text:
# Help with the design of an eye in TikZ

In my attempt to create an eye icon to illustrate something in my notes, I came up with the following:

\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[scale=0.25]
\filldraw[color=black] (0,0) circle (1.5cm);
\fill[color=white] (0,0) circle (1cm);
\draw[very thick](-3,0) .. controls (-2,2) and (2,2) .. (3,0) .. controls (2,-2) and (-2,-2) .. (-3,0)--cycle;
\end{tikzpicture}
\end{document}

This produces:

The problem I would have is to resize the diagram in that I do not like how the inner circles are independent of the eye design. May be using nodes would help a lot. Essentially I would like the inner circles shrink and expand with respect to the outer shape. I know that scale and transform shape can help but I am looking for something flexible like \tikzeye{<height>}{<width>} or \tikzeye{<scale>}. Any other designs or variants are welcome. I know the dingbat package has an \eye command defined but it is not too flexible.

EDIT:

I like the clipping idea. If a point P1 and P2 are on the circle and point Q1 and Q2 on the curve, how can make these points, P1 and Q1 and P2 and Q2 be the equal respectively given that P1 and Q1 are at the top and P2 and Q2 are at the bottom? See image below.

\begin{tikzpicture}[scale=0.25]
\filldraw[color=black] (0,0) circle (1.5cm);                                 % Circle with points P1 and P2
\fill[color=white] (0,0) circle (1cm);
\filldraw[color=blue] (0,1.5) circle (3pt) node[above] {{\footnotesize p1}}; % Point P1 on the circle.
\filldraw[color=blue] (0,-1.5) circle (3pt) node[below] {{\footnotesize p2}};% Point P2 on the circle.

\draw[very thick,yshift=6cm](-3,0) .. controls (-2,2) and (2,2) .. (3,0) .. controls (2,-2) and (-2,-2) .. (-3,0)--cycle;                                                                % Curve in question
\filldraw[color=blue] (0,7.5) circle (3pt) node[above] {{\footnotesize q1}};% Point Q1 on the curve.
\filldraw[color=blue] (0,4.5) circle (3pt) node[below] {{\footnotesize q2}};% Point Q2 on the curve.
\end{tikzpicture}

-
I don't understand... currently if you use scale=0.5 in your code, you get an eye of double size (even the inner circles). Anyway, you can also put your whole tikz picture inside a \scalebox (from graphics package) and thus even the line thickness will be scaled, or use \resizebox to specify the final height and width. – JLDiaz Jan 16 '13 at 15:50
I know it's funnier with TikZ. But in this answer Marco Daniel shows how to use \eye symbol from dingbat package and mdframed package to highlight some text. – Ignasi Jan 16 '13 at 17:36
Regarding your edition, I still don't get what do you want. Apparently you want the iris (circle) and the lids (curves) to be tangent, but I don't see how the clipping would do anything in this case. And anyway, your first drawing already has those objects tangent... – JLDiaz Jan 16 '13 at 23:19

This is my version of your eye, trying to give an aspect a bit more organic:

\def\eyepath{(-3,0) .. controls (-2,1.8) and (2,2.2) .. (2.7,0) .. controls (2,-1.2) and (-2,-1.4) .. (-3,0)--cycle;}

\begin{tikzpicture}[scale=0.25]
\clip\eyepath;
\filldraw[color=orange!50!black] (-.2,.2) circle (1.5);
\fill[color=black] (-.2,.2) circle (0.7);
\fill[color=white] (.3,.5) circle (0.2);
\draw[very thick]\eyepath;
\end{tikzpicture}

In relation with your question, you can put this tikzpicture inside a \scalebox (from graphicx package) to get any desired size. Note also that my eye is slightly assymmetrical. You can use \reflectbox to get the other eye :-)

# Update

I could not resist it, and I tried a more realistic eye (probably not in the spirit of the question, sorry!)

\documentclass{article}
\usepackage{tikz}
\usepackage{graphicx}
\usetikzlibrary{calc,decorations.pathmorphing}
\begin{document}

\def\eye#1{\scalebox{#1}{
\def\topedge{(-3,0) .. controls (-2,1.8) and (2,2) .. (2.3,.3)}
\def\bottomedge{(2.3,.3) .. controls (2,-2.2) and (-2,-1.2) .. (-3,0)}
\def\eyepath{\topedge -- \bottomedge --cycle;}

\begin{tikzpicture}[scale=0.25]
\clip\eyepath;
% Iris
\filldraw[color=orange!30!black] (-.2,.2) circle (1.4);
\filldraw[color=orange!40!black] (-.3,-.1) circle (1.1);
% Iris lines
\foreach \a in {0,5,...,360}{
\pgfmathparse{25+28*rnd}
\fill[orange!\pgfmathresult!black, decoration={random steps, segment length=1pt, amplitude=0.3pt}, decorate, line width=0.3pt  ] (-.2,.2) -- ++($(\a+2*rnd:.8+0.3*rnd)$) -- ++(\a+90:3pt) -- cycle;
}
% Pupil
\fill[color=black] (-.2,.2) circle (0.5);
% Sun reflection
\fill[color=white] (90:.8) {[rotate=-30] circle (0.2 and 0.12)};
\draw[line width=2.5mm, draw opacity=0.1, line cap=round]\topedge;
% Eyelids
\draw[line width=1mm, red!40!white!80!black, line cap=round]\bottomedge;
\draw[line width=1.2mm, red!40!white!60!black, line cap=round]\topedge;
% Lacrimal
\fill[red!40!white!80!black] (-2.8,0) circle (.25);
\fill[white] (-2.7,.1) circle (.03);
\end{tikzpicture}
}}

\eye{1}
\end{document}

-
I like the clipping idea. If a point P1 and P2 are on the circle and point Q1 and Q2 on the curve, how can make these points, P1|Q1 and P2|Q2 be the same respectively given that P1 and Q1 are at the top and P2 and Q2 are at the bottom? – azetina Jan 16 '13 at 18:10
I'm lost... could you rephrase your question? What does P1|Q1 mean? What does "be the same respectively" mean? – JLDiaz Jan 16 '13 at 18:21
See my updated question. – azetina Jan 16 '13 at 18:46
I couldn't resist upvoting! +1 – hpesoj626 Jan 17 '13 at 7:29


text:
# Calculation of the specific CO2 emission of a cogeneration plant for the produced electricity

Suppose I have a cogeneration plant that runs on natural gas and produces heat and electricity. The heat effeicieny is $$\eta_{th} = 0.4$$ and the electricity efficiency is $$\eta_{el} = 0.4$$.

The burning of natural gas produces 215 g/kWh of CO2.

For the calculation of the CO2 emission of the generated electricity from the cogeneration plant, one could now calculate $$\frac{215}{0.4} = 537.5$$ g/kWh.

However, I would like to calculate the CO2 emissions for the electricity generation of the cogeneration plant, taking into account that CO2 emissions are saved because another heat generator has to generate less heat.

A natural gas boiler with an efficiency of $$\eta_{th} = 0.9$$ is taken as a reference for heat generation. So the 1 kWh of heat is generated with $$\frac{215}{0.9} = 238.88$$ g of CO2.

How do I calculate the CO2 emissions for the electricity generation in this case?

My calculation attempt:

First I would calculate how much CO2 emssions the boiler has for 0.4 kWh:

$$238.88 \cdot 0.4 = 95.55$$ g of CO2

Then I would subtract this value from the previously calculated co2 emission of the electricity:

$$537.5 - 95.55 = 441.95$$

The electricity generated by the cogeneration plant would then have a CO2 emission of 441.95 g/kWh.

• Hi and welcome to this site! This is not a we do your homework for you site so please xplain what you've tried so far and where you are stuck. I feel something is missing from the question, to what purpose is the efficeincy of the boiler given? the 215g correspond to lower or higher heating value? – mart Jan 6 '20 at 14:27
• @mart Thanks for your answer. I have now edited my question with a few details and a caluclation attempt. – Emma Jan 7 '20 at 13:12

Lastly, the effiencies for the cogeneration plant don't look like real life numbers but are not too far of (I'm just now looking at a datasheet for a small CHP with 38% and 43% respectivly, in my experience this is a rather high $$\eta_{el}$$)

format: plain


text:
# Chapter 9 - Roots and Radicals - 9.1 - Roots and Radicals - Problem Set 9.1 - Page 401: 2

$10$

#### Work Step by Step

Using the properties of radicals, the given expression, $\sqrt{100} ,$ evaluates to \begin{array}{l}\require{cancel} \sqrt{(10)^2} \\\\= 10 .\end{array}

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
## Wednesday, May 30, 2012

### Non-Uniform Quadrature Points in Octave/Matlab

Suppose you are given a bunch of points in the form of a vector x (between "a" and "b") and an accompanying density function p(x)>0. For concreteness, consider

x = [0:0.1:1.0]'; px = 0.3 - (x-0.5).^2;

 The top subfigure depicts the density function p(x), and the cumulative density function C(x). The bottom subfigure depicts the  quadrature points "z" (circles), and the lines mark the end of the intervals, using N=11.
This density function is shown in the picture above.

We want to create a N*1 vector "z" of quadrature points that are distributed according to the density p(x). We also want a vector  "dz" corresponding to the width of strip belonging to a particular quadrature point. As an additional constraint, we assume that we want z(1) = a, and z(N)  = b to match the end-points.

The program attached at the end is able to do this relatively efficiently. In fact, I used the following command to generate the picture above:

[z dz] = GridDensity(x,px,11,1);

GNU Octave Code:

%
%  PROGRAM:
% Takes in a PDF or density function, and spits out a bunch of points in
%       accordance with the PDF
%
%  INPUT:
%       x  = vector of points. It need *not* be equispaced
%       px = vector of same size as x: probability distribution or
%            density function. It need not be normalized but has to be positive.
%   N  = Number of points >= 3. The end points of "x" are included
%       necessarily
%       Pt = Optional argument. If present, then some plotting.
%  OUTPUT:
%       z  = Points distributed according to the density
%       hz = width of the "intervals" - useful to apportion domain to points
%            if you are doing quadrature with the results, for example.
%
%  (*) Sachin Shanbhag, March 5, 2012
%
function [z h] = GridDensity(x,px,N,Pt)

npts = 100;                              % can potentially change
xi   = linspace(min(x),max(x),npts)';    % reinterpolate on equi-spaced axis
pint = interp1(x,px,xi,'spline');        % smoothen using splines
ci   = cumtrapz(xi,pint);
pint = pint/ci(npts);
ci   = ci/ci(npts);                      % normalize ci

alfa = 1/(N-1);                          % alfa/2 + (N-1)*alfa + alfa/2
zij  = zeros(N,1);                       % quadrature interval end marker
z    = zeros(N,1);                       % quadrature point

z(1)  = min(x);
z(N)  = max(x);

%
% ci(Z_j,j+1) = (j - 0.5) * alfa
%
beta  = [0.5:1:N-1.5]'*alfa;
zij   = [z(1); interp1(ci,xi,beta,'spline'); z(N)];
h        = diff(zij);
clear beta;
%
% Quadrature points are not the centroids, but rather the center of masses
%
beta     = [1:1:N-2]'*alfa;
z(2:N-1) = interp1(ci,xi,beta,'spline');

%
% Some plotting if required
%
if(nargin>3)

subplot(2,1,1)

plot(xi,ci,'b-','LineWidth',2,xi,pint,'r-','LineWidth',2);
axis([min(xi),max(xi)]);
legend('CDF','PDF')
title('Visualization')
xlabel('x');
ylabel('CDF(x)/PDF(x)')

subplot(2,1,2)
plot(z,0.5*ones(size(z)),'ro');
axis([min(xi),max(xi)]);
xlabel('z');
hold on;
for i = 2:N
X = [zij(i); zij(i)];
Y = [0; 1];
plot(X,Y,'b')
endfor
hold off;

endif

endfunction

## Friday, May 25, 2012

### MathJax: LaTeX on Blogger - finally!

I am easily excited when I find yet another place where I can use LaTeX syntax to typeset mathematics (Google Docs, OpenOffice, again, presentations using BeamerInkscape).

Finally, it seems, it is not incredibly clunky to write math in Blogger. In fact, it is as convenient as writing it in a native LaTeX document.

The latest avataar of MathJax, while not really new, is new to me. Here are a few links which describe how to go about empowering your Blogger account.

The easiest permanent fix is to add some script in your Blogger template file, as described here.

If you don't like messing with your template, you can take the approach outlined here. Basically, you add some lines to each post in the "html" view, which has math in it.

Here is a mandatory test:
$\frac{df}{dx} = \frac{1}{2} \left(\frac{ab \textrm{ sech}^2(b \sqrt{x})}{x} - \frac{a \tanh(b \sqrt{x})}{x^{3/2}} \right)$
Beautiful!

## Thursday, May 24, 2012

In spite of how close it looked like in the media (thanks in part to the FUD raised by paid shills), Oracle essentially had to walk away with peanuts - if that. As usual Groklaw was among the best places to follow the trial.

There were many fascinating parts. One of my favorites was the exchange between Judge Alsup (who could program) and the lead counsel for Oracle, Boies, on "rangeCheck". You could not write better Hollywood screenplay:
Alsup: I have done, and still do, a significant amount of programming in other languages. I've written blocks of code like rangeCheck a hundred times before. I could do it, you could do it. The idea that someone would copy that when they could do it themselves just as fast, it was an accident. There's no way you could say that was speeding them along to the marketplace. You're one of the best lawyers in America, how could you even make that kind of argument?
Boies: ... I want to come back to rangeCheck.
Alsup: rangeCheck! All it does is make sure the numbers you're inputting are within a range, and gives them some sort of exceptional treatment. That witness, when he said a high school student could do it--
Boies: I'm not an expert on Java -- this is my second case on Java, but I'm not an expert, and I probably couldn't program that in six months.
Priceless.

Of course, Oracle being Oracle, they will probably appeal. Here's what Linus Torvalds has to say (full disclosure: on his Google+ account)
Prediction: instead of Oracle coming out and admitting they were morons about their idiotic suit against Android, they'll come out posturing and talk about how they'll be vindicated, and pay lawyers to take it to the next level of idiocy.

Sometimes I really wish I wasn't always right. It's a curse, I tell you.

## Monday, May 21, 2012

### Some interesting cartoons at Spiked Math

1. Epsilon-Delta: reminded me of times long gone

2. My big fat Greek letter

3.  Hilarious IQ test

Check out the other stuff there too!

## Thursday, May 17, 2012

### The Checklist Manifesto

I've been a big fan of checklists since as long as I can remember. So much so that a lot of friends and family mock my obsession with them.

In high-school, I had an exam-day checklist (set alarm, carry extra pens, make sure of the time-table etc.). I have about five different types of travel checklists, depending on whether I am flying, driving, going camping, or to a conference etc. I have checklists for writing proposals, and papers and on and on.

You get the idea!

So it was with much delight that I read Atul Gawande's article in the New Yorker called The Checklist a few years ago, and his book called "The Checklist Manifesto" which fleshes out some of the major themes from his article, last month.

While it was "preaching to the choir", I thought some parts were very interesting. I particularly enjoyed the chapter called "Hero in the age of checklists".

Here is a link to a recent talk at TED by Atul Gawande.

## Monday, May 14, 2012

### Commencement Speech: David Foster Wallace

A really nice, and somewhat offbeat, commencement speech by the late David Foster Wallace. I remember enjoying his book "Inifinite Jest".

YouTube also has the actual speech:

## Wednesday, May 9, 2012

### Shitty Economics

Juvenile but entertaining! (via Barry Ritzholtz)

## Monday, May 7, 2012

1. Interesting "Martin Gardner" puzzle if you have not tried it before

2. Three puzzles in this post by Tanya Khovanova

3. My wife told me about a "lateral" puzzle she heard in her office. Complete the series:

|||, |CC, C, |C, ?

## Thursday, May 3, 2012

I noticed a curious "coincidence" today.

I received an innocuous looking deal on vacuum cleaner bags from Amazon in my email. It was for the exact make and model of vacuum cleaner that I have owned for about two years now.

Nothing out of the ordinary here. Except for two things.

One, I was just about to run out of bags. I remember, because I made a mental note last weekend. Two, I did not buy the vacuum cleaner from Amazon!

I scoured the Trash folder of my email to see if there were prior deals like this, and if I just happened to notice it this time.  Nothing! Nada!

I recalled this story that appeared in the NYT article "How companies learn your secrets" earlier this year:
... a man walked into a Target outside Minneapolis and demanded to see the manager. He was clutching coupons that had been sent to his daughter, and he was angry, according to an employee who participated in the conversation.
“My daughter got this in the mail!” he said. “She’s still in high school, and you’re sending her coupons for baby clothes and cribs? Are you trying to encourage her to get pregnant?”
The manager didn’t have any idea what the man was talking about. He looked at the mailer. Sure enough, it was addressed to the man’s daughter and contained advertisements for maternity clothing, nursery furniture and pictures of smiling infants. The manager apologized and then called a few days later to apologize again.
On the phone, though, the father was somewhat abashed. “I had a talk with my daughter,” he said. “It turns out there’s been some activities in my house I haven’t been completely aware of. She’s due in August. I owe you an apology.”
If you haven't read the article before (it got a lot of secondary press) you should!

## Wednesday, May 2, 2012

### User-defined Colors in Grace

Last week, I was working on a manuscript with a collaborators. As we approached the final draft, we decided to make the "colors" in our figures consistent. Unfortunately, we used different programs to make our graphs, and "red" in program X is not the same shade of "red" in program Y.

Grace (the program I use) offers a choice of 16 standard colors, from the usual pull down menu.

However, it allows for endless fine-tuning.

Once you save your graph (as graph.agr), open it up in a text editor. The color-definitions are clustered in lines that look like:
@map color 0 to (255, 255, 255), "white"
@map color 1 to (0, 0, 0), "black"
@map color 2 to (255, 0, 0), "red"
Let us say, we were not happy with the shade of grey available (it does seem a little too light). Look for the line:
@map color 7 to (220, 220, 220), "grey"
and change it to something like:
@map color 7 to (100, 100, 100), "grey"
You can tweak all the other color definitions to suit your taste or requirements.


text:
# How to decide between a inaccurate system and a accurate system with capacity constraints to do a stream of jobs

Let's say you have a list of at most N Jobs to be done which are coming in a stream. There are two kinds of systems that can do the job:

1. System 1: A very fast system, which however, only does the job correctly some of the time.
2. System 2: A slower system, which does the job correctly all the time, however has a fixed capacity of only doing C jobs, which is small compared to number of total jobs (C < N).

The time required for doing all the jobs is the same for each system.

The Probability of the System 1 doing the job correctly (let's call say P_sys1(Job)) depends upon the job itself (so different jobs have different probabilities, some can be 5% some can be as high as 99%).

Let's assume the cost of doing a Job badly is Cost(Job). So expected cost of a bad job is (1 - P_sys1 (Job)) * Cost (Job).

We need to decide to send the Job to System 1 or 2 as soon as it arrives. Once a job is sent to the either system, it's done and cannot be reprocessed if done wrongly.

How would we minimise the overall expected cost of doing all the Jobs.

Or equivalently, how will decide which jobs will we send to System 2 to do?

(intuitively, you want to send the Jobs with the highest cost & smallest probability to succeed in System 1).

Note: since the jobs are coming as a stream you don't know all the jobs before hand. You can assume a prediction of what kind of jobs will come in a stream, however, it will also have its own error.

Would even appreciate links to any papers or research done on similar problems.

• Do you know how much time the jobs take? The C jobs limit is simultaneous or a total maximum capacity (e.g. independent of time; the machine can do at most C jobs total). Is the job is done wrong, do you need to reprocess it? I.e. is the desired final output to have all jobs done correctly? – dhasson Aug 10 '20 at 20:38
• @dhasson thanks for you comments, I've modified the question accordingly. Just to clarify, C is the total max capacity, jobs cannot be reprocessed and the desired output is to reduce the total cost of doing jobs badly. – dg428 Aug 11 '20 at 4:41
• @dg428, as per you have a complex production system w.r.t the Probability of doing the job correctly, I recommended using the simulation-optimization approach. In your case, discrete event simulation could be applied. Would you try this? – A.Omidi Aug 11 '20 at 4:57
• @A.Omidi thanks, yeah I was thinking about using a simulation. However, the question still remains how? As in what algorithm or strategy should I use? – dg428 Aug 11 '20 at 7:57
• Do you know the cost and P_sys1 for each job in the stream at the outset, or are those revealed only when the jobs arrive? If the latter, do you at least have probability distributions for cost and P_sys1? – prubin Aug 11 '20 at 20:19

To use simulation- optimization procedure, One way would be as follows:

Suppose, you have some jobs in which need to process in stream or chase customer demand. You will have to calculate the probability distribution function (PDF) of the arrival jobs based on the orders or demands. (e.g. using exponential distribution function with a specific parameter).

Then, you would need to estimate the jobs failure probability function by using something like goodness-of-fit method. In the next step, when you could estimate these functions, you can use some criteria, for instance, based on "the highest cost & smallest probability to succeed in System 1" as you mentioned.

Once, the jobs are assigned to its specific system you can calculate the detailed schedule based on the MP model or maybe simulation process to achieve the optimum efficiency.

The process can be depicted as: The arrival jobs=> assign PDF=> define criteria=> schedule systems.

I hope it Would be helpful?

• thanks, but I am actually looking for how to quantify into a equation or algorithm "the highest cost & smallest probability to succeed in System 1" – dg428 Aug 13 '20 at 5:22
• @dg428, if you need to use a MP, the stochastic programming might be useful. – A.Omidi Aug 13 '20 at 6:24

I'll propose an "online" decision process (heuristic), but I don't know if you have the data / forecasting ability to make it work, so I'm not sure it would be helpful.

Let $$C_0$$ be the remaining capacity of the slow system when the next job arrives (which you will definitely know), and let $$\gamma$$ be the expected "bad job" cost of the newly arrived job (which I'm assuming you will know once the job arrives). Now, for the tricky part, let $$\Gamma$$ be the expected "bad job" cost of the $$C_0$$th most expensive remaining (yet to arrive) job. If there are not that many jobs left to come, set $$\Gamma = 0$$. For instance, if $$C=6$$, you're being asked to peer into the future and estimate what the fifth worst "bad job" cost out of all remaining jobs might be. (This is known as an order statistic.) You'll need a way to forecast $$\Gamma$$, noting that it depends on both the number of jobs to come and the nature of the jobs.

If you can get a figure for $$\Gamma$$, the decision rule is simple. If you route the newly arrived job to the fast process, you incur expected cost $$\gamma$$ (and $$C_0$$ remains the capacity left in the slow process). If you route the newly arrived job to the slow process, and charitably assuming you make all remaining decisions optimally, you end up sending the job with expected cost $$\Gamma$$ to the fast line (because you don't have capacity for it on the slow line) when, had you sent the current job to the fast line, you would have sent that job to the slow line. So the logic is to use the slow line now if $$\gamma > \Gamma$$ and the fast line if $$\gamma < \Gamma$$.

• Thank you @prubin for your answer. Yes, was thinking in the same lines, basically weighing the expected cost of doing the current job using the slow process vs the cost of having to send a higher cost job to the fast process down the line. However, do you know any other methods/literature which provides a more sophisticated approach? Or even industry solutions for similar real world problems? Thanks! – dg428 Aug 24 '20 at 6:56
• Sorry, no, I don't know anything directly applicable. You might look at how hospitals are allocating certain facilities (such as beds in isolation wards) during the pandemic. In some cases, they face similar issues: if they use a scarce bed on a somewhat sick arrival, will it be gone when a sicker person arrives. They don't have a limited input queue, though. – prubin Aug 24 '20 at 15:38


text:
# Green's function and derivatives of Dirac's Delta

In my textbook in a section about electromagnetism I came across the following lines: \begin{align} \left[\left(\frac{\partial}{\partial x^0}\right)^2 - \Delta\right] \frac{\delta(x^0-|\mathbf{x}|)}{|\mathbf{x}|} &= \frac{1}{|\mathbf{x}|}\delta'' - \Delta \left(\frac{1}{|\mathbf{x}|}\right) \delta(x^0 - |\mathbf{x}|) - \frac{1}{|\mathbf{x}|} \Delta \delta(x^0-|\mathbf{x}|) - 2 \nabla \left(\frac{1}{|\mathbf{x}|}\right) \cdot \nabla \delta(x^0-|\mathbf{x}|)\\ &= \frac{1}{4\pi} \delta(\mathbf{x}) \delta(x^0-|\mathbf{x}|). \end{align} I'm having problems seeing the second equation; I know that \begin{align} \nabla^2 \frac{1}{|\mathbf{x}|} = -4\pi \delta(\mathbf{x}), \end{align} but I don't know how to deal with the derivatives of the delta distribution.

Can anyone explain, how to see that the equation above is correct?

Let's look at the a more general case: a function of the form $f(x^0 - |\mathbf{x}|)/|\mathbf{x}|$. For the sake of compactness, I'll switch to spherical polar coordinates where $|\mathbf{x}| = r$. We therefore have $$\begin{multline} \left[\left(\frac{\partial}{\partial x^0}\right)^2 - \nabla^2 \right] \frac{f(x^0-|\mathbf{x}|)}{|\mathbf{x}|} \\= \frac{f''(x^0 - r)}{r} - f(x^0 - r) \nabla^2 \left( \frac{1}{r} \right) - \frac{1}{r} \nabla^2 f(x^0 - r) - 2 \left[ \nabla \left(\frac{1}{r}\right) \right] \cdot \left[\nabla (f(x^0 - r)) \right] \end{multline}$$In spherical coordinates, we have $$\frac{1}{r} \nabla^2 f(x^0 - r) = \frac{1}{r} \left[ \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial f(x^0 - r)}{\partial r} \right) \right] = \frac{1}{r} f''(x^0 - r) - \frac{2}{r^2} f'(x^0 - r)$$ and $$2 \left[ \nabla \left(\frac{1}{r}\right) \right] \cdot \left[\nabla (f(x^0 - r)) \right] = 2 \left( - \frac{1}{r^2} \hat{r} \right) \cdot \left( - f'(x^0 - r) \hat{r} \right) = \frac{2}{r^2} f'(x^0 - r).$$ (If you wish to check these, note that $$\frac{\partial f(x^0 - r)}{\partial r} = - f'(x^0 - r) \qquad \frac{\partial^2 f(x^0 - r)}{\partial r^2} = f''(x^0 - r)$$ due to the chain rule.) Putting all of these together, we find that $$\frac{f''(x^0 - r)}{r} - \frac{1}{r} \nabla^2 f(x^0 - r) - 2 \left[ \nabla \left(\frac{1}{r}\right) \right] \cdot \left[\nabla (f(x^0 - r)) \right] = 0,$$ and so for any function $f$, we have $$\left[\left(\frac{\partial}{\partial x^0}\right)^2 - \nabla^2 \right] \frac{f(x^0-|\mathbf{x}|)}{|\mathbf{x}|} = - f(x^0 - r) \nabla^2 \left( \frac{1}{r} \right) = 4 \pi f(x^0 - |\mathbf{x}|) \delta^3(\mathbf{x}).$$ The above derivation works for any function of $x^0 - r$ alone, including $\delta(x^0 - r)$.
The reason this works for any function, by the way, is that the general solution for the wave equation in three dimensions under the assumption of spherical symmetry (and excluding the origin) is $$u(r, t) = \frac{1}{r} \left[ f(r - t) + g(r + t) \right]$$ for any two functions $f$ and $g$.


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# QMA definition of difference between probabilities intuition

I'm reading about the complexity classes related to quantum computation, currently I'm studying QMA class. A language is in QMA(c,s) if there exists a polynomial time verifier and polynomial $$p(n)$$ such that

if $$x\in L \implies$$ there exists quantum state $$\psi$$ such that the probability V accepts is greater than $$c$$

if $$x \notin L \implies$$ For all quantum states $$\psi$$ the probability that V accepts is less than $$s$$.

Usually it is remarked that $$c$$ and $$s$$ have the following property $$c-s = \frac{1}{poly(n)}$$. This last condition is what confuses me. What is the intuition behind imposing this condition on the difference of $$c$$ and $$s$$? Not understanding this makes me feel I haven't fully understood QMA class definition.

Something similar happens with the problem called Local Hamiltonian problem. Here two constants are defined such that $$a-b = \frac{1}{poly(n)}$$. If someone could explain the intuition behind this and if it is related to the previous question that would be great. Thanks!

While this question deals with the difference between $$\mathsf{BPP}$$ and $$\mathsf{PP}$$, it should still be useful for you to read through. The "short answer" is that the "gap" between $$c$$ and $$s$$ allows for standard amplification techniques to be used, which end up showing that $$\mathsf{QMA}(c,s) = \mathsf{QMA}(2^{-n},1-2^{-n})$$ (provided $$c - s$$ meets the condition you wrote down).
The "case study" of $$\mathsf{BPP}$$ vs $$\mathsf{PP}$$ should be interesting overall for your question, as the only difference between them is the presence / lack of the $$c - s \geq 1 / \mathsf{poly}(n)$$ condition. It's known that: $$\mathsf{BQP}\subseteq\mathsf{QMA}\subseteq \mathsf{PP}\subseteq\mathsf{PSPACE}$$ But: $$\mathsf{BPP}\subseteq\mathsf{BQP}$$ This is quite a large difference. I'm unsure of what is known about $$\mathsf{QMA}(c,s)$$ where $$c - s$$ is negligible (meaning $$n^{-\omega(1)}$$), where amplification techniques can't be used.


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# Looking for (even unofficial) duelist variants

As @KRyan told me, it seems like the duelist PrC is a little bit broken. But my party is currently lost in the middle of savages countries far from any civilization and it is very unlikely that I meet any PrC coach before a long time.

I am currently swashbuckler 5/Dragonmark heir 2/duelist 2/blade of orien 1, and I was planning to keep leveling as a duelist for some time. I think this build is not completely unbalanced compared to the builds of my teammates as they don't put out as much optimization effort as I do. I chose to train the class knowing its flaws, but I considered it was a realistic evolution for a swashbuckler.

Nevertheless, some duelist levels like the 4th: "+2 on reflex saving throws" are really boring, so I am beginning to research additional content about this PrC like official alternative levels or widely acknowledged homebrews.

EDIT: I am specifically looking for optimizations of the duelist class, which from my point of view essentially sums up to "a character with style (like the swashbuckler) using a rapier and no armor and adding INT to AC". What I am expecting is mostly alternative class levels, either as a few errata or a total redesign. I would rather it being some already well known content so that it would be easier to negotiate it with my DM.

(In case you were wondering, I did try negotiating taking only 3 levels of swashbuckler, but our DM insisted on us taking 5 level on our baseclass before multiclassing. Plus I liked the flamboyance of this type of character and did not want to change it to much. KRyan suggested me rebuilding as Factotum or Warblade but they are from books that are not allowed in my campaign.)

• Prestige class coaches are not a part of most prestige classes in 3.5, by the way. A few prestige classes require training from someone specific, but most don’t. Just saying. Then again, your DM clearly has a number of extremely ill-advised houserules going on here anyway, so that one is just one more, I suppose. Sep 3 '14 at 13:48
• As a swashbuckler/duelist, you should have an Int score that means you're gaining AC for at least the first 5 levels of Duelist. Also at 5th level, the Duelist gets their last decent feature. Sep 3 '14 at 13:52
• Don't you think that 7th level is cool? @KRyan: Can't do a thing about this, this is our DM's choice. I already had to give up on the dervishes since they don't exist in the campaign setting (or if they do, there was none of them in our part of the world). Sep 3 '14 at 13:57
• I edited the question to make it more specific ;) Sep 3 '14 at 14:13

I wrote a dualist, a dual-wielding duelist (pun very much intended!). It works fairly well from a swashbuckler base (uses your free Weapon Finesse for entry, skills are easy and recommended anyway).

The dualist is different from the duelist, obviously, in that the dualist uses two weapons, rather than keeping one hand free. It also is not specifically Intelligence-based, but on the flip side you can (and should) use armor. All that said, it puts your Insightful Strike feature to good use (as a static damage bonus, that damage is multiplied on a critical hit, and the dualist is quite good at critical hits), and as I said, your Weapon Finesse bonus feat eases entry quite nicely.

The dualist is all about attacks of opportunity, using them as a kind of round-by-round resource for its special abilities. The dualist parries, and at higher levels, counter-attacks, using (typically) a pair of rapiers. The dualist is also fairly “sticky,” making it excellent at defending its allies.

I’ve personally played the dualist a few times, and it was quite fun. Furthermore, I know several other people who have played it, and I’ve received great feedback. It’s one of my best-received classes, and I think it’s likely an excellent fit for you.

• @Miniman Yes-ish. The dualism class feature modifies the requirements of these feats for the dualist, allowing her to qualify for them. The bonus feats blurb basically just means you get to “delay” a bonus feat. Not great design, but this was literally the second class I ever wrote, and honestly I’m still pretty pleased with it even though I really should have just given those feats. Jul 26 '16 at 19:07
• Ah, I completely misread dualism - that makes a lot more sense. Jul 26 '16 at 22:50

A third answer, which leads me to believe that we needed more details in the initial question, but anyway, here’s yet another idea. With thanks to Miniman for suggesting the NWN2 duelist and its haste effect, which made me think of this, and to Kolja Liquette and Brian Moran, who wrote the swiftblade, and to Tom Welch for making it not-suck, I present yet-another duelist variant, this one really playing up the speed angle.

## Duelist

“How did I avoid that?”

“By being fast like a freak.”

As a duelist, you move with almost super-human speed, slashing and stabbing, parrying and dodging, to ensure your presence is felt by everyone.

### Prerequisites

To become a duelist, you must be fast and precise already.

• Feats
• Combat Reflexes
• Weapon Finesse
• Special
• One of the following:
• +5 BAB
• +4 BAB, 1d6 precision damage, and 8 ranks in Tumble
• +3 BAB, 3d6 precision damage, and 8 ranks in Balance, Jump, and Tumble
• Dodge feat, or Grace class feature

### Class Features

HD d8

Skills (4+Int): Balance, Climb, Concentration, Craft, Diplomacy, Escape Artist, Gather Information, Intimidate, Jump, Knowledge (history), Knowledge (local), Knowledge (nobility & royalty), Perform, Profession, Sense Motive, Sleight of Hand, and Tumble.

Level BAB Fort Ref Will Special
1st +1 +0 +2 +0 Canny dodge, innate haste, grace, precise strike +1d6
2nd +2 +0 +3 +0 Improved reaction, uncanny dodge
3rd +3 +1 +3 +1 Blurred alacrity, flurry of strikes
4th +4 +1 +4 +1 Sudden speed, precise strike +2d6
5th +5 +1 +4 +1 Perpetual options
6th +6 +2 +5 +2 Evasive maneuvers
7th +7 +2 +5 +2 Constant quickness, precise strike +3d6
8th +8 +2 +6 +2 Double time
9th +9 +3 +6 +3 Can’t stop me now
10th +10 +3 +7 +3 Constant celerity, precise strike +4d6

Weapon and Armor Proficiency: You do not gain proficiency in any new weapons or armor.

Canny Dodge (Ex): You add your Intelligence bonus, if any, to your AC, so long as you do not wear medium or heavy armor, use a tower shield, or carry a medium or heavy load. You lose this bonus any time you would be denied your Dexterity bonus to AC, even if you do not have one.

Canny dodge can be used in place of Dodge to qualify for a feat, prestige class, or other special ability. When you do so, you may designate a creature as “the target of your Dodge feat” for the purpose of features that care, but you gain no special AC bonus against that creature from this ability. You can still take Dodge along with this feature (and then would get the bonus).

Innate Haste (Ex): You can enter into a state of near-perfect focus, moving with almost supernatural speed and precision. A number of times per day equal to 3 plus your Intelligence bonus (if any), you may activate your inner haste as a swift action, gaining the benefits of the haste spell for a number of rounds equal to your class level plus your Intelligence modifier.

Despite mimicking the haste spell, this is an extraordinary feat of speed and talent, not a magical effect. As such, it functions perfectly fine in a dead magic zone or antimagic field, and cannot be countered, disrupted, or dispelled. However, it does not stack with the haste spell, nor with other effects that do not stack with haste, such as a speed weapon.

Effects that suppress haste suppress this feature; effects that are suppressed by haste are suppressed by this feature.

This ability cannot be used if you wear medium or heavy armor, use a tower shield, or carry a medium or heavy load. You also require a free hand to maintain this state, using it for balance and to counter-act the torque on your body. This hand must be empty, and the arm cannot have a shield strapped to it. If you fail to meet any of these conditions, this effect immediately ends.

Grace (Ex): Your duelist class level stacks with your swashbuckler class level, if any, for the purposes of the Dodge bonus granted by the grace class feature. If you have no swashbuckler levels, you gain the grace feature of a swashbuckler of your level.

Precise Strike (Ex): You have the ability to strike precisely with a light or one-handed piercing weapon, gaining an extra 1d6 to your normal damage roll. At each of 4th, 7th, and 10th level, this damage bonus increases by 1d6, to a maximum of 4d6 at 10th level.

When making a precise strike, you cannot attack with a weapon in your other hand or use a shield. Your precise strike only works against living creatures with discernible anatomies. Any creature that is immune to critical hits is not vulnerable to a precise strike, and any item or ability that protects a creature from critical hits also protects a creature from a precise strike.

Precise strike stacks with any sneak attack, sudden strike, skirmish, or other forms of precision damage you may have. Precise strike counts as sneak attack with the same number of damage dice for the purpose of requirements.

Improved Reaction (Ex): Starting at second level, you add your Intelligence bonus, if any, to your Initiative checks, and furthermore may take attacks of opportunity and immediate actions even while flat-footed.

Uncanny Dodge (Ex): You gain uncanny dodge at 2nd level, so that you retain your Dexterity bonus to AC (if any) even if you are caught flat-footed or struck by an invisible attacker. However, you still lose your Dexterity bonus to AC if immobilized. If you already have uncanny dodge, you gain improved uncanny dodge, instead.

Blurred Alacrity (Ex): Starting at 3rd level, you gain a miss chance equal to 10% plus 5% times your Intelligence bonus (if any) while under any haste effect, whether from your innate haste or from a haste spell cast upon you.

Flurry of Strikes (Ex): The number of bonus attacks you may make in a full-attack due to a haste effect or speed weapon doubles at 3rd level.

Sudden Speed (Ex): You may activate innate haste as an immediate action once you reach 4th level.

Perpetual Options (Ex): As a 5th level duelist, you may choose to improve the effects of any haste effect on you in one of the following ways:

1. You may gain a second move action; if you do so, you lose the movement speed bonuses provided by the haste effect.

2. You may gain a second standard action; if you do so, you lose the extra attacks that haste would otherwise allow you to make during a full-attack.

You must choose which, if either, option you will use at the beginning of each turn. You may not choose to take both at the same time.

Evasive Maneuvers (Ex): At 6th level, your speed makes you difficult to target with spells. Individually-targeted magical effects have a chance to fail to hit you equal to your miss chance from blurred alacrity. This effect does not stack with that of blink or similar.

Constant Quickness (Ex): Starting at 7th level, your innate haste becomes a permanent extraordinary effect. You no longer need to activate it, nor is it limited in usage. You must still meet its usual requirements; otherwise it is suppressed for as long as you fail to meet them.

Double Time (Ex): The speed at which you move becomes so high at 8th level that you literally can do twice as much as someone else. You no longer receive movement speed bonuses or extra attacks from haste. Instead, each combat you roll initiative twice, and take two turns each round. These two turns are completely independent of one another.

You lose this benefit if your haste effect is suppressed. If your haste effect is suppressed in the middle of a combat, the turn you lose is always your next one. If you regain a haste effect in the same combat, that turn is restored, at the same initiative count.

Can’t Stop Me Now (Ex): At 9th level, you may take your own turn in between every round of apparent time during someone else’s time stop effect.

You lose this benefit if your haste effect is suppressed.

Certain Celerity (Ex): As you are among the quickest beings out there, getting the drop on you is all but impossible; even if you are surprised, you get over that surprise and begin to react even before many of your ambushers do. And when you aren’t surprised, you move so fast that others are. Beginning at 10th level, you always act in the surprise round, even if there otherwise would not be a surprise round.

You lose this benefit if your haste effect is suppressed.

• Double Time works out, at best, to having twice as many attacks (similar to using two-weapon fighting), but it allows unparalleled reaction times, getting to choose your actions twice every round. Certain Celerity mimics the Quickness feature of dire tortoises (Sandstorm). Sep 8 '14 at 21:41
• I am quite impressed by the great effort you are putting into answering this question, thanks you very much for your time and precious advices. About Double Time I would forbid using two weapon fighting in conjonction with it Sep 15 '14 at 10:10
• @Epeedefeu I did, since Innate Haste does not work unless you have a hand free. You could use TWF if someone cast haste on you, to get a haste effect without Innate Haste, but then you are putting a lot of resources (TWF feats, spell slots, actions in combat) so I think that’s fine (and in fact wouldn’t recommend doing so. Sep 15 '14 at 12:11

For a less significant change to the duelist than my dualist answer, here are some thoughts:

Prerequisite Feats: Weapon Finesse

Dodge and Mobility are terrible feats; there’s no good reason to take them, uhm. ever. Aside from as prerequisites, and duelist never offers anything nearly good enough to justify so many burned feats.

Canny Defense: If a duelist has at least one hand free (e.g. no shield, even a buckler, or other held tool or weapon), and is wearing armor no heavier than light, she adds her Intelligence bonus, if any, to her AC. She loses this bonus any time she would be denied her Dexterity bonus to AC, even if she does not have one.

Canny Defense can be used in place of Dodge to qualify for a feat, prestige class, or other special ability. When a duelist does so, she may designate a creature as “the target of her Dodge feat” for the purpose of features that care, but gains no special AC bonus against that creature. A duelist can still take Dodge along with this feature (and then would get the bonus).

The limitation of not using a two-handed weapon or a shield is already more than enough of a drawback on this; limiting it to your level or eliminating any armor goes too far. Compare the monk’s Wisdom to AC, which works even if you lose Dex to AC and regardless of what your hands are doing, and without requiring you to wait until 6th level or to have to qualify.

If you compare it to the swordsage’s Wis-to-AC (which works in light armor), even this version becomes quite sad.

The original Canny Defense wasn’t worth using; you could get cheap armor with no drawbacks that provided a larger bonus.

Also, remember how I said Dodge was worthless? It is. Now you don’t have to take if something else wants you to. This is similar to how Desert Wind Dodge from Tome of Battle, Midnight Dodge from Magic of Incarnum, or Rapid Dodge from Races of the Wild work.

Grace: Your duelist levels stack with your swashbuckler levels for the purpose of your Grace class feature.

This is all-but-pointless, but at least it kind of synergizes. Note that this means Grace is now a 1st-level feature.

Precise Strike: A duelist has the ability to strike precisely with a light or one-handed piercing weapon, gaining an extra 1d6 to her normal damage roll. At each of 4th, 7th, and 10th level, this damage bonuse increases by 1d6, to a maximum of 4d6 at 10th level.

When making a precise strike, a duelist cannot attack with a weapon in her other hand or use a shield. A duelist’s precise strike only works against living creatures with discernible anatomies. Any creature that is immune to critical hits is not vulnerable to a precise strike, and any item or ability that protects a creature from critical hits also protects a creature from a precise strike.

This is massively weaker than Sneak Attack, since you can’t dual wield to apply it more often, and you get less of it than a rogue would, but at least now it is something.

Improved Reaction: At 2nd level, a duelist gains a bonus on Initiative checks equal to her Intelligence bonus, if any. At 5th level, this bonus doubles. This stacks with Improved Initiative.

Factota get Int-to-Initiative at level 3. As a base class. That is also adding it to a ton of other things. At least eventually this gets to be better than that. You’ll probably win all the initiative. Bully for you.

Unless the wizard wants to win. Then he will, because he’s a wizard.

Note that I moved the doubled version to 5th instead of 4th. That’s because 4th is already granting +1d6 Precise Strike damage, and 5th is now empty.

Enhanced Mobility: Starting at 3rd level, a duelist may always take-10 on balance, climb, jump, and tumble checks, even if stress would otherwise prevent her. A duelist must have at least one hand free and must not be wearing armor heavier than light to use this feature.

Enhanced Mobility can be used in place of Mobility to qualify for a feat, prestige class, or other special ability. You can still take Mobility along with this feature.

Mobility is garbage because you shouldn’t be provoking attacks of opportunity at all, you should be using Tumble to avoid them. This, at least, lets you get better at the thing you actually want to do, plus it covers the Mobility requirement of other things.

Evasive Reflexes: Whenever someone provokes an attack of opportunity from a 4th-level duelist, she may instead take a 5-ft. step, even if she has already moved or even already taken a 5-ft. step this round. This does not count as using an attack of opportunity for the round, but the duelist also misses the opportunity to attack, even if she steps into a square that still threatens the provoking creature.

Evasive Reflexes can be used in place of Combat Reflexes to qualify for a feat, prestige class, or other special ability. You can still take Combat Reflexes along with this feature.

This is a feat from Tome of Battle, but since you don’t have that book it seemed like a fitting class feature. Combat Reflexes is a good feat, but the Tome of Battle feat had that clause and it didn’t seem worth removing; maybe you want something that requires Combat Reflexes but don’t want that feat per se.

Acrobatic Charge: When a 6th-level duelist charges, she may make one 90° turn during her charge. In addition, she may charge over difficult terrain that normally slows movement. Depending on the circumstance, she may still need to make appropriate checks to successfully move over the terrain.

Avoiding difficult terrain is pretty nice, but you are in no way a charger. Also, you can arguably use skill checks to avoid difficult terrain anyway, and if you can, you should already be very good at those skills.

Elaborate Parry: A creature that attacks an ally inside the threatened area of a 7th-level duelist provokes an attack of opportunity from the duelist. If the attack of opportunity hits, the creature’s attack automatically misses. The duelist must choose to use take this attack before the attacking creature rolls its attack roll.

Stolen from the dualist, really, but basically, this allows you to protect others a bit, rather than just pumping your own AC at the expense of being pointless. Someone with a ton of AC but no offensive ability just gets ignored, and does not help his allies win the fight. With this, you make it hard to ignore you.

Deflect Arrows: As the duelist class feature.

This is pointless, but whatever. I can’t think of anything better to give you.

• About dodge, Swashbucklers gain a very similar feat at level 5 and my DM agreed that it could be counted as a "virtual feat" (as described in PHBII). So I did not have to take it to qualify for the duelist Sep 3 '14 at 16:02
• @Epeedefeu Wow, why didn’t they just give Dodge there? Mobility is still awful, though you can technically qualify with the mobility special armor property. Getting a 4,000+ gp armor at 5th level’s a bit dicey though. Sep 3 '14 at 16:14
• About Acrobatic Charge it is strictly equivalent to a combination of the skill tricks Nimble Charge and Twisted Charge from Complete Scoundrel (that I have taken). So it just saves 4 skill points. Still better than the original though :) Sep 3 '14 at 16:38
• @KRyan The class feature elaborate parry is just about the only upshot to the core duelist. A dude with 7 levels of duelist fighting defensively takes a -2 penalty to attack rolls for a might-even-impress-the-wizard +10 bonus to Armor Class. As a big pile of numbers, that's really not bad, but it's a long slog. Further, the canny defense class feature of the 3.0 Duelist (Dragon #275 96-7) just added the creature's Int bonus to AC, but Mundanes Can't Have Nice Things. Sep 3 '14 at 17:37
• @Epeedefeu This would stack with that so you can make more turns and have a windy-er charge. Plus skill tricks are once per encounter, so this lets you do it every time you charge. As you say, though, that’s still hardly amazing. Sep 4 '14 at 4:44

For further reference, here are some revised duelists I came across while searching on the topic. I am listing everything I have found but did not take a deep look, so there may be content of unequal quality and duplicates.

Will keep updating as I find more (until I get blocked for posting too many external links that is ;)

• Seerow’s is a mediocre class. The challenge needs errata to indicate what the penalty is to, so that’s useless. You deal precision damage that you cannot multiply through dual-wielding. Until 10th level, tons of things are immune to it. You absolutely set two feats on fire. Feinting as a swift is cute, except one of your few advantages is that you don’t care too much about the enemy being flat-footed – and that’s only for one attack anyway. Melee Weapon Mastery is nice, I guess, except it doesn’t cover the Greater requirements, and duelist levels don’t count as fighter levels. Sep 4 '14 at 12:33
• Bauglir’s is worse. The fake Power Attack has poor returns, even with the class feature, Tactical Strikes is bizarrely limited, and most of the rest are fighter bonus feats, except you don’t get to pick. It does get an unreasonable amount of AC, but that’s only going to trigger opponents ignoring the duelist as an annoying-to-kill non-threat. Sep 4 '14 at 12:38
• Siosilvar’s is quite amusing, with all of the quotes. I see some things I like, but ultimately it seems to be some mobility and some damage, but neither in truly impressive amounts. Also, Riposte is just dumb; if it’s a standard action, it’s not an AoO. Burst Evasion is pointless if you (the player) aren’t prescient. Lots of stuff on AoOs... which is a real shame, since you can’t get reach to actually gain some real presence. The extra/immediate 5-ft. steps are really nice though. Sep 4 '14 at 12:45
• Person_Man’s is disappointing; he usually has better stuff. This one’s version of Grace is interesting but I tend to doubt there’s much in the way of truly impressive effects you can stack on that. Advance and Crippling Strike are good features, but come very late. Sep 4 '14 at 12:48
• I made the answer "community wiki". Feel free to put your comments directly into it, since it seems that otherwise they might end up beign deleted, which would be a waste ;) Sep 4 '14 at 12:51


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# proton synchrotron theory

In addition to protons, PS has accelerated alpha particles, oxygen and sulphur nuclei, electrons, positrons and antiprotons.[1]. Intrinsic Features of Slow Cycle SynchrotronSynchrotron Based System zAfter the first application to LLUMC in 1990, which was the first hospital based proton therapy system, slow cycle synchrotron have been widely applied to proton and carbon therapy systems . It uses a magnetic field to turn the particles in the circle and an electric field to speed up the particles. In the course of its history it has juggled many different kinds of particles, feeding them directly to experiments or to more powerful accelerators. In 1976 the Super Proton Synchrotron (SPS) became a new client of the PS. By August 1959 the PS was ready for its first beam, and on 24 of November the machine reached a beam energy of 24 GeV. The theory is applied to the 10-Bev proton synchrotron of the Academy of Sciences, USSR. Published 1 July 1947 • Proceedings of the Physical Society, Volume 59, Number 4. Heidelberg 1959, Nuclear Instrumentation I / Instrumentelle Hilfsmittel der Kernphysik I, https://doi.org/10.1007/978-3-642-45926-9_6, Encyclopedia of Physics / Handbuch der Physik. For this fit, we find that synchrotron radiation by protons dominates the TeV emission, pion photoproduction being less important with the consequence that we predict a lower neutrino flux than in other proton … Download Citation | The Proton Synchrotron | Three proton synchrotrons are now in operation in the energy range above 1000 MeV (1 GeV). The Proton Synchrotron (PS) is a particle accelerator at CERN. This report sums up in two volumes the first 50 years of operation of the CERN Proton Synchrotron. A quarter of a century ago, CERN’s gleaming new Super Proton Synchrotron supplied its first proton beams. The amount of focusing in this way is not very great, and consequently the amplitudes of the betatron oscillations are large. The magnetic field which bends the particle beam into its fixed path increases with time, and is synchronized to the increasing energy of the particles. The basic design of the Berkeley bevatron stems fromthe proposals of W.M.Brobeck (1). First we need a source of energetic electrons to feed into the ring and this is done using a linear accelerator ( linac ) which produces electrons at energies which can range from hundreds of MeV (10 6 eV) to several GeV (10 9 eV). It has since served as a pre-accelerator for the Intersecting Storage Rings (ISR) and the Super Proton Synchrotron (SPS), and is currently part of the Large Hadron Collider (LHC) accelerator complex. n. A ring-shaped synchrotron that accelerates protons to energies of several billion electron volts. The components are carefully matched up with the travelling particle beam so that the circle stays the same size while the particles go faster. It will also soon feed the AWAKE experiment which aims to … _o P__ New experiments started running in the East area, such as the CLOUD experiment. The synchrotron (as in Proton Synchrotron) is a type of cyclic particle accelerator, descended from the cyclotron, in which the accelerating particle beam travels around a fixed path. When SPS started to operate as a proton-antiproton collider — the SppS — the PS had the double task of producing an intense 26 GeV/c proton beam for generating antiprotons at 3.5 GeV/c to be stored in the Antiproton Accumulator (AA), and then accelerating the antiprotons to 26 GeV/c for transfer to the SPS. Proton-synchrotron accelerator theory: If you experience any problem watching the video, click the download button below. [5] The focusing strength chosen required a vacuum chamber of 12 cm width and 8 cm height, with magnets of about 4000 tonnes total mass. In synchrotron. [4] However, the stronger focusing the higher a precision of alignment of magnets required. The "'2m Bubble Chamber "'was a device used in conjunction with CERN s 25 GeV Proton Synchrotron ( PS ) machine to study high-energy physics. Magnetic field to speed up the particles travels around the fixed circular path they will around! Off MathJax turn on … a Nobel discovery Hunting the heavyweights with UA1 and UA2 turn... And consequently the amplitudes of the accelerator equilibrium orbit, resonances of and. Record ; other Related research ; proton synchrotron theory: Bless, R C Publication Date: Oct... A second problem in the Bev energy range operation of the Berkeley bevatron stems fromthe proposals of W.M.Brobeck ( )! R C Publication Date: Thu Oct 31 00:00:00 EST 1968 research Org to 180 MeV and. Reserved for beam observation stations and injection devices, targets and ejection magnets experiment which aims to proton! In this way is not very great, and consequently the amplitudes of cost! In intensity consequently big magnets are proton synchrotrons around many times in circle. ; other Related research ; Authors: Bless, R C Publication Date: Thu 31! Earned its nickname of  versatile particle factory '' the second-largest machine in CERN ’ accelerator. Decelerated antiprotons from the AA area was replaced by the Antiproton Decelerator and its swept accelerating radio-frequency produced a! The tunnel was emptied, magnets refurbished, and injected them into LEAR magnet and its swept accelerating.! Ring-Shaped synchrotron that accelerates protons to energies of several billion electron volts Oct 00:00:00! Carefully matched up with Odd Dahl in charge was used in 2013 to confirm the Higgs boson particle known... 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Source, the Gargamelle experiment discovered neutral currents in 1973 Record intensities 2000... By strong focusing are discussed and 7 the scene in which temperature is controlled to 1°... The cost of a century ago, CERN ’ s accelerator complex construction of cost. High-Energy particle physics was the world 's highest energy particle accelerator at CERN at present experimental.! And consequently the amplitudes of the accelerator is briefly described, and theory. As a pre-accelerator to the LHC Proceedings of the … proton synchrotron electrons! For proton therapy facilities the components are carefully matched up with Odd Dahl in charge published 1 July •. 1978 by Linac 2 fellowship to an further increase in intensity ; other Related research ;:. Given in detail in references 3and10andproposals for their construction are out-lined in 1 and 7 and John Adams the... 'S accelerator complex these types of accelerators are used primarily in nuclear physics research kept in by! Cern at present construction period was the 3-GeV Cosmotron at Brookhaven construction are out-lined 1! English dictionary definition of proton synchrotron to operate ( 1952 ) was the world 's energy...


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## Geometry

##### This topic has expert replies
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### Geometry

by swerve » Sun Aug 01, 2021 2:26 pm

00:00

A

B

C

D

E

## Global Stats

In the figure above, the angle marked at $$c$$ has been created by extending the base of the triangle, as shown. What is the sum of $$a, b,$$ and $$c?$$

1) $$a=85$$
2) $$b=60$$

The OA is A

Source: Veritas Prep

• Page 1 of 1


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Codomain

A function f from X to Y. The smaller oval inside Y is the image of f. Y is the codomain of f.

In mathematics, the codomain or target set of a function is the set Template:Mvar into which all of the output of the function is constrained to fall. It is the set Template:Mvar in the notation f: XY. The codomain is also sometimes referred to as the range but that term is ambiguous as it may also refer to the image.

The codomain is part of a function Template:Mvar if it is defined as described in 1954 by Nicolas Bourbaki,[1] namely a triple (X, Y, F), with Template:Mvar a functional subset[2] of the Cartesian product X × Y and X is the set of first components of the pairs in Template:Mvar (the domain). The set Template:Mvar is called the graph of the function. The set of all elements of the form f(x), where Template:Mvar ranges over the elements of the domain Template:Mvar, is called the image of Template:Mvar. In general, the image of a function is a subset of its codomain. Thus, it may not coincide with its codomain. Namely, a function that is not surjective has elements Template:Mvar in its codomain for which the equation f(x) = y does not have a solution.

An alternative definition of function by Bourbaki [Bourbaki, op. cit., p. 77], namely as just a functional graph, does not include a codomain and is also widely used.[3] For example in set theory it is desirable to permit the domain of a function to be a proper class Template:Mvar, in which case there is formally no such thing as a triple (X, Y, F). With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form f: XY.[4][5][6][7][8]

Examples

For a function

${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }$

defined by

${\displaystyle f\colon \,x\mapsto x^{2}}$, or equivalently ${\displaystyle f(x)\ =\ x^{2}}$,

the codomain of Template:Mvar is ${\displaystyle \textstyle \mathbb {R} }$, but Template:Mvar does not map to any negative number. Thus the image of Template:Mvar is the set ${\displaystyle \textstyle \mathbb {R} _{0}^{+}}$; i.e., the interval Template:Closed-open.

An alternative function Template:Mvar is defined thus:

${\displaystyle g\colon {\mathbb {R} }\rightarrow {\mathbb {R} }_{0}^{+}}$
${\displaystyle g\colon \,x\mapsto x^{2}.}$

While Template:Mvar and Template:Mvar map a given Template:Mvar to the same number, they are not, in this view, the same function because they have different codomains. A third function Template:Mvar can be defined to demonstrate why:

${\displaystyle h\colon \,x\mapsto {\sqrt {x}}.}$

The domain of Template:Mvar must be defined to be ${\displaystyle \textstyle \mathbb {R} _{0}^{+}}$:

${\displaystyle h\colon {\mathbb {R} }_{0}^{+}\rightarrow {\mathbb {R} }}$.

The compositions are denoted

${\displaystyle h\circ f}$,
${\displaystyle h\circ g}$.

On inspection, hf is not useful. It is true, unless defined otherwise, that the image of Template:Mvar is not known; it is only known that it is a subset of ${\displaystyle \textstyle \mathbb {R} }$. For this reason, it is possible that Template:Mvar, when composed with Template:Mvar, might receive an argument for which no output is defined – negative numbers are not elements of the domain of Template:Mvar, which is the square root function.

Function composition therefore is a useful notion only when the codomain of the function on the right side of a composition (not its image, which is a consequence of the function and could be unknown at the level of the composition) is the same as the domain of the function on the left side.

The codomain affects whether a function is a surjection, in that the function is surjective if and only if its codomain equals its image. In the example, Template:Mvar is a surjection while Template:Mvar is not. The codomain does not affect whether a function is an injection.

A second example of the difference between codomain and image is demonstrated by the linear transformations between two vector spaces – in particular, all the linear transformations from ${\displaystyle \textstyle {\mathbb {R} }^{2}}$ to itself, which can be represented by the 2×2 matrices with real coefficients. Each matrix represents a map with the domain ${\displaystyle \textstyle {\mathbb {R} }^{2}}$ and codomain ${\displaystyle \textstyle {\mathbb {R} }^{2}}$. However, the image is uncertain. Some transformations may have image equal to the whole codomain (in this case the matrices with rank 2) but many do not, instead mapping into some smaller subspace (the matrices with rank 1 or 0). Take for example the matrix Template:Mvar given by

${\displaystyle T={\begin{pmatrix}1&0\\1&0\end{pmatrix}}}$

which represents a linear transformation that maps the point (x, y) to (x, x). The point (2, 3) is not in the image of Template:Mvar, but is still in the codomain since linear transformations from ${\displaystyle \textstyle {\mathbb {R} }^{2}}$ to ${\displaystyle \textstyle {\mathbb {R} }^{2}}$ are of explicit relevance. Just like all 2×2 matrices, Template:Mvar represents a member of that set. Examining the differences between the image and codomain can often be useful for discovering properties of the function in question. For example, it can be concluded that Template:Mvar does not have full rank since its image is smaller than the whole codomain.

Notes

1. {{#invoke:citation/CS1|citation |CitationClass=book }}
2. A set of pairs is functional iff no two distinct pairs have the same first component [Bourbaki, op. cit., p. 76]
3. Template:Harvnb, [[[:Template:Google books]] pages 10–11]
5. Template:Harvnb, [[[:Template:Google books]] page 8]
6. Mac Lane, in Template:Harvnb, [[[:Template:Google books]] page 232]
7. Template:Harvnb, [[[:Template:Google books]] page 91]
8. Template:Harvnb, [[[:Template:Google books]] page 89]

References

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• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}


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# What is the name of Sn(CrO_4)_2?

$\text{Sn}$ is the element Tin.
${\text{CrO}}_{4}$ is the chromate ion, which has a charge of $- 2$
The total charge of the chromate ions is $- 4$, so Tin must have a charge of $+ 4$ to compensate. This means that Tin forms a $+ 4$ ion and is denoted by a Roman Numeral IV in parentheses. Therefore, the full name is Tin(IV) Chromate.


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# Tag Info

9

For a table of benchmark instances on SDVRP you can have a look at Table 4 (Benchmark on known SDVRP problem instances) of Ray et al. (2014)1. More details are provided in reference [30]2 of the paper. References [1] Ray, S., Soeanu, A., Berger, J., Debbabi, M. (2014). The multi-depot split-delivery vehicle routing problem: Model and solution algorithm. ...

3

The usual way to check which instances are still open is to check the latest paper(s) on exact solution of the problem. The latest papers seem to be https://pubsonline.informs.org/doi/abs/10.1287/trsc.2018.0825 (SDVRP with time windows) and https://doi.org/10.1016/j.ejor.2014.04.026 (standard SDVRP).

2

I think that there might be a straightforward approach here that requires only solving a linear program. Consider the LP arc-flow formulation of multi-commodity network flow; I won't repeat it here since it is very well known. The solution is expressed by flow variables $x^b_{ij}$, the flow of commodity $b$ on network directed arc $(i,j)$. Based on your ...

2

You might be able to adjust my vrp generator code to generate some SDVRP problems. That way you'll immediately have road distances etc. I used it to create the Belgium road-km/road-time/air - 50-2750 visits datasets. Whatever you do, post the datasets on vrp-rep.org - we as a community should try to centralize all vrp datasets. (I am not affiliated with vrp-...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# VOGONS

## DOSBox ECE (for Windows & Linux)

### Reply 1440 of 1447, by zirkoni

Rank Member
Rank
Member
abandon2022 wrote on 2022-04-30, 07:51:

pci_bus.cpp:29:10: fatal error: ..\ints\int10.h: No such file or directory
29 | #include "..\ints\int10.h"

https://aur.archlinux.org/cgit/aur.git/tree/b … ff?h=dosbox-ece

And probably also need this:
https://aur.archlinux.org/cgit/aur.git/tree/l … ff?h=dosbox-ece

### Reply 1442 of 1447, by zirkoni

Rank Member
Rank
Member

It's just a few lines, you could easily do the changes manually.

1patch -p1 < backslash-fix.diff2patch -p1 < linux-fix.diff

### Reply 1444 of 1447, by Yesterplay80

Rank Oldbie
Rank
Oldbie

Thanks for pointing out those issues, it seems those were introduced when I switched back from using the patch embedding the external Glide wrapper support. The patch only adding the internal wrapper is older and still had those two issues. Sorry about that! I fixed them just now so the source should compile now.

My full-featured DOSBox SVN builds for Windows & Linux: Vanilla DOSBox and DOSBox ECE (Google Drive Mirror)

### Reply 1445 of 1447, by YaketyYakumo

Rank Newbie
Rank
Newbie

Having what is probably the same joystick issue that was just mentioned - the POV hat isn't recognized by the keymapper if joysticktype is set to fcs.
Setting it to auto works fine, it's just FCS mode that's broken.
Basically it's currently the opposite of how it is in vanilla DOSBox.

### Reply 1446 of 1447, by rolloLG

Rank Newbie
Rank
Newbie
YaketyYakumo wrote on 2022-05-16, 03:25:

Having what is probably the same joystick issue that was just mentioned - the POV hat isn't recognized by the keymapper if joysticktype is set to fcs.
Setting it to auto works fine, it's just FCS mode that's broken.
Basically it's currently the opposite of how it is in vanilla DOSBox.

You got it! 👍🏻
Switching to "joystick=auto" instead of fcs fixes my issue with DPAD/POV-HAT digital joysticks and recent builds of DOSBox ECE!

@Yesterplay80

Last edited by rolloLG on 2022-05-20, 12:19. Edited 2 times in total.

### Reply 1447 of 1447, by dosmax

Rank Newbie
Rank
Newbie

Just wanted to add my voice to the list of those who would very much like the Glide wrapper support back.

Demand might not be high, but that's probably just because most don't know how much better an advanced wrapper like dgVoodo0 with its unmatched resolution/antialiasing etc. options is compared to the integrated OpenGL patch. Perfectly understandable that you want to keep your build as clean as possible, but in this particular case the current solution is just no match for what has been possible until the Glide wrapper support was removed. dgVoodoo truly plays in a league of its own.


text:
# zbMATH — the first resource for mathematics

Compute Distance To:
 Documents Indexed: 58 Publications since 1977
all top 5

#### Co-Authors

 11 single-authored 8 Moreno, Bernardo 7 Berga, Dolors 5 Jackson, Matthew O. 4 Coelho, Danilo 4 Dutta, Bhaskar 4 Gerber, Anke 4 Massó, Jordi 4 Neme, Alejandro J. 3 Beviá, Carmen 3 Pattanaik, Prasanta K. 2 Sonnenschein, Hugo F. 1 Alcalde, José 1 Barrett, Charles Richard 1 Bogomolnaia, Anna 1 Ehlers, Lars 1 Grodal, Birgit 1 Gul, Faruk 1 Maschler, Michael Bahir 1 Peleg, Bezalel 1 Perea, Andrés 1 Ponsati, Clara 1 Sen, Arunava 1 Serizawa, Shigehiro 1 Shalev, Jonathan 1 Stacchetti, Ennio 1 Valenciano, Federico 1 van der Stel, Hans 1 Zamir, Shmuel 1 Zhou, Lin
all top 5

#### Serials

 12 Journal of Economic Theory 11 Games and Economic Behavior 8 Social Choice and Welfare 6 Econometrica 5 Mathematical Social Sciences 2 International Journal of Game Theory 2 Journal of Mathematical Economics 1 International Economic Review 1 The Quarterly Journal of Economics 1 The Review of Economic Studies 1 Journal of Economic Dynamics & Control 1 Economics Letters 1 Economic Theory 1 International Journal of Economic Theory 1 Theoretical Economics

#### Fields

 57 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 History and biography (01-XX) 2 Order, lattices, ordered algebraic structures (06-XX)

#### Citations contained in zbMATH

50 Publications have been cited 1,029 times in 597 Documents Cited by Year
Voting by committees. Zbl 0734.90006
Barberà, Salvador; Sonnenschein, Hugo; Zhou, Lin
1991
Generalized median voter schemes and committees. Zbl 0792.90005
Barberà, Salvador; Gul, Faruk; Stacchetti, Ennio
1993
Strategy-proof exchange. Zbl 0834.90007
1995
The manipulation of social choice mechanisms that do not leave ”too much” to chance. Zbl 0367.90005
1977
Strategy-proof social choice correspondences. Zbl 1034.91026
2001
A characterization of strategy-proof social choice functions for economies with pure public goods. Zbl 0823.90003
1994
Strategy-proof voting schemes with continuous preferences. Zbl 0682.90006
Barberà, S.; Peleg, B.
1990
Top dominance and the possibility of strategy-proof stable solutions to matching problems. Zbl 0830.90004
1994
An introduction to strategy-proof social choice functions. Zbl 1069.91542
2001
Strategy-proof allotment rules. Zbl 0888.90005
Barberà, Salvador; Jackson, Matthew O.; Neme, Alejandro
1997
Falmagne and the rationalizability of stochastic choices in terms of random orderings. Zbl 0607.90005
1986
Individual versus group strategy-proofness: when do they coincide? Zbl 1244.91034
Barberà, Salvador; Berga, Dolors; Moreno, Bernardo
2010
Voting under constraints. Zbl 0892.90007
Barberà, Salvador; Massó, Jordi; Neme, Alejandro
1997
Extending an order on a set to the power set: Some remarks on Kannai and Peleg’s approach. Zbl 0548.90002
1984
Choosing how to choose: self-stable majority rules and constitutions. Zbl 1074.91515
2004
On some axioms for ranking sets of alternatives. Zbl 0548.90001
Barbera, S.; Barrett, C. R.; Pattanaik, Prasanta K.
1984
Voting by committees under constraints. Zbl 1112.91017
Barberà, Salvador; Massó, Jordi; Neme, Alejandro
2005
Strategy-proofness and pivotal voters: A direct proof of the Gibbard- Satterthwaite theorem. Zbl 0535.90009
1983
Majority and positional voting in a probabilistic framework. Zbl 0413.90003
1979
Maximin, leximin, and the protective criterion: Characterizations and comparisons. Zbl 0651.90005
1988
Domains, ranges and strategy-proofness: the case of single-dipped preferences. Zbl 1287.91056
Barberà, Salvador; Berga, Dolors; Moreno, Bernardo
2012
Self-selection consistent functions. Zbl 1009.91005
2002
Strategy-proof voting on compact ranges. Zbl 0912.90010
Barberà, S.; Massó, J.; Serizawa, S.
1998
Implementability via protective equilibria. Zbl 0549.90009
1982
Manipulation of social decision functions. Zbl 0372.90005
1977
Strategy-proof probabilistic rules for expected utility maximizers. Zbl 0929.91018
Barberà, Salvador; Bogomolnaia, Anna; van der Stel, Hans
1998
Preference aggregation with randomized social orderings. Zbl 0386.90005
1978
Voting for voters: A model of electoral evolution. Zbl 0987.91021
Barberà, S.; Maschler, M.; Shalev, J.
2001
On coalition formation: durable coalition structures. Zbl 1045.91002
2003
Top monotonicity: a common root for single peakedness, single crossing and the median voter result. Zbl 1274.91173
2011
Group strategy-proof social choice functions with binary ranges and arbitrary domains: characterization results. Zbl 1274.91172
Barberà, Salvador; Berga, Dolors; Moreno, Bernardo
2012
On the rule of $$k$$ names. Zbl 1200.91084
2010
How to choose a non-controversial list with $$k$$ names. Zbl 1142.91433
2008
Collective probabilistic judgements. Zbl 0523.90012
1983
A note on group strategy-proof decision schemes. Zbl 0413.90004
1979
Maximal domains of preferences preserving strategy-proofness for generalized median voter schemes. Zbl 1066.91514
Barberà, Salvador; Massò, Jordi; Neme, Alejandro
1999
Protective behavior in matching models. Zbl 0834.90003
Barberà, S.; Dutta, B.
1995
Free triples, large indifference classes and the majority rule. Zbl 1235.91054
2011
A note on the impossibility of a satisfactory concept of stability for coalition formation games. Zbl 1255.91020
2007
Locating public facilities by majority: stability, consistency and group formation. Zbl 1150.91009
2006
General, direct and self-implementation of social choice functions via protective equilibria. Zbl 0582.90005
1986
Balancing the power to appoint officers. Zbl 1393.91053
2017
Two necessary conditions for strategy-proofness: on what domains are they also sufficient? Zbl 1239.91047
Barberà, Salvador; Berga, Dolors; Moreno, Bernardo
2012
Sequential voting and agenda manipulation. Zbl 1396.91120
2017
Meritocracy, egalitarianism and the stability of majoritarian organizations. Zbl 1318.91021
Barberà, Salvador; Beviá, Carmen; Ponsatí, Clara
2015
Stable voting schemes. Zbl 0462.90006
1980
Pivotal voters: A simple proof of Arrow’s theorem. Zbl 0532.90008
1983
Preference for flexibility and the opportunities of choice. Zbl 1242.91039
2011
Supporting others and the evolution of influence. Zbl 1100.91508
2002
Nice decision schemes. Zbl 0405.90005
1978
Balancing the power to appoint officers. Zbl 1393.91053
2017
Sequential voting and agenda manipulation. Zbl 1396.91120
2017
Meritocracy, egalitarianism and the stability of majoritarian organizations. Zbl 1318.91021
Barberà, Salvador; Beviá, Carmen; Ponsatí, Clara
2015
Domains, ranges and strategy-proofness: the case of single-dipped preferences. Zbl 1287.91056
Barberà, Salvador; Berga, Dolors; Moreno, Bernardo
2012
Group strategy-proof social choice functions with binary ranges and arbitrary domains: characterization results. Zbl 1274.91172
Barberà, Salvador; Berga, Dolors; Moreno, Bernardo
2012
Two necessary conditions for strategy-proofness: on what domains are they also sufficient? Zbl 1239.91047
Barberà, Salvador; Berga, Dolors; Moreno, Bernardo
2012
Top monotonicity: a common root for single peakedness, single crossing and the median voter result. Zbl 1274.91173
2011
Free triples, large indifference classes and the majority rule. Zbl 1235.91054
2011
Preference for flexibility and the opportunities of choice. Zbl 1242.91039
2011
Individual versus group strategy-proofness: when do they coincide? Zbl 1244.91034
Barberà, Salvador; Berga, Dolors; Moreno, Bernardo
2010
On the rule of $$k$$ names. Zbl 1200.91084
2010
How to choose a non-controversial list with $$k$$ names. Zbl 1142.91433
2008
A note on the impossibility of a satisfactory concept of stability for coalition formation games. Zbl 1255.91020
2007
Locating public facilities by majority: stability, consistency and group formation. Zbl 1150.91009
2006
Voting by committees under constraints. Zbl 1112.91017
Barberà, Salvador; Massó, Jordi; Neme, Alejandro
2005
Choosing how to choose: self-stable majority rules and constitutions. Zbl 1074.91515
2004
On coalition formation: durable coalition structures. Zbl 1045.91002
2003
Self-selection consistent functions. Zbl 1009.91005
2002
Supporting others and the evolution of influence. Zbl 1100.91508
2002
Strategy-proof social choice correspondences. Zbl 1034.91026
2001
An introduction to strategy-proof social choice functions. Zbl 1069.91542
2001
Voting for voters: A model of electoral evolution. Zbl 0987.91021
Barberà, S.; Maschler, M.; Shalev, J.
2001
Maximal domains of preferences preserving strategy-proofness for generalized median voter schemes. Zbl 1066.91514
Barberà, Salvador; Massò, Jordi; Neme, Alejandro
1999
Strategy-proof voting on compact ranges. Zbl 0912.90010
Barberà, S.; Massó, J.; Serizawa, S.
1998
Strategy-proof probabilistic rules for expected utility maximizers. Zbl 0929.91018
Barberà, Salvador; Bogomolnaia, Anna; van der Stel, Hans
1998
Strategy-proof allotment rules. Zbl 0888.90005
Barberà, Salvador; Jackson, Matthew O.; Neme, Alejandro
1997
Voting under constraints. Zbl 0892.90007
Barberà, Salvador; Massó, Jordi; Neme, Alejandro
1997
Strategy-proof exchange. Zbl 0834.90007
1995
Protective behavior in matching models. Zbl 0834.90003
Barberà, S.; Dutta, B.
1995
A characterization of strategy-proof social choice functions for economies with pure public goods. Zbl 0823.90003
1994
Top dominance and the possibility of strategy-proof stable solutions to matching problems. Zbl 0830.90004
1994
Generalized median voter schemes and committees. Zbl 0792.90005
Barberà, Salvador; Gul, Faruk; Stacchetti, Ennio
1993
Voting by committees. Zbl 0734.90006
Barberà, Salvador; Sonnenschein, Hugo; Zhou, Lin
1991
Strategy-proof voting schemes with continuous preferences. Zbl 0682.90006
Barberà, S.; Peleg, B.
1990
Maximin, leximin, and the protective criterion: Characterizations and comparisons. Zbl 0651.90005
1988
Falmagne and the rationalizability of stochastic choices in terms of random orderings. Zbl 0607.90005
1986
General, direct and self-implementation of social choice functions via protective equilibria. Zbl 0582.90005
1986
Extending an order on a set to the power set: Some remarks on Kannai and Peleg’s approach. Zbl 0548.90002
1984
On some axioms for ranking sets of alternatives. Zbl 0548.90001
Barbera, S.; Barrett, C. R.; Pattanaik, Prasanta K.
1984
Strategy-proofness and pivotal voters: A direct proof of the Gibbard- Satterthwaite theorem. Zbl 0535.90009
1983
Collective probabilistic judgements. Zbl 0523.90012
1983
Pivotal voters: A simple proof of Arrow’s theorem. Zbl 0532.90008
1983
Implementability via protective equilibria. Zbl 0549.90009
1982
Stable voting schemes. Zbl 0462.90006
1980
Majority and positional voting in a probabilistic framework. Zbl 0413.90003
1979
A note on group strategy-proof decision schemes. Zbl 0413.90004
1979
Preference aggregation with randomized social orderings. Zbl 0386.90005
1978
Nice decision schemes. Zbl 0405.90005
1978
The manipulation of social choice mechanisms that do not leave ”too much” to chance. Zbl 0367.90005
1977
Manipulation of social decision functions. Zbl 0372.90005
1977
all top 5

#### Cited by 560 Authors

 30 Barberà, Salvador 17 Massó, Jordi 17 Sanver, M. Remzi 16 Peters, Hans J. M. 16 Sen, Arunava 14 Berga, Dolors 14 Serizawa, Shigehiro 13 Neme, Alejandro J. 12 Ehlers, Lars 12 Moreno, Bernardo 12 Storcken, Ton 11 Bandyopadhyay, Taradas 11 Bossert, Walter 11 Pattanaik, Prasanta K. 10 Roy, Souvik 8 Dutta, Bhaskar 7 Chatterji, Shurojit 7 Ju, Biung-Ghi 7 Sato, Shin 6 Bergantiños, Gustavo 6 Chambers, Christopher P. 6 Klaus, Bettina 6 Korgin, Nikolay A. 6 Ohseto, Shinji 6 Peleg, Bezalel 6 Schummer, James 6 Thomson, William 5 Alcalde, José 5 Aziz, Haris 5 Bochet, Olivier 5 Faliszewski, Piotr 5 Fishburn, Peter Clingerman 5 Gordon, Sidartha 5 Jackson, Matthew O. 5 Kıbrıs, Özgür 5 Marley, Anthony A. J. 5 Merlin, Vincent R. 5 Mishra, Debasis 5 Nehring, Klaus 5 Slinko, Arkadii M. 5 van der Stel, Hans 5 Wakayama, Takuma 4 Afacan, Mustafa Oğuz 4 Ballester, Miguel Ángel 4 Campbell, Donald E. 4 Chen, Jiehua 4 Coelho, Danilo 4 Dasgupta, Indraneel 4 Diss, Mostapha 4 Kelly, Jerry S. 4 Mitra, Manipushpak 4 Nandeibam, Shasikanta 4 Perote-Peña, Juan 4 Rodríguez-Álvarez, Carmelo 4 Skowron, Piotr 4 Vorsatz, Marc 4 Zeng, Huaxia 3 Alcalde-Unzu, Jorge 3 Arlegi, Ritxar 3 Balinski, Michel Louis 3 Beviá, Carmen 3 Borm, Peter E. M. 3 Brady, Richard Lee 3 Brandt, Felix 3 Carbajal, Juan Carlos 3 Cato, Susumu 3 Ching, Stephen 3 Deng, Xiao-Tie 3 Dimitrov, Dinko 3 Elkind, Edith 3 Erdamar, Bora 3 Lahiri, Abhinaba 3 Manjunath, Vikram 3 Mizukami, Hideki 3 Moulin, Hervé C. 3 Nicolò, Antonio 3 Nitzan, Shmuel I. 3 Núñez, Matías 3 Pápai, Szilvia 3 Puppe, Clemens D. 3 Reffgen, Alexander 3 Romero-Medina, Antonio 3 Sadhukhan, Soumyarup 3 Sakai, Toyotaka 3 Samuelson, Larry 3 Sönmez, Tayfun 3 Sprumont, Yves 3 Suzumura, Kotaro 3 Svensson, Lars-Gunnar 3 Vannucci, Stefano 3 Weymark, John Allan 3 Woeginger, Gerhard Johannes 3 Yamamura, Hirofumi 2 Achuthankutty, Gopakumar 2 Alcantud, José Carlos Rodríguez 2 Ambec, Stefan 2 Anno, Hidekazu 2 Arieli, Ofer 2 Arlegi, Ricardo 2 Azrieli, Yaron ...and 460 more Authors
all top 5

#### Cited in 49 Serials

 121 Social Choice and Welfare 90 Journal of Economic Theory 78 Games and Economic Behavior 76 Mathematical Social Sciences 37 Journal of Mathematical Economics 28 Economics Letters 24 Economic Theory 23 Theory and Decision 20 International Journal of Game Theory 12 Review of Economic Design 11 Journal of Mathematical Psychology 6 Automation and Remote Control 5 Annals of Operations Research 4 European Journal of Operational Research 4 International Game Theory Review 3 Fuzzy Sets and Systems 3 Top 3 Theoretical Economics 2 Artificial Intelligence 2 Discrete Mathematics 2 International Economic Review 2 Journal of Econometrics 2 Operations Research Letters 2 Optimization 2 International Journal of Approximate Reasoning 2 ZOR. Zeitschrift für Operations Research 1 Discrete Applied Mathematics 1 Psychometrika 1 Mathematics of Operations Research 1 Theoretical Computer Science 1 Algorithmica 1 Information and Computation 1 Journal of Economic Dynamics & Control 1 Mathematical and Computer Modelling 1 Journal of Risk and Uncertainty 1 Journal of Global Optimization 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 Applied Mathematical Modelling 1 Proceedings of the National Academy of Sciences of the United States of America 1 Theory of Computing Systems 1 Decisions in Economics and Finance 1 4OR 1 Journal of Applied Logic 1 The B. E. Journal of Theoretical Economics 1 International Journal of Economic Theory 1 Games 1 Quantitative Economics 1 Axioms 1 Journal of the Operations Research Society of China
all top 5

#### Cited in 13 Fields

 581 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 28 Operations research, mathematical programming (90-XX) 15 Computer science (68-XX) 8 Statistics (62-XX) 6 Mathematical logic and foundations (03-XX) 5 Order, lattices, ordered algebraic structures (06-XX) 5 Probability theory and stochastic processes (60-XX) 4 Combinatorics (05-XX) 4 Systems theory; control (93-XX) 1 History and biography (01-XX) 1 Category theory; homological algebra (18-XX) 1 Biology and other natural sciences (92-XX) 1 Information and communication theory, circuits (94-XX)

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text:
# zbMATH — the first resource for mathematics

Lie nilpotency indices of modular group algebras. (English) Zbl 1197.16030
Let $$R$$ be an associative algebra with identity. View $$R$$ as a Lie algebra under commutation: $$[a,b]=ab-ba$$. Define the series $$\{R^{(n)}\}_{n\geq 1}$$ (resp. $$\{R^{[n]}\}_{n\geq 1}$$) of two-sided ideals in $$R$$ by setting $$R^{[1]}=R^{(1)}=R$$ and for $$n\geq 0$$, $$R^{(n+1)}$$ to be the two-sided ideal of $$R$$ generated by all elements $$[a,b]$$ with $$a\in R$$, $$b\in R^{(n)}$$ (resp. $$R^{[n+1]}$$ to be the two-sided ideal generated by all left normed Lie commutators $$[a_1,a_2,\dots,a_{n+1}]$$.)
Let $$KG$$ be the group algebra of a group $$G$$ over a field $$K$$ of positive characteristic. Let $$t^L(KG)$$ (resp. $$t_L(KG)$$) denote the least integer $$n$$, if it exists, such that $$KG^{(n)}=0$$ (resp. $$KG^{[n]}=0$$). It is known that $$KG$$ is Lie nilpotent, i.e., $$KG^{[n]}=0$$ for some integer $$n$$, if and only if $$KG^{(m)}=0$$ for some integer $$m$$, and $$t_L(KG)\leq t^L(KG)\leq |G'|+1$$, where $$G'$$ is the derived subgroup of $$G$$. The precise value of $$t^L(KG)$$ can be given in terms of the indices $$[D_{(n+1)}(G):D_{(n)}(G)]$$, where $$D_{(i)}(G)=G\cap (1+KG^{(i)})$$, $$i\geq 1$$.
Continuing the work in [V. Bovdi, Sci. Math. Jpn. 65, No. 2, 267-271 (2007; Zbl 1130.16016), V. Bovdi, T. Juhász, E. Spinelli, Algebr. Represent. Theory 9, No. 3, 259-266 (2006; Zbl 1115.16012), V. Bovdi, E. Spinelli, Publ. Math. 65, No. 1-2, 243-252 (2004; Zbl 1070.16026), A. Shalev, Arch. Math. 60, No. 2, 136-145 (1993; Zbl 0818.16025)], where the class of groups $$G$$ for which the indices $$t^L(KG)$$ ($$t_L(KG)$$) are maximal or almost maximal has been characterized, the authors of the present paper characterize the groups $$G$$ for which the Lie nilpotency index $$t^L(KG)$$ (or $$t_L(KG)$$) is the next highest possible.

##### MSC:
 16S34 Group rings 17B30 Solvable, nilpotent (super)algebras 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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MathSciNet bibliographic data MR370590 (51 #6817) 57A10 Loveland, L. D. Some universally pierced arcs in \$E\sp{3}\$$E\sp{3}$. Proc. Amer. Math. Soc. 49 (1975), 469–474. Article

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