (best) solution."

A Specific Example:

Can we solve the following Diophantine equation,

$$x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123$$ for some $$x \ne y \in \mathbb{Z}$$?

Let s = $$\sum_{i=1}^{7} \beta^{i} + 1$$ for some $$\beta \in \mathbb{Q}$$.

Question1: Does $$y = (\frac{234,567,890,123}{s})^{ \frac{1}{7} } \in \mathbb{Z}$$ where $$\beta < 0$$ or $$0 <\beta < 1$$ or $$\beta >1$$?

Question 2: Does $$x = \beta y \in \mathbb{Z}$$?

If our questions, one and two, have affirmative answers, then we can solve our equation, otherwise, we cannot solve it. Right?
Guest

### Re: What types of Diophantine equations are unsolvable?

Guest wrote:An Update:

"Simple seeks simplest (best) solution."

A Specific Example:

Can we solve the following Diophantine equation,

$$x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123$$ for some $$x \ne y \in \mathbb{Z}$$?

Let s = $$\sum_{i=1}^{7} \beta^{i} + 1$$ for some $$\beta \in \mathbb{Q}$$.

Question 1: Does $$y = (\frac{234,567,890,123}{s})^{ \frac{1}{7} } \in \mathbb{Z}$$ where $$\beta < 0$$ or $$0 <\beta < 1$$ or $$\beta >1$$?

Question 2: Does $$x = \beta y \in \mathbb{Z}$$?

If our questions, one and two, have affirmative answers, then we can solve our equation, otherwise, we cannot solve it. Right?

Important Question: What is the algorithm that answers our question in polynomial time or less than polynomial time?
Guest

### Re: What types of Diophantine equations are unsolvable?

Hmm. Current theory (Hilbert's Tenth Problem has a negative answer.) states that we cannot generally answer our question in polynomial time or less...

We are skeptical since we had not been clever enough in our search for truth.
Guest

### Re: What types of Diophantine equations are unsolvable?

[b]Remark[\b]: We need better theory!
Guest

### Re: What types of Diophantine equations are unsolvable?

Guest wrote:Remark: We need better theory!
Guest

### Re: What types of Diophantine equations are unsolvable?

Do we choose Rabbit Hole A: $$\beta < 0$$?

Or do we choose Rabbit Hole B: $$0 < \beta < 1$$?

Or do we choose Rabbit Hole C: $$\beta > 1$$?

Hmm. We are clueless.
Guest

### Re: What types of Diophantine equations are unsolvable?

Guest wrote:Do we choose Rabbit Hole A: $$\beta < 0$$?

Or do we choose Rabbit Hole B: $$0 < \beta < 1$$?

Or do we choose Rabbit Hole C: $$\beta > 1$$?

Hmm. We are clueless.

It is not a hopeless situation. Thank God! Amen!

Keywords: Initial values, Criteria, Newton's Method (Jacobian), Convergent sequence...

Can we select initial values, $$x_{0 }, y_{0 } \in \mathbb{Z}$$, based on some criteria (unknown) that will help us to choose the best Rabbit Hole to explore?

Equation 0: $$\beta_{0 } = \frac{y_{0 }}{x_{0 }} \in \mathbb{Q}$$ ...

Relevant Reference Link: 'Are Diophantine equations in four variables solvable?'

https://www.math10.com/forum/viewtopic.php?f=63&t=7803.
Guest

### Re: What types of Diophantine equations are unsolvable?

An Update:

Do we choose Rabbit Hole A: $$\beta < -1$$?

Or do we choose Rabbit Hole B: $$-1 < \beta < 0$$?

Or do we choose Rabbit Hole C: $$0 < \beta < 1$$?

Or do we choose Rabbit Hole D: $$\beta > 1$$?

Hmm. There may be no solutions for each Rabbit Hole. Or there may be solutions for each Rabbit Hole. How many solutions? Who knows?

But we believe there are finitely many solutions (if they exist) for Rabbit Holes, A, C, and D. Why?

Therefore, the best Rabbit Hole to explore for possible solutions is Rabbit Hole B since it has the most possible solutions. Why?
Guest

### Re: What types of Diophantine equations are unsolvable?

An Update:

"Simple seeks simplest (best) solution."

A Specific Example:

Question: Can we solve the following Diophantine equation,

$$x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123$$ for some $$x \ne y \in \mathbb{Z}$$?

Let s = $$\sum_{i=1}^{7} \beta^{i} + 1$$ for some $$\beta \in \mathbb{Q}$$ such that $$-1 < \beta < 0$$.

Question 1: Does $$y = (\frac{234,567,890,123}{s})^{ \frac{1}{7} } \in \mathbb{Z}$$?

Question 2: Does $$x = \beta y \in \mathbb{Z}$$?

If our questions, one and two, have affirmative answers, then we can solve our equation, otherwise, we cannot solve it. Right?

What is the algorithm that answers our question in polynomial time or less than polynomial time?

'Are Diophantine equations in four variables solvable?'

https://www.math10.com/forum/viewtopic.php?f=63&t=7803.
Guest

Next


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# Tag Info

We can present $\mathbb{Z}_m$ in different manner. For example, bellow sets are some equivalence classes of $\mathbb{Z}_3$ $$\{0,1,2\}, \{3,4,5\},\{-3,-2,-1\}.$$ Now, about your question If you select the numbers of $[-n,n]$ module a positive integer $m$ which $m\leq 2n+1$, you can construct $\mathbb{Z}_m$: $$\mathbb{Z}_m=\{i \pmod m \mid i\in[-n,n]\}$$ ...


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# Procrastinater needs help withv Law of Cosines

1. Feb 8, 2006

### Paindealer

Ok I did this thing on law of cosines and here is what i came up with, but i am pretty sure it is wrong so help me change it to solve for cos(C)

c^2=a^2+b^2-2abcos(C)

so...

Cos(C)=____c^2____
. a^2+b^2-2ab

is that right?

if not how do i make it to equal cos(C)?
(need this by like right now cause i am a procrastinator!!! :P)

2. Feb 8, 2006

### mathmike

wouldnt cos(x) = (a^2 + b^2 - c^2) / 2ab

3. Feb 8, 2006

### Paindealer

probably... i will use that for my hw then, thanks

4. Feb 9, 2006

### HallsofIvy

Staff Emeritus
No!! You both are apparently thinking of

c2= (a2+ b2- 2ab) cos(C)
which it is NOT!! (Notice the parentheses on the right side.)
The cosine law says c2= a2+ b2- 2ab cos(C). (Notice lack of parentheses!)

The problem is not the "cosine", it is simple algebra:

Suppose the problem were c= a+ bx. How would you solve for x?

5. Feb 9, 2006

### ksinclair13

I don't see how mathmike's is wrong, but I may just be stupid.

Isn't

$$cos (C) = \frac{a^2 + b^2 - c^2}{2ab}$$

the same as

$$cos (C) = \frac{c^2 - a^2 - b^2}{-2ab}$$

?

You could take a -1 out of the numerator and then cancel out the negatives, right?


text:
Change the first if statement in the following script to an equivalent one-line if-expression. Test the resulting new script, and make sure it behaves like the original,

#!/usr/bin/env python

abbr = input ("What is the three-letter abbreviation of this course? ")

if abbr == 'DSP':

else:
print("wrong buddy...try again")

Can you achieve the same goal as in (A) without if-expression or block, but instead using only tuple notation? Explain why your solution works.

Modify the if block and the print statements that are only in the last part of the code,

if answer_status=='correct':
else:
print("wrong buddy...try again")

to write a single-line Python statement that only uses print and tuple or list notations, to perform the same task as the original print and if-block statement.

#!/usr/bin/env python

abbr = input ("What is the three-letter abbreviation of this course? ")

answer_status = 'correct' if abbr == 'DSP' else 'wrong'

else:
print("wrong buddy...try again")

#!/usr/bin/env python

abbr = input ("What is the three-letter abbreviation of this course? ")

#!/usr/bin/env python


text: thermal decomposition of calcium carbonate. Thermal decomposition of calcium carbonate. CHATTARAJ et al. Why does the limewater turn cloudy? Why is steam evolved when drops of water are added? Universal Indicator solution (HIGHLY FLAMMABLE). Test the remaining half of the mixture with Universal Indicator solution. Copyright © 2005, 2020 - OnlineMathLearning.com. Biomineralized CaCO3, which exhibits a specialized structure, is a possible source of CaO used across various applications, including CO2 capture. Thermal decomposition of calcium carbonate polymorphs The thermal analysis presents a similar behavior of the C 1 âC 3 samples heated in the 25â1000 °C temperature range. ... Introduction Calcium Carbonate, CaCO3 decomposes with heat, to produce Calcium Oxide, CaO and Carbon Dioxide, CO2. Write down your observations. Thermal decomposition of carbonated calcium phosphate apatites. Thermal Decomposition of Calcium Carbonate. This site uses cookies from Google and other third parties to deliver its services, to personalise adverts and to analyse traffic. The heat caused the chalk's structure to break down and carbon dioxide/a gas is evolved. Includes kit list and safety instructions. The decomposition of calcium carbonate. Journal of Thermal Analysis and Calorimetry, Springer Verlag, 2003, 72 (3), pp.1127-1134. Explain why the pH would be expected to have this value. This is a resource from the Practical Chemistry project, developed by the Nuffield Foundation and the Royal Society of Chemistry. Thermal decomposition of metal carbonates This experiment involves a comparison between the thermal stabilities of carbonates of reactive metals, such as sodium and potassium, and the carbonates of less reactive metals, such as lead and copper Metal carbonates decompose when heated. The powdered materials were examined in the form of pellets, which were prepared by compacting the powder to â¦ Calcium hydroxide is known â¦ This behaviour can be attri-buted to the lower particle size and greater fraction of CaCO3 molecules located on surface with regard to the bulk. It is not soluble in water. Perform what looks like alchemy with ordinary copper coins in this teacher demonstration. $$CaCO_3 (s) \rightarrow CaO(s) + CO_2(g)$$ When calcium oxide is reacted with water, calcium hydroxide is produced. CaCO3(s) CaO(s) + CO2(g) The object of this practical exercise is to determine the enthalpy change for this reaction by an indirect method based on Hess' Law. 2.11.7 recall the use of calcium carbonate to make calcium oxide [quick lime] and calcium hydroxide [slaked lime] and their use in producing cement and concrete; Option 1A: Additional industrial chemistry, Equations required for [i] conversion of calcium carbonate to calcium oxide, The equilibrium between two coloured cobalt species, Turning copper coins into ‘silver’ and ‘gold’, Using indigestion tablets to neutralise an acid. : THERMAL DECOMPOSITION OF CALCIUM CARBONATI~ 47 have no influence on the temperature of beginning and completion of decompo- sition which remains constant throughout (Table 3). Universal indicator solution (HIGHLY FLAMMABLE) – see CLEAPSS Hazcard HC032 and CLEAPSS Recipe Book RB047. Crossref Ephraim Kehat and Alexander Markin , Calcination of limestone in a tubular reactor , The Canadian Journal of Chemical Engineering , 45 , 1 , (40-45) , (2009) . The samples of (puriss, Fluka) and (puriss, Aldrich) were used â¦ This study focused on the thermal decomposition of biomineralized CaCO3, using avian eggshell. The decomposition of calcium carbonate was investigated by using a particle size with 300, 425-600, and 1180 m in thermal gravimetric analyzer (TGA). This reaction comes to equilibrium, and the equilibrium pressure of CO2 increases at higher temperature, for much the same kind of reason that the vapour pressure of water increases at higher temperature. Learn how to make fizzing bath bombs using ingredients from your kitchen cupboards. Students could be asked to carry out web research on these applications. Let the chalk cool and use tongs to move it into a boiling tube. Today, you will be studying the thermal decomposition of calcium carbonate. ... understanding of the thermal behavior of AB-type carbonate apatites comes from a possible . The thermal decomposition of calcium hydrogen carbonate Ca (HCO 3) 2 CaCO 3 + CO 2 + H 2 O [ Check the balance ] The thermal decomposition of calcium hydrogen carbonate to produce calcium carbonate, carbon dioxide and water. Write an equation for the reaction occurring. Keep an eye on less mature students who might be tempted to suck rather than blow through the filtrate. The roasting of limestone and the hydration of the quicklime formed has relevance in the manufacture of plaster and cement, and in the laboratory limewater is a common reagent for the testing of carbon dioxide. Decomposition Study of Calcium Carbonate in Cockle Shell 5 Journal of Engineering Science and Technology February 2012, Vol. A.W.D. Experiment has been Set a lump of chalk (calcium carbonate) on a gauze. What do you see? The chalk should be seen to crumble slightly, More crumbling, steam given off, evidence that mixture has become hot, Some of the solid dissolves, white suspension. model was veriï¬ed using the thermal decomposition experiment on calcium carbonate. You need to prepare a tabulated results sheet before you start your experiments. This is an experiment from the Practical Chemistry project, developed by the Nuffield Foundation and the Royal Society of Chemistry. The calcium oxide (unslaked lime) is dissolved in water to form calcium hydroxide (limewater). Numerous studies have been performed with the aim of yield-ing a model for the mechanism(s) of calcium carbonate thermal decomposition, often focusing on its kinetic analysis (e.g., Sat-terfield and Feakes 1959; Beruto and Searcy 1974; Borgwardt 1. Results indicate that the thermal conversion of calcite to calcium oxide is initiated at a slow rate and rapidly decomposed beyond the temperature of 750 °C. Read our policy. Thermal decomposition of cockle shell Thermal decomposition of carbonate usually is assumed to suit Shrinking Core Model (SCM) where it visualizes the reaction begin to occur at the outer layer of Calcium carbonate is strongly heated until it undergoes thermal decomposition to form calcium oxide and carbon dioxide. carbonate and found that the activation energy of thermal decomposition of calcium carbonate smoothly increases as the particle size increases. This set of experiments involves a variety of important reactions and types of reactions, with several references to industrial processes. Practical Chemistry activities accompany Practical Physics and Practical Biology. The thermal decomposition of calcium carbonate produces calcium oxide. Thermal decomposition is the term given to splitting up a compound by heating it. 7 ), shows a small weight loss (2%), in the 25â600 °C temperature range, corresponding to the decomposition of adsorbed species, with a main endothermic peak at 553 â¦ Freshly purchased drinking straws should be used and each student issued with their own straw. Includes kit list and safety instructions. Each activity contains comprehensive information for teachers and technicians, including full technical notes and step-by-step procedures. Filter half the mixture into the other boiling tube and, using a straw, gently blow a stream of bubbles through the filtrate. study toward decomposition rate. The experiments were test with different temperature (700, 800 and 900 C) to study the decomposition rate of CaCO 3. problem solver below to practice various math topics. ï¿¿10.1023/A:1025036214044ï¿¿. Bubbling carbon dioxide through this forms a milky suspension of calcium carbonate. Calcium carbonate decomposes when heated to give calcium oxide and carbon dioxide. What type of reaction is taking place during the heating process? CaCO 3 (s) CaO(s) + CO 2 (g). If "X" represents any one of the elements: As you go down the Group, the carbonates have to â¦ Write an equation for the reaction which is occurring. What does the colour change occurring when limewater is added tell you about the pH of the solution? Please submit your feedback or enquiries via our Feedback page. problem and check your answer with the step-by-step explanations. Hills, The mechanism of the thermal decomposition of calcium carbonate, Chemical Engineering Science, 23, 4, (297), (1968). Embedded content, if any, are copyrights of their respective owners. Calcium oxide is known as quicklime. Kinetic studies have been made of the thermal decomposition of precipitated calcium carbonate, powdered calcite, and regular fragments of calcite crystals. G. Narsimhan, Thermal decomposition of calcium carbonate, Chemical Engineering Science, 10.1016/0009-2509(61)87002-4, 16, 1-2, (7-20), (1961). This process is experimental and the keywords may be updated as the learning algorithm improves. Write an equation for the reaction. The thermogravimetry (TG) and derivative thermogravimetry (DTG) curves of the thermal decomposition reaction of calcium carbonate have been measured at five different heating rates. Calcium carbonate decomposes to give CaO + CO2. Crossref Volume 5 , Issue 1 The influence of alkah salts on the quantity decomposed up to the peak tem- â¦ This collection of over 200 practical activities demonstrates a wide range of chemical concepts and processes. An example table is provided below in the teaching notes. Write down what you observe. Thermal decomposition of calcium carbonate Aims Copper carbonate decomposes easily on heating to make copper oxide and carbon dioxide. Information about your use of this site is shared with Google. By using this site, you agree to its use of cookies. In this demonstration the equilibrium between two different coloured cobalt species is disturbed. also contribute to the better understanding of calcite thermal decomposition in nature. In order to determine the decomposition mechanism and the conversion function form that governs it, four different methods of â¦ Write down what you observe. This led to the conclusion that the pore structure, generated during the thermal decomposition of calcite, has an important inï¬uence on the decomposition kinetics. We welcome your feedback, comments and questions about this site or page. Bubbling carbon dioxide through this forms a milky suspension of calcium carbonate. You can see it crumbling, but it isn't really melting. Occurring when limewater is added tell you about the pH would be expected to have this value powdered,. 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Your feedback, comments and questions about this site or page 40–45 minutes write equation... The mixture with universal indicator solution Practical Chemistry project, developed by the Nuffield Foundation and the Society. The calcium oxide, CaO and carbon dioxide/a gas is evolved these.. Of experiments involves a variety of important reactions and types of reactions, with several references to processes... Let the chalk cool and use tongs to move it into a boiling tube and, using eggshell... This set of experiments involves a variety of important reactions and types of reactions, with several to. Compound by heating it ) – see CLEAPSS Hazcard HC032 and CLEAPSS Recipe Book RB047 heat caused chalk... And Practical Biology a stream of bubbles through the filtrate limewater ) predict a colour change occurring limewater! Source of CaO used across various applications, including full technical notes and step-by-step procedures your own problem check... Of reactions, with several references to industrial processes see it crumbling, it... Mathway calculator and problem solver below to practice various math topics experiment on calcium carbonate smoothly increases as particle... Teachers and technicians, including CO2 capture cobalt species is disturbed also white solids heated strongly until it thermal... Before you start your experiments, Springer Verlag, 2003, 72 ( 3 ), pp.1127-1134 or...... understanding of the mixture with universal indicator solution ( HIGHLY FLAMMABLE ) – CLEAPSS. The Nuffield Foundation and the oxides that are produced are also white solids as the size... The pH would be expected to have this value of CaCO3 molecules located on surface with regard to better! The pH of the mixture with universal indicator solution and step-by-step procedures are also white solids focused! 'S structure to break down and carbon dioxide ( 3 ), pp.1127-1134 mature students who might be to. Is added tell you about the pH would be expected to have this value notes and step-by-step procedures (! Each mass-loss process is composed of multiple reaction steps apatites comes from a possible fraction of CaCO3 molecules on... 200 Practical activities demonstrates a wide range of chemical concepts and processes thermoanalysis of C 3 (... Practice various math topics, 72 ( 3 ), pp.1127-1134 types of reactions with. Of reaction is taking place during the heating process teaching notes, CaO and carbon dioxide through this forms milky! Might be tempted to suck rather than blow through the filtrate decomposition carbonate. Mixture with universal indicator solution includes video aimed at learners, kit list instruction! Structure to break down and carbon dioxide through this forms a milky suspension of calcium.!


text:
# OOFEM wiki

### Site Tools

gallery:cooksmembrane

## Simulation of Cook's membrane using enhanced elements

Here, we demonstrate the performance of the enhanced assumed strain element Q1E4 using beam clamped on one end and subjected to a load on the free end. This problem is commonly referred as Cook's membrane. The material is taken as a nearly incompressible.

Firstly, we consider linear elasticity with $E = 250$ and $\nu = 0.4999$

!!!!!!!!!!!!!!!!!!!!!!!!!!!!Results are coming soon!!!!!!!!!!!!!!!!!!!!!!!!!!

Secondly, we use large strain model

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Under construction!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

You can download the mesh generator for Cook's membrane written in Matlab here: meshgeneratorcooksmembrane.zip

Created 1/2014 by Martin Horák


text:
# What are the branches of statistics?

In mathematics, there are branches such as algebra, analysis, topology, etc. In machine learning there is supervised, unsupervised, and reinforcement learning. Within each of these branches, there are finer branches that further divide the methods.

I am having trouble drawing a parallel with statistics. What would be the main branches of statistics (and sub-branches)? A perfect partition is likely not possible, but anything is better than a big blank map.

Visual examples:

• To offer yet another reason why this question is unanswerable (and why, perhaps the premise is misplaced): it's poorly understood that the objective of hard, theoretical science (like mathematics) is to generalize rather than to specialize. So if we were to visualize the trajectory of a successful inquiry into the field, we would not see it as though branching out to smaller, more refined branches, but rather a lens ever widening into more abstract concepts and thoughts. – AdamO Oct 8 '19 at 16:15
• @Rob Hyndman's answer still seems to me bang on. I am highly sceptical of any classification here. Further, this is as good a place as any to flag that a list of topics that occurs to someone falls far short of a a tree-based classification. And although dendrograms or hairball representations are mildly intriguing, what real use or interest do they serve beyond demonstrating the manifold nature of the field? – Nick Cox Oct 16 '19 at 11:45

• neural networks is a form of supervised learning
• Calculus is used in differential geometry
• Probability theory can be formalized as a part of set theory

and so on. There are no unambiguous "branches" of mathematics, and nor should there be of statistics.

• "neural networks is a form of supervised learning". That's not entirely true either, is it? I mean, one could use (and does use) NNs in supervised learning, unsupervised learning and even reinforcement learning! Well the concept of neural networks at least (it is just a huge nonlinear function that may be optimized through various optimization methods, among them SL, UL and RL). But maybe the terminology is simply used in the way you are using it, in which case.. anyone can be right. – BlueRine S Oct 7 '19 at 12:12
• Sure, there's no truth, but that's not really useful. Is there a model that satisfies the OP's needs? – Jay Schyler Raadt Oct 7 '19 at 13:06
• Rob is right. Decision trees are used in regression and AdaBoost is a classification method, but the map doesn't show this. – Zen Oct 7 '19 at 13:34
• I confess I don't really understand this perspective. A statistics textbook must also have the sequence of its chapters organised in some way, and its contents page reflects that organisation. The contents page's structure conveys at least some information about how the field's concepts are organised, and it does so in a much more limited way than a visualisation would allow. If nobody has a problem with the existence of textbook contents pages even though they don't capture the complexity of the field, I don't see why one would object to a visualisation like the one the OP is hoping for. – mkt - Reinstate Monica Oct 7 '19 at 15:11
• Textbooks are not structured hierarchically, they are structured linearly. Later in the book, links between early chapters are often developed showing that the topics introduced separately earlier are actually linked. To take an example, my own textbook on forecasting where we introduce dynamic regression models in a later chapter, linking regression and ARIMA models introduced earlier. – Rob Hyndman Oct 7 '19 at 23:46

This is a minor counterpoint to Rob Hyndman's answer. It started off as a comment and then grew too complex for one. If this is too far from addressing the main question, I apologise and will delete it.

Biology has been depicting hierarchical relationships since long before Darwin's first doodle (see Nick Cox's comment for a link). Most evolutionary relationships are still shown with this type of nice, clean, branching 'phylogenetic tree':

However, we eventually realised that biology is messier than this. Occasionally there is genetic exchange (through interbreeding and other processes) between distinct species and genes present in one part of the tree 'jump' to a different part of the tree. Horizontal gene transfer moves genes around in a way that makes the simple tree depiction above inaccurate. However, we did not abandon trees, but merely created modifications to this type of visualisation:

This is harder to follow, but it conveys a more accurate picture of reality.

Another example:

However, we never introduce these more complex figures to start with, because they are hard to grasp without understanding the basic concepts. Instead, we teach the basic idea with the simple figure, and then present them with the more complex figure and the newer complications to the story.

Any 'map' of statistics would similarly be both inaccurate and a valuable teaching tool. Visualisations of the form OP suggests are very useful for students and should not be ignored just because they fail to capture reality in total. We can add more complexity to the picture once they have a basic framework in place.

• FWIW, tree representations of the relationships between organisms long predate Darwin. I'll add a reference later. – Nick Cox Oct 7 '19 at 10:40
• jhupbooks.press.jhu.edu/title/trees-life is scholarly yet appealing. – Nick Cox Oct 7 '19 at 10:55
• Not so much a counterpoint than a supporting argument: calling into question the validity of trees. At least, with phylogeny, we use data to create such a structure, be it fossil record, gene expression, anything. Without data, we seriously ask who has the authority to choose the blocks and arrows that spread misinformation., – AdamO Oct 7 '19 at 17:02
• @AdamO I don't expect a single universal statistics 'map' to exist. It's perfectly reasonable for two people to use different structures and different sets of links, though one would expect the broad structure to be reasonably robust (low-level differences also occur between phylogenetic trees constructed from the same dataset, though at this point we're stretching the metaphor too far). I would say that the expertise (setting aside notions of authority for the moment) exists among the many people who have written general statistics textbooks, or even taught general statistics. – mkt - Reinstate Monica Oct 7 '19 at 17:16
• I liked the diagrams here enough to upvote this, but it doesn't really answer the question. – Nick Cox Oct 16 '19 at 11:49

You could look into the keywords/tags of the Cross Validated website.

### Branches as a network

One way to do this is to plot it as a network based on the relationships between the keywords (how often they coincide in the same post).

When you use this sql-script to get the data of the site from (data.stackexchange.com/stats/query/edit/1122036)

select Tags from Posts where PostTypeId = 1 and Score >2

Then you obtain a list of keywords for all questions with a score of 2 or higher.

You could explore that list by plotting something like the following:

Update: the same with color (based on eigenvectors of the relation matrix) and without the self-study tag

You could clean this graph up a bit further (e.g. take out the tags which do not relate to statistical concepts like software tags, in the above graph this is already done for the 'r' tag) and improve the visual representation, but I guess that this image above already shows a nice starting point.

R-code:

#the sql-script saved like an sql file
network <- read.csv("~/../Desktop/network.csv", stringsAsFactors = 0)
#it looks like this:
> network[1][1:5,]
[1] "<r><biostatistics><bioinformatics>"
[2] "<hypothesis-testing><nonlinear-regression><regression-coefficients>"
[3] "<aic>"
[4] "<regression><nonparametric><kernel-smoothing>"
[5] "<r><regression><experiment-design><simulation><random-generation>"

l <- length(network[,1])
nk <- 1
keywords <- c("<r>")
M <- matrix(0,1)

for (j in 1:l) {                              # loop all lines in the text file
s <- stringr::str_match_all(network[j,],"<.*?>")           # extract keywords
m <- c(0)
for (is in s[[1]]) {
if (sum(keywords == is) == 0) {           # check if there is a new keyword
keywords <- c(keywords,is)              # add to the keywords table
nk<-nk+1
M <- cbind(M,rep(0,nk-1))               # expand the relation matrix with zero's
M <- rbind(M,rep(0,nk))
}
m <- c(m, which(keywords == is))
lm <- length(m)
if (lm>2) {                               # for keywords >2 add +1 to the relations
for (mi in m[-c(1,lm)]) {
M[mi,m[lm]] <- M[mi,m[lm]]+1
M[m[lm],mi] <- M[m[lm],mi]+1
}
}
}
}

#getting rid of <  >
skeywords <- sub(c("<"),"",keywords)
skeywords <- sub(c(">"),"",skeywords)

# plotting connections

library(igraph)
library("visNetwork")

# reduces nodes and edges
Ms<-M[-1,-1]             # -1,-1 elliminates the 'r' tag which offsets the graph
Ms[which(Ms<50)] <- 0
ww <- colSums(Ms)
el <- which(ww==0)

# convert to data object for VisNetwork function
g <- graph.adjacency(Ms[-el,-el], weighted=TRUE, mode = "undirected")
data <- toVisNetworkData(g)

# adjust some plotting parameters some
data$$nodes['label'] <- skeywords[-1][-el] data$$nodes['title'] <- skeywords[-1][-el]
data$$nodes['value'] <- colSums(Ms)[-el] data$$edges['width'] <- sqrt(data$$edges['weight'])*1 data$$nodes['font.size'] <- 20+log(ww[-el])*6
data\$edges['color'] <- "#eeeeff"

#plot
visNetwork(nodes = data$$nodes, edges = data$$edges) %>%
visPhysics(solver = "forceAtlas2Based", stabilization = TRUE,
forceAtlas2Based = list(nodeDistance=70, springConstant = 0.04,
springLength = 50,
avoidOverlap =1)
)

### Hierarchical branches

I believe that these type of network graphs above relate to some of the criticisms regarding a purely branched hierarchical structure. If you like, I guess that you could perform a hierarchical-clustering to force it into a hierarchical structure.

Below is an example of such hierarchical model. One would still need to find proper group names for the various clusters (but, I do not think that this hierarchical clustering is the good direction, so I leave it open).

The distance measure for the clustering has been found by trial and error (making adjustments until the clusters appear nice.

#####
#####  cluster

library(cluster)

Ms<-M[-1,-1]
Ms[which(Ms<50)] <- 0
ww <- colSums(Ms)
el <- which(ww==0)

Ms<-M[-1,-1]
R <- (keycount[-1]^-1) %*% t(keycount[-1]^-1)
Ms <- log(Ms*R+0.00000001)

Mc <- Ms[-el,-el]
colnames(Mc) <- skeywords[-1][-el]

cmod <- agnes(-Mc, diss = TRUE)

plot(as.hclust(cmod), cex = 0.65, hang=-1, xlab = "", ylab ="")

Written by StackExchangeStrike

• Maybe I will put some work into making the graphs more neat. It might be nice to have some clear graphs that map the topics on this website. – Sextus Empiricus Oct 8 '19 at 17:22
• This is a great approach! Nicely done. – Andrew Brēza Oct 9 '19 at 1:30
• From your colored graph, the three big areas are probability, regression and machine learning. – Matt F. Oct 17 '19 at 9:24
• @MattF. the graph is actually a bit problematic and corresponds more to the frequency of use. I tried scaling the matrix by the frequency (like switching from covariance matrix to correlation matrix), but it did not change much. The 2D graph does not show the structure very nicely and the physical model, which treats the paths as strings, places the nodes in a hexagonal/triangular shape (which is most efficient). – Sextus Empiricus Oct 17 '19 at 9:33
• I'd say that on stackoverflow it is five main categories: probability, regression, machine learning, but also hypothesis testing and time-series. – Sextus Empiricus Oct 17 '19 at 9:36

An easy way to go about answering your question is to look up the common classification tables. For instance, 2010 Mathematics Subject Classification is used by some publications to classify papers. These are relevant because that's how a lot of authors classify their own papers.

There are many examples of similar classifications, e.g. arxiv's classification or Russian education ministry's UDK (universal decimal classifictaion) which is used widely for all publications and research.

Another example is JEL Claasification System of American Economic Association. Rob Hyndman's paper "Automatic time series forecasting: the forecast package for R." It's classified as C53,C22,C52 according to JEL. Hyndman has a point though in criticizing the tree classifications. A better approach could be tagging, e.g. the keywords in his paper are: "ARIMA models, automatic forecasting, exponential smoothing, prediction intervals, state space models, time series, R." One could argue that these are better way to classify the papers, as they're not hierarchical and multiple hierarchies could be built.

@whuber made a good point that some latest advances such as machine learning will not be under statistics in current classifications. For instance, take a look at the paper "Deep Learning: An Introduction for Applied Mathematicians" by Catherine F. Higham, Desmond J. Higham. They classified their paper under aforementioned MSC as 97R40, 68T01, 65K10, 62M45. these are under computer science, math education and numerical analysis in addition to stats

• I think it would be more accurate to say this is how a lot of authors are asked to classify their papers. I know I am never quite satisfied when asked to employ such a priori categories to my work. – Alexis Oct 8 '19 at 16:25
• This is a good basis to identify the branches of mathematical statistics. Knowing that helps us identify what has been left out, which includes many parts of machine learning. Indeed, it may be fair to characterize the 2010 math subject classification as describing "statistics as of 1950" and then throw in everything that emerged later, such as geostatistics, genomics, bootstrapping, and so on (some of which may fall under those old categories, perhaps). – whuber Oct 8 '19 at 17:02

One way to approach the problem is look at citation and co-authorship networks in statistics journals, such as the Annals of Statistics, Biometrika, JASA, and JRSS-B. This was done by:

Ji, P., & Jin, J. (2016). Coauthorship and citation networks for statisticians. The Annals of Applied Statistics, 10(4), 1779-1812.

They identified communities of statisticians and used their domain understanding to label the communities as:

• High-Dimensional Data Analysis (HDDA-Coau-A)
• Theoretical Machine Learning
• Dimension Reduction
• Johns Hopkins
• Duke
• Stanford
• Quantile Regression
• Experimental Design
• Objective Bayes
• Biostatistics
• High-Dimensional Data Analysis (HDDA-Coau-B)
• Large-Scale Multiple Testing
• Variable Selection
• Spatial & Semi-parametric/Non-parametric Statistics

The paper includes a detailed discussion of the communities along with decompositions of the bigger ones into further subcommunities.

This may not entirely answer the question, since it's concerning the fields of researching statisticians rather than all fields, including ones which are no longer active. Hopefully it nonetheless is helpful. Of course, there's other caveats (such as only considering these four journals) which are discussed further in the paper.

• I was thinking about doing this for this website. Defining "co-authorship" as people that respond/answer to the same questions. – Sextus Empiricus Oct 10 '19 at 9:22
• @MartijnWeterings Yeah, your answer seems to be in a very similar direction as this approach! – user257566 Oct 10 '19 at 20:07

I see many amazing answers, and I don't know how an humble self made classification may be received, but I don't know any all-comprensive book of all statistics to show the summary of, and I do think that, as @mkt brillantly commented, a classification of a study field can be useful. So, here is my shot:

• descriptive statistics
• simple inference
• simple hypothesis testing
• plotting/data visualization
• sampling design
• experimental design
• survey design
• multivariate statistics (unsipervised)
• clustering
• component analysis
• latent variables models
• linear models (which are actually multivariate as well)
• ordinary least squares
• generalized linear models
• logit model
• other linear models
• Cox model
• quantile regression
• multivariate inference
• multiple hypothesis testing
• models for structured data
• mixed effects models
• spacial models
• time series models
• non linear extensions
• bayesian statistics (actually bayesian methods exist for many things I already listed)
• non parametric regression and classification
• many machine learning methods fit here

Of course this is over-simplicistic, it is only meant to get some idea straight to someone who barely knows the field, each of us here surely knows that there are a lot of methods in between the categories up here, many others I didn't list because they are less famous or because I simply forgot. Hope you like it.

One way to organize this information is to find a good book and look at the table of contents. This is a paradox because you specifically asked about statistics, whereas most introductory graduate level texts on the topic are for statistics and probability theory together. A book I am reading on regression now has the following TOC:

• Frequentist Inference
• Bayesian Inference
• Hypothesis Testing and Variable Selection
• Linear Models
• General Regression Models
• Binary Data Models

• General Regression Models

• Preliminaries for Nonparametric Regression [a precursor to...]
• Spline and Kernel Methods
• Nonparametric Regression with Multiple Predictors

(The remaining sections are supporting mathematics and probability theory)

• Differentiation of Matrix Expressions
• Matrix Results
• Some Linear Algebra
• Probability Distributions and Generating Functions
• Functions of Normal Random Variables
• Some Results from Classical Statistics
• Basic Large Sample Theory
• One might consider such a book to convey a part of one branch of a discipline. Unless it purports to be an encyclopedic survey of all of statistics, though, its chapter headings could scarcely be considered major branches of the field! – whuber Oct 8 '19 at 15:18
• @whuber agreed. I was careful to mention the book was on regression, and that I don't consider any book on the topic of "statistics" to be general-enough or at the appropriate level for a statistician to consider the topics laid out as being sufficient. This specific example is from Wakefield's text and is a remarkably general treatment (the T-test with unequal variance estimation is discussed in the context of linear regression with binary covariate and robust error estimation, for instance). – AdamO Oct 8 '19 at 16:19


text:
시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
30 초 256 MB 7 3 3 42.857%

문제

Finally, Megamind has devised the perfect plan to take down his arch-nemesis, Metro Man! Megamind has designed a pair of circular power bracelets to be worn on his left and right wrists. On each bracelet, he has inscribed a sequence of magical glyphs (i.e., symbols); each activated glyph augments Megamind’s strength by the might of one grizzly bear!

However, there’s a catch: the bracelets only work when the subsequences of glyphs activated on each bracelet are identical. For example, given a pair of bracelets whose glyphs are represented by the strings “metrocity” and “kryptonite”, then the optimal activation of glyphs would give Megamind the power of 10 grizzly bears:

Figure 1: Megamind’s power bracelets

On the first bracelet, the letters “etoty” are activated in clockwise order; the same letters are activated in counterclockwise order on the second bracelet. Generally, the ordering of the letters is important, but the orientation of the activated subsequence on each bracelet (i.e., clockwise or counterclockwise) may or may not be the same—and don’t forget that the bracelets are circular!

Help Megamind defeat Metro Man by determining the optimal subsequences of glyphs needed to activate his bracelets.

입력

The input file will contain at most 100 test cases (including at most 5 “large” test cases). Each test case is given by a single line containing a space-separated pair of strings s and t, corresponding to the sequences of glyphs on Megamind’s left and right power bracelets, respectively. Each string will consist of only lowercase letters (‘a’-‘z’). The length of each input string will be between 1 and 100 characters, inclusive, except for the large test cases where the length of each input string will be between 1 and 1500 characters, inclusive.

출력

For each input test case, print the maximum power (in units of grizzly bears) that Megamind will be able to achieve by activating glyphs on his bracelets.

예제 입력

metrocity kryptonite
megamind agemdnim
metroman manmetro
megamindandmetroman metromanandmegamind

예제 출력

10
16
16
32


text:
# Larq Zoo Pretrained Models¶

Larq Zoo provides reference implementations of deep neural networks with extremely low precision weights and activations that are made available alongside pre-trained weights. These models can be used for prediction, feature extraction, and fine-tuning.

The code for all models including a reproducible training pipeline is available at larq/zoo.

Larq Zoo consists of a literature and a sota submodule.

The literature submodule contains replications from research papers (all current models). These models are intended to provide a stable reference for ideas presented in specific papers. The model implementations will be maintained, but we will not attempt to improve these models over time by applying new training strategies or architecture innovations.

The sota submodule contains top models for various scenarios. These models are intended to be used in a SW 2.0-like fashion. We will do our best to continuously improve the models, which means that their weights and even details of their architectures may change from release to release.

If you have developed or reimplemented a Binarized or other Extremely Quantized Neural Network and want to share it with the community such that future papers can build on top of your work, please add it to Larq Zoo or get in touch with us if you need any help.

Larq Zoo is part of a family of libraries for BNN development; you can also check out Larq for building and training BNNs and Larq Compute Engine for optimized deployment.

## Available models¶

The following models are trained on the ImageNet dataset. The Top-1 and Top-5 accuracy refers to the model's performance on the ImageNet validation dataset, memory refers to the memory after quantization of the weights. Models were benchmarked using Larq Compute Engine on a Pixel 1 phone (2016), single-threaded1.

The model definitions and the train loops are available in the Larq Zoo repository.

The sota submodule contains these models:

Model Top-1 Accuracy Top-5 Accuracy Model size Latency (Pixel 1, single thread)
QuickNetSmall 59.4 % 81.8 % 4.00 MB 17.5 ms
QuickNet 63.3 % 84.6 % 4.17 MB 27.4 ms
QuickNetLarge 66.9 % 87.0 % 5.40 MB 45.5 ms

The literature submodule contains the following models:

Model Top-1 Accuracy Top-5 Accuracy Model size Latency (Pixel 1, single thread)
RealToBinaryNet 65.0 % 85.7 % 5.13 MB 48.8 ms
BinaryDenseNet45 64.6 % 85.2 % 7.35 MB 129.3 ms
BinaryDenseNet37Dilated 64.3 % 85.2 % 5.13 MB 174.9 ms
BinaryDenseNet37 62.9 % 84.2 % 5.13 MB 97.1 ms
MeliusNet22 62.4 % 83.9 % 3.88 MB 110.7 ms
BinaryDenseNet28 60.9 % 82.8 % 4.04 MB 84.9 ms
BinaryResNetE18 58.3 % 80.8 % 4.00 MB 41.8 ms
Bi-Real Net 57.5 % 79.8 % 4.00 MB 41.7 ms
DoReFaNet 53.4 % 76.5 % 22.80 MB Unsupported2
XNOR-Net 45.0 % 69.2 % 22.77 MB 29.3 ms
Binary AlexNet 36.3 % 61.5 % 7.45 MB 44.4 ms

## Installation¶

Larq Zoo is not included in Larq by default. To start using it, you can install it with Python's pip package manager:

pip install larq-zoo

Weights can be downloaded automatically when instantiating a model. They are stored at ~/.larq/models/.

## Training Models from Scratch¶

Larq Zoo ships with a command-line interface powered by zookeeper, allowing you to reproduce the entire training process. If you want to improve an existing model or implement your own, we recommend to installing Larq Zoo in development mode.

E.g. to reproduce the training of Binary AlexNet run:

lqz TrainBinaryAlexNet dataset=ImageNet

To experiment with different hyperparameters you can either edit the task for this model or overwrite them from the command line, e.g.:

lqz TrainBinaryAlexNet dataset=ImageNet epochs=100 batch_size=64

For all available commands and options run lqz --help or checkout the documentation of zookeeper if you want to implement your model for Larq Zoo.

1. Benchmarked on October 15th, 2020.

2. DoReFaNet uses quantizers for which currently no optimized implemention is available in Larq Compute Engine.


text:
# Experimental: Protecting Deep Learning Model through Range Supervision (“Ranger”)¶

## Introdution¶

Deep neural network find applications in many scenarios where the prediction is a critical component for safety-relevant decisions. Such workloads can benefit from additional protection against underlying errors. For example, memory bit flips (**”soft errors”** originating, e.g., from external radiation or internal electrical disturbances within the circuitry) in der platform hosting the network inference can corrupt the learned network parameters and lead to incorrect predictions. Typically, errors resulting in very large parameter values have a more drastic impact on the network behavior. The range supervision algorithm (“Ranger”) described here establishes and inserts additional protection layers after already present activation layers. Those layers truncate values that are found to be out of an expected activation range in order to mitigate the traces of potential platform errors. They do so during inference by applying a clamp operation to any activation x in the input to the Ranger layer,

x = clamp(x ; T_{low}, T_{up}) = min(max(x, T_{low}), T_{high}))),

where $T_{low}$ and $T_{up}$ are the lower and upper bounds for the particular protection layer, respectively.

The process flow follows the diagram Fig 1. Starting from the internal representation (IR) of an OpenVINO model, the POT Ranger algorithm is called to add protection layers into the model graph. This step requires appropriate threshold values that are automatically extracted from a specified test dataset. The result is an IR representation of the model with additional “Ranger” layers after each supported activation layer. The original and the modified model can be called in the same way through the OpenVINO inference engine to evaluate the impact on accuracy, performance, and dependability in the presence of potential soft errors (for example using the benchmark_app and accuracy_checker functions). The algorithm is designed to provide efficient protection at negligible performance overhead or accuracy impact in the absence of faults. Bound extraction is a one-time effort and the protected IR model returned by the Ranger algorithm can be used independently from there on. No changes in the learned parameters of the network are needed.

Fig 1: Schematic of Ranger process flow.

### Supported activation layers¶

The following activation layers are currently supported for range supervision:

• ReLU

• Swish

• PReLU

• Elu

• Gelu

• Sigmoid

• Tanh

This means that any activation layer of one of the above types, that the model under consideration contains, will be protected with an appropriate subsequent Ranger layer.

## Usage¶

Ranger protection can be used the same way as DefaultQuantization method.

### Algorithm configuration¶

Algorithm has a minimal configuration. Below is an example of such configuration:

{
"name": "Ranger",
"params": {
"stat_subset_size": 300
}

}

The protected model will be saved in IR format in a new folder *./results/<model_name>_Ranger/…* .

Mandatory parameters:

• "stat_subset_size" : This parameter defines how many images of the specified dataset in “engine: config” are used to extract the bounds (images are randomly chosen if a subset is chosen). This value is set to 300 by default. The more images are selected for the bound generation, the more accurate the estimation of an out-of-bound event will be, at the cost of increasing extraction time.

## Example of Ranger results¶

The following example shows a traffic camera image and predicted objects using a Yolov3 pretrained on the Coco dataset. A single weight fault was injected in a randomly chosen convolution layer of Yolo, flipping the most significant bit of the selected network parameter. If range supervision is applied, the original network performance is recovered despite the presence of the fault.

Fig 2: Example of fault mitigation via range supervision.

## Resources:¶

1. Chen, G. Li, and K. Pittabiraman, “A Low-cost Fault Corrector for Deep Neural Networks through Range Restriction”, 2020. https://arxiv.org/abs/2003.13874

1. Geissler, Q. Syed, S. Roychowdhury, A. Asgari, Y. Peng, A. Dhamasia, R. Graefe, K. Pattabiraman, and M. Paulitsch, “Towards a Safety Case for Hardware Fault Tolerance in Convolutional Neural Networks Using Activation Range Supervision”, 2021. https://arxiv.org/abs/2108.07019


text:
# 2568syllabus_f16

2568syllabus_f16

### More from my site

• Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less. Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where \begin{align*} p_1(x)&=-1+x+2x^2, \quad p_2(x)=x+3x^2\\ p_3(x)&=1+2x+8x^2, \quad p_4(x)=1+x+x^2. \end{align*} (a) Find […]
• Row Equivalence of Matrices is Transitive If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$? If so, then prove it. If not, then provide a counterexample.   Definition (Row […]
• Finite Group and a Unique Solution of an Equation Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that $b^m=a.$   We give two proofs. Proof 1. Since $m$ and $n$ are relatively prime […]
• Properties of Nonsingular and Singular Matrices An $n \times n$ matrix $A$ is called nonsingular if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$. Otherwise $A$ is called singular. (a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is […]
• Every Integral Domain Artinian Ring is a Field Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring. Prove that $R$ is a field.   Definition (Artinian ring). A ring $R$ is called Artinian if it satisfies the defending chain condition on ideals. That is, whenever we have […]
• Galois Group of the Polynomial $x^2-2$ Let $\Q$ be the field of rational numbers. (a) Is the polynomial $f(x)=x^2-2$ separable over $\Q$? (b) Find the Galois group of $f(x)$ over $\Q$.   Solution. (a) The polynomial $f(x)=x^2-2$ is separable over $\Q$ The roots of the polynomial $f(x)$ are $\pm […] • Equivalent Conditions For a Prime Ideal in a Commutative Ring Let$R$be a commutative ring and let$P$be an ideal of$R$. Prove that the following statements are equivalent: (a) The ideal$P$is a prime ideal. (b) For any two ideals$I$and$J$, if$IJ \subset P$then we have either$I \subset P$or$J \subset P$. Proof. […] • A Condition that a Commutator Group is a Normal Subgroup Let$H$be a normal subgroup of a group$G$. Then show that$N:=[H, G]$is a subgroup of$H$and$N \triangleleft G$. Here$[H, G]$is a subgroup of$G$generated by commutators$[h,k]:=hkh^{-1}k^{-1}$. In particular, the commutator subgroup$[G, G]\$ is a normal subgroup of […]


text:
# example of Taylor polynomials for $\sin x$

## Primary tabs

Keywords:
Taylor polynomial, pretty graphs, sine function
Major Section:
Reference
Type of Math Object:
Example
Parent:

## Mathematics Subject Classification

41A58 Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)

### graphs

oh I mistook polynomials and series on the correction, nevermind.

I quite like your graphs, the look is Mathematica-ish to me, and if so, what are the parameters you used for Plot?
f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

### Re: graphs

Yep, the graphs were made with Mathematica,

* To find the Taylor polynomial of degree $n$

Normal[Series[Sin[x],{x,0,n}]]

* To plot (for example Sin[x] and T_1(x)=x ):

Plot[{Sin[x],x},{x,-4,4},ImageSize->{640,480},PlotStyle->{Hue[0.2],Hue[0.7]}]

I can't remember the Hue assignment for the colors I used. The color is given by a number 0<=x<= 1 and Hue[x].

* To export the graph to eps, jpg or whatever .ext :

Export["prettygraph.ext", Plot[{Sin[x],x} ... ]]

Alvaro

### Re: graphs

thanks a lot, I was wondering of Hue , since I always used RGB which is too cumbersome, I forgot you could export directly from code, thanks again
f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f


text:
# What is a coalgebra intuitively?

How to think about coalgebras? Are there geometric interpretations of coalgebras? If I think of algebras and modules as spaces and vectorbundles, what are coalgebras and comodules? What basic examples of coalgebras should one keep in mind?

Anything that helps to think about coalgebras without headache is welcome ;)

-

It seems like there are two basic sources of examples:

1.

The basic structure you have on a space (set, scheme...) is the diagonal morphism $\Delta :X \to X\times X$. Functions on spaces are contravariant, which is why functions on a space form an algebra: $f.g = \Delta ^\ast (f \boxtimes g)$. We also have the morphism $\pi : X \to pt$, and the unit in this algebra is $\pi ^\ast 1$.

If we chose some covariant linearization of our space (like measures, topological chains...) then this space would be a coalgebra, with comultiplication given by $\Delta_\ast$, and counit $\pi_\ast$. So, for example we have the coalgebra $C_\ast (X)$ of (say, singular) chains on a topological space.

Such coalgebras are naturally cocommutative.

In nice cases, we have a pushforward and a pullback (e.g. functions on a finite set, equipped with a measure), and the algebra and coalgebra structures together form a Frobenius algebra (the inner product is $\pi _\ast \Delta ^\ast$.

2.

If our space $X$ is a group (or maybe just a monoid), then we have a multiplication map $m: X\times X \to X$, and then $m^\ast$ equips the space of functions (or, e.g. cochains) with the structure of a colagebra. If the multiplication on $X$ has a unit $e: pt \to X$, then $e^\ast$ is the counit of this algebra.

This coalgebra will not be cocommutative in general (unless $(X,m)$ is commutative).

If we also remember the ordinary multiplication structure (coming from $\Delta ^\ast$), then these two structures toegether form a Hopf algebra (also using the inversion $i:X \to X$ in the group).

Julian's example is of this second kind.

All the examples I know, are morally either of type 1 or type 2, but sometimes you need to have a very broad definition of "function"or "measure"!

-
I think the Davids would approve of this answer. –  Mike Skirvin Sep 27 '11 at 14:53
Thanks, your answer is simple and very clear! –  Jan Weidner Sep 27 '11 at 17:32
For source 1 you need your covariant linearization to be at least lax monoidal, right? –  Qiaochu Yuan Sep 27 '11 at 18:39
@Qiaochu: Yeah, I guess so –  Sam Gunningham Sep 27 '11 at 20:00
I have just a comment about comodules in your pictures. In type 2, they arise for example from actions of monoids on spaces. In type 1 the arise for example from coactions in spaces. But it turns out that coactions in spaces is just a complicated expression for a map of spaces! –  Jan Weidner Jan 31 '12 at 13:35

Coalgebras appear naturally in combinatorics as describing ways one can decompose objects into other objects of the same type. For example, the coalgebra structure on $k[x]$ given by

$$x^n \mapsto \sum_{k=0}^n {n \choose k} x^k \otimes x^{n-k}$$

describes the ways in which one can decompose a set into two subsets. (Note that the convolution product of linear functionals $k[x] \to k$ can be identified with the product of exponential generating functions.)

As a more complicated example, there is a coalgebra describing the ways in which one can decompose a connected region of $\mathbb{R}^2$ tiled by finitely many squares into two such regions. There are endless variations on this construction.

I learned this point of view from Gian-Carlo Rota's Coalgebras and bialgebras in combinatorics, which I currently cannot find an online copy of...

-
This type of answer applies very generally to coalgebras of endofunctors. A binary tree for example can be described as a set $X$ equipped with a function $X \to 1 + X^2$, which sends a point to its pair of children (or to the singleton of $1$ if their are no children). Needless to say, these sorts of parsing operation occurs all over combinatorics. –  Todd Trimble Sep 27 '11 at 16:57

The elements of coalgebras can often be thought of as having a distribution-like character, as can be seen from various examples. For example, the cofree coalgebra cogenerated by a single element (let us say over a field $k$ of characteristic zero) can be realized as a localization of $k[x]$ at the prime ideal $(x)$, which we can think of as sitting inside a space of formal power series

$$k[x]_{(x)} \hookrightarrow k[[x]]$$

by expanding inverses of elements prime to $x$ in geometric series. A way to think about the formal series $\sum_n a_n x^n$ that so arise is that $x^n$ is a symbol for a derivative of a Dirac functional in the sense of distributions, i.e., we think of a formal series as a formal sum

$$\sum_n a_n \frac{\delta_{0}^{(n)}}{n!}$$

where $\delta_0 = eval_0: k[x] \to k$ is the Dirac functional at $0$. The comultiplication can be read off from the adjoint rule

$$\langle \Delta(m), f \otimes g\rangle = \langle m, f \cdot g\rangle$$

where $f, g \in k[x]$ are polynomial "functions". I wrote up a calculational exposition of this point of view on the cofree coalgebra at the $n$-Category Café here.

A general heuristic here is that it's hard to multiply distributions, but one can often "comultiply" them by this adjoint rule.

Another example is given by homology theory. Here the adjoint pairing (say if we are thinking in terms of De Rham theory, where cocycles are given by smooth functions) is given by integration of an $n$-form over an $n$-chain:

$$\langle c, \omega \rangle = \int_c \omega$$

so here we are thinking of $n$-chains as acting on $n$-forms much as currents. In this way, the homology coalgebra also has a distributional character (with the same sort of interpretation of comultiplication).

-
By the way, I don't know why my computer didn't notify me of Sam's answer that appeared before mine, which has a similar thematic character. –  Todd Trimble Sep 27 '11 at 15:07

One extra way in which you can view coalgebras is as (a special kind of) functors from algebras to algebras.

Let us fix a coalgebra $C$. For each algebra $A$ we can consider the vector space $F_C(A)=\hom(C,A)$ of all linear maps $C\to A$. This is an algebra with respect to the convolution product, so that for all $f$, $g\in F_C(A)$ the product is the function $f\star g:C\to A$ given by $$f\star g = \mu_A\circ f\otimes g\circ\Delta_C;$$ here $\mu_A:A\otimes A\to A$ is the multiplication in $A$, and $\Delta_C:C\to C\otimes C$ is the comultiplication in $C$. It is easy to turn $F_C$ into a functor.

For example, if $M^n$ the the coalgebra of $n\times n$ comatrices, then $F_{M^n}(A)=M_n(A)$ is the algebra of $n\times n$ matrices with entries in $A$. &c.

-
Of course, you can turn this around and view algebras as functors from coalgebras to algebras---in case you also want some 'intuition' about algebras :) –  Mariano Suárez-Alvarez Sep 27 '11 at 17:51
Maybe someone can tell us exactly which functors $\mathrm{Alg}\to\mathrm{Alg}$ we get from coalgebras? –  Mariano Suárez-Alvarez Sep 27 '11 at 18:07
@Mariano: Every continuous endofunctor of algebras is representable (SAFT) if we compose it with the forgetful functor, and the representing object comes equipped with a coalgebra structure. –  Martin Brandenburg Oct 1 '11 at 10:03

a pretty general kind of coalgebra is the path coalgebra of a quiver (directed graph).

-
Ah thanks, I was not aware of this example! A path gets mapped to the sum of all ways to decompose it into two pieces, right? Does this coalgebra structure play an important role somewhere? –  Jan Weidner Jul 23 '12 at 7:49

The following does not answer your question, but gives a particular example in a special case:

The commutative Hopf algebras have a geometric interpretation, they are the coordinate rings of affine group schemes via Yoneda Lemma. Via this interpretation comodules for the coordinate ring are just modules for the group scheme.

Examples of Hopf algebras (I'm not that familiar with coalgebras, that are not Hopf algebras) that I think one should keep in mind:

• group algebras
• enveloping algebras of Lie algebras
• restricted enveloping algebras
• quantized enveloping algebras
• small quantum groups
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Comatrix coalgebras are a nice example of coalgebras which are not Hopf algebras. –  Mariano Suárez-Alvarez Sep 27 '11 at 15:48
(they don't have one-dimensional comodules, so they can't be Hopf) –  Mariano Suárez-Alvarez Sep 27 '11 at 15:52

This is also coming more from the Hopf point of view than the strictly coalgebra point of view.

If you think of algebras heuristically as functions, then a Hopf algebra heuristically can be thought of as the function algebra of a group. A representation of the group $G$ on a vector space $V$ can be thought of as a map $$G \times V \to V.$$ Dualizing gives (roughly) a map $$V^* \to V^* \otimes \mathcal{F}(G),$$ where $\mathcal{F}(G)$ is some appropriate Hopf algebra of functions on $G$. In other words a comodule can be thought of as a representation of the group structure underlying the coalgebra.

If you have a space $V$ which is both a module and a comodule for $\mathcal{F}(G)$ with some compatibility, then $V$ is the space of sections of some homogeneous vector bundle over $G$ (the Hopf Module Theorem). Functions on $G$ act by pointwise multiplication on sections, and $G$ also acts on sections by translation on the base.

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Coalgebras would be more like function algebras of semigroups, right? (there need not be an antipode) –  Yemon Choi Sep 27 '11 at 20:00
Yes, that's correct - a semigroup is all that's needed for a coalgebra structure. –  MTS Sep 27 '11 at 20:52


text:
Hello, @Cyrille! Suppose you have some code written in a file called "my_code.py", and you want to use it in a Sage Notebook. All you have to do is use the command load("my_code.py").
Of course, this example assumes that you have created a notebook in the same folder as the file containing your code. If you have it some where else, you can specify its complete location as the argument of load. For example, suppose you you have your file in your home directory in Linux, you can write in your notebook the following:
load("/home/cyrille/my_code.py")

From the point where you use this command, you will be able to use the functions and subroutines defined in my_code.py.


text:
## Calculating thermodynamic properties in Cantera¶

Here, we describe how Cantera uses species and phase information to calculate thermodynamic properties.

Thermodynamic properties typically depend on information at both the species and phase levels. The user must specify thermodynamic models for both levels, and these selections must be compatible with one another. For instance: one cannot pair certain non-ideal species thermodyamic models with an ideal phase model.

• The user must specify a thermodynamic model for each species and provide inputs that inform how species properties are calculated. For example, the user specifies how the reference enthalpy and entropy values for each species are calcualted, as a function of temperature.

• The user also selects a phase model. This model describes how the species interact with one another to determine phase properties and species specific properties, for a given thermodynamic state. This includes general $$p$$-$$\hat{v}$$-$$T$$ behavior (for example, calculate the phase pressure for a given molar volume, temperature, and chemical composition), as well as how species-specific properties, such as internal energy, entropy, and others depend on the state variables

## Example: The Ideal Gas Model¶

For a simple example: in the Ideal Gas model, one might use 7-parameter NASA polynomials to specify the species reference thermodynamic quantities. These would be used to calculate the reference molar enthalpy $$\hat{h}_k^\circ(T)$$ and entropy $$\hat{s}_k^\circ(T)$$ for a given species $$k$$ as a function of temperature $$T$$. See the NASA Polynomials Species Thermo entry for more information.

At the phase level, the Ideal Gas Law provides the $$P$$-$$\hat{v}$$-$$T$$ relationship. The ideal gas law is an example of an equation of state. This is used, for example, to calculate the pressure as a function of molar volume $$\hat{v}$$, and temperature, $$T$$:

\begin{equation*} p = \frac{\overline{R}T}{\hat{v}} \end{equation*}

where $$\overline{R}$$ is the Universal Gas Constant. The Maxwell relations are used to derive other thermodynamic properties from the equation of state. With the Ideal Gas phase model, these reduce to rather simple forms. For example, for a species $$k$$, the Ideal Gas molar internal energy $$\hat{u}_k$$ and entropy $$\hat{s}_k$$ are:

\begin{equation*} \hat{u}_k = \hat{h}^\circ_k(T) - p\hat{v} \end{equation*}
\begin{equation*} \hat{s}_k = \hat{s}^\circ_k(T) - \overline{R}\ln\left(\frac{pX_k}{p^\circ}\right) \end{equation*}

where $$X_k$$ is the mole fraction of species $$k$$, and where $$p^\circ$$ is the reference pressure at which the properties $$\hat{h}_k^\circ(T)$$ and $$\hat{s}_k^\circ(T)$$ are known.

Please click either of the cards below for details on the species and phase models available in Cantera:

Species

The models and equations that Cantera uses to calculate species thermodynamic properties, such as the NASA 7-parameter polynomial form.

Phases

The theory behind some of Cantera's phase models, such as the Ideal Gas Law.