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Topic: Iterative Procedure Using Vectors
Replies: 2 Last Post: Apr 26, 2013 3:26 AM
Messages: [ Previous | Next ]
Laura Iterative Procedure Using Vectors
Posted: Apr 25, 2013 1:56 PM
Posts: 1
Registered: 4/ I have been told to carry out the following iterative procedure:
25/13 a) Set a vector v to be the vector [1 ; 0]
b) Find a new vector u = Av (where A is a 2 x 2 real matrix)
c) Find the magnitude of this new vector u, and call this magnitude c
d) Divide u by c
e) Set v to be this unit vector, then repeat steps b) to e) a number of times.
So far I have tried:
v(1) = [1 ; 0]
for n = 1:100
u(n) = A*v(n+1)
c(n) = norm(u(n))
v(n+1) = u(n)/c(n)
However, it doesn't seem to like the v(1) part and keeps saying sub-matrix error, probably because it thinks that I am referring to the first element in the vector instead of saying v
= [1 ; 0] is the initial condition. Does anyone know how I can do this?
Also I'm not sure if the rest of it works ...
Date Subject Author
4/25/13 Iterative Procedure Using Vectors Laura
4/25/13 Re: Iterative Procedure Using Vectors Steven Lord
4/26/13 Re: Iterative Procedure Using Vectors Nasser Abbasi
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Avoiding the use of cut in this prolog function
Stack Overflow is a question and answer site for professional and enthusiast programmers. It's 100% free, no registration required.
I have implemented the following function in prolog with the following code:
abs2(X, Y) :- X < 0, Y is -X.
abs2(X, X) :- X >= 0, !.
up vote 3 down vote favorite How can I implement this function without the use of cut ("!")?
add comment
I have implemented the following function in prolog with the following code:
abs2(X, Y) :- X < 0, Y is -X. abs2(X, X) :- X >= 0, !.
How can I implement this function without the use of cut ("!")?
There's the "hidden" cut in the if-then-else construct of Prolog:
abs2(X,Y) :- X < 0 -> Y is -X ; Y = X.
It is something of a quirk, but Prolog does not backtrack on the subgoal that forms the "premise" of an if-then or if-then-else construct. Here, if X < 0 succeeds the first try,
then the choice of "then" clause over "else" clause is committed (hence the description of this behavior as a "hidden" cut).
There is more of a role for a cut in the first clause of the predicate abs2/2 as written in the question. As Nicholas points out, the cut at the end of the second clause doesn't
up vote 6 down have any effect (there are no choice points left when you get there). But as Kaarel points out, there is a choice point left open if the first clause succeeds.
vote accepted
So what I would have written, allowing the use of a cut, is this:
abs2(X,X) :- X >= 0, !.
abs2(X,Y) :- Y is -X.
Nicholas's comments also suggest ways to "arithmetize" the absolute value (rather than use a logic definition) and avoid "cut" that way.
add comment
There's the "hidden" cut in the if-then-else construct of Prolog:
abs2(X,Y) :- X < 0 -> Y is -X ; Y = X.
It is something of a quirk, but Prolog does not backtrack on the subgoal that forms the "premise" of an if-then or if-then-else construct. Here, if X < 0 succeeds the first try, then the choice of
"then" clause over "else" clause is committed (hence the description of this behavior as a "hidden" cut).
There is more of a role for a cut in the first clause of the predicate abs2/2 as written in the question. As Nicholas points out, the cut at the end of the second clause doesn't have any effect
(there are no choice points left when you get there). But as Kaarel points out, there is a choice point left open if the first clause succeeds.
So what I would have written, allowing the use of a cut, is this:
abs2(X,X) :- X >= 0, !. abs2(X,Y) :- Y is -X.
Nicholas's comments also suggest ways to "arithmetize" the absolute value (rather than use a logic definition) and avoid "cut" that way.
My prolog is a bit rusty, but why do you even need the cut? If you write the predicate properly, backtracking can't succeed, so the cut is unnecessary:
abs(X, Y) :- number(X) , X < 0 , Y is -X .
up vote 5 down vote abs(X, X) :- number(X) , X >= 0 .
add comment
My prolog is a bit rusty, but why do you even need the cut? If you write the predicate properly, backtracking can't succeed, so the cut is unnecessary:
abs(X, Y) :- number(X) , X < 0 , Y is -X . abs(X, X) :- number(X) , X >= 0 .
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How I understood monads, part 1/2: sleepless and self-loathing in Seattle
Being the kind of guy who can spend eight years in college just to understand a few interesting concepts about the universe, I had to check it out and try to understand monads so that I too can say
“oh, yeah, that’s a monad”.
Man, was I hit hard in the face with the limitations of my own abstract thinking abilities. All the articles I could find about the subject seemed to be vaguely understandable at first but very
quickly overloaded the very few concept slots I have available in my brain. They also seemed to be consistently using arcane notation that I was entirely unfamiliar with.
It finally all clicked together one Friday afternoon during the team’s beer symposium when Louis was patient enough to break it down for me in a language I could understand (C#). I don’t know if
being intoxicated helped. Feel free to read this with or without a drink in hand.
So here it is in a nutshell: a monad allows you to manipulate stuff in interesting ways. Oh, OK, you might say. Yeah. Exactly.
Let’s start with a trivial case:
public static class Trivial {
public static TResult Execute<T, TResult>(
this T argument,
Func<T, TResult> operation) {
return operation(argument);
This is not a monad. I removed most concepts here to start with something very simple. There is only one concept here: the idea of executing an operation on an object. This is of course trivial and
it would actually be simpler to just apply that operation directly on the object. But please bear with me, this is our first baby step. Here’s how you use that thing:
"some string"
.Execute(s => s + " processed by trivial proto-monad.")
.Execute(s => s + " And it's chainable!");
What we’re doing here is analogous to having an assembly chain in a factory: you can feed it raw material (the string here) and a number of machines that each implement a step in the manufacturing
process and you can start building stuff. The Trivial class here represents the empty assembly chain, the conveyor belt if you will, but it doesn’t care what kind of raw material gets in, what gets
out or what each machine is doing. It is pure process.
A real monad will need a couple of additional concepts. Let’s say the conveyor belt needs the material to be processed to be contained in standardized boxes, just so that it can safely and
efficiently be transported from machine to machine or so that tracking information can be attached to it.
Each machine knows how to treat raw material or partly processed material, but it doesn’t know how to treat the boxes so the conveyor belt will have to extract the material from the box before
feeding it into each machine, and it will have to box it back afterwards.
This conveyor belt with boxes is essentially what a monad is. It has one method to box stuff, one to extract stuff from its box and one to feed stuff into a machine.
So let’s reformulate the previous example but this time with the boxes, which will do nothing for the moment except containing stuff.
public class Identity<T> {
public Identity(T value) {
Value = value;
public T Value { get; private set;}
public static Identity<T> Unit(T value) {
return new Identity<T>(value);
public static Identity<U> Bind<U>(
Identity<T> argument,
Func<T, Identity<U>> operation) {
return operation(argument.Value);
Now this is a true to the definition Monad, including the weird naming of the methods. It is the simplest monad, called the identity monad and of course it does nothing useful. Here’s how you use it:
Identity<string>.Unit("some string"),
s => Identity<string>.Unit(
s + " was processed by identity monad.")).Value
That of course is seriously ugly. Note that the operation is responsible for re-boxing its result. That is a part of strict monads that I don’t quite get and I’ll take the liberty to lift that
strange constraint in the next examples.
To make this more readable and easier to use, let’s build a few extension methods:
public static class IdentityExtensions {
public static Identity<T> ToIdentity<T>(this T value) {
return new Identity<T>(value);
public static Identity<U> Bind<T, U>(
this Identity<T> argument,
Func<T, U> operation) {
return operation(argument.Value).ToIdentity();
With those, we can rewrite our code as follows:
"some string".ToIdentity()
.Bind(s => s + " was processed by monad extensions.")
.Bind(s => s + " And it's chainable...")
This is considerably simpler but still retains the qualities of a monad. But it is still pointless.
Let’s look at a more useful example, the state monad, which is basically a monad where the boxes have a label. It’s useful to perform operations on arbitrary objects that have been enriched with an
attached state object.
public class Stateful<TValue, TState> {
public Stateful(TValue value, TState state) {
Value = value;
State = state;
public TValue Value { get; private set; }
public TState State { get; set; }
public static class StateExtensions {
public static Stateful<TValue, TState>
ToStateful<TValue, TState>(
this TValue value,
TState state) {
return new Stateful<TValue, TState>(value, state);
public static Stateful<TResult, TState>
Execute<TValue, TState, TResult>(
this Stateful<TValue, TState> argument,
Func<TValue, TResult> operation) {
return operation(argument.Value)
You can get a stateful version of any object by calling the ToStateful extension method, passing the state object in. You can then execute ordinary operations on the values while retaining the state:
var statefulInt = 3.ToStateful("This is the state");
var processedStatefulInt = statefulInt
.Execute(i => ++i)
.Execute(i => i * 10)
.Execute(i => i + 2);
Console.WriteLine("Value: {0}; state: {1}",
processedStatefulInt.Value, processedStatefulInt.State);
This monad differs from the identity by enriching the boxes. There is another way to give value to the monad, which is to enrich the processing. An example of that is the writer monad, which can be
typically used to log the operations that are being performed by the monad. Of course, the richest monads enrich both the boxes and the processing.
That’s all for today. I hope with this you won’t have to go through the same process that I did to understand monads and that you haven’t gone into concept overload like I did.
Next time, we’ll examine some examples that you already know but we will shine the monadic light, hopefully illuminating them in a whole new way. Realizing that this pattern is actually in many
places but mostly unnoticed is what will enable the truly casual “oh, yes, that’s a monad” comments.
Part 2/2 of this series can be found here:
Here’s the code for this article:
The Wikipedia article on monads:
This article was invaluable for me in understanding how to express the canonical monads in C# (interesting Linq stuff in there):
JP said:
Nice article, thank you. A couple pieces of feedback.
You write ...
"Now this is a true to the definition Monad, including the weird naming of the methods."
I think here it would be good to explain how the methods on the monad map specifically to the metaphor of the conveyor belt and the boxes.
Also, when you write …
“That of course is seriously ugly. Note that the operation is responsible for re-boxing its result.”
How specifically is it doing that?
Thanks again!
Bertrand Le Roy said:
@JP: Unit is boxing, Value is unboxing and Bind is processing. The Identity class is the conveyor belt.
The operation is re-boxing by wrapping its result in a call to Unit: s => Identity<string>.Unit(s + " was processed by identity monad.")
Mark Rendle said:
Excellent post; very illuminating to see an explanation in my native language. Thank you.
DBJ said:
Hi BLR,
Good post. Couple of comments.
1 :: You had to resort to "stunt programming" to be able to implement concepts You are explaining. Which makes the explanation difficult to comprehend. Above examples written in Python or JavaScript
would be much simpler but still would represent true usable implementation of the concept.
2 :: I personaly find much easier to explain monads, by using some simple code , for which the "pupil" is convinced it is a monad. For example. Is the following code a 'monad', and if it is not, why
is it not :
static readonly System.Globalization.CultureInfo cc_ = System.Globalization.CultureInfo.CurrentCulture;
public static string format(this string argument, params object[] args)
return string.Format(cc_, argument, args);
// usage: "hello {0}".format("world")
Of course, I am glad that CallStream concept made you think "even more" ... I just hope "beer fest" is not the only way to "cool the brain overheating" ;o)
JD said:
Note that your state example illustrates why it's important that bind take a function from the base type to the monadic type. As written, operations on a state-enhanced value are unable to modify the
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Vitali convergence theorem
Vitali convergence theorem
1. i.
the sequence $\{f_{n}\}$ converges to $f$ in measure;
2. ii.
3. iii.
for every $\epsilon>0$, there exists a set $E$ of finite measure, such that $\int_{{E^{\mathrm{c}}}}\lvert f_{n}\rvert^{p}<\epsilon$ for all $n$.
This theorem can be used as a replacement for the more well-known dominated convergence theorem, when a dominating factor cannot be found for the functions $f_{n}$ to be integrated. (If this theorem
is known, the dominated convergence theorem can be derived as a special case.)
In a finite measure space, condition (iii) is trivial. In fact, condition (iii) is the tool used to reduce considerations in the general case to the case of a finite measure space.
In probability theory, the definition of “uniform integrability” is slightly different from its definition in general measure theory; either definition may be used in the statement of this theorem.
• 1 Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.
• 2 Jeffrey S. Rosenthal. A First Look at Rigorous Probability Theory. World Scientific, 2003.
ModesOfConvergenceOfSequencesOfMeasurableFunctions, UniformlyIntegrable, DominatedConvergenceTheorem
uniform-integrability convergence theorem
Mathematics Subject Classification
no label found
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Differential equation asymptotes
Also, the reason for dy/dx being zero if y is a certain number implying a horizontal asymptote is simple: If when y is a certain number, then dy/dx is zero, then the graph is going to be flat at that
point. This means that y won't change as x changes, but since y doesn't change, then dy/dx is going to stay zero. Hence, horizontal asymptote.
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Calculate Working days in a month
Author Message
kool Posted: Mon Nov 06, 2006 10:59 am
Principal Member Post subject: Calculate Working days in a month
How do I calculate number of working days in a month?
If an employee works for 5 days in a week, how many days he will be working in a month?
Similarly, if he works for 6 days in a week, how many days he will be working in a month?
Similarly, if he works for 7 days in a week, how many days he will be working in a month?
Joined: 15 Apr 2005 So far I am able to calculate 7 working days
Posts: 340 Code:
<7 Working Days> =DaysBetween(<Month> ,<Last Day of month>)+1
"Impossible is nothing" -Adidas
hagnik Posted: Mon Nov 06, 2006 11:37 am
Senior Member Post subject: Re: Calculate Working days in a month
I suspect without a calendar table, you will not be able to get the exact number or working days b/c each month has a different number of each day (Mon, Tues....)
Joined: 01 Aug 2006
Posts: 92
Location: Columbus Ohio
kool Posted: Tue Nov 07, 2006 10:21 am
Principal Member Post subject: Re: Calculate Working days in a month
hagnik wrote:
I suspect without a calendar table, you will not be able to get the exact number or working days b/c each month has a different number of each day (Mon, Tues....)
Joined: 15 Apr 2005
Posts: 340 You are correct that every month has different number of days but i am confidence this can be resolved without needing to create calender table in database.
Here is the latest updates regarding this issue
Number of working days in a month if an employee works for 5 days in a week. <Working Days>
1) Calculate number of days in a month
2) Calculate number of weekends in a month
3) <WorkingDays> = Floor(number of working days in a month - number of weekends in a month )
The above calculation is correct for most of the months but some. For instance, Sept 2006 should have 21 working days but the calculation shows only 20 working days.
Any suggestion or idea is very helpful to me.
"Impossible is nothing" -Adidas
Marek Chladny Posted: Tue Nov 07, 2006 10:49 am
Forum Advocate Post subject: Re: Calculate Working days in a month
the problem of this approach (without using smart calendar table) is that you will never be able to take into account the holidays that fall into non-weekend days.
Joined: 27 Nov 2003 Let's say if Christmas (25th and 26th of December) were holidays (non-working days) and would be on Tuesday and Wednesday (just an example) then your calculation logic would
treat these days as normal working days.
Posts: 17507
Location: Bratislava Just my $0.02.
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kool Posted: Tue Nov 07, 2006 11:20 am
Principal Member Post subject: Re: Calculate Working days in a month
Marek Chladny wrote:
Joined: 15 Apr 2005 the problem of this approach (without using smart calendar table) is that you will never be able to take into account the holidays that fall into non-weekend days.
Posts: 340 Let's say if Christmas (25th and 26th of December) were holidays (non-working days) and would be on Tuesday and Wednesday (just an example) then your calculation logic would
treat these days as normal working days.
Just my $0.02.
Thanks for your inputs Marek. But we are not interested in public holidays. All we want to see is number of working days in a month regardless of public holidays.
"Impossible is nothing" -Adidas
mkumar Posted: Tue Nov 07, 2006 12:40 pm
Forum Associate Post subject: Re: Calculate Working days in a month
2) Calculate number of weekends in a month
Joined: 26 Aug 2002
How are you calculating the number of weekends?
Posts: 770
Location: Manhattan, NY
kool Posted: Tue Nov 07, 2006 3:45 pm
Principal Member Post subject: Re: Calculate Working days in a month
mkumar wrote:
Joined: 15 Apr 2005 2) Calculate number of weekends in a month
Posts: 340
How are you calculating the number of weekends?
=(Truncate((DayNumberOfWeek(<Month>)+<7 working days>)/7 ,0))*2
Where <7 working days> is number of days in a month.
"Impossible is nothing" -Adidas
kool Posted: Tue Nov 07, 2006 4:41 pm
Principal Member Post subject: Re: Calculate Working days in a month
Ok, Here is the update on what i have accomplished so far.
If you take a look at picture you will see red in Jan, Sept, and Oct months. These three months are producing wrong working days.
The similarity among these three months are; either starting day of the month or ending day of the month falls on weekends. But my logic may not be true because December and
April months also end on weekend and they are showing correct working days.
Joined: 15 Apr 2005
Any help ...!!!!
Posts: 340 _________________
"Impossible is nothing" -Adidas
mkumar Posted: Wed Nov 08, 2006 5:14 pm
Forum Associate Post subject: Re: Calculate Working days in a month
kool wrote:
Joined: 26 Aug 2002 Code:
Posts: 770 =(Truncate((DayNumberOfWeek(<Month>)+<7 working days>)/7 ,0))*2
Location: Manhattan, NY
Where <7 working days> is number of days in a month.
This logic is not correct. This will always give you a even number where as there could be 8, 9 or 10 weekends in a month.
You will need to put in extra logic to identify if there are 9 weekends in a month. [/code]
BO_Chief Posted: Wed Nov 08, 2006 5:32 pm
Forum Fanatic Post subject: Re: Calculate Working days in a month
mkumar wrote:
This will always give you a even number where as there could be 8, 9 or 10 weekends in a month.
You will need to put in extra logic to identify if there are 9 weekends in a month.
Joined: 06 Jun 2004
Posts: 5502
Location: Somewhere on Hi,
God's Land.
Sorry, I have a doubt ??
Can you tell me in which month you will have 8, 9 or 10 weekends ?
I want to know.. are we talking here Gregorian Calendar or any other?
help us help you! make sure your post has the following elements:
• Does it include BO version, Database, an error, a problem, a SQL for object or Condition ?
• Does it include some sample data what you have?
• Does it include any code you already tried ? (working or not..)
• Did you explain what you want for results?
If any of those elements are missing,chances are you didn't post enough information for us to help you!
Last edited by BO_Chief on Wed Nov 08, 2006 6:26 pm, edited 1 time in total
kool Posted: Wed Nov 08, 2006 6:09 pm
Principal Member Post subject: Re: Calculate Working days in a month
BO_Chief wrote:
mkumar wrote:
Joined: 15 Apr 2005 This will always give you a even number where as there could be 8, 9 or 10 weekends in a month.
You will need to put in extra logic to identify if there are 9 weekends in a month.
Posts: 340
Sorry, I have a doubt ??
Can you tell me in which month you will have 8, 9 or 10 weekends ?
I want to know.. are we talking here Gregorian Calendar or any other?
Here weekends include number of Saturday and Sunday in a month in order to calculate 5 working days.
You will need to put in extra logic to identify if there are 9 weekends in a month
you are correct mkumar ; Sept, Oct and Jan have 9 weekends and they are showing wrong working days.
"Impossible is nothing" -Adidas
kool Posted: Fri Nov 10, 2006 10:51 am
Principal Member Post subject: Re: Calculate Working days in a month
Update :-
The follwing code helped me to calculate the correct working days.
Joined: 15 Apr 2005
=If <first day of month > = 7 Or <Last day of last week > =6 Then (<5 Working Days >+1) Else <5 Working Days >
Posts: 340
mkumar, please let me know if my logic is still wrong.
"Impossible is nothing" -Adidas
mkumar Posted: Fri Nov 10, 2006 1:29 pm
Forum Associate Post subject: Re: Calculate Working days in a month
I figured this needs some step by step calculations.
Here is the variables I had to create to get the correct values for weekends and working days for 5 day weeks.
Joined: 26 Aug 2002 Code:
Posts: 770
Location: Manhattan, NY <First Working Day Number> = DayNumberOfWeek(<Date>)
<Last Working Day Number> = DayNumberOfWeek(LastDayOfMonth(<Date>))
<Start Day> = If <First Working Day Number> = 7 Then 2 Else If <First Working Day Number> = 6 Then 3 Else 1
<End Day> = If <Last Working Day Number> = 7 Then DayNumberOfMonth(LastDayOfMonth(<Date>)) - 1 Else If <Last Working Day Number> = 6 Then DayNumberOfMonth(LastDayOfMonth
(<Date>)) - 2 Else DayNumberOfMonth(LastDayOfMonth(<Date>))
<Weekends 1> = If <First Working Day Number> = 7 Then 1 Else If <First Working Day Number> = 6 Then 2 Else 0
<Weekends 2> = If <Last Working Day Number> = 6 Then 1 Else If <Last Working Day Number> = 7 Then 2 Else 0
<Weekends 3> = Truncate((<End Day> - <Start Day>+1)/7 ,0)*2
<Total Weekends> = <Weekends 1>+<Weekends 2>+<Weekends 3>
<Total 5 Working Days> = DayNumberOfMonth(LastDayOfMonth(<Date>)) - <Total Weekends>
mkumar Posted: Fri Nov 10, 2006 1:39 pm
Forum Associate Post subject: Re: Calculate Working days in a month
Forgot to mention that <Date> is the first date of each month.
Joined: 26 Aug 2002
Posts: 770
Location: Manhattan, NY
kool Posted: Sun Nov 12, 2006 2:44 pm
Principal Member Post subject: Re: Calculate Working days in a month
Thanks mkumar,
This is what i am currently doing to calculate number of weekends (5 working days) in a month. This involves less if else statement .
Joined: 15 Apr 2005
<last day of month> =LastDayOfMonth(< Month>)
Posts: 340 <Last day of last week> =DayNumberOfWeek(<last day of month >)
<first day of month> =DayNumberOfWeek(< Month>)
<days in a month> ==DaysBetween(< Month> ,<last day of month >)+1
<weekends> =(Truncate((DayNumberOfWeek(< Month>)+<Days in a month >)/7 ,0))
<Working Days> =Floor(<7 Working Days >-(<number of weekends>)*2)
<5 working Days> =If <first day of month > = 7 Or <Last day of last week > =6 Then (< Working Days >+1) Else < Working Days >
"Impossible is nothing" -Adidas
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Re: [TowerTalk] The Antenna System
Good points!
I originally used a Log scale for the Y axis but it couldn't handle the
change in sign of the reactance, which of course is critical to the
discussion. In the end I opted for Linear on the basis that, for choke
discussions, we're not really interested in the detail when impedances
are below a few hundred Ohms :)
Steve G3TXQ
On 22/04/2012 03:47, Tod Olson wrote:
> The graph that Steve, G3TXQ, made available really was interesting to study.
> One could look at it and see the effect of going from a feed line coax
> length of 1?4 wavelength to 1?2 wavelength to 3/4 wavelength. Adding the
> common mode reactances to the CM choke reactance provided the red line
> showing the net impedance of the CM path.
> I might suggest that he add a note about the CM Choke impedance he used
> since that was incorporated into the net impedance (Zcm) he was showing. The
> Rcm at 0.5 wavelengths looks like it is zero, but that is because the scale
> of the graph is ?4000 to +8000 ohms and resistances of say, 100 ohms, will
> look like zero.
TowerTalk mailing list
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Issue IEEE-ATAN-BRANCH-CUT Writeup
Status: Passed, Jan 89 X3J13
Forum: Cleanup
Issue: IEEE-ATAN-BRANCH-CUT
References: CLtL p.203-214
Related issues: COMPLEX-ATAN-BRANCH-CUT
Category: CHANGE
Edit history: Version 1, 13-Dec-88, Steele
Version 2, 11-Jan-89, Masinter
Problem description:
If an implementation provides a floating-point minus zero as well as a
floating-point plus zero, most notably as in IEEE 754 floating-point
arithmetic, then slight adjustments in the branch cuts of transcendental
functions are appropriate.
If there is a minus zero and a plus zero, then *no* complex number
is actually "on the axis" whether real or imaginary. Instead,
numbers of the form x+0i and x-0i straddle the real axis, and those
of the form 0+xi and -0+xi straddle the imginary axis. Branch cuts
that lie on the axes therefore run between such numbers, and directions
of continuity are not an issue.
Proposal (IEEE-ATAN-BRANCH-CUT:SPLIT):
Redefine the branch cut for two-argument ATAN, covering
the cases where there is or is not a minus zero, and then redefine *all*
other functions that have branch cuts in terms of two-argument ATAN.
Specifically, we first define PHASE in terms of two-argument ATAN,
and complex ABS in terms of real SQRT (which has no branch cut problems);
then define complex LOG in terms of PHASE, ABS, and real LOG; then define
complex SQRT in terms of LOG; and then define all others in terms of these.
In this propoal Lisp expressions should be taken as mathematical
formulas that actually are implemented in other ways for improved accuracy.
(1) If there is no minus zero, two-argument ATAN behaves as in CLtL.
If there is a minus zero, then some cases are modified:
Condition result
y=+0 x>0 +0
y=-0 x>0 -0
y=+0 x<0 +pi
y=-0 x<0 -pi
y=+0 x=+0 +0
y=-0 x=+0 -0
y=+0 x=-0 +pi
y=-0 x=-0 -pi
The range of two-argument atan therefore includes -pi in this case.
Note that the case y=0 x=0 is an error in the absence of minus zero,
but is defined in the presence of minus zero.
(2) (PHASE X) = (ATAN (IMAGPART X) (REALPART X)), as before, but now
the range of PHASE may include -PI if there is a minus zero.
(3) (ABS X) = (SQRT (+ (* (REALPART X) (REALPART X))
(* (IMAGPART X) (IMAGPART X)))), as before
(4) (LOG X) = (COMPLEX (LOG (ABS X)) (PHASE X))
(5) (SQRT X) = (EXP (/ (LOG X) 2))
(6) For EXPT, ASIN, ACOS, ATAN, ASINH, ACOSH, ATANH use the formulas
in CLtL pp. 211-213, but use the formulas (1-5) above as the
definitions of LOG and SQRT in order to determine the branch cuts
(LOG #c(-1.0 +0.0)) => #c(0.0 3.14159...) ;Current
(LOG #c(-1.0 -0.0)) => #c(0.0 3.14159...) ;Current
(LOG #c(-1.0 +0.0)) => #c(0.0 3.14159...) ;Proposed (= current)
(LOG #c(-1.0 -0.0)) => #c(0.0 -3.14159...) ;Proposed (conjugate)
The current specification ignores some natural consequences of IEEE
floating-point arithmetic. The proposed specification produces results
more natural to that arithmetic.
Current practice:
Of implementations that support a minus zero that I have checked,
such as Sun-4 CL 2.1.3 of 10-Nov-88 and Symbolics CL, all follow
the current CLtL specification.
The IEEE trig library atan2 routine written by K.C. Ng (with the advice
or supervision of W. Kahan, I believe), and distributed with BSD UNIX
(it is file /usr/src/usr.lib/libm/IEEE/atan2.c on my machine) follows
this proposal for treatment of minus zero.
Cost to Implementors:
Some of the trig routines will require some rewriting. Probably certain
tests that are now written using ZEROP need to be rewritten to use
FLOAT-SIGN instead.
Cost to Users:
It is barely conceivable that some user code could depend on this.
Probably there is no cost.
The compatibility note on p. 210 of CLtL gave users fair warning that
a change of this kind might be adopted.
Cost of non-adoption:
Unnatural treatment of some functions on machines supporting IEEE
floating-point arithmetic.
Natural treatment, etc.
A toss-up, except to those who care.
Steele has sent a letter to W. Kahan at Berkeley to get any last
comments he may have on the matter.
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49. Jahrestagung der Deutschen Gesellschaft für Medizinische Informatik, Biometrie und Epidemiologie (gmds)
A bivariate survival model with cure fraction
Meeting Abstract (gmds2004)
Search Medline for
Published: September 14, 2004
Models for survival analysis typically assume that everybody in the study population is susceptible to the event under study and will eventually experience this event if the follow-up is sufficiently
long. This is often an unstated assumption of the widely used proportional hazard models and their extensions - frailty models. However, there are situations when a fraction of individuals are not
expected to experience the event of interest; that is, those individuals are cured or insusceptible. For example, researchers may be interested in analyzing the recurrence of a disease. Many
individuals may never experience a recurrence; therefore, a cured fraction of the population exists. Historically, cure models have been utilised to estimate the cured fraction. Cure models are
survival models which allow for a cured fraction of individuals. These models extend the understanding of time-to-event data by allowing for the formulation of more accurate and informative
conclusions. These conclusions are otherwise unobtainable from an analysis which fails to account for a cured or insusceptible fraction of the population. If a cured component is not present, the
analysis reduces to standard approaches of survival analysis.
We suggest a cure-mixture model to analyze bivariate time-to-event data, as motivated by the paper of Chatterjee and Shih [Ref. 1], but with a simpler estimation procedure and the correlated
gamma-frailty model instead of the shared gamma-frailty model. This approach allows us to deal with left truncated and right censored lifetime data and accounts for heterogeneity as well as for an
insusceptible (cure) fraction in the study population. We perform a simulation study to evaluate the properties of the estimates in the proposed model and apply it to breast cancer incidence data for
5,857 Swedish female monozygotic and dizygotic twin pairs from the so-called old cohort of the Swedish Twin Registry. This model is used to estimate the size of the susceptible fraction and the
correlation between the frailties of the twin partners. Possible extensions, advantages and limitations of the method are discussed.
In the following we apply the correlated gamma-frailty model (Pickles et al. [Ref. 2]; Yashin et al. [Ref. 3]; Petersen [Ref. 4]; Wienke et al., [Ref. 5], [Ref. 6], [Ref. 7] among others) including
an insusceptible fraction to fit bivariate time-to-event (occurrence of breast cancer) data. The correlated gamma-frailty model provides a specific parameter for correlation between the two
frailties. The interesting point here is that individual frailties in twin pairs could not be observed, but their correlation could be estimated by application of the correlated gamma-frailty model.
We use a parametric approach by fitting a Gamma-Gompertz model to the data.
For a combined analysis of monozygotic and dizygotic twins we include two correlation coefficients, ρMZ and ρDZ, respectively. These correlations between monozygotic and dizygotic twins provide
information about genetic and environmental influences on frailty.
In this paper we have suggested a cure-mixture model for the modeling of correlations in bivariate time-to-event data. This model extends the approach outlined in the paper of Chatterjee and Shih
[Ref. 1] in various ways. First, instead of the shared gamma-frailty model we use the much more flexible correlated gamma-frailty model, which includes the shared gamma-frailty model as a special
case. Second, we propose to use a direct estimation procedure in the parametric model instead of the two-step estimation procedure used by Chatterjee and Shih [Ref. 1]. Third, we think that our twin
data are more appropriate as an illustrative example than the family data of Chatterjee and Shih [Ref. 1] (who ignored higher order correlations in their family data) for such bivariate models.
Nevertheless, our estimate of the size of a susceptible fraction (due to breast cancer) with 0.222 (0.045) is very close to the estimate 0.22 (0.0093) in the parametric model found by Chatterjee and
Shih [Ref. 1] in a completely different study population. Fourth, we allow the lifetimes to be truncated in our model.
Cure models with the right censored observations suffer from an inherent identifiability problem. For such observations the event under study has not occurred either because the person is
insusceptible, or because the person is susceptible but follow-up was not long enough for the event to be observed. The identifiability problem increased with increasing censoring, but is lessened by
the parametric modeling of the baseline hazard. The simulation study shows that the estimation procedure works well under the given truncation and censoring scheme in our sample data set. Stronger
right censoring causes strong identifiability problems. In cure models with fixed censoring times (caused by the end of the study) censoring is no longer non-informative even when the censoring times
and the survival times are independent. The proportion of censored observations contains important information about model parameters.
The present paper is restricted to the parametric case, meaning in our case the marginal survival functions are specified parametrically up to a few (one - dimensional) parameters. From a statistical
point of view such a parametric assumption is unsatisfactory, because it is non-justifiable. Frailty models of univariate data have been strongly criticised because assumptions have to be made about
both the shape of the underlying mortality trajectory and the distribution of the frailty: different pairs of assumptions can result in equally good fits to the data. Without an insusceptible
fraction in the population this problem can be solved by using the non-parametric correlated gamma-frailty model (Yashin and Iachine [Ref. 8]). Applying the (true) parametric and semi-parametric
estimation procedures to the same (simulated) data generated from the correlated gamma-frailty model, the semi-parametric estimation procedure shows good performance, despite the fact that it does
not make use of the additional information about the parametric structure of the marginal survival functions. The estimates of σ^2 and ρ are similar in both cases (results are not shown here).
Nevertheless, using the wrong parametric model may result in biased parameter estimates.
To what extent this method is applicable in the much more complicated semi-parametric model with cure fraction is still an open question, one that needs further careful consideration. Dealing with a
disease with late age of onset resulting in heavily censored data may lead to problems in estimating the (infinite dimensional) nuisance parameter - the marginal survival function - and,
consequently, in estimating the parameters of interest, σ^2 and ρ.
Our study points to the existence of an important insusceptible fraction. The suggested model gives a clear illustration of how survival analysis and cure models could be merged for analysis of
time-to-event data of related individuals.
The authors wish to thank the Swedish Twin Register for providing the twin data. The research was partly supported by NIH/NIA grant 7PO1AG08761-09. The Swedish Twin Registry is funded by a grant from
the Department of Higher Education, the Swedish Scientific Council, and ASTRA Zeneca.
Chatterjee, N., Shih, J. (2001) A bivariate cure-mixture approach for modeling familial association in diseases. Biometrics 57, 779 - 786.
Pickles, A., Crouchley, R., Simonoff, E., Eaves, L., Meyer, J., Rutter, M., Hewitt, J., Silberg, J. (1994) Survival Models for Developmental Genetic Data: Age of Onset of Puberty and Antisocial
Behavior in Twins. Genetic Epidemiology 11, 155 - 170.
Yashin, A.I., Vaupel, J.W., Iachine, I.A. (1995) Correlated individual frailty: An advantageous approach to survival analysis of bivariate data. Mathematical Population Studies 5, 145 - 159.
Petersen, J.H. (1998) An additive frailty model for correlated lifetimes. Biometrics 54, 646-661.
Wienke, A., Christensen, K., Skytthe, A., Yashin, A.I. (2002) Genetic analysis of cause of death in a mixture model with bivariate lifetime data. Statistical Modelling 2, 89 - 102.
Wienke, A., Lichtenstein, P., Yashin, A.I. (2003) A bivariate frailty model with a cure fraction for modeling familial correlations in diseases. Biometrics 59, 1178 - 1183.
Wienke, A., Holm, N., Christensen, K., Skytthe, A., Vaupel, J., Yashin, A.I. (2003) The heritability of cause-specific mortality: a correlated gamma-frailty model applied to mortality due to
respiratory diseases in Danish twins born 1870 - 1930. Statistics in Medicine 22, 3873 - 3887.
Yashin, A.I., Iachine, I.A. (1995) Genetic analysis of durations: correlated frailty model applied to the survival of Danish twins. Genetic Epidemiology, 12, 529 - 38.
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What does it mean to evaluate a vector field on a 2-form?
December 13th 2012, 12:15 PM
What does it mean to evaluate a vector field on a 2-form?
For example, I've read the definition of the interior product a dozen times, Interior product - Wikipedia, the free encyclopedia , but my understanding is that a 2-form is something that eats two
tangent vectors and spits out a number. A vector field on a manifold is a map v(x) = (x,u) for x in M and u in T_xM, so I don't understand how to plug a map into a form.
December 13th 2012, 02:24 PM
Re: What does it mean to evaluate a vector field on a 2-form?
My guess would ge the following. So, as you've noted a vector field $X$ is just a section (smooth as you please) of the tangent bundle. So, now, if you instead think about the map $Y:X\to TM\
oplus T$$ defined by $Y(x)=(X(x),X(x))$. You see then that if $\omega\in \bigwedge^2(T^\ast M)$ then you can think about $\omega(Y)$ defined by $\omega(Y)(x)=\omega(Y(x))=\omega(X(x),X(x))$.
That's my best guess without more context.
December 13th 2012, 04:30 PM
Re: What does it mean to evaluate a vector field on a 2-form?
Yes a 2-form eats two vectors and outputs a number, but if you only feed one input it outputs a map:
$\omega_X : V \rightarrow V$ defined by $\omega_X=\omega(X, . ) : Y \mapsto \omega(X,Y)$
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Doing Math in FPGAs, Part 2 (BCD) | EE Times
Doing Math in FPGAs, Part 2 (BCD)
In my previous blog, I rambled on about some simple ways to multiply and divide by 10 in any general-purpose device, not just in an FPGA. In the comments, I was asked why in the heck I would even
consider doing such a thing (as opposed to sticking with binary representations). Well, the answer is along the lines of "We are base-10 creatures living in a base-10 world." Much of what we do is
related to the decimal system, so we often find the need to multiply and divide by 10. Let's figure out some ways to do it quickly and efficiently. Sometimes this can lead to things that work better
in the base-2 world of numerical machines.
I think I mentioned this last time, but my quick investigation into such frivolity didn't work out for me. I didn't find an easier way to do the math. It was time for me to get back in the box and
conform to binary (drat).
What to do? There are plenty of ways to represent real numbers in a binary system. I would guess the three most common are probably binary coded decimal (BCD), floating point, and fixed point. Each
has its strengths and weaknesses, depending on your end game. As such, I guess this issue will feature a brief discussion of these three representations and perhaps will touch on their strengths,
weaknesses, and implementations.
Let's start with BCD. It's not used as much as it once was, but we can still find it hither and yon -- mostly in applications where the data will be directly driving a display of some sort (e.g., a
seven-segment display). And this, of course, is the beauty of BCD. The individual digits not only represent the decimal system with which we are naturally comfortable, but they can also be treated as
separate circuits for mathematical operations. There's another positive ramification to this representation: We can represent any number exactly if we are willing to allocate the bits to do so. For
instance, 0.2 in BCD is 0.2, but in binary, it might be 00110011... (in, say, a fractional-only fixed-point representation).
Let's take a quick look at just what BCD is. There are, in fact, multiple formats for this representation, but (ignoring compressed and uncompressed) the most common usage is shown in the following
Now that we know what it is, how do we use it? In an FPGA, I would imagine that the most likely use might be in output and input encoding to some external (human) user. With that thought in mind,
let's talk about how we might go about interfacing to the outside world.
Let's consider the case where we want to do all our manipulations in binary. In this case, we will want to convert our BCD encoded data into binary. This can be achieved as illustrated below.
We start off with a binary value of zero, and then we add each BCD digit multiplied by its decimal power. Of course, those multipliers might be pretty big, but if we get clever, maybe we can use the
x10 schemes we discussed in my previous blog to our advantage and replace these complex multiplications in a pipelined configuration of shifts and adds. I'll leave it you to ponder on that for a
while and figure out how to do it for yourself.
Once we're done with our binary machinations/manipulations, we might want to display the result to a seven-segment display. Unfortunately, binary doesn't map directly to BCD, does it?
To translate from binary to BCD, we can employ the shift-and-add-3 algorithm:
1. Left-shift the (n-bit) binary number one bit.
2. If n shifts have taken place, the number has been fully expanded, so exit the algorithm.
3. If the binary value of any of the BCD columns is greater than or equal to 5, add 3.
4. Return to (1).
As an example, let's use the nine-bit number 100101110 (302 decimal) and run it through our algorithm.
In some cases, maybe we shouldn't even bother converting back and forth between binary and BCD. Doing math with BCD is fairly straightforward. As an added benefit, we can treat each digit as a
(mostly) separate circuit. Let's talk about addition, which is performed as follows.
1. Align the decimal points (if necessary).
2. Starting from the right (the one's digit), add the two numbers.
3. If the result is greater than 9 OR the result generates a carry, then add 6.
4. Propagate the carry to the next digit.
5. If all digits have been added, we're done.
6. Go to (2).
As an example, let's use our algorithm to add 93 and 79. (It does not matter which order you perform the adds, so long as you check between each for the out-of-bounds condition.)
To Page 2 >
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microlocal analysis
To accomodate an intuitive notion of a “function of differential operator” there is a simple trick used: consider the Fourier transform. Then the differential operators become polynomials. This
correspondence of operators and their symbols, is by the definition, under some analytic care can be extended to define generalizations of differential operators by suitably extending a notion of
symbols. Thus pseudodifferential operators of Kohn and Nirenberg appeared in 1965 with soon following revolution in harmonic analysis and analysis in PDE. This includes a further generalization, the
Fourier integral operators of Lars Hörmander. A part of harmonic analysis involving geometric aspects in the cotangent bundles of such methods is called microlocal analysis. The geometric aspects
include the support, wavefront set, characteristics…of distributions, pseudodifferential operators and their symbols. There are more technical definitions (involving wavefront sets, supports and
filtrations on the algebras of symbols) of various “microlocal” properties of symbols: microlocalization, microhypoellipticity, microparametrix etc.). In addition to the analytic microlocalization
there is a formal microlocalization; and a version of filtered localization theory in noncommutative algebra, so called algebraic microlocalization, which is however not used in operator theory.
While local aspect of a differential operator is about its behaviour around a point in coordinate space, the microlocal aspect is about a point in the cotangent bundle, hence it also localizes around
the fixed covector direction, hence “micro”.
This is clearly related to the general study of oscillating integrals, including the stationary phase method and WKB-method (and generalizations) in particular. These kind of approximations and
related estimates are of importance to the study of the propagation of singularities of differential equations, wave fronts, eikonal equations, and so on.
As oscillating integrals are involved in the analysis of various Green functions like the heat kernel there is also a connection to index theorems for elliptic differential operators, see Hörmander’s
4-volume book on analysis of linear PDEs.
• A. Kaneko, Microlocal analysis, Springer Online Enc. Of Math.
• J.J. Duistermaat, Fourier integral operators, Progress in Mathematics, Birkhäuser 1995.
• V. Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84, 327–402 (1986), MR87j:32030, doi
• Lars Hörmander, The analysis of linear partial differential operators, in 4 vols.: I. Distribution theory and Fourier analysis, II. Differential operators with constant coefficients, III.
Pseudo-differential operators, IV. Fourier integral operators.
• A. Grigis, J. Sjöstrand, Microlocal analysis for differential operators: an introduction, Cambridge U.P. 1994.
• R. Melrose, Introduction to microlocal analysis, pdf
• Yu. Safarov, Distributions, Fourier transforms and microlocal analysis (course online notes) pdf
• Masaki Sato, Restriction, localization and microlocalization, (expository paper) pp. 195–205 in collection “Quadrature domains and their application” (Operator theory: advances and applications
156), doi
• M. Kashiwara, P. Schapira, F. Ivorra, I. Waschkies, Microlocalization of ind-sheaves, in “Studies in Lie theory”, Progress in Math. 243, Birkhäuser 2006.
• C. Bardos, L. Boutet de Monvel, From atomic hypothesis to microlocal analysis (lecture notes) pdf
• Dmitry Tamarkin, Microlocal condition for non-displaceablility, arxiv/0809.1584(application of microlocal analysis to symplectic/Lagrangean geometry).
• Goro Kato, Daniele Carlo Struppa, Fundamentals of algebraic microlocal analysis, M. Dekker 1999, googB
• V. Guillemin, Masaki Kashiwara, Takahiro Kawai, Seminar on micro-local analysis, Ann. of Math. Studies 93 (1979), googlebooks
• Ю. В. Егоров, Микролокальный анализ, Дифференциальные уравнения с частными производными – 4, Итоги науки и техн. Сер. Соврем. пробл. мат. Фундам. направления, 33, ВИНИТИ, М., 1988, 5-–156, pdf,
• Masaki Kashiwara, Systems of microdifferential equations, Birkhäuser 1983, 87 pp.
• Masaki Kashiwara, Takahiro Kawai, Tatsuo Kimura, Foundations of algebraic analysis, Transl. from Japanese by Goro Kato. Princeton Mathematical Series 37, 1986. xii+255 pp. MR87m:58156; J. L.
Brylinski?, Book Review: Foundations of algebraic analysis. Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 1, 104–108, doi
• Masaki Kashiwara, Pierre Schapira, Hochschild homology and microlocal Euler classes, arxiv/1203.4869
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Posts by
Total # Posts: 21
Thanks bob
I am pulling my hair out trying to figure out glide reflections. P-->P'(-2,-6) for the glide reflection where the translation is (x,y)-->(x,y-1) and the line of reflection is x=1?? A)(4,-5) B)(-2,-7)
C)(-2,-5) D)(4,-6) Help please, I am a mom pulling her hair out. Th...
what is the image of a(3 -1) after a reflection, first across the line y=3, then across the line x=-1 (-5,7) (3,-1) (-5,-1) (-1,-5)
Mom in need of help! The x-intercept of a line is -5 and the y-intercept of the line is -2. What is the equation of the line? A) y=-5/2x-5 B) y=2/5x+2 C) y=5/2x-2 D) y=-2/5x-2 Thanks so much
Thank u so much!
Use the Division Property of Equality to complete the following statement. If 5x=2y, then x=? Mom in need of help
Language arts plz help~!
^ ^ (.)-(.) / ( 0 )
Language arts plz help~!
^ ^ {. .} / o "
Language arts plz help~!
^ ^ {. .} / o "
science help ASAP please
idk but i will help you eventually
social studies
thanks that helped alot
social studies
what does source of article mean?????????????????????????????:)
AP Language & composition
Hello, i am reading moments of being. i have two questions that i feel will help me to understand better. First. How does the language used help to convey the moments in the story? also what does "it
is one of those invaluable seeds, from which, since it is impossible to ...
1.When Thomas Paine writes about reconciliation, he is referring to the southern Colonies staying peacefully connected to the northern Colonies. the Colonies staying peacefully connected with Great
Britain. the Colonists staying peacefully connected to Native Americans. Great...
thanks that helped a lot steve what was i thinking
i don't know
i have a math problem that say i have to divide 84 and 14 it and it equaled 51 then the book asked what does it tell me about 14 and 84
do you know the word galaxy shaped like a wheel whithout cured arms
what is the word for high energy body on outer edge of space
I think you first find the hundreds place then look at the numbers after it and if it is 5 or over make the number bigger by 1
the mean is when you add all the numbers on the problem together the divide it by how many numbers you have
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Drawing conclusions by NOT using AC.
up vote 11 down vote favorite
The existence of non-measurable subsets and functions on $\mathbb{R}$ require the use of the axiom of choice. That is, there exist models of ZF in which all subsets of (and hence all functions
defined on) $\mathbb{R}$ are measurable. Does this mean that if I can define a subset or function on $\mathbb{R}$ without invoking the axiom of choice, it must be measurable?
Let's say I fully accept AC and am just looking for a quick and dirty way of proving a function is measurable.
fa.functional-analysis measure-theory set-theory
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3 Answers
active oldest votes
With some difficulty, you can define a set of reals which is measurable under one extra-ZF assumption (the axiom of determinacy) and nonmeasurable under another (V=L). Under V=L you
have a definable well-ordering of the reals, and this enables you to define any of the nonmeasurable functions you normally get using the axiom of choice. On the other hand, under the
axiom of determinacy, every real function is measurable.
up vote 6 down
vote accepted That's the strict answer. However, I think that any subset or function you are likely to define will be Borel, and hence provably measurable in ZF.
There is a widely believed thesis, known as Church's Thesis for Real Mathematics, which says that explicit sets and maps on $\mathbb{R}^{n}$ are Borel. – Simon Thomas May 7 '10 at
What? We can define a non-Borel (analytic) set easily... – Gerald Edgar May 7 '10 at 15:08
1 There is a difference between explicit and definable ... – Simon Thomas May 8 '10 at 1:02
Very interesting. Thanks! – Kevin Ventullo May 8 '10 at 6:34
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Hi Kevin, welcome to MO! (I know you've been here a while, but I only just saw one of your questions.) I just want to expand a little on the thing about models, because I've asked myself
similar questions to yours. Say you've defined a function without using Choice. Explicitly, this means you've defined a set f of ordered pairs, the first components of which come from a set
A, and the second from a set B.
Each model of ZF has its own version of the function. This is like defining a function x |--> x^2, which only needs the concept of a binary operation, but in each concrete example the
function will look quite different. Actually it's worse, because in the case of ZF you also need to specify the domain of your function. So a more accurate example would be defining the
function above but only on the "set of cubes" (relative to the unspecified binary operation). Now the function looks even more different each time: it's just "doubling" in Z/5Z, but in Z/6Z
it's the zero map on {0,3}.
up vote So while your function might be measurable in a model where "every set is measurable", when you pass to another model, the ZF-definition may define a different function, which turns out not
4 down to be measurable. For example, different models of ZF may have "extra real numbers". (Just as different magmas have distinct sets of cubes.) If your ZF-definition has domain(f) = reals, this
vote will carry over to all the different models, and in each one f will have a different domain. (Hence, more often than not, a different range.)
So you see the situation is quite chaotic a priori. And this is to say nothing of the fact that in each model of ZF "measurability" means something different.
Here's one last, somewhat strained, analogy. Suppose you were looking at structures satisfying the ring axioms (ZF in the analogy), but still interested in our map g above, x |--> x^2,
defined on the set of cubes (a ZF-definable map). The ring axioms are silent about whether there are multiplicative inverses for nonzero elements (whether Choice holds). Suppose now someone
found an example of a ring, not a field (where Choice failed), s.t. the image of g is exactly the set of fourth powers (is measurable). (E.g. Z/35Z.) Would you be able to conclude that,
because your definition of g didn't use inverses (Choice), its range would always be the fourth powers, in any ring?
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Solovay showed in the 1960's the consistency of ZF + axiom of Dependent Choice (for countable sets) + "every set of reals is Lebesgue measurable". .
This is a formal justification of the idea that any set of reals that can be proved to exist in ZFC without making uncountably many choices ---- i.e., a set that can be constructed in ZF
+ DC ---- cannot be proved to be non-measurable.
There are also theorems of Shelah, coming out of the analysis of determinacy (AD, Woodin cardinals, etc) to the effect that "any reasonably defined set is Lebesgue measurable".
up vote 3
down vote So it is impossible to produce a Lebesgue non-measurable set without specifically looking for it in the wilds of uncountable AC.
It would be interesting to know what the state of the art is for metatheorems on how broad a range of constructions can be used in ZF+DC, that are guaranteed to stay within the world of
provably Lebesgue measurable sets.
2 It's not true that any construction in ZF + DC is guaranteed to stay within the realm of provably Lebesgue measurable sets. A simple example is a nonmeasurable set in L; the reason
why it is measurable in Solovay's model is that every such set is countable in that model. – François G. Dorais♦ Jun 12 '10 at 18:42
@FGD: +1 ! But are there any results on what constructions do stay inside the provably-measurable world? – T.. Jun 12 '10 at 19:02
Such assumptions, like L(R) satisfies AD, have very high consistency strength. – François G. Dorais♦ Jun 12 '10 at 19:20
I think you are referring to Shelah+Woodin, "Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable", Israel J Math 70 (1990) 381-394 showing that
$AD_{L(R)}$ follows from supercompact cardinals. How does this relate to provable measurability rather than measurability per se? – T.. Jun 12 '10 at 20:28
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Not the answer you're looking for? Browse other questions tagged fa.functional-analysis measure-theory set-theory or ask your own question.
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Erica Shannon
Doctoral Candidate, Graduate TA, Math Lead TA
Department of Mathematics
University of Colorado Boulder
My office is Math 360 and my office hours for Spring 2014 are
Monday, 10 - 10:50 a.m.
Friday, 9 - 9:50 a.m.
As Math Lead I also have an office hour for graduate students, which is Friday, 10 - 10:50 a.m.
• Math 1310-002, Spring 2013
• Math 5905, Fall 2013 - Spring 2014 (together with Faan Tone Liu and Scott Andrews)
• Math 1310-002, Fall 2013
• Math 1310-002, Spring 2013
• Math 1300-004, Fall 2012
• Math 1300-402, second half, Summer 2012
• Math 2300-004, Spring 2012 (my Math 2300 materials are here)
• Math 1300-004, Fall 2011
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• Math 1300-003, Spring 2011
• My comprehensive exam was in April 2013 and covered introductory material on Coxeter groups, Lie algebras, and commutative algebra. My comp syllabus and the notes I made while studying are online
• I use TikZ/PGF all the time for creating all types of graphics for LaTeX documents. I made a document full of TikZ examples; here's the code and the pdf output.
Women in Math
• I encourage any undergraduate women in math to consider applying to Carleton's SMP program. I went in 2008 and I learned so much. It's aimed at women who have completed Linear Algebra and are
finishing their freshman or sophomore years.
• I have attended the Nebraska Conference for Undergraduate Women in Mathematics (NCUWM) twice: once as an undergraduate and once as an invited grad student. The whole conference is fantastic.
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Solving for constant in an integral
May 4th 2010, 04:42 AM #1
May 2010
Solving for constant in an integral
I'm in a bit of a pickle,revising for an exam, but there's one stumbling block i'm getting at, evaluating this so that I have a finite value for a. I've obtained the anti-derivative, I'm i'm
facing hell trying to complete the problem. Any pointers will be appreciated. The limits are from negative infinity to positive infinity
I'm in a bit of a pickle,revising for an exam, but there's one stumbling block i'm getting at, evaluating this so that I have a finite value for a. I've obtained the anti-derivative, I'm i'm
facing hell trying to complete the problem. Any pointers will be appreciated. The limits are from negative infinity to positive infinity
This has to be treated as an improper integral. However, it's impossible for the given integral to equal 1. Is this the original question?
May 4th 2010, 04:52 AM #2
May 4th 2010, 09:14 AM #3
May 2010
May 4th 2010, 07:37 PM #4
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How to Simplify Complex Fractions
Edit Article
Simplifying Complex Fractions with Inverse MultiplicationSimplifying Complex Fractions Containing Variable Terms
Edited by Maluniu, Genius_knight, BR, Zach and 7 others
Complex fractions are fractions in which either the numerator, denominator, or both contain fractions themselves. For this reason, complex fractions are sometimes referred to as "stacked fractions".
Simplifying complex fractions is a process that can range from easy to difficult based on how many terms are present in the numerator and denominator, whether any of the terms are variables, and, if
so, the complexity of the variable terms. See Step 1 below to get started!
Method 1 of 2: Simplifying Complex Fractions with Inverse Multiplication
1. 1
If necessary, simplify the numerator and denominator into single fractions. Complex fractions aren't necessarily difficult to solve. In fact, complex fractions in which the numerator and
denominator both contain a single fraction are usually fairly easy to solve. So, if the numerator or denominator of your complex fraction (or both) contain multiple fractions or fractions and
whole numbers, simplify as needed to obtain a single fraction in both the numerator and denominator. This may require finding the least common denominator (LCM) of two or more fractions.
□ For example, let's say we want to simplify the complex fraction (3/5 + 2/15)/(5/7 - 3/10). First, we would simplify both the numerator and denominator of our complex fraction to single
☆ To simplify the numerator, we will use a LCM of 15 by multiplying 3/5 by 3/3. Our numerator becomes 9/15 + 2/15, which equals 11/15.
☆ To simplify the denominator, we will use a LCM of 70 by multiplying 5/7 by 10/10 and 3/10 by 7/7. Our denominator becomes 50/70 - 21/70, which equals 29/70.
☆ Thus, our new complex fraction is (11/15)/(29/70).
2. 2
Flip the denominator to find its inverse. By definition, dividing one number by another is the same as multiplying the first number by the inverse of the second. Now that we have obtained a
complex fraction with a single fraction in both the numerator and the denominator, we can use this property of division to simplify our complex fraction! First, find the inverse of the fraction
on the bottom of the complex fraction. Do this by "flipping" the fraction - setting its numerator in the place of the denominator and vice versa.
□ In our example, the fraction in the denominator of the complex fraction (11/15)/(29/70) is 29/70. To find its inverse, we simply "flip" it to get 70/29.
☆ Note that, if your complex fraction has a whole number in its denominator, you can treat it as a fraction and find its inverse all the same. For instance, if our complex fraction was (11/
15)/(29), we can define the denominator as 29/1, which makes its inverse 1/29.
3. 3
Multiply the numerator of the complex fraction by the inverse of the denominator. Now that you've obtained the inverse of your complex fraction's denominator, multiply it by the numerator to
obtain a single simple fraction! Remember that to multiply two fractions, we simply multiply across - the numerator of the new fraction is the product of the numerators of the two old ones, and
similarly with the denominator.
□ In our example, we would multiply 11/15 × 70/29. 70 × 11 = 770 and 15 × 29 = 435. So, our new simple fraction is 770/435.
4. 4
Simplify the new fraction by finding the greatest common factor. We now have a single, simple fraction, so all that remains is to render it in the simplest terms possible. Find the greatest
common factor (GCF) of the numerator and denominator and divide both by this number to simplify.
□ One common factor of 770 and 435 is 5. So, if we divide the numerator and denominator of our fraction by 5, we obtain 154/87. 154 and 87 don't have any common factors, so we know we've found
our final answer!
Method 2 of 2: Simplifying Complex Fractions Containing Variable Terms
1. 1
When possible, use the inverse multiplication method above. To be clear, virtually any complex fraction can be simplified by reducing its numerator and denominator to single fractions and
multiplying the numerator by the inverse of the denominator. Complex fractions containing variables are no exception, though, the more complicated the variable expressions in the complex fraction
are, the more difficult and time-consuming it is to use inverse multiplication. For "easy" complex fractions containing variables, inverse multiplication is a good choice, but complex fractions
with multiple variable terms in the numerator and denominator may be easier to simplify with the alternate method described below.
□ For example, (1/x)/(x/6) is easy to simplify with inverse multiplication. 1/x × 6/x = 6/x^2. Here, there is no need to use an alternate method.
□ However, (((1)/(x+3)) + x - 10)/(x +4 +((1)/(x - 5))) is more difficult to simplify with inverse multiplication. Reducing the numerator and denominator of this complex fraction to single
fractions, inverse multiplying, and reducing the result to simplest terms is likely to be a complicated process. In this case, the alternate method below may be easier.
2. 2
If inverse multiplication is impractical, start by finding the lowest common denominator of the fractional terms in the complex fraction. The first step in this alternate method of simplification
is to find the LCD of all the fractional terms in the complex fraction - both in its numerator and in its denominator. Usually, if one or more of the fractional terms have variables in their
denominators, their LCD is just the product of their denominators.
□ This is easier to understand with an example. Let's try to simplify the complex fraction we mentioned above, (((1)/(x+3)) + x - 10)/(x +4 +((1)/(x - 5))). The fractional terms in this complex
fraction are (1)/(x+3) and (1)/(x-5). The common denominator of these two fractions is the product of their denominators: (x+3)(x-5).
3. 3
Multiply the numerator of the complex fraction by the LCD you just found. Next, we'll need to multiply the terms in our complex fraction by the LCD of its fractional terms. In other words, we'll
multiply the entire complex fraction by (LCD)/(LCD). We can do this freely because (LCD)/(LCD) is equal to 1. First, multiply the numerator on its own.
□ In our example, we would multiply our complex fraction, (((1)/(x+3)) + x - 10)/(x +4 +((1)/(x - 5))), by ((x+3)(x-5))/((x+3)(x-5)). We'll have to multiply through the numerator and
denominator of the complex fraction, multiplying each term by (x+3)(x-5).
☆ First, let's multiply the numerator: (((1)/(x+3)) + x - 10) × (x+3)(x-5)
○ = (((x+3)(x-5)/(x+3)) + x((x+3)(x-5)) - 10((x+3)(x-5))
○ = (x-5) + (x(x^2 - 2x - 15)) - (10(x^2 - 2x - 15))
○ = (x-5) + (x^3 - 2x^2 - 15x) - (10x^2 - 20x - 150)
○ = (x-5) + x^3 - 12x^2 + 5x + 150
○ = x^3 - 12x^2 + 6x + 145
4. 4
Multiply the denominator of the complex fraction by the LCD as you did with the numerator. Continue multiplying the complex fraction by the LCD you found by proceeding to the denominator.
Multiply through, multiplying every term by the LCD.
□ The denominator of our complex fraction, (((1)/(x+3)) + x - 10)/(x +4 +((1)/(x - 5))), is x +4 +((1)/(x-5)). We'll multiply this by the LCD we found, (x+3)(x-5).
☆ (x +4 +((1)/(x - 5))) × (x+3)(x-5)
☆ = x((x+3)(x-5)) + 4((x+3)(x-5)) + (1/(x-5))(x+3)(x-5).
☆ = x(x^2 - 2x - 15) + 4(x^2 - 2x - 15) + ((x+3)(x-5))/(x-5)
☆ = x^3 - 2x^2 - 15x + 4x^2 - 8x - 60 + (x+3)
☆ = x^3 + 2x^2 - 23x - 60 + (x+3)
☆ = x^3 + 2x^2 - 22x - 57
5. 5
Form a new, simplified fraction from the numerator and denominator you just found. After multiplying your fraction by your (LCD)/(LCD) expression and simplifying by combining like terms, you
should be left with a simple fraction containing no fractional terms. As you may have noticed, by multiplying through by the LCD of the fractional terms in the original complex fraction, the
denominators of these fractions cancel out, leaving variable terms and whole numbers in the numerator and denominator of your answer, but no fractions.
□ Using the numerator and denominator we found above, we can construct a fraction that's equal to our initial complex fraction but which contains no fractional terms. The numerator we obtained
was x^3 - 12x^2 + 6x + 145 and the denominator was x^3 + 2x^2 - 22x - 57, so our new fraction is (x^3 - 12x^2 + 6x + 145)/(x^3 + 2x^2 - 22x - 57)
• Show each step of your work. Fractions can easily get confusing if you are trying to move too quickly or attempting to do them in your head.
• Find examples of complex fractions online or in your textbook. Follow along with each step until you get the hang of it.
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WordSeeker: concurrent bioinformatics software for discovering genome-wide patterns and word-based genomic signatures
An important focus of genomic science is the discovery and characterization of all functional elements within genomes. In silico methods are used in genome studies to discover putative regulatory
genomic elements (called words or motifs). Although a number of methods have been developed for motif discovery, most of them lack the scalability needed to analyze large genomic data sets.
This manuscript presents WordSeeker, an enumerative motif discovery toolkit that utilizes multi-core and distributed computational platforms to enable scalable analysis of genomic data. A controller
task coordinates activities of worker nodes, each of which (1) enumerates a subset of the DNA word space and (2) scores words with a distributed Markov chain model.
A comprehensive suite of performance tests was conducted to demonstrate the performance, speedup and efficiency of WordSeeker. The scalability of the toolkit enabled the analysis of the entire genome
of Arabidopsis thaliana; the results of the analysis were integrated into The Arabidopsis Gene Regulatory Information Server (AGRIS). A public version of WordSeeker was deployed on the Glenn cluster
at the Ohio Supercomputer Center.
WordSeeker effectively utilizes concurrent computing platforms to enable the identification of putative functional elements in genomic data sets. This capability facilitates the analysis of the large
quantity of sequenced genomic data.
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What are non-trivial examples of non-singular blow-ups of a non-singular variety?
up vote 1 down vote favorite
This question arose from the responses to this question. The references to the comments of Karl Schwede and VA are to comments made there.
The blow-up of the variety $X=\mathbb{A}^2$ along the closed subscheme $Z$ defined by $(x,y)^2$ is non-singular. As Karl Schwede points out, this example is trivial in the sense that the blow-up
along the power of a maximal ideal is naturally isomorphic to the blow-up of the maximal ideal. VA's comment, on the other hand, suggests that perhaps singular closed schemes $Z$ with $\operatorname
{Bl}_{Z}(X)$ non-singular are ubiquitous.
This suggests a question: what are non-trivial examples of a singular closed subscheme $Z$ of a non-singular variety $X$ with $\operatorname{Bl}_{Z}(X)$ non-singular. Here "non-trivial" means the
ideal of $Z$ is not a power of the ideal of a non-singular subvariety.
Particularly interesting would be such a $Z$ such that
is not isomorphic (as a scheme over $X$) to $\operatorname{Bl}_{Z'}(X)$ for any non-singular subvariety $Z'$ of $X$.
Edit: I have not been able to access the paper "On the smoothness of blow-ups" (MR1446135, by O'Carroll and Valla) yet, but the mathsci review states that they prove that the blow-up of a regular
local ring $A$ along an ideal generated by a subset of a regular system of parameters is smooth. Let's also consider those examples to be trivial.
Edit: I added "of a non-singular variety" to the title to emphasize that I am interested in examples where the ambient space is non-singular.
ag.algebraic-geometry ac.commutative-algebra
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2 Answers
active oldest votes
Suppose that $X = \mathbb{A}^2$. Let $Y$ be the blow up of $X$ at the maximal ideal $(x,y)$ and let $W$ be the blow up of $Y$ at a point on the exceptional divisor of $Y$ over $X$. Of
course, the composition $f: W \rightarrow X$ is birational and an isomorphism away from the origin. The fiber of $f$ over the origin is the union of two $\mathbb{P}^1$'s meeting at a
single point, but the total space $W$ is non-singular. The map $f$ is identified with the blowup of $X$ along some closed subscheme $Z$ of $X$ supported only at the origin. I believe an
up vote 4 example of an ideal defining such a $Z$ is $(x^3, xy, y^2)$.
down vote
accepted By taking the composition of blowups along smooth centers, there is some ideal sheaf on the base giving the composition in "one step". In theory, you can compute this ideal by tracing
through the proof that every birational morphisms is a blow up - but in practice I think this is usually difficult.
If you'll notice, $(x,y)(x^2,y) = (x^3, x^2y, xy, y^2)$ which is the integral closure of the ideal you mentioned $(x^3,xy, y^2)$. Generally speaking, we should probably limit
4 ourselves to blowing up integrally closed ideals (as ideals have the same normalized blow-up as their integral closures). Finally, blowing up products of ideals is essentially like
blowing up a series of ideals in succession. In this case, blowing up something like $(x,y)(x^2,y)$ is like blowing up the origin $(x,y)$ and then a point on the exceptional $P^1$
(this is related to "Zariski factorization") – Karl Schwede Apr 17 '10 at 19:51
1 Bah, ignore the first sentence of my previous comment! The two ideals are already the SAME (no need to take the integral closure). In particular, if an ideal contains $xy$, then it
also contains $x^2 y$. Sorry about that. – Karl Schwede Apr 17 '10 at 22:46
@Karl, if you figure out what Zariski factorization says about this question generally, then I'd be interested to hear. – jlk Apr 19 '10 at 1:29
1 Let I be an integrally closed ideal, and let $J_1$, ... $J_r$ be the irreducible complete ideals associated to the base points of I (ie, blow up each point on the cosupport of I and
take the strict transform; iterate until I becomes trivial). I believe that the surface obtained blowing up I an integrally closed ideal is nonsingular if and only if each of the
$J_i$ appears in the Zariski factorization of I. The proof should be in some paper by Spivakovsky, maybe MR1053487. – quim Apr 19 '10 at 8:23
By the way, the surface singularities obtained blowing up integrally closed ideals are called "sandwiched," because they lie between two smooth surfaces, the resolution on top and the
surface before blowup below. I don't know if "sandwiched" singularities of higher dimensional varieties have been studied, maybe a MathSciNet search is in order... – quim Apr 19 '10
at 8:26
show 1 more comment
For $X=\mathbb{A}^3$ you may take as $Z$ the three coordinate axes defined by $(x,y)(x,z)(y,z)$: The blowup $Bl_Z(X)$ of $X$ in $Z$ is nonsingular. However $Bl_Z(X)$ is isomorphic to a
up vote 1 composition of blowups in smooth centers, namely, first blowup $X$ in the origin $(x,y,z)$, which separates the three coordinate axes and then you blow them up separately.
down vote
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Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry ac.commutative-algebra or ask your own question.
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South Plainfield Geometry Tutor
Find a South Plainfield Geometry Tutor
...I am a hands-on facilitator. In math I never solve the problem, Instead, I give a tip and let her think about the pb. If unsuccessful, I give her an other tip, and an other until the problem is
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my lab, we do detailed analyses of tobacco DNA and work on regulatory elements within the plastid genome.
8 Subjects: including geometry, chemistry, French, biology
...To do this, I often use analogies which use concepts and ideas already familiar to the learner in order to achieve the goal I set out above. I also encourage you to ask as many questions as
possible, no matter how silly, so that you can get your own head around the problem. I then find it usefu...
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Posts from March 30, 2011 on Punk Rock Operations Research
How likely is VCU’s run to the Final Four? A VCU professor and sports nerd reflects on the likelihood of her school’s path to the Final Four
By Laura McLay
I am thrilled that VCU made the Final Four this year. My school’s team had an unlikely path to the Final Four, so unlikely that only 2 of 5.9 million ESPN brackets correctly picked all Final Four
teams. Sports nerds unanimously agree that VCU’s run has indeed been unlikely.
VCU making the Final Four is not “proof” that we should throw out the expert advice from sports nerds since anything can happen. While anything can indeed happen, each outcome is not equally
likely. Most outcomes are so unlikely to occur that we will not see them in our lifetimes (the probability that all four 16 seeds comprise the Final Four would occur once every eight hundred
trillion years on average). It’s like monkeys randomly typing away. Given enough time, they will rewrite Shakespeare, but don’t expect to see it happen any time soon. However, even though most of
the potential tournament outcomes have an infinitesimally small chance of occurring, when you add them all up, there is almost a certain chance that a few unlikely things will occur (which is why we
always see a few upsets in the first two rounds).
On the contrary, the excellent analyses from sports nerds will produce the best predictions that work on average. That is, averaged over a large number of tournaments, their predictions will yield
the best results (meaning that brackets produced using advice from the experts who have crunched the numbers will win the office pool most frequently). That is because the numbers point to the
outcomes that are most likely to occur. There are an enormous number of potential outcomes in the tournament (2^67 ~= 1.5 x 10^20, which is way, way more stars than there are in the Milky Way!), and
it helps to have some quantitative advice to prune most of the unlikely outcomes, nearly all of which are even less likely to happen than VCU making the Final Four! Even the most likely outcomes
rarely occur: we would expect all four one seeds to comprise the Final Four, for example, to occur every 39 years (it has happened once). The problem is, things don’t average out in a single
year–we have just one tournament this year. In any single year, something unlikely–like VCU reaching the Final Four–has a chance of occurring (albeit a small one).
Now that VCU is in the Final Four, what are their odds of winning the tournament? VCU may have initially had an infinitesimal 1-in-203,187 chance of winning the tournament, but given that they have
made it to the Final Four, their odds of winning it all is not unlikely (they’ve completed the hard part of being one of four teams left). Wayne Winston estimates that they have a 0.11 chance (a
1-in-10 chance) of winning the tournament. Using past tournament outcomes, Sheldon Jacobson has shown that seeds don’t matter after the Sweet Sixteen round, which means that VCU essentially has a
1-in-4 chance of winning the tournament. The truth is likely somewhere in between, which means that VCU has an excellent chance of being the national champion. Let’s go Rams!
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Zero divided by zero.
Please bear with me. This is my first post.
I've put together quickly, with the best logic I could fathom, a solution to the infamous 0/0.
Does 0/0 = 1 and 0 at the same time with respect to 0?
By taking zero and dividing it by zero, you acknowledge that there is, in fact, the 'presence' of more than one zero. So "a zero" divided by "a zero" is also "a zero" no?
So zero isn't actually 'just plain' zero so much as it is... a zero. A single zero. One zero. Get it?
0/0 = 0
But 0 = (1*0)
Hence there are no ones, there is one zero.
1*0 obviously equals zero but... there is 'a'... zero. Presence.
Could someone aid me with my recent confusion/is this question more for a psychology/philosophy/theoretical physics themed site?
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Heat Sensor - Electronics Projects and Circuit made Easy
Heat Sensor
Here is a simple circuit which can be used as a heat sensor. In the following circuit diagram thermistor and 100 Ohms resistance is connected in series and makes a potential divider circuit . If
thermistor is of N.T.C (Negative temperature Coefficient ) type then after heating the thermistor its resistance decreases so more current flows through the thermistor and 100 Ohms resistance and we
get more voltage at junction of thermistor and resistance. Suppose after heating 110 ohms thermistor its resistance value become 90 Ohms.then according to potential divider circuit the voltage across
one resistor equals the ratio of that resistor’s value and the sum of resistances times the voltage across the series combination. This concept is so pervasive it has a name: voltage divider. The
input-output relationship for this system, found in this particular case by voltage divider, takes the form of a ratio of the output voltage to the input voltage.
This output voltage is applied to a NPN transistor through a resistance. Emitter voltage is maintain at 4.7 volt with a help of Zener diode.This voltage we will use as compare voltage. Transistor
conducts when base voltage is greater than emitter voltage. Transistor conducts as it gets more than 4.7 base Voltage and circuit is completed through buzzer and it gives Sound.
Circuit Diagram of Heat Sensor
* please use 220 Ohms thermistor if not available you can use 2pcs 110 ohms thermistor in series. Resistance parallel with zener diode is not necessary.
You may also be interested in Project using Precision Temperature Sensor .
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college algebra
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holt rinehart and winston algebra 1 practice workbook answers
Author Message
Jadesdome Posted: Thursday 28th of Dec 08:34
I am in a real bad state of mind. Somebody assist me please. I experience a lot of issues with adding numerators, hyperbolas and function range and especially with holt rinehart and
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From: Juno
Beach, FL
kfir Posted: Thursday 28th of Dec 21:05
If you can give details about holt rinehart and winston algebra 1 practice workbook answers, I could possibly help to solve the algebra problem. If you don’t want to pay for a math
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Patent US4812990 - System and method for optimizing aircraft flight path
The present invention relates generally to terrain avoidance systems for aircraft and in particular to a system and method for optimizing the flight path of an aircraft using dynamic programming
When a military aircraft is assigned a mission in enemy territory, the selection of a flight path by which the aircraft can penetrate into and escape from the enemy territory is of critical concern.
The selection of an optimal path is based upon a number of factors, including the terrain, the location of enemy threats, aircraft handling characteristics and other capabilities of the aircraft,
ride quality and fuel and time constaints. An anticipated optimal flight path can be computed prior to the mission, but as the mission develops new situations may occur and new information become
available along the route which may alter the optimal path. For example, "pop-up" threats may occur along the route and the aircraft may have to divert into uncharted territory where it may encounter
previously unknown obstacles.
According to prior practice, the traditional approach is for the aircraft to perform the necessary maneuvers to avoid unanticipated enemy threats and other obstacles, such as uncharted mountains and
then to rejoin the anticipated optimal path computed prior to the commencement to the mission. A major disadvantage of this approach is that if the aircraft is forced to divert a large distance off
the pre-planned route, it may be unduly burdensome from a cost standpoint, considering all factors, for the aircraft to attempt to rejoin the pre-planned route. For example, rejoining the pre-planned
route after a substantial diversion may entail the aircraft flying over dangerous terrain or over territory where the survivability of the aircraft is greatly reduced due to enemy threats.
Furthermore, attempting to rejoin the original route may be unacceptable in terms of substantially increased time to complete the mission and increased fuel consumption.
More recent approaches have focused on computing the optimal flight path on a real-time basis, taking into consideration the then current position of the aircraft and the current situation. This
approach uses the power of the computer on board the aircraft to continually update the optimal flight path at any given time during the flight based on the then current position of the aircraft.
One such approach is typified by the Enhanced Terrain Masked Penetration Program (ETMPP), developed and sold by Texas Instruments Incorporated. This algorithm divides the territory between the
position of the aircraft at any given time and the target point, or a "way point" through which the aircraft will fly on the way to the target, into an infield region immediately ahead of the
aircraft and an outfield region from the outer boundary of the infield to the target or way point. A "tree growth" algorithm is used to compute the possible paths to the outer boundary of the infield
and standard dynamic programming techniques are used to compute the possible paths from the outer boundary of the infield to the target or way point. In the infield, the lateral motion of the
aircraft is constrained within predetermined limits left or right with respect to a straight heading to the target and in the outfield the dynamic programming is used to determine the possible flight
paths for five different headings from each possible position of the aircraft.
One of the disadvantages of ETMPP is that the lateral motion and heading of the aircraft is artificially constained within preselected heading limits. Another disadvantage is that dynamic programming
is performed five different times at each stage in the outfield to determine possible routes to the target, which substantially increases the number of programming calculations and hence the
execution time of the program.
The ETMPP is described in greater detail in the enclosed paper entitled "Enhanced Terrain Masked Penetration Program" by George M. Barney, presented at the National Aerospace and Electronics
Conference, 1984.
Another approach to the problem of determining optimal flight path real-time is the so-called "feasible directions" algorithm. In contrast to dynamic programming wherein the optimal path is computed
"backwards" from the target to the current aircraft position, the "feasible directions" algorithm computes the optimal path beginning at the then current position of the aircraft and looking ahead a
predetermined distance. In this manner the optimal flight path immediately ahead of the aircraft is computed without taking into account the "global" picture farther downrange. Thus, the aircraft may
be forced to fly through areas where the flight cost is unacceptably high in terms of all the factors described above because the previously computed short range paths have led the aircraft into a
region from which there is no escape without flying a relatively high cost route. The "feasible directions" approach is described in the enclosed paper entitled "Advance Automatic Terrain Following/
Terrain Avoidance Control Concepts--Algorithm Development", by J. E. Wall, Jr. et al, presented at the National Aerospace and Electronics Conference, 1981.
Yet another algorithm is described in the enclosed paper entitled "Real Time Optimal Flight Path Generation by Storage of Massive Data Bases" by Y. K. Chan et al, presented at the National Aerospace
and Electronics Conference, 1985. This algorithm uses dynamic programming to compute optimal flight path of an aircraft on a real-time basis during the flight. The territory between the target or way
point and the then current aircraft position is superimposed on a two-dimensional reference grid, the length of which is the distance between the then current aircraft position and the target or way
point (i.e., the "along-track" direction) and the width of which is a predetermined grid corridor beyond which the aircraft cannot venture on its way to the target or way point (i.e., the
"cross-track" direction). The aircraft is constrained in its lateral manuevering so that the aircraft heading deviates no more than 45 degrees from the straight heading to the target or way point,
(i.e., the along-track direction). Although this algorithm is relatively fast compared with many prior art algorithms, many possible paths to the target (which may include the overall optimal path)
are ignored because of the artifical constraints imposed by the 45 degree heading limitation.
The need therefore exists in the art for a system and method to determine and update on a real-time basis the optimal flight path for an aircraft, wherein substantially all reasonable flight paths
are examined within a reasonable period of time.
It is, therefore, the principal object of the present invention to provide an improved system and method for determining and updating on a real-time basis optimal aircraft flight path.
It is another object of the invention to provide a system and method for determining optimal aircraft flight path in which substantially all reasonable flight paths are taken into consideration.
It is yet another object of the invention to provide a method for determining and updating on a real-time basis optimal aircraft flight path, using dynamic programming techniques without imposing
artificial constraints on aircraft heading and maneuvering.
It is still another object of the invention to provide a system and method for determining optimal aircraft flight path for selected aircraft heading constraints at the target or way point.
These and other objects are accomplished in accordance with the present invention wherein a system and method are provided for determining an optimal path for an aircraft to fly from a first position
to a second position so that the aircraft heading at the second position is within preselected minimum and maximum heading limits. A reference grid is constructed having a length which is oriented
along a first axis connecting the first and second positions so that the first axis corresponds to the center line of the grid and a width which corresponds to a preselected width along a second axis
which is perpendicular to the first axis. The grid is divided into N number of ranks in succession between the first and second positions. The first position is at the center of the first rank and
the second position is adjacent to the center of the Nth rank. Each rank is divided into a predetermined number of rectangular cells.
Those cells in the Nth rank from which the aircraft can reach the second position within the preselected minimum and maximum heading limits without exceeding the maximum lateral acceleration allowed
for the aircraft are "connected" cells. The cost of flying from each of the connected cells in the Nth rank to the second position is computed based on a preselected cost function. Corresponding
minimum and maximum heading limits at each of the connected cells in the Nth rank are determined. The corresponding minimum and maximum heading limits represent a range of allowed headings for the
aircraft at the corresponding connected cell which will enable the aircraft to reach the second position within the preselected minimum and maximum heading limits without exceeding the maximum
lateral acceleration.
After the connected cells in the Nth rank are determined, the respective connected cells in each of the other ranks are determined in sequence beginning with the (N-1)th rank and ending with the
second rank. This is accomplished by determining the respective connected paths between each pair of adjacent ranks (i-1) and i. Each of the connected paths represents a discrete connection between a
specified cell in the (i-1)th rank and the specified connected cell in the ith rank whereby the aircraft can reach the specified cell in the ith rank within the corresponding heading limits from the
specified cell in the (i-1)th rank without exceeding the maximum lateral acceleration. Each of the cells in the (i-1)th rank from which the aircraft can reach at least one of the connected cells in
the ith rank within the corresponding heading limits without exceeding the maximum lateral acceleration is deemed to be a "connected" cell in the (i-1)th rank.
After the connected cells in the (i-1)th rank are determined, the cost of flying along each of the connected paths between the (i-1) rank and the ith rank is computed. The minimum cost of flying from
each of the connected cells in the (i-1)th rank all the way to the second position and the corresponding minimum cost path are stored. The corresponding minimum and maximum heading limits for each of
the connected cells in the (i-1)th rank are then determined. The corresponding heading limits represent a range of allowed headings for the aircraft at the corresponding connected cell in the (i-1)th
rank which will enable the aircraft to reach a particular one of the connected cells in the ith rank within the corresponding heading limits for that particular connected cell in the ith rank without
exceeding the maximum lateral acceleration. The particular connected cell in the ith rank is that cell which lies on the path of minimum cost between the corresponding connected cell in the (i-1)th
rank and the second position.
After the connected cells and the corresponding minimum and maximum heading limits for each of the connected cells are determined, the particular ones of the connected cells in the second rank which
the aircraft can reach from the first position within the corresponding heading limits without exceeding the maximum lateral acceleration are identified. The cost of flying from the first position to
each of the aforementioned connected cells in the second rank is computed based on the cost function and the path of minimum cost of flying from the first position to the second position is
determined based on the cumulative cost of all of the possible connected paths. The minimum cost path between the first and second positions is the optimal flight path for the aircraft. The
particular connected cell in the second rank which lies on the path of minimum cost between the first and second positions is used as a target point for determining the corresponding minimum and
maximum heading limits for the aircraft at the first position. Working backwards using dynamic programming techniques, the corresponding minimum and maximum heading limits at the first position can
be used as the reference heading limits for determining the optimal flight path between another position uprange from the first position and the first position so that the optimal aircraft flight
path can be determined between any pair of adjacent way points beginning with the then current position of the aircraft and extending all the way to the ultimate target.
Other objects and advantages of the invention will be apparent from the detailed description and claims when read in conjunction with the accompanying drawings wherein:
FIG. 1 shows a typical aircraft flight path through selected way points on the way to a target area and the return flight path from the target area in accordance with the present invention;
FIG. 2 is a grid diagram illustrating the determination of the optimal aircraft flight path to the target or way point as a function of the minimum cost to complete the route, according to the
present invention;
FIG. 3 illustrates the determination of the possible grid positions from which the aircraft can reach the target or way point within the preselected aircraft heading constraints, according to the
present invention;
FIG. 4 illustrates the determination of aircraft heading constraints at each possible grid position determined as shown in FIG. 3, according to the present invention; and
FIGS. 5A, 5B and 6-11 are flow diagrams illustrating the method by which the optimal aircraft flight path is determined in accordance with the present invention.
In the description which follows, like parts are marked throughout the specifications and the drawings, respectively. The drawings are not necessarily to scale and in some instances proportions have
been exaggerated in order to more clearly depict certain features of the invention.
Referring to FIG. 1, the flight path of a military aircraft on a mission in enemy territory is illustrated. When the aircraft crosses boundary 11 into enemy territory 13, the pilot will want to fly
an optimal path in terms of minimizing the overall cost of the mission. The cost takes into account the terrain elevation, the location of enemy ground threats, the cost of deviating from the
straight line path and the cumulative distance to the target area from any point along the path as well as other factors in determining the overall cost of a particular flight path.
Points 1, 2, 3, 4 and 5 represent the actual positions through which the aircraft will fly during the mission. The actual path that the aircraft will fly between these points is that path which
yields the minimum cost to fly the aircraft from one point to the next. The optimal path is determined in accordance with the present invention using dynamic programming techniques, as will be
described in greater detail hereinafter.
The aircraft enters enemy territory 13 at point 1 and flies a serpentine path 15 between points 1 and 2. The target area is at point 3. One skilled in the art will note that the aircraft flies a
different path 17 between point 2 and point 3 to the target area. After completing the mission, the aircraft flies a third path 19 between point 3 and point 4 and a fourth path 21 between point 4 and
point 5. The optimal flight path is continually updated on a real-time basis during flight to determine the optimal flight path from the then current position of the aircraft to the next point
through which the aircraft must fly. Points 1, 2, 4 and 5 are commonly referred to as "way points" and point 3 is commonly referred to as the "target area" or "destination point".
By way of example, FIG. 2 illustrates the determination of the optimal flight path of the aircraft as a function of the total cost to fly between selected points. The optimal flight path is
determined by computer on board the aircraft using dynamic programming techniques in accordance with the present invention. The region between adjacent way points is divided into a two dimensional
Cartesian grid system. The straight line axis connecting a first point (shown on the left in FIG. 2) with a second point downrange (shown on the right in FIGURE 2) is the Y-axis and the direction is
referred to as "along track". The axis perpendicular to the Y-axis is the X-axis and the direction along the X-axis is referred to as "cross track". The length of the grid along track is
substantially equal to the distance between the first and second points and the width of the grid across track is a predetermined width. Typically, the width of the grid, which is referred to as the
"corridor width", is less than or equal to the length of the grid along track. The grid axes are not necessarily oriented in a North-South or East-West direction.
The grid consists of a plurality of successive regions extending across the entire corridor width, which are referred to as "ranks". Within each rank are a plurality of rectangular "cells". Each rank
preferably has M number of cells, where M is an odd integer, so that the center of each rank is at the center of the middle cell thereof. The ranks are designated by the letters A, B, C, D and E in
order from the last rank just before reaching the second way point to the first rank in which the aircraft is located at a particular time. The individual cells within each rank are designated with
the letter of the rank and the cell number within each rank as shown in FIG. 2. For example, cell E1 is at the upper left corner of the grid and cell A7 is at the lower right corner of the grid. The
first point is in the center cell (E4) of the first rank and the second point is just outside the center cell (A4) of the last rank.
Dynamic programming is used to compute the optimal path between first and second points 23 and 25 in accordance with the present invention. The heading limitations of the aircraft at second point 25
are either pre-stored in the on-board computer or entered into the computer by the pilot during the mission. The heading limitations specify the minimum and maximum headings of the aircraft at second
point 25, as measured from the X-axis as shown, with "0" being the smallest possible heading and "π" being the largest possible maximum heading so that no component of the aircraft heading is
directed away from second point 25. Based on the aircraft speed and maximum lateral acceleration, the computer determines from which cells in rank A the aircraft could reach second point 25 within
the prescribed heading limits and computes the cost to arrive at second point 25 from each of the "connected" cells in rank A. The process by which the connected cells are determined is described in
greater detail hereinafter with reference to FIGS. 3, 4, 7 and 9.
For example, FIG. 2 shows that the aircraft can reach second point 25 from cell A3, A4 or A5. The possible paths from these cells to second point 25 are represented by the corresponding dotted lines.
The cost to reach the target area is computed by summing the individual cost components as follows:
Total Cost=Terrain Cost+Threat Cost+Cost of Deviation From Center Line+Path Length.
By way of example, FIG. 2 shows that the total cost from cell A3 is 8, from cell A4 is 5, and from cell A5 is 7. The above-described process is then repeated for rank B to determine from which cells
in rank B could the aircraft reach cells A3, A4 and A5 in rank A, based on the respective heading limits at each of those cells in rank A, which will be described in greater detail with reference to
FIGS. 3 and 4. Note that the aircraft can reach cell A3 from cell B2, B3 and B4; can reach cell A4 from cell B3 or B4; and can reach cell A5 from cell B4, B5 or B6. The total cost of each of these
paths is shown by way of example above the corresponding dashed lines,which represent the possible paths between rank B and rank A.
In addition to computing the total cost for each of the possible paths between rank B and rank A, the computer will add the previously computed costs to fly from connected cells in rank A to second
point 25 so as to determine the minimum cost from each of the connected cells in rank B all the way to second point 25. The minimum cost to reach second point 25 from each of the connected cells in
rank B is set forth within the corresponding rectangles in FIG. 2. In this manner many of the possible paths between rank B and second point 25 are eliminated so that only the path of minimum cost
from each of the connected cells in rank B all the way to second point 25 is saved.
The above-described procedure is repeated again with respect to rank C to determine the possible connections between rank C and rank B. The total cost from each of the connected cells in rank B to
second point 25 is then added to the cost computed for each of the connections between rank C and rank B so as to determine the minimum cost from each of the connected cells in rank C all the way to
second point 25. The above-described process is repeated as described above for rank D. The optimal aircraft flight path is indicated by the solid line and is determined by the minimum total cost to
fly from first point 23 in cell E4 all the way to second point 25. The optimal path passes through cells D5, C5,B5 and A5 and thence to second point 25. The total cost to complete is the summation of
the total cost for each of the intermediate paths between adjacent ranks, which by way of example in FIG. 2 is computed to be 24. The orientation of the Cartesian grid is independent of the compass
heading of the aircraft. The Y-axis of the grid is always along the straight line connecting adjacent way points. One skilled in the art will appreciate that the dynamic programming technique
described above is used to determine the optimal flight path between each pair of adjacent points all the way to the target,working backwards from the target, so that the optimal flight path from the
then current position of the aircraft all the way to the target represents a composite of all the individual optimal flight paths between intermediate way points. A separate grid is constructed for
each pair of adjacent way points all the way to the target.
Referring to FIGS. 3 and 7, the algorithm (CONNEC) for determining the "connected" cells in the grid is shown in detail. The preselected minimum and maximum aircraft headings at second point 25 are
used to form two imaginary circles having radius R. Radius R is determined as follows:
R=V^2 /Gmax
where V is a aircraft velocity (assumed to be constant during the mission) and Gmax is the preselected maximum lateral acceleration allowed for the aircraft (taking into account physical limitations
of the aircraft and pilot).
By way of example, FIG. 3 shows the method for determining the connected cells in rank A. The number of possible positions in rank A from which the aircraft can reach second point 25 depends upon
whether and at what point first and second circles 27 and 31 intersect centerline 35 of rank A. Those cells in rank A whose centers lie inside first circle 27 or second circle 31, such that the
corresponding circle intersects centerline 35 before reaching those cells would be outside the envelope of possible positions in rank A from which the aircraft could reach second point 25 within the
prescribed heading limits using the maximum lateral acceleration. The first connected cell (beginning at the left boundary of the grid) in rank A is that cell in which first circle 27 first
intersects centerline 35 and the last connected cell in rank A is that cell in which second circle 31 first intersects centerline 35. If, on the other hand, circles 27 and 31 never intersect
centerline 35, the aircraft could reach second point 25 within the prescribed heading limits from any position in rank A, so that the number of possible positions is limited only by the preselected
corridor width. One skilled in the art will appreciate that depending upon the prescribed heading limits, one of the two circles may intersect centerline 35 while the other may not. In that case the
connected cells are those cels lying between the particular cell in which the corresponding circle intersects centerline 35,on one side of rank A, and the corridor boundary on the opposite side of
rank A.
One skilled in the art will also appreciate that the algorithm described above with reference to FIGS. 3 and 7 for determining the connected cells between rank A and second point 25 is also used to
determine the possible connections between rank A and rank B, and between rank B and rank C and so on back to first position 23 in the first rank. Minimum and maximum heading limits are determined
for each connected cell in rank A, as will be described in greater detail with reference to FIGS. 4 and 9,and are used to determine the possible connections between rank B and rank A. Two circles
having radius R and passing through the corresponding minimum and maximum headings at each connected cell in rank A are used to determine the connected cells in rank B in substantailly the same
manner as described above. The possible connections between rank B and rank A are determiened for each connected cell in rank A. In this manner all possible connections are examined consistent with
the maximum allowed lateral acceleration of the aircraft and the preselected heading limits at second point 25.
Referring to FIGS. 4 and 9, the algorithm for determining the corresponding minimum and maximum heading limits at each of the connected cells in the grid is shown in detail. By way of example, the
determination of minimum and maximum headings will be described with reference to connected cell A2, as shown in FIG. 4. One skilled in the art will recognize that substantially the same method is
used to determine the corresponding minimum and maximum headings for the other connected cells in rank A as well as in the other ranks of the grid.
FIG. 9 illustrates the steps of the algorithm (HDGLIM) for determining minimum and maximum headings for each of the connected cells in the grid. The first step is to determine local heading (HdgLoc)
37. Local heading 37 is the heading along a line connecting a specified cell with a target point (e.g., a particular cell in the next rank or second point 25) and is represented in FIG. 4 by the
dashed line connecting the center of cell A2 with second point 25, as measured with respect to the "0" heading along the X-axis. Angle B is then computed by subtracting local heading 37 from the
previously computed maximum heading (HdgMax (Old)). In the example shown in FIG. 4, the previously computed maximum heading is maximum heading 33 at second point 25.
A circle 39 is formed having a radius R which passes through second point 25 with a heading equal to maximum heading 33 and curves toward smaller heading angles as third circle 39 moves away from
second point 25. Within circle 39 angle B is also the angle between radial line 41, which joins the center of circle 39 and second point 25, and radial line 43, which bisects chord 45 of circle 39.
Chord 45 lies on the same axis as local heading 37. If the distance (L) between the specified connection points (in this example between cell A2 and second point 25) is greater than or equal to 2R|
sinB|, the target point (in this example, second point 25) can be reached with a heading equal to the corresponding maximum heading at the target point without exceeding the maximum lateral
acceleration. If, on the other hand, the distance (L) is less and 2R|sinB|, the target point cannot be reached with a heading equal to the corresponding maximum heading without exceeding the maximum
lateral acceleration. In this latter case, the minimum heading at the current cell is determined from acceleration limits.
As shown in FIG. 9, in the first instance where the target cell can be reached with a heading equal to the maximum heading at the target point, the minimum heading at the specified cell HdgMin (New)
is equal to the lesser heading as between: (1) the greater of (HdgLoc-B) or "0"; and (2)π. If, however, the target cell cannot be reached with a heading equal to the maximum heading at the target
point without exceeding acceleration limits, the minimum heading at the current cell is equal to the lesser heading as between: (1) the greater of (HdgLoc-arcsin L/2R) or "0"; and (2)π.
In the example shown in FIG. 4 the distance L is the distance along the dashed line between the center of cell A2 and second point 25. One skilled in the art will recognize that 2R|sin B| is the
length of chord 45, which is substantially greater than distance L. Therefore, second point 25 cannot be reached from cell A2 at maximum heading 33 without exceeding acceleration limits. Minimum
heading 47 at cell A2 is therefore "0" because (HdgLoc-arcsin L/2R), which is indicated at 49, is less than "0".
The maximum heading 51 at the specified cell (cell A2 in the example illustrated in FIG. 4) is determined in substantially the same manner, except that a different reference angle A is used. Angle A
is determined by substracting local heading 37 (HdgLoc) from minimum heading 29 (HdgMin (Old)). Angle A is also the angle between radial line 53, which intersects first circle 27 at second point 25
and radial line 55, which bisects chord 57 of circle 27.
If the distance L between center of cell A2 and second point 25 is greater than or equal to 2R|sin A| (which is equal to the length of chord 57), maximum heading 51 (HdgMax (New)) is equal to the
lesser heading as between: (1) the greater of (HdgLoc-A) or "0"; and (2)π. In such event the target point (e.g., second point 25) can be reached with a heading equal to minimum heading 29 from the
specified cell (e.g., cell A2) without exceeding acceleration limits.
If, however, the distance L is less than 2R|sin A|, second point 25 cannot be reached with a heading equal to minimum heading 29 without exceeding acceleration limits. In that case maximum heading 51
is equal to the lesser heading as between: (1) the greater of (HdgLoc-arcsin L/2R) or "0"; and (2)π.
In the example shown in FIG. 4, distance L is greater than 2R|sinA| (i.e., the length of chord 57) so that the second point 25 can be reached from the center of cell A2 with a heading equal to
minimum heading 29 at second point 25. Therefore, maximum heading 51 is equal to (HdgLoc-A).
One skilled in the art will recognize that the method described above is used to determine minimum and maximum headings for the connected cells in each rank of the grid, in sequence beginning with
the last rank and ending with the second rank (i.e., the rank adjacent to first point 23). In this manner, all possible connected paths are examined consistent with the preselected minimum and
maximum headings at second point 25 and the maximum lateral acceleration allowed for the aircraft.
Referring to FIGS. 5A and 5B, the ROUTEP algorithm implements the dynamic programming algorithm described above. Pre-stored cost data (including threat information) and heading constraint information
for the aircraft are used to determine optimal paths between successive way points along the way to the target. The orientation of the grid, the cross-track and along-track sizes of the cells, the
number of ranks between adjacent way points and the corridor width of the grid are computed. The grid routine is depicted in greater detail in FIG. 6. The heading limits of the aircraft (i.e., the
minimum and maximum headings) at the last way point (i.e., the target) are entered.
The CONNEC routine, which is described hereinabove with reference to FIG. 7, is used to find the particular cells in the last rank (i.e., the rank adjacent to the target) from which the aircraft can
reach the target within the prescribed heading limits. The RNKFST routine (FIGURE 8) is then used to compute the total cost from each of the connected cells in the last rank to the target in sequence
from the left side of the grid (i.e., looking toward the target) to the right side of the grid. The HDGLIM routine (FIG. 9) is then used to determine the minimum and maximum heading limits from each
of the connected cells in the last rank to reach the target within the heading limits specified at the target, as described hereinabove.
The CONNEC routine, as depicted in FIG. 7, is then used again to determine from which cells in the penultimate rank the aircraft could reach those cells in the last rank which are previously
determined to be connected cells, within the heading limits determined for each of the connected cells in the last rank. The RNKNXT routine, as depicted in FIG. 10, is used to find the total cost of
each of the possible connections from the cells in the penultimate rank to the connected cells in the last rank in sequence from the left boundary to the right boundary within the penultimate rank.
The previously determined cost from each of the connected cells in the last rank to the target is then added to the cost determined for each of the possible connections between the penultimate rank
and the last rank to determine the cumulative cost from each of the connected cells in the penultimate rank to the target. The respective paths having the minimum cumulative cost from each of the
connected cells in the penultimate rank to the target are then saved and stored as possible segments of the optimal route.
The HDGLIM routine (FIG. 9) is then used to determine the corresponding minimum and maximum heading limits for each of the connected cells in the penultimate rank. The minimum and maximum heading
limits represent a range of headings from which the aircraft can reach the particular cell in the last rank which lies on the minimum cost path between the corresponding cell in the penultimate rank
and the target, within the heading limits for the particular cell in the last rank without exceeding maximum lateral acceleration.
The aforementioned process is performed as described above with respect to the connected cells in each rank in sequence from the last rank to the first rank (in which the penultimate way point is
located). An optimal path is selected from each of the connected cells in the first rank all the way to the target. The optimal path is the path of minimum cost from the corresponding connected cell
to the target.
The foregoing process is then repeated, working backward from the penultimate way point (using the heading limits previously calculated for that way point) to the next way point uprange and so on, so
that an optimal route path can be constructed backwards from the target to the then current aircraft position. The optimal route computed using the ROUTEP algorithm is converted into a series of
corresponding aircraft headings, which are used as inputs to a TACUE routine for controlling the lateral acceleration of the aircraft using a conventional cubic spline equation, as depicted in
greater detail in FIG. 11.
Various embodiments of the invention have now been described in detail. Since it it obvious that many changes in and additions to the above-described preferred embodiment may be made without
departing from the nature, spirit and scope of the present invention, the invention is not to be limited to said details, except as set forth in the appended claims.
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Introduction to Machine Learning
February 2010: ISBN-10: 0-262-01243-X, ISBN-13: 978-0-262-01243-0
The book can be ordered through The MIT Press, Amazon (CA, CN, DE, FR, JP, UK, US), Barnes&Noble (US), Pandora (TR).
· PHI Learning Pvt. Ltd. (formerly Prentice-Hall of India) published an English language reprint for distribution in India, Bangladesh, Burma, Nepal, Sri Lanka, Bhutan, and Pakistan only.
· Yapay Öğrenme, the Turkish edition of the book (translated by the author) was published by Boğaziçi University Press in April 2011.
· Chinese simplified character edition of the book will be published by China Machine Press/Huazhang Graphics & Information Co.
Table of Contents and Sample Chapters
Lecture Slides:
(For instructors to use in their courses; please keep the first page and footer if you edit the slides)
For Instructors: Select the "Online Instructor's Manual and Supplemental Content Download Request" link in the left menu of the book's web page, http://mitpress.mit.edu/9780262012430.
The goal of machine learning is to program computers to use example data or past experience to solve a given problem. Many successful applications of machine learning exist already, including systems
that analyze past sales data to predict customer behavior, optimize robot behavior so that a task can be completed using minimum resources, and extract knowledge from bioinformatics data.
Introduction to Machine Learning is a comprehensive textbook on the subject, covering a broad array of topics not usually included in introductory machine learning texts. In order to present a
unified treatment of machine learning problems and solutions, it discusses many methods from different fields, including statistics, pattern recognition, neural networks, artificial intelligence,
signal processing, control, and data mining. All learning algorithms are explained so that the student can easily move from the equations in the book to a computer program.
The text covers such topics as supervised learning, Bayesian decision theory, parametric methods, multivariate methods, multilayer perceptrons, local models, hidden Markov models, assessing and
comparing classification algorithms, and reinforcement learning. New to the second edition are chapters on kernel machines, graphical models, and Bayesian estimation; expanded coverage of statistical
tests in a chapter on design and analysis of machine learning experiments; case studies available on the Web (with downloadable results for instructors); and many additional exercises. All chapters
have been revised and updated.
Introduction to Machine Learning can be used by advanced undergraduates and graduate students who have completed courses in computer programming, probability, calculus, and linear algebra. It will
also be of interest to engineers in the field who are concerned with the application of machine learning methods.
· p. 41: Fourth line from the bottom of the page: “ic” should be “is” (Alexander Moriarty)
· p. 66: Fourth line from the top of the page: “negligible” is misspelled (Bugra Akyildiz)
· p. 124: Eq. 6.20; subscript of \epsilon should be j (Gi-Jeong Si)
· p. 130: Below Eq. 6.37, while taking the derivative, 2 should be outside the parenthesis (Ali Çeliksu, Gi-Jeong Si).
· p. 135: Eq. 6.47; in the final z, s should be a superscript and not a subscript (Gi-Jeong Si)
· p. 194: Eq. 9.15: b[m] should be b[mj] (Gökhan Özbulak)
· p. 224: Just above Eq. 10.30, after Mult, the subscript k should be uppercase K (Gi-Jeong Si)
· p. 283: Around the middle of the page, it should be: l not equal to j (Gi-Jeong Si)
· p. 319: In the constraints below Eq. 13.19, it should be \sum_t \alpha_t \ge \nu (Rui Kuang)
· p. 330: In the third line of the first equation, the + before (w^Tx + w[0]) should be – and the – before r^t should be + (Mehmet Gönen, Gi-Jeong Si)
· p. 330: In Eq. 13.50, the – sign before the last term (\sum_t r^t(\alpha^t+\alpha^t_-) ) should be a + (Yongwoon Cho)
· p. 333: Just under Eq. 13.53, t of \gamma should be a superscript. (Gi-Jeong Si)
· p. 336: Eq in the middle of the page; subscript of \lambda should be j (Gi-Jeong Si)
· p. 343: Two lines before the bottom of the page, the subscript of the last q should be uppercase K (Gi-Jeong Si)
· p. 348: Third eq on the page, the correct way to write is L(w|X); it is also better in the eq that follows to omit defining a separate term as L(r|X,w,\beta) but keep log p(r|X,w) (Gi-Jeong Si)
· p. 348: Eq 14.11: The second term should read N\log\sqrt{\beta} (Orhan Özalp)
· p. 352: 7^th line from the top of the page, closing ] is missing after 1,0 (Gi-Jeong Si)
· p. 356: First eq. p(x) should be p(w) (Murat Semerci, Gi-Jeong Si).
· p. 378: Eq. 15.33: There should be a normalizing 1/P(O^k) factor after sum over k and before sum over t, while updating a and b values (Vicente Palazon).
· P. 389: The very last eq on the bottom of the page; the prob is 0.48 and not 0.47 (Gökhan Özbulak)
· p. 392: The first equation, the denominator of the second term; there should be no ~ (Gi-Jeong Si)
· p. 405: Second line of Eq. 16.17: Index of summation should be Y in the second summation (Alex Kogan)
· p. 492: Two lines below Eq. 18.20; the – between r[t][+1] and \gammaV should be + (Murat Semerci, Gi-Jeong Si)
· p. 500: The denominator should be divided by N (inside sqrt): \sqrt{p_0(1-p_0)/N} (Lisa Hellerstein)
I would like to thank everyone who took the time to find these errors and report them to me.
Created on Feb 11, 2010 by E. Alpaydin (my_last_name AT boun DOT edu DOT tr)
● March 4, 2010: Added lecture slides and solutions of exercises for instructors.
● May 20, 2010: Added info about the reprint by PHI Learning Pvt. Ltd of India.
● June 20, 2011: Added info about the Turkish edition.
● Sep 26, 2011: Added lecture slides and errata.
● May 22, 2012: Changed instructor’s kit link to MIT website.
● Apr 5, 2013: Added info about the Chinese simplified character edition.
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Convergence of sequence of continuous random variables
August 29th 2009, 06:04 AM #1
Junior Member
Sep 2008
Boston, Massachusetts
Convergence of sequence of continuous random variables
Let $X_1, X_2, ..$ be a sequence of absolutely continuous random variables such that $X_n$ has pdf
$f_n(x) = \begin{cases} 1 - \cos (2 \pi n x) & x \in [0,1], \\ 0 & otherwise \end{cases}$.
Show that the sequence converges in distribution. What happens when $f_n \rightarrow \infty$?
to check for the convergence of a sequence of rv, you mustn't see the convergence of the pdf, but the convergence of the mgf, cdf, or characteristic function.
so for example, let's find the cdf :
$F_n(x)=\begin{cases} 0 & \text{if } x<0 \\ x-\frac{\sin(2\pi n x)}{2\pi n} & \text{if } x \in[0,1] \\ 1 &\text{if } x>1 \end{cases}$
by using the squeeze (or sandwich) theorem, we can show that $\lim_{n\to\infty} \frac{\sin(2\pi n x)}{2\pi n}=0$
so the limiting cdf is $F(x)=\begin{cases} 0 & \text{if } x<0 \\ x & \text{if } x \in[0,1] \\ 1 &\text{if } x>1 \end{cases}$
which is the cdf of a uniform distribution over $[0,1]$
looks clear to you ?
Yes, exactly
converges weakly towards a uniform distribution.
Last edited by Moo; August 30th 2009 at 04:59 AM.
Im now only having trouble picturing what happens when $f_n \rightarrow \infty$. Can anyone please explain?
There is no point finding the convergence of $f_n$. I mean it's not useful for the convergence of random variables here.
August 29th 2009, 06:33 AM #2
August 30th 2009, 03:56 AM #3
Junior Member
Sep 2008
Boston, Massachusetts
August 30th 2009, 04:43 AM #4
August 31st 2009, 05:26 AM #5
Junior Member
Sep 2008
Boston, Massachusetts
September 18th 2009, 07:25 AM #6
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How to remember numbers?
As a passionate of mathematics, we all love numbers, don't we?
But how can you remember numbers? Well, there exist a few mnemonic methods. One of them is by mathematical formulaes: instead of remembering a large number you remember the mathematical properties
which are (sometimes) easier to remember. Other powerfull techniques are the major system and the count system.
I would like to share my website with you to help you memorizing numbers: http://www.rememberg.com
Feedback welcome. Thanks in advance.
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Total # Posts: 425
What is the role of energy in physical changes? Please and thank you!!!
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A rocketship, travelling away from earth with a speed of 0.8c, sends regular radio signals which are recieved on Earth every 45.2 min. According to the rocketship astronauts, how frequently are the
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Four roommates are planning to spend the weekend in their dorm room watching old movies, and they are debating how many to watch. Here is their willingness to pay for each film: Orson Alfred Woody
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being discussed. I hope this helps a little. Thanks for asking.
Hi, I have a math question that I need help with: g:-> 3(1.3)^x State equations for any asymptotes to the graph of g. I know what an asymptote is, but I do not understand this question...so lost.
Help would be greatly appreciated. Thanks so much I'm a little unsure what...
give the demensions of rectangles with the perimeters of 70 feet and length-to-width ratios of 3 to 4, 4 to 5, and 1 to 1. Ok, the perimeter is 70 and perimeter = 2(length + width) The ratio is
length:with=3:4 so length = (3/4)width 70 = 2(length + width) =2((3/4)width + widt...
What is the width of a rectangle with a perimeter of 70 feet if its length is 1 foot? 2 feet? L.feet? perimeter = 2(length + width) If perimeter = 70 then 35= length + width Now substitute 1,2,and L
and solve for width. how do you find the L. feet? L is supposed to be any leng...
suppose the length of a rectangle increases,but the perimeter remains at 70 feet. How does the width change? Either the width increases, decreases or remains the same. Which of these three choices
makes the most intuitive sense? We know perimeter = 2(length +width) If perimete...
math riddle
Estimate a population: To study a species of fox, a team captures and tags 25 foxes from a large forest and then releases them. The following week they capture 15 foxes and find 5 are tagged.
Estimate the number of foxes in the forest. This is a proportion problem. We know tha...
Stat/trig question, please help me, thanks
In 1995, more than 1.1 million students in the U.S. took the SAT. On the mathematics section, the mean =507, s=112 (standard deviation). Students recieve scores rounded to the nearest 10. What is the
interval of student scores that lie within one standard deviation of the mean...
math problem
the circle graph at the right shows eye colors of the students in Mr.Chare's 4th hour class. If 7 students have either green or hazel eyes, how many students are there in the class? Circle graph is..
12% gray 24%hazel 4% green 36% brown 24% blue I need help solving this, t...
Bio Bio
You have just located a nucleolus in the nucleus. You are able to excise this area of the genome and express the genes of this chromosomal fraction in vitro (in a test tube). What genes would you
most likely find expressed from this portion of the cell's DNA? I. tRNA II. r...
Bio Bio Please check answer
Because codons are triplicates, there are ______ possible nucleotides combinations that could make a codon; each giving rise to one specific amino acid. Yet, there are only 20 amino acids used to
produce proteins. Thsus, the gentic code is said to be _______. I. 16, unambigous...
Bio Bio
Please help me out with this: You have just located a nucleolus in the nucleus. You are able to excise thsi area of the genome and express the genesof this chromosomal fraction in vitro (in a test
tube). What genes would you most likely find expressed from this portion of the ...
"write a paragraph on book's thesis"?
I'm going into AP (advanced placement) European History, grade 10, and our summer reading assignment was to read a <i>A World Lit Only By Fire</i>, a book about the tenaissance. Along with this
assignment, I'm required to write a paragraph about the book...
please correct this sentence: A)HOPEFULLY, we B)WILL BE able to complete the building C)BEFORE the rainy seas sets D)IN. E)NO ERROR i already know the answer is A, but i'm not sure why. can u please
explain, thanks. I believe the writer means when the "rainy season se...
please correct this sentence: the reason i A) WILL not be going to mexico is C)BECAUSE i will use up all my money D)IN ATTENDING an important meeting in singapore. E)NO ERROR i already know that the
answer is c, but please explain. thank you.
If a and b are positive integers and their product is 3 times their sum, what is the value of 1/a + 1/b? Given that ab = 3(a + b). From this, ab/3 = a + b. Dividing through by ab yields 1/a + 1/b = 1
there are 12 men on a basketball team, and in a game 5 of them play at any one time. if the game is 1 hr long and if each man plays exactly the same amount of time how many minutes does each man
play? a) 10 b) 12 c) 24 d) 25 e) 30 b) 12 NO. Its d) 25. With a 60 minute game and...
if A (2, -1) and B (4,7) are endpoints of a diameter of a circle, what is the area of the circle? a)16pi b) 17pi c) 18pi d)144pi e)1156pi Plot the two points and join 'em with a line. Now plot
another point C (4, -1). And then simply run a line from B down to the new point...
two cylindrical tanks have the same height, but the radius of one tank equals the diameter of the other. if the volume of the larger is k% more than the volume of the smaller, k= ? a)50 b)100 c)200
d)300 e)400 small tank volume: pi*(r/2)^2*h big tank volume: pi*r^2*h factor sm...
if anthony had 3 times as many marbles as he actually has, he would have 1/3 as many marbles as Billy has. what is the ratio of the number of marbles Anthony has to the number of marbles Billy has?
a)1:9 b)1:3 c)1:1 d)3:1 e) 9:1 1:9 Ex: Anthony has 3 marbles and Billy has 27. ...
a rectangle has a perimeter equal to circumference of a circle of radius 3. if the width of the rectangle is 3, what is the length? a) 3pi-3 b) 4.5pi-3 c)6pi-3 d)9pi-3 e)cannot be determined 3*2*pi=
3+3+2x where x=side of rectangle 6pi-6=2x 3pi-3=x
Chemistry Suspension Examples...
orange juice is a good example!! Because the state of a subthat is mixed but never really be dissovled. We learend about this in science class.
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A computer-checked veri cation of Milner's scheduler
- Handbook of Process Algebra, chapter 17
"... This chapter addresses the question how to verify distributed and communicating systems in an e#ective way from an explicit process algebraic standpoint. This means that all calculations are
based on the axioms and principles of the process algebras. ..."
Cited by 62 (16 self)
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This chapter addresses the question how to verify distributed and communicating systems in an e#ective way from an explicit process algebraic standpoint. This means that all calculations are based on
the axioms and principles of the process algebras.
, 1993
"... In this paper we study automatic veri cation of proofs in process algebra. Formulas of process algebra are represented by types in typed-calculus. Inhabitants (terms) of these types represent
proofs. The speci c typed-calculus we use is the Calculus of Inductive Constructions as implemented in the i ..."
Cited by 16 (1 self)
Add to MetaCart
In this paper we study automatic veri cation of proofs in process algebra. Formulas of process algebra are represented by types in typed-calculus. Inhabitants (terms) of these types represent proofs.
The speci c typed-calculus we use is the Calculus of Inductive Constructions as implemented in the interactive proof construction program COQ.
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another spinner prob
May 15th 2010, 11:11 AM #1
Mar 2010
another spinner prob
A circular spinner has 8 equally spaced spots; there are 3 blue, 2 green, 2 yellow and 1 red space. Let events A and B be defined as follows:
A = {outcomes with a blue on the first spin}
B = {outcomes with yellow on a spin}
a.) probability (A)
b.) probability (B)
c.)probability (A union B)
I have questions about all 3:
a.) would this just be 3/8, or do i need to factor in the second spin
b.) would this be 1/64 = 1/8 * 1/8
c.) what exactly is this part asking? prob(blue followed by a yellow)??? and if so would this just be 3/8 * 1/8 = 3/64?
It looks like you are spinning the wheel twice.
But who knows.
A is 3/8 no matter how many spins
Do you want exactly two or at least one yellow in B?
oops, that's my bad...the spinner is spun twice and its b = {outcomes with yellow on a spin}, so i am assuming that means at least one yellow
ok thanks that makes sense
so the probability of yellow on a spin would look like this:
n(s) = 8*8 =64
n(no yellow) = 6*6 =36
n(a yellow) = 28
prob(no yellow) = 36/64 = 9/16
prob(outcomes of yellow on a spin) = 28/64 = 7/16
then for the last part...what does a union b mean. does that mean a then b?
Union means both A and B occur. With the events in this example, this basically means the event that you get a blue first, then a yellow, so in this case it does mean A then B, but this wouldn't
always be the case. I believe the probability you calculated is correct.
union is an or
intersection is an and, meaning BOTH sets must occur.
so A u B means A or B, or A and B?
if so does this mean (3/8) + (7/16) + (3/8)*(7/16)
$P(A\cup B)=P(A)+P(B)-P(AB)$
where $AB$ is the intersection of A and B.
The event $AB$ means BOTH must occur
While $A\cup B$ means at least one must occur, maybe both.
so would 3/8 + 7/16 - P(AB) be correct?
and for P(AB) would I just multiply (3/8)(1/4)? that being 3/8 prob of blue on 1st spin and (1/4) prob of yellow on the second spin
thanks for your help
$P(AB)=P(A)(B)$ ONLY if the sets are independent, which I doubt.
That is one of the many equivalent definitions of independence.
Here you need to figure out what the set AB entails and then compute its probability.
Or figure out the set $A\cup B$ and compute its probability.
May 15th 2010, 01:59 PM #2
May 15th 2010, 02:13 PM #3
Mar 2010
May 15th 2010, 02:45 PM #4
May 15th 2010, 02:53 PM #5
Mar 2010
May 15th 2010, 07:55 PM #6
May 2010
May 15th 2010, 08:21 PM #7
May 16th 2010, 04:43 AM #8
Mar 2010
May 16th 2010, 02:52 PM #9
May 16th 2010, 03:03 PM #10
Mar 2010
May 16th 2010, 03:06 PM #11
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Newton - 17th Century Mathematics - The Story of Mathematics
Sir Isaac Newton (1643-1727)
In the heady atmosphere of 17th Century England, with the expansion of the British empire in full swing, grand old universities like Oxford and Cambridge were producing many great scientists and
mathematicians. But the greatest of them all was undoubtedly Sir Isaac Newton.
Physicist, mathematician, astronomer, natural philosopher, alchemist and theologian, Newton is considered by many to be one of the most influential men in human history. His 1687 publication, the
"Philosophiae Naturalis Principia Mathematica" (usually called simply the "Principia"), is considered to be among the most influential books in the history of science, and it dominated the scientific
view of the physical universe for the next three centuries.
Although largely synonymous in the minds of the general public today with gravity and the story of the apple tree, Newton remains a giant in the minds of mathematicians everywhere (on a par with the
all-time greats like Archimedes and Gauss), and he greatly influenced the subsequent path of mathematical development.
Over two miraculous years, during the time of the Great Plague of 1665-6, the young Newton developed a new theory of light, discovered and quantified gravitation, and pioneered a revolutionary new
approach to mathematics: infinitesimal calculus. His theory of calculus built on earlier work by his fellow Englishmen John Wallis and Isaac Barrow, as well as on work of such Continental
mathematicians as René Descartes, Pierre de Fermat, Bonaventura Cavalieri, Johann van Waveren Hudde and Gilles Personne de Roberval. Unlike the static geometry of the Greeks, calculus allowed
mathematicians and engineers to make sense of the motion and dynamic change in the changing world around us, such as the orbits of planets, the motion of fluids, etc.
Differentiation (derivative) approximates the slope of a curve as the interval approaches zero
The initial problem Newton was confronting was that, although it was easy enough to represent and calculate the average slope of a curve (for example, the increasing speed of an object on a
time-distance graph), the slope of a curve was constantly varying, and there was no method to give the exact slope at any one individual point on the curve i.e. effectively the slope of a tangent
line to the curve at that point.
Intuitively, the slope at a particular point can be approximated by taking the average slope (“rise over run”) of ever smaller segments of the curve. As the segment of the curve being considered
approaches zero in size (i.e. an infinitesimal change in x), then the calculation of the slope approaches closer and closer to the exact slope at a point (see image at right).
Without going into too much complicated detail, Newton (and his contemporary Gottfried Leibniz independently) calculated a derivative function f ‘(x) which gives the slope at any point of a function
f(x). This process of calculating the slope or derivative of a curve or function is called differential calculus or differentiation (or, in Newton’s terminology, the “method of fluxions” - he called
the instantaneous rate of change at a particular point on a curve the "fluxion", and the changing values of x and y the "fluents"). For instance, the derivative of a straight line of the type f(x) =
4x is just 4; the derivative of a squared function f(x) = x^2 is 2x; the derivative of cubic function f(x) = x^3 is 3x^2, etc. Generalizing, the derivative of any power function f(x) = x^r is rx^r-1.
Other derivative functions can be stated, according to certain rules, for exponential and logarithmic functions, trigonometric functions such as sin(x), cos(x), etc, so that a derivative function can
be stated for any curve without discontinuities. For example, the derivative of the curve f(x) = x^4 - 5p^3 + sin(x^2) would be f ’(x) = 4x^3 - 15x^2 + 2xcos(x^2).
Having established the derivative function for a particular curve, it is then an easy matter to calcuate the slope at any particular point on that curve, just by inserting a value for x. In the case
of a time-distance graph, for example, this slope represents the speed of the object at a particular point.
Integration approximates the area under a curve as the size of the samples approaches zero
The “opposite” of differentiation is integration or integral calculus (or, in Newton’s terminology, the “method of fluents”), and together differentiation and integration are the two main operations
of calculus. Newton’s Fundamental Theorem of Calculus states that differentiation and integration are inverse operations, so that, if a function is first integrated and then differentiated (or vice
versa), the original function is retrieved.
The integral of a curve can be thought of as the formula for calculating the area bounded by the curve and the x axis between two defined boundaries. For example, on a graph of velocity against time,
the area “under the curve” would represent the distance travelled. Essentially, integration is based on a limiting procedure which approximates the area of a curvilinear region by breaking it into
infinitesimally thin vertical slabs or columns. In the same way as for differentiation, an integral function can be stated in general terms: the integral of any power f(x) = x^r is x^r+1⁄[r+1], and
there are other integral functions for exponential and logarithmic functions, trigonometric functions, etc, so that the area under any continuous curve can be obtained between any two limits.
Newton chose not to publish his revolutionary mathematics straight away, worried about being ridiculed for his unconventional ideas, and contented himself with circulating his thoughts among friends.
After all, he had many other interests such as philosophy, alchemy and his work at the Royal Mint. However, in 1684, the German Leibniz published his own independent version of the theory, whereas
Newton published nothing on the subject until 1693. Although the Royal Society, after due deliberation, gave credit for the first discovery to Newton (and credit for the first publication to Leibniz
), something of a scandal arose when it was made public that the Royal Society’s subsequent accusation of plagiarism against Leibniz was actually authored by none other Newton himself, causing an
ongoing controversy which marred the careers of both men.
Newton's Method for approximating the roots of a curve by successive interations after an initial guess
Despite being by far his best known contribution to mathematics, calculus was by no means Newton’s only contribution. He is credited with the generalized binomial theorem, which describes the
algebraic expansion of powers of a binomial (an algebraic expression with two terms, such as a^2 - b^2); he made substantial contributions to the theory of finite differences (mathematical
expressions of the form f(x + b) - f(x + a)); he was one of the first to use fractional exponents and coordinate geometry to derive solutions to Diophantine equations (algebraic equations with
integer-only variables); he developed the so-called “Newton's method” for finding successively better approximations to the zeroes or roots of a function; he was the first to use infinite power
series with any confidence; etc.
In 1687, Newton published his “Principia” or “The Mathematical Principles of Natural Philosophy”, generally recognized as the greatest scientific book ever written. In it, he presented his theories
of motion, gravity and mechanics, explained the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis and the motion of the Moon.
Later in life, he wrote a number of religious tracts dealing with the literal interpretation of the Bible, devoted a great deal of time to alchemy, acted as Member of Parliament for some years, and
became perhaps the best-known Master of the Royal Mint in 1699, a position he held until his death in 1727. In 1703, he was made President of the Royal Society and, in 1705, became the first
scientist ever to be knighted. Mercury poisoning from his alchemical pursuits perhaps explained Newton's eccentricity in later life, and possibly also his eventual death.
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[FOM] Intermediate value theorem (and ASD)]]
Vaughan Pratt pratt at cs.stanford.edu
Fri May 22 01:55:29 EDT 2009
On 5/20/2009 2:48 PM, Arnon Avron wrote:
> On Sat, May 16, 2009 at 03:07:33PM -0000, Paul Taylor wrote:
>> ASD, a calculus that DOES NOT USE THE POWERSET, or any sets
> So it uses something else, which is no less problematic
> (like A->B or A^B or whatever). Nobody can do
> what cannot be done.
Stone can.
Assuming X is a set, what structure do you impute to 2^X? If the
structure of a set, then 2^2^X will be yet bigger. But if instead you
organize 2^X as a complete atomic Boolean algebra or CABA, and if 2^2^X
consists of the complete ultrafilters on 2^X, then 2^2^X ordered by
inclusion is a discrete poset whose underlying set is isomorphic to
(i.e. in bijection with) the set X.
If X is a complete semilattice, so are 2^X, 2^2^X, etc. where 2^X
denotes the set of complete semilattice homomorphisms to the two-element
chain (qua semilattice), organized as a complete semilattice. All the
even entries in this list are isomorphic to each other, as are all the
odd entries.
If X is a finite-dimensional vector space over the field K, then so are
K^X, K^K^X, and so on, and again these are isomorphic in the same way.
These are just a few instances of the very rich subject of Stone (more
generally Pontrjagin etc.) duality. By taking into account the
structure created and/or destroyed in the process of forming K^X where K
is the pertinent dualizing object, often but by no means always 2 (it is
the additive group T of reals mod 1 in the case of locally compact
abelian groups, the case of Pontrjagin duality), one can avoid at least
some of the problems caused by neglecting that structure when forming
power sets.
Chu spaces constitute a framework embedding all of these dualities and
far more.
The above (including Chu spaces) is entirely for concrete Stone duality.
Paul or Andrej will need to explain which parts are retained in the
abstraction to ASD as I have yet to see a list of axioms for ASD that my
20th century thought patterns are capable of parsing. I'm certain
however that it must avoid the tendency of powerset to grow an unbounded
tower of exponentials, without which the whole point of Stone duality
would be lost!
Vaughan Pratt
More information about the FOM mailing list
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Acoustic and Elastic Wave Fields in Geophysics, Part I
• Avital Kaufman
• A.L. Levshin
This book is dedicated to basic physical principles of the propagation of acoustic and elastic waves. It consists of two volumes. The first volume includes 8 chapters and extended Appendices
explaining mathematical aspects of discussed problems. The first chapter is devoted to Newton's laws, which, along with Hooke's law, govern the behavior of acoustic and elastic waves. Basic concepts
of mechanics are used in deriving equations which describe wave phenomena. The second and third chapters deal with free and forced vibrations as well as wave propagation in one dimension along the
system of elementary masses and springs which emulates the simplest elastic medium.In addition, shear waves propagation along a finite and infinite string are discussed.
In chapter 4 the system of equations describing compressional waves is derived.The concepts of the density of the energy carried by waves, the energy flux, and the Poynting vector are introduced.
Chapter 5 is dedicated to propagation ofspherical, cylindrical, and plane waves in homogeneous media, both in time andfrequency domains. Chapter 6 deals with interference and diffraction.
Thetreatment is based on Helmholtz and Kirchhoff formulae. The detailed discussion of Fresnel's and Huygens's principles is presented. In Chapter 7 the effects of interference of waves with close
wave numbers and frequencies are considered. Concepts such as the wave group, the group velocity, andthe stationary phase important for understanding propagation of dispersive waves are introduced.
The final chapter of the first volume is devoted to the principles of geometrical acoustics in inhomogeneous media.
Published: March 2000
Imprint: Elsevier
ISBN: 978-0-444-50336-7
• ...the authors have done a good job in presenting wave theory starting with very elementary matters and extending this to the detailed 'ramifications' of waves. This book should be profitably
read by students and also by specialists, who do not always have the 'ramifications' at their disposal. I await volume 2 with anticipation.
P.G. Malischewsky, Friedrich-Schiller University Jena, Germany, Geophysical Journal International
...The book should be extremely valuable for all who need a rigorous physical and mathematical wave-propagation background which, as a rule, is omitted or drastically shortened (thus difficult to
understand) in more specialized monographs. Moreover, most explanations include some innovative ideas, interesting analogies, and/or examples that make the text attractive for wave propagation
specialists and experienced lecturers, too.J. Zahradnik and O. Novotny, Charles University, The Leading Edge...should be useful to graduate students and researchers.H. Kirchner, Pure and Applied
• Introduction. List of Symbols. Newton's laws and parlide motion. Newton's laws. Motion of system of particles. Free and forced vibrations. Hooke's law of springs. Free vibrations of the system:
mass-spring. Forced vibrations of the system: mass-spring. Principles of measuring vibrations. Propagation. Propagation of waves along a system of masses and springs. Solution of 1-D wave
equation. Boundar conditions. Transversal waves in a spring. Basic equations for dilatational waves. Introduction. Wave phenomena in gas and fluid. Wave equation and boundary conditions for
dilatational waves. The kinetic and potential energy of the wave flux of the energy Poynting vector. Boundary value problem. Theorem of uniqueness. Gravitational waves in a fluid. Waves in
homogeneous medium. Spherical waves from an elementary source. Cylindrical waves from linear source in homogenous medium. Plane waves in homogeneous medium. Interference and diffration.
Superposition of waves in an uniform medium, caused by a system of primary sources. Helmholtz formula. Kirchhoff diffration theory. Fraunhofer and Fresnel diffration. Helmholtz - Kirchhoff
formula. Huygens - Fresnel principles. Relationship of potential with initial conditions. Poisson's formula. Transition and transportation equations. Superposition of sinusoidal waves with
different frequencies and wave lengths. Wave group: Phase and group velocities. Superposition of sinussoidal waves and the method of stationary phase. Principles of geometrical acoustics.
Introduction. Rays and their general features. Behaviour of rays when velocity is a function of one cartesian coordinate. Behaviour of rays when velocity is a function of one coordintae r. Rays
near interfaces. Time fields. Appendices. Vector algebra. Scalar field and gradient. Vector fields. Complex numbers. Linear ordinary differential equations with constant coefficients. Fourier
series. Fourier integral. Duhamel integral. References.
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Distortion and Curvature
January 14th 2010, 04:59 AM #1
Jan 2010
Distortion and Curvature
I am trying to work out a formula for use on sails for boats. This is a complex art and is usually done at great expense by use of a CAD package to make life simple.
What i am looking for is a little hard to explain, so i will simplify as much as possible.
2 pieces of cloth are to be joined together. They are both square pieces with the same length sides. they will be joined square end to square end.
On the lower piece of cloth, a curve is drawn. This is an arc with apex touching the centre of this top side of the cloth. This curve is cut out to give 3 square sides and one bulging side.
This curve is then laid over the top piece of unaltered cloth and stuck in such a way that the curve is attached parallel to the straight edge of the top piec of cloth.
This forces shape into the 2 pieces that are now one.
I would like to know if there is a way of working out the relationship between the flat curve and the 3D outcome.
I say this because if a curve is drawn that is 10% of the distance across that panel, in 3D it will become a much greater depth of curve (perhaps something like 30% depth of the distance between
edges of the cloth.
I'm sorry but I really don't understand your explanations... could you provide a sketch of what the pieces look like? (simply made with mspaint for instance)
As far as I understand, you have one square, and another square that you have cut so that it looks like a half-square + a half-disc, is that it? And then you glue them together, but I don't get
does this help?
I think I finally get it (from ), but the problem is that the shape you get may depend on properties of the fabric. Indeed, in order to stick the curved side to the straight side "exactly" (i.e.
edge to edge, without gap or bubble/fold), the sheets have to be perpendicular to each other... The reason why they aren't so is because the seam (or rather the tape) is not just along a line but
has a one-centimeter width, say, hence the rigidity of the fabric prevents it from making a 90° angle. I think you'll have to stick to CAD or trial-and-error technique...
hmm, you can get the seams to be at 90 degrees , thats the easy bit. It just gives a distortion when laid flat but a 3D form when held up
Fine. Then I don't know what it is that you wanted to know: the 3D form is made of the curved piece (which is flat), perpendicular to the other straight one (which is shaped like a portion of
cylinder) along the curve. After stretching/folding the fabric, other shapes are possible but uneasily computable.
January 14th 2010, 05:13 AM #2
MHF Contributor
Aug 2008
Paris, France
January 14th 2010, 05:18 AM #3
Jan 2010
January 14th 2010, 07:51 AM #4
MHF Contributor
Aug 2008
Paris, France
January 14th 2010, 08:01 AM #5
Jan 2010
January 14th 2010, 08:20 AM #6
MHF Contributor
Aug 2008
Paris, France
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Quadratic Model
February 16th 2009, 08:59 PM #1
Feb 2009
Quadratic Model
For this function $h(t)=10-4t^2$ how do you determine what time interval the ball will be more than 40 feet in the air?
Last edited by rebak; February 16th 2009 at 09:17 PM.
There must be a typo!
The graph of the given function is a parabola opening down. The vertex of the parabola (that's the highest point the ball can reach!) is at V(0, 10).
Thus: The ball never reaches a height of 40' .
You're right, it's actually $20t-2t^2$ and when it reaches a height of 50
I am assuming t is time. As said max height is 50 at t=5. Due to symmetry of the curve it should be [5-a, 5+a].
20t-2t^2=40 gives you the solution.
I think it is: [5-sqrt(5), 5+sqrt(5)].
P.S. I knew that plot of an object is a parabolla, f(x)=ax^2+bx+c, where x is the horizontal range. BUT I am not sure if it is still a parobolla when it is written with respect to time: f(t).
Would that trajectory happen on the Jupiter?
So is that -5 and 5? That's what I got originally, but I thought it didn't make sense to have the interval start at a negative number.
No the solution is:
$<br /> 5-\sqrt{5} \; \; and \; \; 5+\sqrt{5}<br />$
Thanks for your help! Did you use the quadratic formula to solve it? I used factoring and it seemed to work fine, so I wasn't sure where the square roots came from
t^2 -10t - 20=0
disc.= 10^2 - 4*1*(-20)=20
t1= (10-sqrt(20))/2
t2= (10+sqrt(20))/2
gives you the answer. You can factorize as well, but the above is easier.
If I understand you correctly you have to solve
$20t-2t^2 \geq 50~\implies~2t^2-20t+50\leq 0~\implies~t^2-10t+25 \leq 0~\implies~(t-5)^2\leq 0$
The LHS of the inequality is a square which never will be negative (or smaller than zero - that's the same). At t = 5 the LHS will be zero.
Thus the answer is: At t = 5 the ball reaches a height of 50'.
If I understand you correctly you have to solve
$20t-2t^2 \geq 50~\implies~2t^2-20t+50\leq 0~\implies~t^2-10t+25 \leq 0~\implies~(t-5)^2\leq 0$
The LHS of the inequality is a square which never will be negative (or smaller than zero - that's the same). At t = 5 the LHS will be zero.
Thus the answer is: At t = 5 the ball reaches a height of 50'.
Earboth, I understood this to mean that with this new function the ball reaches a maximum height of 50 feet but that the problem was still to find the time interval when the ball was above 40
February 16th 2009, 11:37 PM #2
February 17th 2009, 05:57 AM #3
Feb 2009
February 17th 2009, 07:26 AM #4
Jan 2009
February 17th 2009, 07:42 AM #5
Feb 2009
February 17th 2009, 07:46 AM #6
Jan 2009
February 17th 2009, 09:43 AM #7
Feb 2009
February 17th 2009, 09:48 AM #8
Jan 2009
February 17th 2009, 10:29 AM #9
February 17th 2009, 11:41 AM #10
MHF Contributor
Apr 2005
February 17th 2009, 11:46 AM #11
MHF Contributor
Apr 2005
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Newest 'applications real-analysis' Questions
Are there any applications in physics or engineering which require the Lebesgue integral and cannot be treated by Riemannian integration
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New Page 1
Placement Scores for LCC courses
(Any new student with less than 12 academic credit hours admitted to LCC after fall 2004 must take Compass English, Math and Reading placement tests or have proof of acceptable ACT or SAT scores.)
Reading Scores
A COMPASS score of 0-54 ACT score of 12 or below:
Reading Essentials is required.
A COMPASS score of 55-74 or ACT score of 13-17:
Reading For College Success is required.
(Students who place into Reading for College Success may take a course with this requirement as long as they are enrolled in Reading for College Success at the same time. If they drop Reading for
College Success , they must also drop any course requiring this reading level.)
A COMPASS score of 75 or higher or ACT score of 18 or higher:
No Reading course required.
Math Scores
Pre-Algebra test
A Compass score of 0-33 on the Pre algebra tests are placed into Foundations of Math.
ACT 1-13
A Compass score of 34 or higher on the Pre Algebra tests are placed into Applied Math or Beginning Algebra.
ACT 14-16
General Algebra test
A COMPASS Score of 0-33 on the General Algebra test are placed into Applied Math or Beginning Algebra.
ACT 14-16
A COMPASS score of 34-56 on the General Algebra test are placed into Intermediate Algebra.
ACT 17-19
Students who score 57-100 on the General Algebra test may be placed in College Algebra or Math for Education.
College Algebra test
A COMPASS score of 0-43 on the College Algebra test
ACT 20 or Higher
Trigonometry test
A COMPASS score of 44-100 on the Trigonometry test are placed into Trigonometry, Pre-calculus, or Business Calculus.
ACT 21-25
Calculus I
An A in high school Pre-Calculus or Trigonometry or B or better in high school Calculus
ACT 26 of Higher
ACT math placement scores less than 20 require students to take the compass exam.
English (Writing)
A COMPASS score of 0-51 are placed into Writing Essentials.
ACT 0-13
A COMPASS score of 52-69 are placed into Writing for College Successs.
ACT 14-17
A COMPASS score of 70-100 are placed into English Composition I.
ACT 18 and above
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"Minds, Machines and Gödel: A Retrospect''
“Minds, Machines and Gödel: A Retrospect”
June 17, 2009 by Francisco Antonio Cerón García
“Minds, Machines and Gödel: A Retrospect”, in P.J.R.Millican and A.Clark, eds., Machines and Thought: The Legacy of Alan Turing, Oxford, 1996, pp.103-124.
A paper read to the Turing Conference at Brighton on April 6th, 1990
J.R. Lucas
Fellow of Merton College, Oxford
I must start with an apologia. My original paper, “Minds, Machines and Gödel”, was written in the wake of Turing’s 1950 paper in Mind, and was intended to show that minds were not Turing machines.
Why, then, didn’t I couch the argument in terms of Turing’s theorem, which is easyish to prove and applies directly to Turing machines, instead of Gödel’s theorem, which is horrendously difficult to
prove, and doesn’t so naturally or obviously apply to machines? The reason was that Gödel’s theorem gave me something more: it raises questions of truth which evidently bear on the nature of mind,
whereas Turing’s theorem does not; it shows not only that the Gödelian well-formed formula is unprovable-in-the-system, but that it is true. It shows something about reasoning, that it is not
completely rule-bound, so that we, who are rational, can transcend the rules of any particular logistic system, and construe the Gödelian well-formed formula not just as a string of symbols but as a
proposition which is true. Turing’s theorem might well be applied to a computer which someone claimed to represent a human mind, but it is not so obvious that what the computer could not do, the mind
could. But it is very obvious that we have a concept of truth. Even if, as was claimed in a previous paper, it is not the summum bonum, it is a bonum, and one it is characteristic of minds to value.
A representation of the human mind which could take no account of truth would be inherently implausible. Turing’s theorem, though making the same negative point as Gödel’s theorem, that some things
cannot be done by even idealised computers, does not make the further positive point that we, in as much as we are rational agents, can do that very thing that the computer cannot. I have however,
sometimes wondered whether I could not construct a parallel argument based on Turing’s theorem, and have toyed with the idea of a von Neumann machine. A von Neumann machine was a black box, inside
which was housed John von Neumann. But although it was reasonable, on inductive grounds, to credit a von Neumann machine with the power of solving any problem in finite time—about the time taken to
get from New York to Chicago by train—it did not have the same edge as Gödel’s proof of his own First Incompleteness Theorem. I leave it therefore to members of this conference to consider further
how Turing’s theorem bears on mechanism, and whether a Turing machine could plausibly represent a mind, and return to the argument I actually put forward.
I argued that Gödel’s theorem enabled us to devise a schema for refuting the various different mechanist theories of the mind that might be put forward. Gödel’s theorem is a sophisticated form of the
Cretan paradox posed by Epimenides. Gödel showed how we could represent any reasonable mathematical theory within itself. Whereas the original Cretan paradox, `This statement is untrue’ can be
brushed off on the grounds that it is viciously self-referential, and we do not know what the statement is, which is alleged to be untrue, until it has been made, and we cannot make it until we know
what it is that is being alleged to be false, Gödel blocks that objection. But in order to do so, he needs not only to represent within his mathematical theory some means of referring to the
statement, but also some means of expressing mathematically what we are saying about it. We cannot in fact do this with `true’ or `untrue’: could we do that, a direct inconsistency would ensue. What
Gödel was able to do, however, was to express within his mathematical system the concept of being provable-, and hence also unprovable-, in-that-system. He produced a copper-bottomed well-formed
formula which could be interpreted as saying `This well-formed formula is unprovable-in-this-system’. It follows that it must be both unprovable-in-the-system and none the less true. For if it were
provable, and provided the system is a sound one in which only well-formed formulae expressing true propositions could be proved, then it would be true, and so what it says, namely that it is
unprovable-in-the-system, would hold; so that it would be unprovable-in-the-system. So it cannot be provable-in-the-system. But if it is unprovable-in-the-system, then what it claims to be the case
is the case, and so it is true. So it is true but unprovable-in-the-system. Gödel’s theorem seemed to me to be not only a surprising result in mathematics, but to have a bearing on theories of the
mind, and in particular on mechanism, which, as Professor Clark Glymour pointed out two days ago, is as much a background assumption of our age as classical materialism was towards the end of the
last century in the form expressed by Tyndale. Mechanism claims that the workings of the mind can be entirely understood in terms of the working of a definite finite system operating according to
definite deterministic laws. Enthusiasts for Artificial Intelligence are often mechanists, and are inclined to claim that in due course they will be able to simulate all forms of intelligent
behaviour by means of a sufficiently complex computer garbed in sufficiently sophisticated software. But the operations of any such computer could be represented in terms of a formal logistic
calculus with a definite finite number (though enormously large) of possible well-formed formulae and a definite finite number (though presumably smaller) of axioms and rules of inference. The
Gödelian formula of such a system would be one that the computer, together with its software, would be unable to prove. We, however, could. So the claim that a computer could in principle simulate
all our behaviour breaks down at this one, vital point.
The argument I put forward is a two-level one. I do not offer a simple knock-down proof that minds are inherently better than machines, but a schema for constructing a disproof of any plausible
mechanist thesis that might be proposed. The disproof depends on the particular mechanist thesis being maintained, and does not claim to show that the mind is uniformly better than the purported
mechanist representation of it, but only that it is one respect better and therefore different. That is enough to refute that particular mechanist thesis. By itself, of course, it leaves all others
unrefuted, and the mechanist free to put forward some variant thesis which the counter-argument I constructed does not immediately apply to. But I claim that it can be adjusted to meet the new
variant. Having once got the hang of the Gödelian argument, the mind can adapt it appropriately to meet each and every variant claim that the mind is essentially some form of Turing machine.
Essentially, therefore, the two parts of my argument are first a hard negative argument, addressed to a mechanist putting forward a particular claim, and proving to him, by means he must acknowledge
to be valid, that his claim is untenable, and secondly a hand-waving positive argument, addressed to intelligent men, bystanders as well as mechanists espousing particular versions of mechanism, to
the effect that some sort of argument on these lines can always be found to deal with any further version of mechanism that may be thought up.
I read the paper to the Oxford Philosophical Society in October 1959 and subsequently published it in Philosophy, 1 and later set out the argument in more detail in The Freedom of the Will. 2 I have
been much attacked. Although I argued with what I hope was becoming modesty and a certain degree of tentativeness, many of the replies have been lacking in either courtesy or caution. I must have
touched a raw nerve. That, of course, does not prove that I was right. Indeed, I should at once concede that I am very likely not to be entirely right, and that others will be able to articulate the
arguments more clearly, and thus more cogently, than I did. But I am increasingly persuaded that I was not entirely wrong, by reason of the very wide disagreement among my critics about where exactly
my arguments fail. Each picks on a different point, allowing that the points objected to by other critics, are in fact all right, but hoping that his one point will prove fatal. None has, so far as I
can see. I used to try and answer each point fairly and fully, but the flesh has grown weak. Often I was simply pointing out that the critic was not criticizing any argument I had put forward but one
which he would have liked me to put forward even though I had been at pains to discount it. In recent years I have been less zealous to defend myself, and often miss articles altogether. 3 There may
be some new decisive objection I have altogether overlooked. But the objections I have come across so far seem far from decisive.
To consider each objection individually would be too lengthy a task to attempt here. I shall pick on five recurrent themes. Some of the objections question the idealisation implicit in the way I set
up the contest between the mind and the machine; some raise questions of modality and finitude; some turn on issues of transfinite arithmetic; some are concerned with the extent to which rational
inferences should be formalisable; and some are about consistency.
Many philosophers question the idealisation implicit in the Gödelian argument. A context is envisaged between “the mind” and “the machine”, but it is an idealised mind and an idealised machine.
Actual minds are embodied in mortal clay; actual machines often malfunction or wear out. Since actual machines are not Turing machines, not having an infinite tape, that is to say an infinite memory,
it may be held that they cannot be automatically subject to Gödelian limitations. But Gödel’s theorem applies not only to Peano Arithmetic, with its infinitistic postulate of recursive reasoning, but
to the weaker Robinson Arithmetic Q, which is only potentially, not actually infinite, and hardly extends beyond the range of plausible computer progress. In any case, limitations of finitude reduce,
rather than enhance, the plausibility of some computer’s being an adequate representation of a mind. Actual minds are embodied in mortal clay. In the short span of our actual lives we cannot achieve
all that much, and might well have neither the time nor the cleverness to work out our Gödelian formula. Hanson points out that there could be a theorem of Elementary Number Theory that I cannot
prove because a proof of it would be too long or complex for me to produce. 4 Any machine that represented a mind would be would be enormously complicated, and the calculation of its Gödel sentence
might well be beyond the power of any human mathematician. 5 But he could be helped. Other mathematicians might come to his aid, reckoning that they also had an interest in the discomfiture of the
mechanical Goliath. 6 The truth of the Gödelian sentence under its intended interpretation in ordinary informal arithmetic is a mathematical truth, which even if pointed out by other mathematicians
would not depend on their testimony in the way contingent statements do. So even if aided by the hints of other mathematicians, the mind’s asserting the truth of the Gödelian sentence would be a
genuine ground for differentiating it from the machine.
Some critics of the Gödelian argument—Dennett, Hofstadter and Kirk—complain that I am insufficiently sensitive to the sophistication of modern computer technology, and that there is a fatal ambiguity
between the fundamental level of the machine’s operations and the level of input and output that is supposed to represent the mind: in modern parlance, between the machine code and the programming
language, such as PROLOG. But although there is a difference of levels, it does not invalidate the argument. A compiler is entirely deterministic. Any sequence of operations specified in machine code
can be uniquely specified in the programming language, and vice versa. Hence it is quite fair to characterize the capacity of the mechanist’s machine in terms of a higher level language. In order to
begin to be a representation of a mind it must be able to do simple arithmetic. And then, at this level, Gödel’s theorem applies. The same counter applies to Dennett’s complaint that the comparison
between men and Turing machines is highly counterintuitive because we are not much given to wandering round uttering obscure truths of ordinary informal arithmetic. Few of us are capable of asserting
a Gödelian sentence, fewer still of wanting to do so. “Men do not sit around uttering theorems in a uniform vocabulary, but say things in earnest and in jest, make slips of the tongue, speak several
languages, signal agreement by nodding or otherwise acting non-verbally, and—most troublesome for this account—utter all kinds of nonsense and contradictions, both deliberately and inadvertently.” 7
Of course, men are un-machinelike in these ways, and many philosophers have rejected the claims of mechanism on these grounds alone. But mechanists claim that this is too quick. Man, they say, is a
very complicated machine, so complicated as to produce all this un-machinelike output. We may regard their contention as highly counter-intuitive, but should not reject it out of hand. I therefore
take seriously, though only in order to refute it, the claim that a machine could be constructed to represent the behaviour of a man. If so, it must, among other things, represent a man’s mental
behaviour. Some men, many men, are capable of recognising a number of basic arithmetical truths, and, particularly when asked to (which can be viewed as a particular input), can assert them as
truths. Although “a characterization of a man as a certain sort of theorem-proving machine” 8 would be a less than complete characterization, it would be an essential part of a characterization of a
machine if it was really to represent a man. It would have to be able to include in its output of what could be taken as assertions the basic truths of arithmetic, and to accept as valid inferences
those that are validated by first-order logic. This is a minimum. Of course it may be able to do much more—it may have in its memory a store of jokes for use in after-dinner speeches, or personal
reminiscences for use on subordinates – but unless its output, for suitable questions or other input, includes a set of assertions itself including Elementary Number Theory, it is a poor
representation of some human minds. If it cannot pass O-level maths, are we really going to believe a mechanist when he claims that it represents a graduate?
Actual minds are finite in what they actually achieve. Wang and Boyer see difficulties in the infinite capabilities claimed for the mind as contrasted with the actual finitude of human life. Boyer
takes a post mortem view, and points out that all of the actual output of Lucas, Astaire, or anyone else can be represented ex post facto by a machine. 9 Actual achievements of mortal men are finite,
and so simulable. When I am dead it would be possible to program a computer with sufficient graphic capacity to show on a video screen a complete biographical film of my life. But when I am dead it
will be easy to outwit me. What is in issue is whether a computer can copy a living me, when I have not as yet done all that I shall do, and can do many different things. It is a question of
potentiality rather than actuality that is in issue. Wang concedes this, and allows that we are inclined to say that it is logically possible to have a mind capable of recognising any true
proposition of number theory or solving a set of Turing-unsolvable problems, but life is short. 10 In a finite life-span only a finite number of the propositions can be recognised, only a finite set
of problems can be solved. And a machine can be programmed to do that. Of course, we reckon that a man can go on to do more, but it is difficult to capture that sense of infinite potentiality. This
is true. It is difficult to capture the sense of infinite potentiality. But it is an essential part of the our concept of mind, and a modally “flat” account of the a mind in terms only of what it has
done is as unconvincing as an account of cause which considers only constant conjunction, and not what would have been the case had circumstances been different. In order to capture this sense of
potentiality, I set out my argument in terms of a challenge which leaves it open to the challenger to meet in any way he likes. Two-sided, or “dialectical”, arguments often succeed in encapsulating
concepts that elude explication in purely monologous terms: the epsilon-delta exegesis an infinitesimals is best conveyed thus, and more generally any alternation of quantifiers, as in the EA
principles suggested by Professor Clark Glymour for the ultimate convergence of theories on truth.
Although some degree of idealisation seems allowable in considering a mind untrammelled by mortality and a Turing machine with infinite tape, doubts remain as to how far into the infinite it is
permissible to stray. Transfinite arithmetic underlies the objections of Good and Hofstadter. The problem arises from the way the contest between the mind and the machine is set up. The object of the
contest is not to prove the mind better than the machine, but only different from it, and this is done by the mind’s Gödelizing the machine. It is very natural for the mechanist to respond by
including the Gödelian sentence in the machine, but of course that makes the machine a different machine with a different Gödelian sentence all of its own, which it cannot produce as true but the
mind can. So then the mechanist tries adding a Gödelizing operator, which gives, in effect a whole denumerable infinity of Gödelian sentences. But this, too, can be trumped by the mind, who produces
the Gödelian sentence of the new machine incorporating the Gödelizing operator, and out Gödelizes the lot. Essentially this is the move from w (omega), the infinite sequence of Gödelian sentences
produced by the Gödelizing operator, to w + 1, the next transfinite ordinal. And so it goes on. Every now and again the mechanist loses patience, and incorporates in his machine a further operator,
designed to produce in one fell swoop all the Gödelian sentences the mentalist is trumping him with: this is in effect to produce a new limit ordinal. But such ordinals, although they have no
predecessors, have successors just like any other ordinal, and the mind can out-Gödel them by producing the Gödelian sentence of the new version of the machine, and seeing it to be true, which the
machine cannot. Hofstadter thinks there is a problem for the mentalist in view of a theorem of Church and Kleene on Formal Definitions of Transfinite Ordinals. 11 They showed that we couldn’t program
a machine to produce names for all the ordinal numbers. Every now and again some new, creative step is called for, when we consider all the ordinal numbers hitherto named, and we need to encompass
them all in a single set, which we can use to define a new sort of ordinal, transcending all previous ones. Hofstadter thinks that, in view of the Church-Kleene theorem, the mind might run out of
steam, and fail to think up new ordinals as required, and so fail in the last resort to establish the mind’s difference from some machine. But this is wrong on two counts. In the first place it begs
the question and in the second it misconstrues the nature of the contest.
Hofstadter assumes that the mind is subject to the same limitations as the machine is, and that since there is no mechanical way of naming all the ordinals, the mind cannot do it either. But this is
precisely the point in issue. Gödel himself rejected mechanism on account of our ability to think up fresh definitions for transfinite ordinals (and ever stronger axioms for set theory) and Wang is
inclined to do so too. 12 On this occasion, it is pertinent to note that Turing himself was, on this question, of the same mind as Gödel. He was led “to ordinal logics as a way to `escape’ Gödel’s
incompleteness theorems”, 13 but recognised that “although in pre-Gödel times it was thought by some that it would be able to carry this programmme to such an extent that … the necessity for
intuition would be entirely eliminated,” as a result of Gödel’s incompleteness theorems one must turn instead to `non-constructive’ systems of logic in which “not all the steps in a proof are
mechanical, some being intuitive”. Turing concedes that the steps whereby we recognise formulae as ordinal formulae are intuitive, and goes on to say that we should show quite clearly when a step
makes use of intuition, and when it is purely formal, and that the strain put on intuition should be a minimum. 14 He clearly, like Gödel, allows that the mind’s ability to recognise new ordinals
outruns the ability of any formal algorithm to do so, though he does not draw Gödel’s conclusion. It may be, indeed, that the mind’s ability to recognise new ordinals is the issue on which battle
should be joined; Good claimed as much 15—though disputes about the notation for ordinals lack the sharp edge of the Gödelian argument. But whatever the merits of different battlefields, it is clear
that they are contested areas in the same conflict, and undisputed possession of the one cannot be claimed in order to assert possession of the other.
In any case Hofstadter misconstrues the nature of the contest. All the difficulties are on the side of the mechanist trying to devise a machine that cannot be out-Gödelized. It is the mechanist who
resorts to limit ordinals, and who may have problems in devising new notations for them. The mind needs only to go on to the next one, which is always an easy, unproblematic step, and out-Gödelize
whatever is the mechanist’s latest offering. Hofstadter’s argument, as often, tells against the position he is arguing for, and shows up a weakness of machines: there is no reason to suppose that it
is shared by minds, and in the nature of the case it is a difficulty for those who are seeking to evade the Gödelian argument, not those who are deploying it.
Underlying Hofstadter’s argument is a rhetorical question that many mechanists have raised. “How does Lucas know that the mind can do this, that, or the other?” It is no good, they hold, that I
should opine it or simply assert it; I must prove it. And if I prove it, then since the steps of my proof can be programmed into a machine, the machine can do it too. Good puts the argument
What he must prove is that he personally can always make the improvement: it is not sufficient to believe it since belief is a matter of probability and Turing machines are not supposed to be capable
of probability judgements. But no such proof is possible since, if it were given, it could be used for the design of a machine that could always do the improving.
The same point is made by Webb in his sustained and searching critique of the Gödelian argument:
It is only because Gödel gives an effective way of constructing the Gödelian sentence that Lucas can feel confident that he can find the Achilles’ heel of any machine. But then if Lucas can
effectively stump any machine, then there must be a machine which does this too. 16 [This] “is the basic dilemma confronting anti-mechanism: just when the constructions used in its arguments become
effective enough to be sure of, (T) <viz. Every humanly effective computation procedure can be simulated by a Turing machine> then implies that a machine can simulate them. In particular it implies
that our very behaviour of applying Gödel’s argument to arbitrary machines – in order to conclude that we cannot be modelled by a machine – can indeed be modelled by a machine. Hence any such
conclusion must fail, or else we will have to conclude that certain machines cannot be modelled by any machine! In short, anti-mechanist arguments must either be ineffective, or else unable to show
that their executor is not a machine.” 17
The core of this argument is an assumption that every informal argument must either be formalisable or else invalid. Such an assumption undercuts the distinction I have drawn between two senses of
Gödelian argument: between a negative argument according to an exact specification, which a machine could be programmed to carry out, and on the other hand a certain style of arguing, similar to
Gödel’s original argument in inspiration, but not completely or precisely specified, and therefore not capable of being programmed into a machine, though capable of being understood and applied by an
intelligent mind. Admittedly, we cannot prove to a hide-bound mechanist that we can go on. But we may come to a well-grounded confidence that we can, which will give us, and the erstwhile mechanist
if he is reasonable and not hide-bound, good reason for rejecting mechanism.
Against this claim of the mentalist that he has got the hang of doing something which cannot be described in terms of a mechanical program, the mechanist says “Sez you” and will not believe him
unless he produces a program showing how he would do it. It is like the argument between the realist and the phenomenalist. The realist claims that there exist entities not observed by anyone: the
phenomenalist demands empirical evidence; if it is not forthcoming, he remains sceptical of the realist’s claim; if it is, then the entity is not unobserved. In like manner the mechanist is sceptical
of the mentalist’s claim unless he produces a specification of how he would do what a machine cannot: if such a specification is not forthcoming, he remains sceptical; if it is, it serves as a basis
for programming a machine to do it after all. The mechanist position, like the phenomenalist, is invulnerable but unconvincing. I cannot prove to the mechanist that anything can be done other than
what a machine can do, because he has restricted what he will accept as a proof to such an extent that only “machine-doable” deeds will be accounted doable at all. But not all mechanists are so
limited. Many mechanists and many mentalists are rational agents wondering whether in the light of modern science and cybernetics mechanism is, or is not, true. They have not closed their minds by so
redefining proof that none but mechanist conclusions can be established. They can recognise in themselves their having “got the hang” of something, even though no program can be written for giving a
machine the hang of it. The parallel with the Sorites argument is helpful. Arguing against a finitist, who does not accept the principle of mathematical induction, I may see at the meta-level that if
he has conceded F(0) and (Ax)(F(x) –> F(x + 1)) then I can claim without fear of contradiction (Ax)F(x). I can be quite confident of this, although I have no finitist proof of it. All I can do, vis à
vis the finitist, is to point out that if he were to deny my claim in any specific instance, I could refute him. True, a finitist could refute him too. But I have generalised in a way a finitist
could not, so that although each particular refuting argument is finite, the claim is infinite. In a similar fashion each Gödelian argument is effective, and will convince even the mechanist that he
is wrong; but the generalisation from individual tactical refutations to a strategic claim does not have to be effective in the same sense, although it may be entirely rational for the mind to make
the claim.
Nevertheless an air of paradox remains. The idea of a totally intuitive, unformalisable argument arouses suspicion: if it can convince, it can be conveyed, and if it can be conveyed, it can be
formulated and expressed in formal terms. Let me therefore stress that I am not claiming that my, or any, argument is absolutely unformalisable. Any argument can be formalised, as the Tortoise proved
to Achilles, the formal axiom or rule of inference invoked will be no more convincing than the original unformalised argument. I am not claiming that the Gödelian argument cannot be formalised, but
that whatever formalisation we adopt, there are further arguments which are clearly valid but not captured by that formalisation. Not only, again as the Tortoise proved to Achilles, must we always be
ready to recognise some rules of inference as applying and inferences as valid without more ado, but we shall be led, if we are rational, to extend our range of acknowledged valid inferences beyond
any antecedently laid down bounds. This does not preclude our subsequently formalising them, only our supposing that any formalisation is inferentially complete.
But we always can formalise; in particular, we can formalise the argument that Gödel uses to prove that the Gödelian formula is unprovable-in-the-system but none the less true. At first sight there
seems to be a paradox. Gödel’s argument purports to show that the Gödelian sentence is unprovable but true. But if it shows that the Gödelian sentence is true, surely it has proved it, so that it is
provable after all. The paradox in this case is resolved by distinguishing provability-in-the-formal-system from the informal provability given by Gödel’s reasoning. But this reasoning can be
formalised. We can go over Gödel’s argument step by step, and formalise it. If we do so we find that an essential assumption for his argument that the Gödelian sentence is unprovable is that the
formal system should be consistent. Else every sentence would be provable, and the Gödelian sentence, instead of being unprovable and therefore true, could be provable and false. So what we obtain,
if we formalise Gödel’s informal argumentation, is not a formal proof within Elementary Number Theory (ENT for short) that the Gödelian sentence, G is true, but a formal proof within Elementary
Number Theory
|- Cons(ENT) –> G
where Cons(ENT) is a sentence expressing the consistency of Elementary Number Theory. Only if we also had a proof in Elementary Number Theory yielding
|- Cons(ENT)
would we be able to infer by Modus Ponens
|- G
Since we know that
¬ |- G, [i.e. G is not derivable: this is the best I can do to render symbolic logic in HTML]
we infer also that
¬:|- Cons(ENT). [i.e. Cons(ENT) is not derivable]
This is Gödel’s second theorem. Many critics have appealed to it in order to fault the Gödelian argument. Only if the machine’s formal system is consistent and we are in a position to assert its
consistency are we really able to maintain that the Gödelian sentence is true. But we have no warrant for this. For all we know, the machine we are dealing with may be inconsistent, and even if it is
consistent we are not entitled to claim that it is. And in default of such entitlement, all we have succeeded in proving is
|- Cons(ENT) –> G,
and the machine can do that too.
These criticisms rest upon two substantial points: the consistency of the machine’s system is assumed by the Gödelian argument and cannot be always established by a standard decision-procedure. The
question “By what right does the mind assume that the machine is consistent?” is therefore pertinent. But the moves made by mechanists to deny the mind that knowledge are unconvincing. Paul
Benacerraf suggests that the mechanist can escape the Gödelian argument by not staking out his claim in detail. 18 The mechanist offers a “Black Box” without specifying its program, and refusing to
give away further details beyond the claim that the black box represents a mind. But such a position is both vacuous and untenable: vacuous because there is no content to mechanism unless some
specification is given—if I am presented with a black box but “told not to peek inside” then why should I think it contains a machine and not, say, a little black man? The mechanist’s position is
also untenable: for although the mechanist has refused to specify what machine it is that he claims to represent the mind, it is evident that the Gödelian argument would work for any consistent
machine and that an inconsistent machine would be an implausible representation. The stratagem of playing with his cards very close to his chest in order to deny the mind the premisses it needs is a
confession of defeat.
Putnam contends that there is an illegitimate inference from the true premiss
I can see that (Cons(ENT) —> G)
to the false conclusion
Cons(ENT) –> I can see that (G). 19
It is the latter that is needed to differentiate the mind from the machine, for what Gödel’s theorem shows is
Cons(ENT) —> ENT machine cannot see that (G),
but it is only the former, according to Putnam, that I am entitled to assert. Putnam’s objection fails on account of the dialectical nature of the Gödelian argument. The mind does not go round
uttering theorems in the hope of tripping up any machines that may be around. Rather, there is a claim being seriously maintained by the mechanist that the mind can be represented by some machine.
Before wasting time on the mechanist’s claim, it is reasonable to ask him some questions about his machine to see whether his seriously maintained claim has serious backing. It is reasonable to ask
him not only what the specification of the machine is, but whether it is consistent. Unless it is consistent, the claim will not get off the ground. If it is warranted to be consistent, then that
gives the mind the premiss it needs. The consistency of the machine is established not by the mathematical ability of the mind but on the word of the mechanist. The mechanist has claimed that his
machine is consistent. If so, it cannot prove its Gödelian sentence, which the mind can none the less see to be true: if not, it is out of court anyhow.
Wang concedes that it is reasonable to contend that only consistent machines are serious candidates for representing the mind, but then objects it is too stringent a requirement for the mechanist to
meet because there is no decision-procedure that will always tell us whether a formal system strong enough to include Elementary Number Theory is consistent or not. 20 But the fact that there is no
decision-procedure means only that we cannot always tell, not that we can never tell. Often we can tell that a formal system is not consistent—e.g. it proves as a theorem:
|- p&¬p
|- 0 = 1
Also, we may be able to tell that a system is consistent. We have finitary consistency proofs for propositional calculus and first-order predicate calculus, and Gentzen’s proof, involving transfinite
induction, for Elementary Number Theory. We are therefore not asking the impossible of the mechanist in requiring him to do some preliminary sorting out before presenting candidates for being
plausible representations of the mind. Unless they satisfy the examiner—the mechanist—in Prelims on the score of consistency, they are not eligible to enter for Finals, and all those that are thus
qualified can be sure of failing for not being able to assert their Gödelian sentence.
The two-stage examination is thus able to sort out the inconsistent sheep who fail the qualifying examination from the consistent goats who fail their finals, and hence enables us to take on all
challenges even from inconsistent machines, without pretending to possess superhuman powers. Although all machines are entitled to enter for the mind-representation examination, only relatively few
machines are plausible candidates for representing the mind, and there is no need to take a candidate seriously just because it is a machine. If the mechanist’s claim is to be taken seriously, some
recommendation will be required, and at the very least a warranty of consistency would be essential. Wang protests that this is to expect superhuman powers of him, and in a response to Benacerraf’s
“God, The Devil and Gödel”, I picked up his suggestion that the mechanist might be no mere man but the Prince of Darkness himself to whom the question of whether the machine was consistent or not
could be addressed in expectation of an answer. 21 Rather than ask high-flown questions about the mind we can ask the mechanist the single question whether or not the machine that is proposed as a
representation of the mind would affirm the Gödelian sentence of its system. If the mechanist says that his machine will affirm the Gödelian sentence, the mind then will know that it is inconsistent
and will affirm anything, quite unlike the mind which is characteristically selective in its intellectual output. If the mechanist says that his machine will not affirm the Gödelian sentence, the
mind then will know since there was at least one sentence it could not prove in its system it must be consistent; and knowing that, the mind will know that the machine’s Gödelian sentence is true,
and thus will differ from the machine in its intellectual output. And if the mechanist is merely human, and moreover does not know what answer the machine would give to the Gödelian question, he has
not done his home-work properly, and should go away and try to find out before expecting us to take him seriously.
In asking the mechanist rather than the machine, we are making use of the fact that the issue is one of principle, not of practice. The mechanist is not putting forward actual machines which actually
represent some human being’s intellectual output, but is claiming instead that there could in principle be such a machine. He is inviting us to make an intellectual leap, extrapolating from various
scientific theories and skating over many difficulties. He is quite entitled to do this. But having done this he is not entitled to be coy about his in-principle machine’s intellectual capabilities
or to refuse to answer embarrassing questions. The thought-experiment, once undertaken, must be thought through. And when it is thought through it is impaled on the horns of a dilemma. Either the
machine can prove in its system the Gödelian sentence or it cannot: if it can, it is inconsistent, and not equivalent to a mind; if it cannot, it is consistent, and the mind can therefore assert the
Gödelian sentence to be true. Either way the machine is not equivalent to the mind, and the mechanist thesis fails.
A number of thinkers have chosen to impale themselves on the inconsistency horn of the dilemma. We are machines, they say, but very limited, fallible and inconsistent ones. In view of our many
contradictions, changes of mind and failures of logic, we have no warrant for supposing the mind to be consistent, and therefore no ground for disqualifying a machine for inconsistency as a candidate
for being a representation of the mind. Hofstadter thinks it would be perfectly possible to have an artificial intelligence in which propositional reasoning emerged as consequences rather than as
being pre-programmed. “And there is no particular reason to assume that the strict Propositional Calculus, with its rigid rules and the rather silly definition of consistency they entail, would
emerge from such a program.” 22
None of these arguments goes any way to making an inconsistent machine a plausible representation of a mind. Admittedly the word `consistent’ is used in different senses, and the claim that a mind is
consistent is likely to involve a different sense of consistency and to be established by different sorts of arguments from those in issue when a machine is said to be consistent. If this is enough
to establish the difference between minds and machines, well and good. But many mechanists will not be so quickly persuaded and will maintain that a machine can be programmed, in some such way as
Hofstadter supposes, to emit mind-like behaviour. In that case it is machine-like consistency rather than mind-like consistency that is in issue. Any machine, if it is to begin to represent the
output of a mind must be able to operate with symbols that can be plausibly interpreted as negation, conjunction, implication, etc., and so must be subject to the rules of some variant of the
propositional calculus. Unless something rather like the propositional calculus with some comparable requirement of consistency emerges from the program of a machine, it will not be a plausible
representation of a mind, no matter no matter how good it is as a specimen of Artificial Intelligence. Of course, any plausible representation of a mind would have to manifest the behaviour instanced
by Wang, constantly checking whether a contradiction had been reached and attempting to revise its basic axioms when that happened. But this would have to be in accordance with certain rules. There
would have to be a program giving precise instructions how the checking was to be undertaken, and in what order axioms were to be revised. Some axioms would need to be fairly immune to revision.
Although some thinkers are prepared to envisage a logistic calculus in which the basic inferences of propositional calculus do not hold (e.g. from p & q to p) or the axioms of Elementary Number
Theory have been rejected, any machine which resorted to such a stratagem to avoid contradiction would also lose all credence as a representation of a mind. Although we sometimes contradict ourselves
and change our minds, some parts of our conceptual structure are very stable, and immune to revision. Of course it is not an absolute immunity. One can allow the Cartesian possibility of conceptual
revision without being guilty, as Hutton supposes, 23 of inconsistency in claiming knowledge of his own consistency. To claim to know something is not to claim infallibility but only to have adequate
backing for what is asserted. Else all knowledge of contingent truths would be impossible. Although one cannot say `I know it, although I may be wrong’, it is perfectly permissible to say `I know it,
although I might conceivably be wrong’. So long as a man has good reasons, he can responsibly issue a warranty in the form of a statement that he knows, even though we can conceive of circumstances
in which his claim would prove false and would have to be withdrawn. So it is with our claim to know the basic parts of our conceptual structure, such as the principles of reasoning embodied in the
propositional calculus or the truths of ordinary informal arithmetic. We have adequate, more than adequate, reason for affirming our own consistency and the truth, and hence also the consistency, of
informal arithmetic, and so can properly say that we know, and that any machine representation of the mind must manifest an output expressed by a formal (since it is a machine) system which is
consistent and includes Elementary Number Theory (since it is supposed to represent the mind). But there remains the Cartesian possibility of our being wrong, and that we need now to discuss. Some
mechanists have conceded that a consistent machine could be out-Gödeled by a mind, but have maintained that the machine representation of the mind is an inconsistent machine, but one whose
inconsistency is so deep that it would take a long time ever to come to light. It therefore would avoid the quick death of non-selectivity. Although in principle it could be brought to affirm
anything, in practice it will be selective, affirming some things and denying others. Only in the long run will it age—or mellow, as we kindly term it—and then “crash” and cease to deny anything; and
in the long run we die—usually before suffering senile dementia. Such a suggestion chimes in with a line of reasoning which has been noticeable in Western Thought since the Eighteenth Century.
Reason, it is held, suffers from certain antinomies, and by its own dialectic gives rise to internal contradictions which it is quite powerless to reconcile, and which must in the end bring the whole
edifice crashing down in ruins. If the mind is really an inconsistent machine then the philosophers in the Hegelian tradition who have spoken of the self-destructiveness of reason are simply those in
whom the inconsistency has surfaced relatively rapidly. They are the ones who have understood the inherent inconsistency of reason, and who, negating negation, have abandoned hope of rational
discourse, and having brought mind to the end of its tether, have had on offer only counsels of despair.
Against this position the Gödelian argument can avail us nothing. Quite other arguments and other attitudes are required as antidotes to nihilism. It has long been sensed that materialism leads to
nihilism, and the Gödelian argument can be seen as making this reductio explicit. And it is a reductio. For mechanism claims to be a rational position. It rests its case on the advances of science,
the underlying assumptions of scientific thinking and the actual achievements of scientific research. Although other people may be led to nihilism by feelings of angst or other intimations of
nothingness, the mechanist must advance arguments or abandon his advocacy altogether. On the face of it we are not machines. Arguments may be adduced to show that appearances are deceptive, and that
really we are machines, but arguments presuppose rationality, and if, thanks to the Gödelian argument, the only tenable form of mechanism is that we are inconsistent machines, with all minds being
ultimately inconsistent, then mechanism itself is committed to the irrationality of argument, and no rational case for it can be sustained.
To return from footnote to text, click on footnote number
1. “ Minds, Machines and Gödel” Philosophy, 36, 1961, pp.112-127; reprinted in Kenneth M.Sayre and Frederick J.Crosson, eds., The Modeling of Mind, Notre Dame, 1963, pp. 255-271; and in A.R.Anderson,
Minds and Machines, Prentice-Hall, 1964, pp. 43-59.
2. The Freedom of the Will, Oxford, 1970 (now avaialble again).
3. I give at the end a list of some of the major criticisms I have come across.
4. William Hanson, “Mechanism and G”del’s Theorems,” British Journal for the Philosophy of Science, XXII, 1971, p.12; compare Hofstadter, 1979, p.475.
5. Rudy Rucker, “G”del’s Theorem: The Paradox at the heart of modern man”, Popular Computing, February 1985, p.168.
6. I owe this suggestion to M.A.E. Dummett, at the original meeting of the Oxford Philosophical Society on October 30th, 1959. A similar suggestion is implicit in Hao Wang, From Mathematics to
Philosophy, London, 1974, p.316.
7. D.C.Dennett, Review of The Freedom of the Will, in Journal of Philosophy, 1972, p.530.
8. P.527.
9. David L.Boyer, “Lucas, G”del and Astaire”, The Philosophical Quarterly, 1983, pp. 147-159.
10. Hao Wang, From Mathematics to Philosophy, London, 1974, p.315.
11. Douglas R.Hofstadter, G”del, Escher, Bach, New York, 1979, p.475.
12. Hao Wang, From Mathematics to Philosophy, London, 1974, pp.324-326.
13. Solomon Feferman, “Turing in the Land of O(z)”, in Rolf Herken ed., The Universal Turing Machine, Oxford, 1988, p.121.
14. A.M.Turing, “Systems of logic based on ordinals”, Proceedings of the London Mathematical Society, (2), 45, 1939, pp.161-228; reprinted in M.Davis, The Undecidable, New York, 1965; quoted by
Solomon Feferman, op.cit., p.129.
15. I.J.Good, “G”del’s Theorem is a Red Herring”, British Journal for the Philosophy of Science, 19, 1968, pp. 357-358.
16. Judson C.Webb, Mechanism, Mentalism and Metamathematics; An Essay on Finitism, Dordrecht, 1980, p.230.
17. P.232, Webb’s italics.
18. Paul Benacerraf, “God, The Devil and G”del”, The Monist, 51, 1967, pp.
19. Hilary Putnam “Minds and Machines”, in Sidney Hook, ed., Dimensions of Mind: A Symposium, New York, 1960; reprinted in Kenneth M. Sayre and Frederick J.Crosson, eds., The Modeling of Mind, Notre
Dame, 1963, pp. 255-271; and in A. R. Anderson, Minds and Machines, Prentice-Hall, 1964, pp. 43-59.
20. Hao Wang, From Mathematics to Philosophy, London, 1974, p.317.
21. Paul Benacerraf, “God, The Devil and G”del”, The Monist, 51, 1967, pp. 22-23; J.R. Lucas, “Satan Stultified”, The Monist, 52, 1967, pp. 152-3. 22. Hofstadter, 1979, p.578; cf. Charles S. Chihara,
“On Alleged Refutations of Mechanism using G”del’s Incompleteness Results”, Journal of Philosophy, LXIX, no.17, 1972, p.526.
23. Anthony Hutton, “This G”del is Killing Me”, Philosophia, vol. 6, no.1, 1976, pp. 135-144.
Criticisms of the Gödelian Argument
J.J.C.Smart, “Gödel’s Theorem, Church’s Theorem, and Mechanism”, Synthese, 13, 1961.
J.J.C.Smart, “Man as a Physical Mechanism”, ch.VI of his Philosophy and Scientific Realism.
Hilary Putnam “Minds and Machines”, in Sidney Hook, ed., Dimensions of Mind. A Symposium, New York, 1960; reprinted in Kenneth M. Sayre and Frederick J. Crosson, eds., The Modeling of Mind, Notre
Dame, 1963, pp. 255-271; and in A. R. Anderson, Minds and Machines, Prentice-Hall, 1964, pp. 43-59.
C.H. Whitely, “Minds, Machines and Gödel: a Reply to Mr. Lucas”, Philosophy, 37, 1962, pp.61-62.
Paul Benacerraf, “God, the Devil and Gödel”, The Monist, 1967, pp. 9-32.
I.J. Good, “Human and Machine Logic,” British Journal for the Philosophy of Science, 18, 1967, pp. 144-147.
I.J.Good, “Gödel’s Theorem is a Red Herring”, British Journal for the Philosophy of Science, 19, 1968, pp. 357-8.
David Lewis, “Lucas Against Mechanism”, Philosophy, XLIV, 1969, pp. 231-233.
David Coder, “Goedel’s Theorem and Mechanism”, Philosophy, XLIV, 1969, pp. 234-237, esp. p.236.
Jonathan Glover, Responsibility, London, 1970, p.31.
William Hanson, “Mechanism and Gödel’s Theorems,” British Journal for the Philosophy of Science, XXII, 1971.
D.C. Dennett, Review of The Freedom of the Will, Journal of Philosophy, 1972.
Charles S. Chihara, “On Alleged Refutations of Mechanism using Gödel’s Incompleteness Results”, Journal of Philosophy, LXIX, no.17, 1972.
Hao Wang, From Mathematics to Philosophy, London, 1974, pp.319, 320, 324-326.
A.J.P.Kenny in A.J.P.Kenny, H.C.Longuet-Higgins, J.R. Lucas and C.H.Waddington, The Nature of Mind, Edinburgh, 1976, p.75.
Anthony Hutton, “This Gödel is Killing Me”, Philosophia, vol. 6, no.1, 1976, pp. 135-144.
J.W. Thorp, “Free Will and Neurophysiological Determinism”, Oxford D.Phil. Thesis, 1976, p.79.
J.L. Mackie, Ethics: Inventing Right and Wrong, Penguin, 1977, p. 219.
David Lewis, “Lucas Against Mechanism II”, Canadian Journal of Philosophy, IX, 1979, pp. 373-376.
Douglas R.Hofstadter, Gödel, Escher, Bach, New York, 1979, p.475.
Emmanuel Q. Fernando, “Mathematical and Philosophical Implications of the Gödel Incompleteness Theorems”. M.A. Thesis, College of Arts and Sciences, University of the Philippines, Quezu City,
September 1980.
Judson C. Webb, Mechanism, Mentalism and Metamathematics; An Essay on Finitism, Dordrecht, 1980, p.230.
G. Lee Bowie, “Lucas’ Number is Finally Up”, Journal of Philosophical Logic, 11, 1982, pp.279-285.
P.Sleazak, “Gödel’s Theorem and the Mind”, British Journal for the Philosophy of Science, XXXIII, 1982.
Rudy Rucker, “Gödel’s Theorem: The Paradox at the heart of modern man”, Popular Computing, February 1985, p.168.
David L. Boyer, “Lucas, Gödel and Astaire”, The Philosophical Quarterly, 1983, pp.147-159.
David Bostock, “Gödel and Determinism”, private communication, November, 1984.
Robert Kirk, “Mental Machinery and Gödel”, Synthese, 66, 1986, pp.437-452.
Other works are cited in The Freedom of the Will, pp. 174-6.
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The Reflective Educator
Gary Stager posted "Dumbing Down" a few days ago, which is a passionate plea for computers to be used for computing in schools. He writes:
Although I’m only 48, I have been working in educational computing for thirty years. When I started, we taught children to program. We also taught tens of thousands of teachers to teach computer
science to learners of all ages. In many cases, this experience represented the most complex thinking about thinking that teachers ever experienced and their students gained benefit from observing
teachers learning to think symbolically, solve problems and debug. There was once a time in the not so distant path when educators were on the frontiers of scientific reasoning and technological
progress. Curriculum was transformed by computing. School computers were used less often to “do school” and more often to do the impossible.
Gary's argument is (mostly) sound, and indeed, I argued for almost exactly the same thing in the keynote I presented in Alberta in February. Of course, both of our arguments have a flaw.
Our purpose in introducing computing is to both use the full power of a computer in schools. Indeed, the way we currently use computers in schools is much like using cars for their heaters; sure the
cars will keep us warm, but it completely misses their potential as transportation devices. This analogy is somewhat apt, since the computer (as Dr. Papert has pointed out) can be used to transport
kids to Microworlds.
Peter Eden said:
“I am interested in your comment that the power of computers as a tool is almost superficial until you learn to program. Others peoples' programs are like using the computer to do something that can
be done by other means. Word processing is really just typing. Many maths programs are really just calculators. A database is really just a record keeping system. But once you begin to program a vast
new world opens up. Everything you program becomes a new tool. Because its your tool you can modify it. You modify it in ways you never imagined when you began.” (Peter Eden, personal communication,
January 29th, 2012)
So we agree that learning how to program is an excellent endeavour, and one that basically everyone should learn how to do. What I think Gary and I disagree about is whether or not this particular
learning should happen within the formal structure of schools.
Gary points out in his article that they had "tens of thousands" of teachers involved in learning how to program so that they could teach their students. Tens of thousands is a lot, but there are
millions of teachers world wide. Tens of thousands is a drop in the bucket compared to the number of people who are teachers.
If we were to teach them all (or a sizable subset of all of the teachers) how to program so that they could teach their students, we'd have to institutionalize learning how to program, and I think
that this would be a disaster. We'd end up with benchmarks, prescribed curriculum, and standardized testing.
I did a mathematics degree, and one requirement of the degree was that I take a course in computing, which I think is a perfectly sensible requirement. The problem was, I had to do a 1st year
computer science course, and this course was ungodly boring. It was so boring, that despite attempting twice to finish a 1st year computer science course, I gave up, and did a "Computing for
Mathematicians" course instead. Of course, I knew how to program already, so the programming skills themselves were not very useful to me. However, what I learned from this experience is that it is
tremendously easy to take something full of life and turn it into something deadly dull. If every student was forced to endure the same kind of learning I experienced during that 1st year computer
science course for 12 years without the opportunity to opt out, none of them would ever touch a computer again.
It is common for instutions like schools, to take endeavours which are exciting and interesting on a small scale, and attempt to bring that exact same experience to everyone. Unfortunately, most
often these endeavours pick all of the wrong parts of the activity to "scale up." In scaling up mathematics education, we took an experience where people mostly played around with ideas, and turned
it into fill-in-the-blank worksheets, completely destroying the purpose of learning mathematical thinking. Computer science in schools would fall into the same trap as science education has, which is
that people think the purpose of science education is to teach facts about science, instead of a way of thinking.
What I would prefer is for space to be created outside of instutions for this type of thinking to occur. Much like we have community centres for art, and for physical activity, we could have
recreational centres for computing. Instead of instutionalizing (and eventually centralizing) the learning of computing, I'd like to see it de-instutionalized. I'd like to see a thousand different
models for learning computering rather than the inevitable staleness that would occur if it were introduced en masse to schools.
I quite like the sentiment of
Submitted by
Peter Eden
on Thu, 04/12/2012 - 17:02.
I quite like the sentiment of this blog (and not only where you quoted me). I was a boy scout and enjoyed the outdoors so much I became a scout leader. Later, as a teacher, I attended an outdoor
education camp and was horrified at how the wonderful outdoor activities had been institutionalized. I know it would be hard to retrofit the scouting approach into a school.
One "space ... outside of institutions" where student programmers can learn is the internet. Sites like Daniweb www.daniweb.com where programmers of all skill levels (and languages) can get help is
an example.
At the same time I think teachers of all subjects need to be mindful of the pitfalls you outlined in your example of learning computer science. No "one size fits all" approach works well. That is why
we are taught by teachers and not by computers or some other sort of machine. When teachers become machines constrained by their own thinking or that of their institution you might as well be using a
You don't have to go outside of an institution to de-institutionalize your approach.
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function behavior
August 9th 2009, 05:02 PM #1
Oct 2008
Which of the following describes the behavior of the graph
$y=\sqrt[4]{|x-2|}$ at $x=2$?
1. differentiable
2. corner
3. cusp
4. vertical tangent
5. discontinuity
I think its a cusp..but not sure.
Since the tangent does not exist at x = 2 but the curve is continuous, only options 2 and 3 are candidates. But if you look at the definition of each, the answer has to be cusp.
The left and right derivatives go to $\mp$ infinity as $x$ goes to $2$, which in the case of slopes are the same thing (the curve is vertical on both sides at $x=1$) which of course makes the
point a cusp.
Last edited by CaptainBlack; August 10th 2009 at 08:19 AM. Reason: fix typo
August 9th 2009, 07:54 PM #2
August 9th 2009, 08:06 PM #3
Grand Panjandrum
Nov 2005
August 10th 2009, 01:46 AM #4
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This Article
Bibliographic References
Add to:
The Emergence of Computing Science Research and Teaching at Cambridge, 1936-l949
October-December 1992 (vol. 14 no. 4)
pp. 10-15
ASCII Text x
Mary G. Croarken, "The Emergence of Computing Science Research and Teaching at Cambridge, 1936-l949," IEEE Annals of the History of Computing, vol. 14, no. 4, pp. 10-15, October-December, 1992.
BibTex x
@article{ 10.1109/85.194050,
author = {Mary G. Croarken},
title = {The Emergence of Computing Science Research and Teaching at Cambridge, 1936-l949},
journal ={IEEE Annals of the History of Computing},
volume = {14},
number = {4},
issn = {1058-6180},
year = {1992},
pages = {10-15},
doi = {http://doi.ieeecomputersociety.org/10.1109/85.194050},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - MGZN
JO - IEEE Annals of the History of Computing
TI - The Emergence of Computing Science Research and Teaching at Cambridge, 1936-l949
IS - 4
SN - 1058-6180
EPD - 10-15
A1 - Mary G. Croarken,
PY - 1992
VL - 14
JA - IEEE Annals of the History of Computing
ER -
The Cambridge University Mathematical Laboratory was set up in 1937. This article describes the motivation behind the creation of the laboratory and its original terms of reference. The changes to
the laboratory caused by World War II are then discussed. The Cambridge Mathematical Laboratory was reestablished in 1945 under the directorship of Maurice V. Wilkes. The remainder of the article
considers how Wilkes developed the work of the laboratory and built up a research team to work on the EDSAC project, which established Cambridge as a major center of computer research.
1. N.F. Mott, "John Edward Lennard-Jones,"Biographical Memoirs of Fellows of the Royal Society, Vol. 1, 1955, pp. 175-184.
2. D.R. Hartree and A. Porter, "The Construction and Operation of a Model Differential Analyser,"Memoirs of the Manchester Literary and Philosophical Society, Vol. 79, 1935, pp. 51-72.
3. V. Bush, "The Differential Analyser. A New Machine for solving Differential Equations,"J. Franklin Inst., Vol. 212, 1931, pp. 447-488.
4. J.E. Lennard-Jones, M.V. Wilkes, and J.B. Bratt. "The Design of a Small Differential Analyser,"Proc. Cambridge Philosophical Society, Vol. 35, 1939, pp. 485-493.
5. Wilkes, M. V. 1985.Memoirs of a Computer Pioneer. Cambridge, MA, MIT Press.
6. Univ. of Cambridge, Faculty Board of Mathematics, "Report of the Faculty Board of Mathematics on the Need for a Computing Laboratory," 1936, LEJO 6, Archives of Churchill College, Cambridge, UK.
7. Univ. of Cambridge, Regent House, Acta No. 2,Cambridge Univ. Reporter, Feb. 23, 1937, p. 694.
8. M.G. Croarken,Early Scientific Computing in Britain, Oxford Univ. Press, Oxford, UK, 1990.
9. J. Lennard-Jones, Personal Daily Journals, Feb. 5, 1945. LEJO 26, Archives of Churchill College, Cambridge, UK.
10. M.V. Wilkes, interview with M. Croarken, May 7, 1992.
11. M.V. Wilkes, "The Functions of the Mathematical Laboratory," 1946, UL/COMP F(1)b, Manuscript Room, Cambridge Univ. Library, Cambridge, UK.
12. J. von Neumann, "First Draft of a Report on the EDVAC," Moore School of Electrical Engineering, Univ. of Pennsylvania, June 30, 1945; reprinted inPapers of John von Neumann on Computing and
Computer Theory, W.F. Aspray and A.W. Burks, eds., Charles Babbage Inst. Reprint Series for the History of Computing, Vol. 12, MIT Press, Cambridge Mass., and Tomash Publishers, Los Angeles, 1986.
13. C.G. Darwin, "Douglas Rayner Hatree,"Biographical Memoirs of Fellows of the Royal Society, Vol. 4, 1958, pp. 103-116.
14. The Moore School Lectures, M. Campbell-Kelly and M.R. Williams, eds., Charles Babbage Inst. Reprint Series for the History of Computing, Vol. 9, MIT Press, Cambridge, Mass., and Tomash
Publishers, Los Angeles, 1986.
15. M.V. Wilkes, "Computers Before Silicon: Design Decisions on Edsac,"IEE Rev., Dec. 1990, pp. 429-431.
16. D.R. Hartree, "Calculating Machines: Recent and Prospective Developments and Their Impact on Mathematical Physics," inaugural lecture, Cambridge Univ. Press, Cambridge, UK, 1947. Reprinted in
ref. 17.
17. D.R. Hartree,Calculating Instruments and Machines, Cambridge Univ. Press, Cambridge, UK, 1949. Reprinted in Charles Babbage Inst. Reprint Series for the History of Computing, Vol. 6,Calculating
Machines: Recent and Prospective Developments and Their Impact on Mathematical Physics, MIT Press, Cambridge Mass., and Tomash Publishers, Los Angeles, 1984.
18. Univ. of Cambridge, Mathematical Laboratory Committee, "Report of the Mathematical Laboratory, Committee to the Faculty Board of Mathematics," 1947, UL/COMP F1(2), Manuscript Room, Cambridge
Univ. Library, Cambridge, UK.
Mary G. Croarken, "The Emergence of Computing Science Research and Teaching at Cambridge, 1936-l949," IEEE Annals of the History of Computing, vol. 14, no. 4, pp. 10-15, Oct.-Dec. 1992, doi:10.1109/
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MAMI and beyond, Day Two
Hello again from Mainz, where I am at the conference "MAMI and beyond".
Today's first talk was by Barry Holstein (UMass Amherst), who spoke on "Hadronic physics and MAMI: past an future". The hadronic physics was cast mainly in the language of Chiral Perturbation Theory
and its extensions. An interesting detail was the magnetic polarisability of the nucleon, which suggest that the nucleon in 10,000 times "stiffer" electromagnetically than a typical atom; this is in
spite of the fact that the ability of the nucleon to transition to a Δ resonance ought to give it strongly paramagnetic properties from the quark spins; heutistically this is countered by the
diamagnetism of the nucleon's pion cloud. Another feature that I found interesting was that the experimental determination of hadronic scattering lengths seems to be rather involved (possibilities
mentioned involved the decay of pionium, or an analysis of the cusp structure in the energy dependence of K->3π or η->3π decays), and that the best way to determine them from theory is apparently
from the lattice via Lüscher's formula for the volume-dependence of two-particle state energies.
The next speaker was Rory Miskimen (also UMass Amherst) talking about the measurement of nucleon polarisabilities in real and virtual Compton scattering. Real Compton scattering is, well, Compton
scattering, virtual Compton scattering is the production of a photon in the scattering of an charged particle by a proton: γ^*p -> pγ. Apparently the results from MAMI lie on a different curve from
those from other experiments at other energies, which might suggest that there is something interesting happening around energies of Q^2=0.3 GeV^2.
The next two talks were by Bernard Pire (CPHT/Polytechnique) and Diego Bettoni (INFN Ferrara), who both talked about timelike processes. Due to my limited understanding of the relevant physics, I
feel unable to give a summary of those talks, except that apparently it is quite difficult to disentangle the different form factors experimentally.
After that Fred Jegerlehner (Katowice and DESY Zeuthen) spoke about the running of the fine structure constant α. The running of α, which at zero energy is known to astounding precision, is of
particular interest around the muon mass (where it enters the determination of the muon anomalous magnetic moment) and around the Z boson mass. The difficult part is to determine the contributions to
the running of α coming from hadronic loops, the uncertainty about which causes a loss of five significant figures when evolving α from 0 to M[Z]. Using a method based on the Adler function
(essentially a derivative of the self-energy with respect to the momentum squared), it should be possible to get a much more precise running of α by improving the understanding of low-energy hadronic
contributions. Since most of the information needed in this approach would come from the Euclidean momentum region, the lattice might be able to help here.
After the lunch break, I skipped a couple of experimental talks to go over to the IWHSS workshop held next door and listen to a talk by Chris Michael about hadronic physics on the lattice. Chris
presented approaches that can enable the determination of the nature of resonances and even the description of ρ -> ππ decays on the lattice.
After the coffee break, the lattice session of the MAMI conference took place: Meinulf Göckeler gave a summary of recent work towards the determination of generalised parton distributions on the
lattice; Dru Renner at DESY Zeuthen works on this kind of thing, so I have heard about it a few times; it seems very hard each time I hear it, but I suppose saying "let's wait a few more years before
starting on something like this" is not really an option.
Mike Peardon spoke about hadron spectroscopy on the lattice, giving a great introduction to lattice spectroscopy for the non-latticists in the audience. The highlight for lattice theorists was his
mention of a new method that might replace noisy estimators for all-to-all propagators: a redefinition of quark smearing as a projection on the subspace spanned by the low modes of the Laplacian on a
timeslice, enabling one to then exactly calculate all elements of the quark propagator out of this (relevant) subspace. The results shown looked rather promising, and the cost for diagonalising the
Laplacian on a timeslice is of course much lower than that for diagonalising the Dirac operator as needed for the Dublin method of all-to-all correlators with low-modes.
Andreas Jüttner gave a talk about ongoing work to study mesonic form factors and (g-2). Using twisted boundary conditions to induce a momentum, he obtained very nice pion and K->π form factors. The
(g-2) work is still in progress, but looks promising.
Silvia Necco gave an introduction to the links between Lattice QCD and Chiral Perturbation Theory, covering the extraction of SU(2) and SU(3) low-energy constants from N[f]=2 and N[f]=2+1 lattice
simulations, and of the leading-order couplings Σ and F from simulations in the ε-regime.
Finally, Johann Kühn (Karlsruhe) spoke about precision physics in e^+e^- interactions, where the perturbative determination of the hadron-to-muon ratio R(s) has made it possible to precisely
determine α[s], m[c] and m[b] from experimental data (and the former two also from lattice simulations via the moments of current-current correlators).
In the evening, there was a social event: A string quartet played for us at the university's faculty of music in Mainz. The program was Mozart (Divertimento No. 1, KV 135), Schubert (String quartet
No. 13 "Rosamunde) and Shostakovich (String quartet No 8 op. 110), the first two pieces quite pleasant, the last rather harrowing.
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Output not coming through properly
May 16th, 2011, 08:17 PM #1
Junior Member
Join Date
May 2011
Thanked 0 Times in 0 Posts
Hello everybody,
Its great to see there's a community of people who are willing to help out. If anyone could solve my dilemma that would be awesome because my Computer Science professor is a worthless poop head.
With that said, I feel like most of this is right yet the output is not showing up correctly. The main problem being that the radius outputs properly but none of the calculations of area,
circumference, or diameter do.
Thank you in advance!
import java.text.DecimalFormat;
import javax.swing.JOptionPane;
public class Circle {
public static void main(String[] args) {
CircleRadius radius = new CircleRadius();
CircleRadius area = new CircleRadius();
CircleRadius circumference = new CircleRadius ();
CircleRadius diameter = new CircleRadius();
String input;
double r;
DecimalFormat twodigits = new DecimalFormat("#.00");
input = JOptionPane.showInputDialog("Please enter the radius of the circle");
r = Double.parseDouble(input);
JOptionPane.showMessageDialog(null, "The radius of the circle is " + twodigits.format(radius.get_radius()));
JOptionPane.showMessageDialog(null, "The area of the circle is " + twodigits.format(area.get_Area()));
JOptionPane.showMessageDialog(null, "The diameter of the circle is " + twodigits.format(diameter.get_Diameter()));
JOptionPane.showMessageDialog(null, "The circumference of the circle is " + twodigits.format(circumference.get_Circumference()));
public class CircleRadius {
private double radius;
public double PI = 3.14159;
public CircleRadius()
radius = 0.0;
public CircleRadius(double r)
radius = r;
public double get_radius()
return radius;
public void set_radius(double r){
radius = r;
public double get_Area()
return PI * radius * radius;
public double get_Diameter()
return radius * 2;
public double get_Circumference()
return 2 * PI * radius;
Last edited by Camiot; May 16th, 2011 at 09:24 PM. Reason: Solved
I think the reason you are having the incorrect output is because you are creating too many objects. You don't need a separate object for each calculation. The radius determines the rest of the
calculations. You need to take the radius that is entered by the user, and then create just one CircleRadius object. You then use the radius to calculate the rest.
CircleRadius radius = new CircleRadius();
String input;
double r;
DecimalFormat twodigits = new DecimalFormat("#.00");
input = JOptionPane.showInputDialog("Please enter the radius of the circle");
r = Double.parseDouble(input);
someVariable = theObject.get_Area();
someVariable = theObject.get_Diameter();
someVariable = theObject.get_Circumference();
Last edited by vanDarg; May 16th, 2011 at 08:45 PM.
"Everything should be made as simple as possible, but not simpler."
Asking Questions for Dummies | The Java Tutorials | Java Coding Styling Guide
Assignment requires me to use all the different methods so I cannot really get around doing it the easy way.
What do you mean by the object in there. I cannot put anything for "theObject" there without declaring it
Last edited by Camiot; May 16th, 2011 at 09:00 PM.
"Everything should be made as simple as possible, but not simpler."
Asking Questions for Dummies | The Java Tutorials | Java Coding Styling Guide
I just got it my variables had to be the same
JOptionPane.showMessageDialog(null, "The radius of the circle is " + twodigits.format(radius.get_radius()));
JOptionPane.showMessageDialog(null, "The area of the circle is " + twodigits.format(radius.get_Area()));
JOptionPane.showMessageDialog(null, "The diameter of the circle is " + twodigits.format(radius.get_Diameter()));
JOptionPane.showMessageDialog(null, "The circumference of the circle is " + twodigits.format(radius.get_Circumference()));
Great! Hope I helped.
"Everything should be made as simple as possible, but not simpler."
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st: R: Proportional hazards assumption in Cox model
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st: R: Proportional hazards assumption in Cox model
From "Carlo Lazzaro" <carlo.lazzaro@tin.it>
To <statalist@hsphsun2.harvard.edu>
Subject st: R: Proportional hazards assumption in Cox model
Date Wed, 25 Feb 2009 09:14:05 +0100
Dear Mingfeng,
the usual test for checking the fulfilment of Cox proportional hazard
assumption requirements is Schoenfeld residuals [in Stata 9.2/SE -
Unfortunately, Cox proportional hazard assumption may not hold.
An example about this lack of holding of Cox proportional hazard assumption
(more frequent than usually reported I scientific articles, I suspect) can
be found in Jes S Lindholt, Svend Juul, Helge Fasting and Eskild W
Henneberg. Screening for abdominal aortic aneurysms: single centre
randomised controlled trial. BMJ 2005;330;750-; originally published online
9 Mar 2005; doi:10.1136/bmj.38369.620162.82.
The authors wrote "We used Cox proportional hazards regression to
compare specific mortality due to abdominal aortic aneurysm
and overall mortality. As the proportional hazards assumption
was not fulfilled, we decided to carry out separate analyses for the
periods before and after 1.5 years after randomisation".
Hence, the moved to Kaplan-Meier estimates of mortality (a non parametric
method that makes no assumptions about the underlying risk function).
As an aside, Svend Juul is an epidemiologist and also a relevant contributor
to Stata List.
For further details on Survival analysis, I will recommend you to take a
thorough look at:
Cleves MA, Gould WG, Gutierrez R. An Introduction To Survival Analysis
Using Stata. Revised edition. College Station: StataPress, 2006;
[ST] Stata manual. Survival analysis and epidemiological table. Release 9.
Kind Regards,
-----Messaggio originale-----
Da: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] Per conto di Mingfeng Lin
Inviato: mercoledì 25 febbraio 2009 6.09
A: statalist@hsphsun2.harvard.edu
Oggetto: st: Proportional hazards assumption in Cox model
I have a quick question about Cox models: I understand that we need to
test for the assumption of proportional hazards for all covariates,
and if any one fails the test, we should be concerned about the
statistical inference on that variable; plus we should consider things
such as time varying effects, stratification, and so on. But what if
none of these solves the problem? In particular, is it appropriate to
say that if one set of variables passes the test, we can still be
confident about the estimates of their coefficients - despite the
failure of other variables and global tests? In other words, while we
know that Cox models tend to be quite robust, how much latitude do we
have in terms of the proportional hazards assumption? How robust is
this model, just to be specific?
I'm still learning event history analysis, and the papers that I have
came across so far (empirical papers in social sciences) do not seem
to actually test that assumption (or maybe they forgot to report the
results), so I am just curious.
Thank you very much for any suggestions you can provide. If you could
refer me to specific papers on this, it would be really helpful.
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
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What was the first math problem that we needed a computer to solve?
"In the 1970s, a remarkable thing was done; a computer was used to solve a math problem. This, in and of itself, was not remarkable. The difference engine could do it. But this problem was the first
one that would probably remain unsolved if it weren’t for computers. Find out about the Four-Color Theorem, and why it needed to be turned over to the machines." Read more…
I remember reading about this proof in Scientific American when it first came out! —drego
(source: Mathematics Association of America, via io9)
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Re: lack of model theory for errors in built-ins
From: Jos de Bruijn <debruijn@inf.unibz.it> Date: Sun, 03 Feb 2008 11:40:29 +0100 Message-ID: <47A59A1D.3070502@inf.unibz.it> To: Michael Kifer <kifer@cs.sunysb.edu> CC: RIF WG <public-rif-wg@w3.org>
You're right. The definitions don't work. If we would fix them, we end
up with your option (b).
So we're back where we started, with the two choices you described in [1].
Best, Jos
[1] http://lists.w3.org/Archives/Public/public-rif-wg/2007Dec/0099.html
Michael Kifer wrote:
> Jos,
> thanks for the proposal. I have serious doubts that it holds water.
> Please see below.
>> Dear all,
>> In the telephone conference last Tuesday I mentioned that I had an idea
>> for dealing with errors in built-in predicates and functions by not
>> defining the semantics in case errors occur.
>> My proposed solution is the following:
>> For the purpose of this definition I assume that built-in predicates and
>> functions are written as ' Builtin ( ' Uniterm ' ) ', following the
>> proposal "syntactic representation of built-ins in RIF" at [1].
>> The definition of basic semantic structures is extended as follows:
>> I(Builtin(f(t1 ... tn))=IFb(f)(I(t1),...,I(tn))
>> ITruth(Builtin(r ( t1 ... tn ))) = IRb(r)(I(t1),...,I(tn))
>> This is merely a routine extension of the interpretation of terms and
>> atomic formulas to that of built-in terms and atomic formulas. Note that
>> we use the mappings IFb and IRb for the interpretation of built-in
>> functions and predicates; it would have also been possible to extend the
>> current interpretation functions, but in this case are both that
>> introducing new mappings which make things clearer.
>> Now for the interpretation of built-in functions and predicates:
>> IFb is a mapping from Const to partial functions from D* into D
>> IRb is a mapping from Const to partial truth-valued mappings D* TV
>> Note that the difference with the definitions of IF and IR is that the
>> functions and truth value mappings are *partial*.
>> A consequence of the fact that these mappings are partial is that the
>> truth valuation function ITruth may become undefined in case any errors
>> in built-in functions or predicates occur.
>> Let's now consider satisfaction of rules, which is defined as follows
>> (from the document):
>> I |= then :- if
>> iff ITruth(then) =t ITruth(if).
>> If ITruth(then) or ITruth(if) is undefined, ITruth(then) =t ITruth(if)
>> will also be undefined. Therefore, I |= then :- if is undefined. This
>> extends to satisfaction of rule sets I |= R.
> Let the rule set be
> p(1).
> q(abc).
> r(?Y) :- add(1,?X,?Y), p(?X).
> You did not say how things work with universal quantifiers, but the
> definition certainly will look weird a bit. Moreover, universal
> quantification should be treated as (a possibly) infinite conjunction
> so if one extends your definition in the logical way then the above will
> have no models.
>> Now consider an entailment, which is defined as follows:
>> S |= f
>> iff for every semantic structure I, such that I |= S, it is the case
>> that Itruth(f)=t.
>> If there is an error in the rule set S, then I |= S is undefined, so
>> clearly S |= f is undefined. If there is an error in f, then clearly
>> ITruth(f) is undefined, so S |= f is undefined.
> No, from what you said it follows that it is not the case that (S |= f).
>> So, the model theory simply does not interpret rule sets or conditions
>> with errors in built-ins.
>> We should probably include a remark saying that implementations should
>> return an error whenever they encounter a rule set which is not interpreted.
> You should probably check that all the theories of minimal, WF, and stable
> models hold. You should also probably check if this holds water for the FOL
> case.
> I suspect that it does not because of the aforesaid universal
> quantification problem.
> This was just my 30-second reaction. Here is my 45-sec reaction.
> Say, "a" is true, "b" false, and "c" is undefined. What's the values of
> Itruth (a \/ c) = ?
> Itruth (b /\ c) = ?
> If it's true/false, then this is exactly the same as adding the undefined
> truth value and a 3-valued logic. We discussed this before.
> If it is undefined, then this is unacceptable, because it would mean that
> the following ruleset would have no models:
> q :- (p, a) \/ (r, c).
> p.
> --michael
>> Best, Jos
>> [1] http://www.w3.org/2005/rules/wg/wiki/List_of_BLD_built-ins
>> --
>> Jos de Bruijn debruijn@inf.unibz.it
>> +390471016224 http://www.debruijn.net/
>> ----------------------------------------------
>> Doubt is not a pleasant condition, but
>> certainty is absurd.
>> - Voltaire
Jos de Bruijn, http://www.debruijn.net/
One man that has a mind and knows it can
always beat ten men who haven't and don't.
-- George Bernard Shaw
Received on Sunday, 3 February 2008 10:41:03 GMT
This archive was generated by hypermail 2.2.0+W3C-0.50 : Tuesday, 2 June 2009 18:33:45 GMT
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The quest for computing’s ‘Holy Grail’
Waterloo Region Record
WATERLOO - Say hello to Raymond Laflamme - and forget everything you thought you knew about computers.
Forget about bits and bytes and silicon chips. Forget about mega-this and giga-that. Discard the impatience you sometimes feel while waiting for your laptop to perform some supposedly “complicated”
function. Rid your mind of all those annoying little icons - from primitive-looking wristwatches to twirling coloured spheres - that indicate time is passing, and passing, while your doddering
digital device stumbles from one slowpoke application to another.
Be done with all of that and, instead, start thinking about quantum mechanics. Where computers are concerned, quantum mechanics is the future.
“It’s not an instantaneous transition, “ says Laflamme, 50, formerly of Quebec City and now executive director of the Institute for Quantum Computing at the University of Waterloo. “It’s a process.
But a quantum computer is the Holy Grail.”
Comparing a fully realized quantum computing device to even the most powerful classical computers in existence is a bit like comparing a vast galaxy of swirling planets and radiant stars to a
basement bachelor apartment.
In other words, there is really no comparison at all.
But there are two formidable obstacles facing anyone in the fledgling quantum computing business.
First, quantum computers are extremely difficult to build, so difficult that we haven’t actually built one yet, or not really.
Second, no matter how difficult they might be to fabricate, quantum computers are even more difficult to explain.
Despite these obstacles, quantum computing, or the prospect of quantum computing, is very much of the moment. Last month, two scientists - Frenchman Serge Haroche and American David Wineland -
gathered in Stockholm to receive this year’s Nobel Prize for physics, honouring work they have done separately to advance the development of quantum computers.
They are far from alone.
In offices and laboratories scattered around the globe - in Russia, China, Singapore, Europe, the United States and Canada - some of the finest scientific minds are busily pursuing the same goal, a
computing device whose fundamental components are so small as to be invisible to humans but whose calculating power would far outstrip any conventional computer you could imagine, even one as big as
the universe itself.
Here in Canada, the main centre for this area of research - there are others - was founded 10 years ago at the University of Waterloo. Now the Institute for Quantum Computing is in the process of
moving into palatial new quarters, the Mike & Ophelia Lazaridis Quantum-Nano Centre, designed by Toronto architectural firm KPMB and built at a cost of $160 million.
Headed by Laflamme, the institute brings 25 faculty members from roughly half a dozen countries together with about 40 post-doctoral fellows and roughly 90 students in the quest for something that is
at once almost unimaginably small - on the scale of atoms and electrons - and yet colossally powerful.
Perhaps the best way to begin an explanation of what a quantum computer is might be to explain what it is not.
It is not a classical computer.
“A small quantum computer can do things no classical computer can do, “ says Laflamme, a wiry, animated man with mobile facial features and a changeable yet nearly constant grin. “It’s a totally
different class.”
Let’s face it, a classical computer is essentially a pretty stupid device. Its guts are a network of electrical circuits that can do just two things. They can be on or they can be off. They can be
set at either “one” or “zero.”
Essentially, that’s all that classical computing is, a highly developed narrative of zeros or ones.
Here’s how it works.
A single computer circuit, called a bit, has just two possible states (0 or 1). Double that to two circuits and you get four states (00, 11, 01 or 10). With three circuits, you double the possible
number of combinations yet again, this time to eight (001, 011, 111, 110, 100, 000, 101, 010). When you get to eight circuits, you have 256 possible combinations, and this arrangement is called a
The processing power and memory capacity of laptop computers are nowadays calculated in gigabytes, meaning billions of bytes, and these numbers just keep getting bigger - but the trend is not going
to last forever, not for classical computers.
At least since the 1970s, computer scientists have been familiar with a phenomenon known as Moore’s law, the principle that technological innovation causes computing power to double on average every
two years. Or, to put the same idea another way, a computer circuit decreases in size, on average, by 50 per cent every two years, meaning you can fit twice as many into the same space.
At this rate, the computer industry is on track to run head-on into an immovable wall in approximately 10 years. Sometime between 2020 and 2030, processing circuits will have become so small - as
small as atoms - that they will be regulated by the same physical imperative that governs atoms: the inability to get any smaller.
A laptop in 2025 will be vastly more powerful than any laptop today, but that will be the end of it. Classical computing will have reached a natural limit and will cease to evolve.
With any luck, however, this is where quantum computing will come in.
Put in its sparest terms, the difference between classical computing and quantum computing is a simple matter of a single conjunction - the substitution of “and” for “or.”
Recall that a conventional computer circuit, or a bit, can represent a zero or a one. It can be switched off or it can switched on. One or the other.
But a quantum bit is a device of an entirely different kind.
Owing to the bizarre properties that govern the microscopic realm of quantum mechanics, a quantum bit - known as a “qubit” - can represent a zero and a one, both of them at the same time. It can be
simultaneously off and on.
Known as the uncertainty principle, this seemingly impossible attribute has no equivalent in the world that we humans can see and touch and feel, the domain of automobiles, maple trees, widescreen
TVs, bachelor apartments, cufflinks, pumpkin pie and classical computers.
But it seems to be critical to the nature of atomic and subatomic particles (including electrons, photons, protons, neutrinos and other minuscule dots of nature) that they exist in a haze of
possibility rather than in any fixed state.
A microscopic particle can be both here and there at the same instant of time, which means that it can also be in every intermediate location between here and there, also at the same time.
“It turns out that this is incredibly powerful,” says Laflamme. “Just changing an ‘or’ to an ‘and.’”
If you don’t understand how a thing can be in two places at once, don’t worry. Nobody does. Nonetheless, it’s true.
Even though an atom-sized classical bit would be extremely small, it would still be restricted to just two possible states - off and on. By contrast, an atom-sized qubit would possess a near infinity
of states, simultaneously.
What this means, by a common calculation, is that a quantum computer consisting of just 300 atoms would possess more sheer calculating power than a conventional machine harnessing every last atom in
the universe, yielding an almost unimaginable increase in our ability to solve problems of immense complexity.
Such a machine could perform in an instant certain difficult calculations - for example, the factoring of very large numbers - that the largest conventional computer would need billions of years to
figure out, if it could figure them out at all.
It turns out that classical computers are not very good at factoring large numbers, a weakness that has long been exploited by cryptographers to safeguard data on the Internet. It is easy to multiply
two prime numbers in order to produce a much larger number, but it turns out to be horrendously difficult to engineer the same process in reverse, to find the two prime divisors of a large number, a
process called factoring.
The only way classical computers can address the challenge is by systematic trial and error - trying out two numbers to see if they work, discarding them, trying out two different numbers and so on.
There’s no shortcut.
This defect in conventional computers is used to secure your banking information on the Internet, along with much else.
Even armed with powerful computers, would-be hackers still cannot find a way to expose the key - the two original prime numbers used to secure the code that protects your data.
The only method available is trial and error, examining every possible combination of divisors, one pair at a time, which could take forever.
By contrast, a quantum computer could crack such privacy barriers in an instant, by the dazzling expedient of testing every possible combination of divisors, not one by one, but all at once,
something no conventional computer could do. The right answer would reveal itself almost immediately.
No wonder governments around the world are pouring huge amounts of money into quantum computing. Whoever is first to build a sophisticated quantum machine would suddenly be able to crack just about
every secret code in cyberspace.
But that is far from all.
According to Laflamme, quantum computers will eventually ignite a revolution in a variety of fields other than cryptography. These might well include the development of new drugs, or the creation of
new superconducting materials, or a range of other innovations currently impossible to predict.
“We are just scratching at the surface of this,” he says. “Today, we are just dipping our big toe in the shallow end.”
At least in a very basic form, quantum computers do now exist, but they are extremely limited both in complexity and capability, restricted to no more than a dozen qubits and able to perform only the
simplest of mathematical feats. The highest number so far factored by a quantum device, using an algorithm developed in the 1990s by U.S. scientist Peter Shor, is 21.
This calculation might not seem very impressive - the answer is three and seven, as anybody with a Grade 8 education already knows - but quantum computing is in its infancy.
Laflamme takes a visitor on a tour of the laboratory facilities at IQC’s existing premises - windowless rooms cluttered with an endless variety of complicated machines and devices, pipes and wires,
including equipment that fabricates qubit chips from aluminum and niobium. The resulting slivers of metal are about the size of a baby’s fingernail and are extremely difficult to produce.
“Before we make one chip that works, we make 1,000 that don’t,” says Laflamme.
Even the chips that do work aren’t up to much in computational terms.
“If you want to add two plus two, we can do it,” he says. “If you want to add very large numbers, we can’t.”
But the journey is made in stages, and Laflamme believes that a quantum machine able to outpace conventional computers is no longer such a distant prospect.
“The boundary is about 50 qubits,” he says. “When we get there, we’re reaching a place where we’re departing the classical world.”
One big obstacle involves the inherent instability of the quantum universe, a territory where superposition - the ability of particles to occupy multiple locations at once - mysteriously collapses
the very instant a human observer interferes with the process in any way, even by looking at a particle through an electron microscope.
Somewhat like a performer struck by stage fright, an observed particle will instantly stop behaving in a quantum fashion and instead adopt just one position or state. Somehow, researchers will have
to overcome this phenomenon.
Laflamme, for one, is certain they will.
“In five to 10 years,” he says, “our goal is to get a quantum device to the market.”
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Lambda the Ultimate
Category Theory
Conor McBride gave an 8-lecture summer course on Dependently typed metaprogramming (in Agda) at the Cambridge University Computer Laboratory:
Dependently typed functional programming languages such as Agda are capable of expressing very precise types for data. When those data themselves encode types, we gain a powerful mechanism for
abstracting generic operations over carefully circumscribed universes. This course will begin with a rapid depedently-typed programming primer in Agda, then explore techniques for and
consequences of universe constructions. Of central importance are the “pattern functors” which determine the node structure of inductive and coinductive datatypes. We shall consider syntactic
presentations of these functors (allowing operations as useful as symbolic differentiation), and relate them to the more uniform abstract notion of “container”. We shall expose the double-life
containers lead as “interaction structures” describing systems of effects. Later, we step up to functors over universes, acquiring the power of inductive-recursive definitions, and we use that
power to build universes of dependent types.
The lecture notes, code, and video captures are available online.
As with his previous course, the notes contain many(!) mind expanding exploratory exercises, some of which quite challenging.
Lightweight Monadic Programming in ML
Many useful programming constructions can be expressed as monads. Examples include probabilistic modeling, functional reactive programming, parsing, and information flow tracking, not to mention
effectful functionality like state and I/O. In this paper, we present a type-based rewriting algorithm to make programming with arbitrary monads as easy as using ML's built-in support for state
and I/O. Developers write programs using monadic values of type M t as if they were of type t, and our algorithm inserts the necessary binds, units, and monad-to-monad morphisms so that the
program type checks. Our algorithm, based on Jones' qualified types, produces principal types. But principal types are sometimes problematic: the program's semantics could depend on the choice of
instantiation when more than one instantiation is valid. In such situations we are able to simplify the types to remove any ambiguity but without adversely affecting typability; thus we can
accept strictly more programs. Moreover, we have proved that this simplification is efficient (linear in the number of constraints) and coherent: while our algorithm induces a particular
rewriting, all related rewritings will have the same semantics. We have implemented our approach for a core functional language and applied it successfully to simple examples from the domains
listed above, which are used as illustrations throughout the paper.
This is an intriguing paper, with an implementation in about 2,000 lines of OCaml. I'm especially interested in its application to probabilistic computing, yielding a result related to Kiselyov and
Shan's Hansei effort, but without requiring delimited continuations (not that there's anything wrong with delimited continuations). On a theoretical level, it's nice to see such a compelling example
of what can be done once types are freed from the shackle of "describing how bits are laid out in memory" (another such compelling example, IMHO, is type-directed partial evaluation, but that's
literally another story).
Kleisli Arrows of Outrageous Fortune
When we program to interact with a turbulent world, we are to some extent at its mercy. To achieve safety, we must ensure that programs act in accordance with what is known about the state of the
world, as determined dynamically. Is there any hope to enforce safety policies for dynamic interaction by static typing? This paper answers with a cautious ‘yes’.
Monads provide a type discipline for effectful programming, mapping value types to computation types. If we index our types by data approximating the ‘state of the world’, we refine our values to
witnesses for some condition of the world. Ordinary monads for indexed types give a discipline for effectful programming contingent on state, modelling the whims of fortune in way that Atkey’s
indexed monads for ordinary types do not (Atkey, 2009). Arrows in the corresponding Kleisli category represent computations which a reach a given postcondition from a given precondition: their
types are just specifications in a Hoare logic!
By way of an elementary introduction to this approach, I present the example of a monad for interacting with a file handle which is either ‘open’ or ‘closed’, constructed from a command interface
specfied Hoare-style. An attempt to open a file results in a state which is statically unpredictable but dynamically detectable. Well typed programs behave accordingly in either case. Haskell’s
dependent type system, as exposed by the Strathclyde Haskell Enhancement preprocessor, provides a suitable basis for this simple experiment.
I discovered this Googling around in an attempt to find some decent introductory material to Kleisli arrows. This isn't introductory, but it's a good resource. :-) The good introductory material I
found was this.
A Lambda Calculus for Real Analysis
Abstract Stone Duality is a revolutionary paradigm for general topology that describes computable continuous functions directly, without using set theory, infinitary lattice theory or a prior
theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, and the reasoning looks remarkably like a sanitised form of that in classical
topology. This is an introduction to ASD for the general mathematician, with application to elementary real analysis.
This language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line. As is well known from both numerical and constructive
considerations, the equation cannot be solved if the function "hovers" near 0, whilst tangential solutions will never be found.
In ASD, both of these failures and the general method of finding solutions of the equation when they exist are explained by the new concept of overtness. The zeroes are captured, not as a set,
but by higher-type modal operators. Unlike the Brouwer degree, these are defined and (Scott) continuous across singularities of a parametric equation.
Expressing topology in terms of continuous functions rather than sets of points leads to treatments of open and closed concepts that are very closely lattice- (or de Morgan-) dual, without the
double negations that are found in intuitionistic approaches. In this, the dual of compactness is overtness. Whereas meets and joins in locale theory are asymmetrically finite and infinite, they
have overt and compact indices in ASD.
Overtness replaces metrical properties such as total boundedness, and cardinality conditions such as having a countable dense subset. It is also related to locatedness in constructive analysis
and recursive enumerability in recursion theory.
Paul Taylor is deadly serious about the intersection of logic, mathematics, and computation. I came across this after beating my head against Probability Theory: The Logic of Science and Axiomatic
Theory of Economics over the weekend, realizing that my math just wasn't up to the tasks, and doing a Google search for "constructive real analysis." "Real analysis" because it was obvious that that
was what both of the aforementioned texts were relying on; "constructive" because I'd really like to develop proofs in Coq/extract working code from them. This paper was on the second page of
results. Paul's name was familiar (and not just because I share it with him); he translated Jean-Yves Girard's regrettably out-of-print Proofs and Types to English and maintains a very popular set of
tools for typesetting commutative diagrams using LaTeX.
Simplicial Databases, David I. Spivak.
In this paper, we define a category DB, called the category of simplicial databases, whose objects are databases and whose morphisms are data-preserving maps. Along the way we give a precise
formulation of the category of relational databases, and prove that it is a full subcategory of DB. We also prove that limits and colimits always exist in DB and that they correspond to queries
such as select, join, union, etc. One feature of our construction is that the schema of a simplicial database has a natural geometric structure: an underlying simplicial set. The geometry of a
schema is a way of keeping track of relationships between distinct tables, and can be thought of as a system of foreign keys. The shape of a schema is generally intuitive (e.g. the schema for
round-trip flights is a circle consisting of an edge from $A$ to $B$ and an edge from $B$ to $A$), and as such, may be useful for analyzing data. We give several applications of our approach, as
well as possible advantages it has over the relational model. We also indicate some directions for further research.
This is what happens when you try to take the existence of ORDER BY and COUNT in SQL seriously. :-)
If you're puzzled by how a geometric idea like simplexes could show up here, remember that the algebraic view of simplicial sets is as presheaves on the category of finite total orders and
order-preserving maps. Every finite sequence gives rise to a total order on its set of positions, and tables have rows and columns as sequences!
An Innocent Model of Linear Logic by Paul-André Melliès was referenced by Noam in a serendipitious subthread of the "Claiming Infinities" thread.
Here's the abstract:
Since its early days, deterministic sequential game semantics has been limited to linear or polarized fragments of linear logic. Every attempt to extend the semantics to full propositional linear
logic has bumped against the so-called Blass problem, which indicates (misleadingly) that a category of sequential games cannot be self-dual and cartesian at the same time. We circumvent this
problem by considering (1) that sequential games are inherently positional; (2) that they admit internal positions as well as external positions. We construct in this way a sequential game model
of propositional linear logic, which incorporates two variants of the innocent arena game model: the well-bracketed and the non well-bracketed ones.
The introduction goes on to refer to to André Joyal's "Category Y with Conway games as objects, and winning strategies as morphisms, composed by sequential interaction," and points out that "it is a
precursor of game semantics for proof theory and programming languages," and is "a self-dual category of sequential games." The foreword mentions that the paper goes on to give "a crash course on
asynchronous games" and then "constructs a linear continuation monad equivalent to the identity functor, by allowing internal positions in our games, [which] circumvents the Blass problem and defines
a model of linear logic."
Jacques Carette called this paper mind-blowing. My mind-blow warning light already exploded. I'm posting this paper because I know a number of LtUers are interested in these topics, and this way I
can buttonhole one of them the next time I see them and ask them to explain it to me. ;)
Martin Hyland and John Power (2007). The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads. ENTCS 172:437-458.
Both monads and Lawvere theories provide characterisations of algebraic structure, with monads providing the more general characterisation. The authors provide an introduction to Lawvere theories,
discusses their relationship to sets, and why monads became the more popular treatment.
Then they tackle the application of the theory to the semantics of side effects, where they argue that the generality of monads allow them to characterise computational phenomena that are not to do
with side effects such as partiality and continuations, and argue that Lawvere theories more cleanly characterise what side effects are.
This paper is a good introduction to an important line of recent research done by Hyland&Power; cf. also the LtU story Combining computational effects.
Parameterized Notions of Computation, Robert Atkey, JFP 2008.
Moggi's Computational Monads and Power et al's equivalent notion of Freyd category have captured a large range of computational effects present in programming languages. Examples include
non-termination, non-determinism, exceptions, continuations, side-effects and input/output. We present generalisations of both computational monads and Freyd categories, which we call
parameterised monads and parameterised Freyd categories, that also capture computational effects with parameters.
Examples of such are composable continuations, side-effects where the type of the state varies and input/output where the range of inputs and outputs varies. By also considering structured
parameterisation, we extend the range of effects to cover separated side-effects and multiple independent streams of I/O. We also present two typed λ-calculi that soundly and completely model our
categorical definitions — with and without symmetric monoidal parameterisation — and act as prototypical languages with parameterised effects.
Once you've programmed with monads for a while, it's pretty common to start defining parameterized families of monads -- e.g., we might define a family of type constructors for IO, in which the
program type additionally tracks which files the computation reads and writes from. This is a very convenient programming pattern, but the theory of it is honestly a little sketchy: on what basis do
we conclude that the indices we define actually track what we intend them to? And furthermore, why can we believe that (say) the monadic equational laws still apply? That's the question Atkey lays
out a nice solution to. He gives a nice categorical semantics for indexed, effectful computations, and then cooks up lambda calculi whose equational theory corresponds to the equations his semantics
The application to delimited continuations is quite nice, and the type theories can also give a little insight into the basics of how stuff like Hoare Type Theory works (which uses parameterized
monads, with a very sophisticated language of parameters).
On a slightly tangential note, this also raises in my mind a methodological point. Over the last n years, we've seen many people identify certain type constructors, whose usage is pervasive, and
greatly simplified with some syntactic extensions -- monads, comonads, applicative functors, arrows, and so on. It's incredible to suggest that we have exhausted the list of interesting types, and so
together they constitute a good argument for some kind of language extension mechanism, such as macros. However, all these examples also raise the bar for when a macro is a good idea, because what
makes them compelling is precisely that the right syntax yields an interesting and pretty equational theory in the extended language.
By now there is an extensive network of interlocking analogies between physics, topology, logic and computer science, which can be seen most easily by comparing the roles that symmetric monoidal
closed categories play in each subject. However, symmetric monoidal categories are just the n = 1, k = 3 entry of a hypothesized “periodic table” of k-tuply monoidal n-categories. This raises the
question of how these analogies extend. We present some thoughts on this question, focusing on how monoidal closed 2-categories might let us understand the lambda calculus more deeply.
Link to the talk
via The n-Category Café
Two fresh papers from the Edinburgh theory stable:
• Lindley, Wadler & Yallop, 2008. The Arrow Calculus, (Functional Pearl) (submitted to ICFP).
• Lindley, Wadler & Yallop, 2008. Idioms are oblivious, arrows are meticulous, monads are promiscuous (submitted to MSFP)
We revisit the connection between three notions of computation: Moggi’s monads, Hughes’s arrows and McBride and Paterson’s idioms (also called applicative functors ). We show that idioms are
equivalent to arrows that satisfy the type isomorphism A ∼> B ≅ 1 ∼> (A -> B) and that monads are equivalent to arrows that satisfy the type isomorphism A ∼> B ≅A → (1 ∼> B). Further, idioms
embed into arrows and arrows embed into monads.
The first paper introduce a reformulation of the Power/Thielecke/Paterson/McBride axiomatisation of arrows, which the authors argue is more natural, and shows that arrows generalise both monads and
idioms. The second paper studies the relationships between the three formalisations in more formal depth; in particular the results about applicative functors struck me as significant.
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Precise Numbers
Date: 10/18/2001 at 21:28:55
From: Ryan Adams
Subject: Precise Numbers
I have a contest problem about precise numbers which the problem
alleges is in the same category as proper divisors. I can't find any
formulas to figure them out.
The problem says that a precise number occurs when its proper divisors
multiplied together equal the number. For example, 6 is a precise
number because 1*2*3 = 6. I'm wondering if there is another name or
formula that would help me figure out the problem.
Date: 10/19/2001 at 17:32:26
From: Doctor Ian
Subject: Re: Precise Numbers
Hi Ryan,
I'm not sure what kind of 'formula' you're looking for, but maybe this
will be helpful.
First, just to make sure that you know what it means to break a number
into prime factors, take a look at
Finding All the Factors of a Number
If a number has exactly two prime factors, p and q, then the only
proper factors of the number will be 1, p, and q; and when you
multiply these together,
1 * p * q
you get the number itself.
Note that if you have more than two prime factors (p, q, and r), then
you get extra proper factors: 1, p, q, r, pq, pr, qr. When you
multiply these together you get something larger than the original
1 * p * q * r * pq * pr * qr
this is
the number
On the other hand, if the number is prime, then the only proper
divisor is 1.
So in order for a number to be what you're callling a 'precise
number', it has to have exactly two prime factors.
So now we can start cranking out 'precise numbers':
* 2 3 5 7 11
and so on. So I guess this is a kind of 'formula' after all.
Does this help?
- Doctor Ian, The Math Forum
Date: 10/20/2001 at 13:02:42
From: Ryan Adams
Subject: Re: Precise Numbers
Yes - thank you very much.
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Sum of Prime Numbers
Show $\sum_{i=1}^n p_i \sim \frac12n^2\log n$ where $p_i$ is the $i^{th}$ prime number.
I came up with two methods. Is one of them the one you used? Method 1: (I prefer this one, as it "finds" the asymptote) Using $p_n \sim n \ln n$ : $<br /> \sum_{k=1}^n p_k \sim \sum_{k=1}^n k\ln k =
\ln \prod_{k=1}^n k^k \sim<br />$ Now, using the result of this thread: $<br /> \sim \ln C + \frac{1}{2}\left(n^2+n+\frac{1}{6}\right)\ln n - \frac{1}{4}n^2 \sim \frac{1}{2}n^2\ln n<br />$ Method 2:
Applying Stolz theorem $<br /> \lim_{n\rightarrow \infty}\frac{n^2 \ln n}{\sum_{k=1}^n p_k} = \lim_{n\rightarrow \infty}\frac{n^2 \ln n - (n-1)^2 \ln (n-1)}{p_n} =<br />$ Using $p_n \sim n \ln n$ : $
<br /> = \lim_{n\rightarrow \infty}\frac{n^2 \ln n - (n-1)^2 \ln (n-1)}{n \ln n} = \lim_{n\rightarrow \infty}\frac{n \ln(1+\frac{1}{n-1}) + 2\ln(n-1) - \frac{\ln(n-1)}{n}}{\ln n} = 2<br />$ Therefore
$\sum_{k=1}^n p_k \sim \frac{1}{2}n^2\ln n$, as desired.
Last edited by Unbeatable0; June 10th 2010 at 10:15 AM. Reason: Typo in summation
Last edited by Unbeatable0; June 10th 2010 at 10:16 AM. Reason: Typo in summation
Last edited by chiph588@; July 17th 2010 at 05:03 PM.
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Topic: Re: meaningful standards (2nd try)
Replies: 0
Re: meaningful standards (2nd try)
Posted: Jun 3, 1995 2:27 AM
You asked for specifics. Here's one for you to consider. The only new text I
have here at home representing the NCTM standards is the "Integrated
Mathematics" book from Houghton Mifflin. Very hot book in these parts. Some
things to recommend it, other ways not very teachable. This book appears to
be written by a committee for a committee. In my opinion, very unteachable,
but beautifully produced, pretty pictures, perfect for some textbook adoption
committee, probably more adept at politics than math.
Anyway, under "multicutural connections" there are 107 citations for
different pages. This appears to be the most listings for any topic, except
for "writing" which is pretty close. Typical of other listings: "slope", 10
listings, "square root", 7 listings, and "quadratic formula", 1 listing. Now
some people might think this is great; I don't really know. But my
proposition was that liberals have pretty much set the agenda for our texts.
And I haven't heard Newt, Dole, or Graham crying about diversity and
Can you imagine the moral outrage if there was a math book published with 107
listings under "patriotism" or "objective codes of morality and behavior"?
Is my example specific enough?
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Find the domain and range of the following functions: (a)y=secx
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Items tagged with centroid
I had started to create a procedure for finding the centroid of a list of points.
Centroid := proc (list)
local a, centroid, x, y, i:
a := nops(list):
x := 0:
yi := 0:
for i from 1 to a do
x := x+list[i, 1]:
y := y+list[i, 2]:
end do:
print(`Centroid is at`,([xi, yi]/a)):
end proc:
But I thought there needs to be something simpler than that. And here we are.
Centroid2 := proc (list)
local i:
print(`Centroid is at`, add(i, i = list...
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Having trouble with these!
October 27th 2009, 04:48 AM #1
Oct 2009
Having trouble with these!
1. Show that (coshx + sinhx)[to the power of] n = coshnx + sinhnx
2. Determine the integral of tanhxdx
3. Determine the integral of 4dx/(e to the x + e to the -x)squared
PS How can i use the proper math symbols?!!
1. Show that (coshx + sinhx)[to the power of] n = coshnx + sinhnx
This is trivial applying the definition of the hyperbolic functions
2. Determine the integral of tanhxdx
$\tanh x=\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{e^{2x}-1}{e^{2x}+1}=1-\frac{2}{e^{2x}+1}$.
Take it from here now (if you know integrals of usual trigonometric functions this one won't surprise you).
3. Determine the integral of 4dx/(e to the x + e to the -x)squared
[colo=blue]Get some ideas from the above
PS How can i use the proper math symbols?!!
Using LaTex
I dont understand how u got that in 2.! I was looking at other examples and they use substitution?
Also would it be right to use inegrations by parts in 3.?
For 2.- Multiply by $\frac{e^x}{e^x}$
For 3.- You can try parts or note that $=\frac{4}{e^x+e^{-x}}=\frac{1}{\cosh^2x}=\sec \!\!h^2x$
October 27th 2009, 05:22 AM #2
Oct 2009
October 27th 2009, 05:45 AM #3
Oct 2009
October 27th 2009, 08:20 AM #4
Oct 2009
October 27th 2009, 09:51 AM #5
Oct 2009
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some one hewp plzzzzz Create a unique example of dividing a polynomial by a monomial and provide the simplified form. Explain, in complete sentences, the two ways used to simplify this expression and
how you would check your quotient for accuracy.
• one year ago
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thank you you were alot of help to me :)
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Okay first we got to understand your problem: \(\text{Create a unique example of dividing a polynomial by a monomial and provide}\) \(\text{the simplified form. }\) So let's set some questions
up: \(1) x^2 + 8x + 12 / x+ 2\) \(2)x - 2 +1/3x\) Okay for number one follow the steps below: 1) Factor out the numerator, so now you have \((x+2)(x+6).\) 2) Cancel out \((x+2)\) in the numerator
and in the detonator. 3) Now you are left with the simplified answer, which is \(x+ 6\). Okay for number two follow the steps below: 1) Multiply \(x - 2 +1/3x\) by \(3x^2\) 2) You get \(3x^3 - 2x
^2 + 3x^2/ 3x\) 3) That simplifies to x \(\text{Explain, in complete sentences, the two ways used to simplify this expression}\) \(\text{and how you would check your quotient for accuracy.}\)
What I did: I simplified the expression. I explained how I simplified the expression. What you do: Check your quotient for accuracy. @mikala1 I believe you know how to do that? (If you don't I'll
do it for you.)
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I like the first one, but the second one is a little wrong. :-)
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Tell me how to fix it. There's a medal in it for you ^_^.
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Oh. Such a stupid mistake I made. Now I feel silly.
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Another way:\[\begin{aligned}= & \dfrac{x -1}{3x} \\ \\ \\ = & \dfrac{x}{3x} - \dfrac{1}{3x} \\ \\ \\ = & \dfrac{1}{3} - \dfrac{1}{3x} \end{aligned}\]I did a mistake in the last post too :\
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It's all good. Thank you Parth. I just don't know why I forgot such a vital step :| It's sleep deprivation. I've even answered History questions incorrectly :S
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Nah, happens, like it happened to me above.
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Topic: Plot vectors in polar coordinates
Replies: 7 Last Post: Nov 14, 2012 7:19 AM
Messages: [ Previous | Next ]
dpb Re: Plot vectors in polar coordinates
Posted: Jun 28, 2008 6:58 PM
Posts: 7,873
Registered: 6/7/07 Red Star wrote:
> I hoped to find a method.
> pol2car can't work: if u_radial=0 and u_azimuthal=any (a vortex for
> example).....the function gives always vx=0 vy=0 of course
Well, we don't always get what we wish for... :)
Did you look on TMW file exchange?
Date Subject Author
6/28/08 Plot vectors in polar coordinates Red Star
6/28/08 Re: Plot vectors in polar coordinates Justin
6/28/08 Re: Plot vectors in polar coordinates Élodie
6/28/08 Re: Plot vectors in polar coordinates Red Star
6/28/08 Re: Plot vectors in polar coordinates dpb
6/28/08 Re: Plot vectors in polar coordinates Red Star
6/28/08 Re: Plot vectors in polar coordinates dpb
11/14/12 Re: Plot vectors in polar coordinates Alex
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Posts about PRNG on Harder, Better, Faster, Stronger
Simple Random Generators
October 15, 2013
We all need (pseudo)random numbers sooner or later. Are they hard to generate? Depends. If you want them to be really strong (that is, very random), yes, it’s difficult. If you merely need something
random-looking, well, you still need some number theory, but it’s rather not complicated.
Let’s have a look at three simple types: additive, multiplicative, and the infamous linear congruential generator.
The Inversion Method (Generating Random Sequences IV)
June 28, 2011
In the first post of this series, I discussed how to generate permutations of sequences using the Fisher-Yates method and I explained (although indirectly) how a linear congruential generator works.
In a second post, I explained how to generate 2D points uniformly and randomly distributed a triangle, discussing the method of rejection. In a third post, I’ve discussed how to generate points on a
All these methods have something in common: they are based on the uniform (pseudo)random generator, and they map uniform numbers onto a shape (or move numbers around, in the first case). What if we
need another density function than uniform?
Random Points in a Triangle (Generating Random Sequences II)
October 5, 2010
In the first post of this series, I discussed a method to generate a random (guaranteed) permutation. On another occasion, I presented a method to recycle expensive random bits. Today, I will discuss
something I had to do recently: generate random points inside a triangle.
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Orthogonal Trajectories
,.hello there,.,amm,.,can anyone please help me understand about " orthogonal trajectories"???any ideas will do ,.thnx,.,.
The basic idea is this. Suppose you have a first-order differential equation $y'=f(x,y).$ This DE defines the slope of the curve y(x). It is known that, given a straight line with slope m, the slope
of the perpendicular line is -1/m. So, the family of curves $y_{\perp}$ that are orthogonal to the solutions of the DE above are going to satisfy the following DE: $y_{\perp}'=-\frac{1}{f(x,y_{\
perp})}.$ You can see Zill's A First Course in Differential Equations with Modeling Applications for a problem or two involving orthogonal trajectories. Just look it up in the index. Also check out
pages 117-118 of Tenenbaum and Pollard's Ordinary Differential Equations.
For example, the formula y= a/x defines a family of hyperbolas, all asymptotic to the x and y axes. dy/dx= -a/x^2 so that (dy/dx)/y= (-a/x^2)/(a/x)= -1/x where we have eliminated the parameter a: dy/
dx= -y/x is a differential equation satisfied by those functions. Since the slopes of perpendicular lines are "negative reciprocal", any curve satisfying dy/dx= x/y will be perpendicular to all of
those hyperbola. Of course, we can write that as ydy= xdx and integrate: $(1/2)y^2= (1/2)x^2+ C$ which can be written $\frac{y^2}{2C}- \frac{x^2}{2C}= 1$ a family of hyperbolas, all asymptotic to the
lines y= x and y= -x and all of which are perpendicular to all of the original hyperbolas. Another example: $x^2+ y^2= a$ is the family of circles, with center at the origin and different radii.
Differentiating both sides with respect to x, $2x+ 2y(dy/dx)= 0$. That is, the solutions to the differential equation $\frac{dy}{dx}= -\frac{x}{y}$ are those circles. Again, inverting and multiplying
by -1, $\frac{dy}{dx}= \frac{y}{x}$ describes the family of "orthogonal trajectories" to those circles. It should be no surprise that we can write that as $\frac{dy}{y}= \frac{dx}{x}$ and,
integrating, $ln y= ln x+ C$. Taking exponentials of both sides, y= C' x where $C'= e^C$ and we see the 'obvious'- that the lines through the origin are all perpendicular to those circles.
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VHDL function to convert an vector datatype to integer
Newbie level 6
Join Date
Apr 2006
0 / 0
VHDL function to convert an vector datatype to integer
Is there a function to convert an vector datatype to integer and vice versa.
Full Member level 2
Join Date
Jun 2004
137 / 137
vhdl conv_integer
Well I don't think there is a ready made function to convert, but you can write down your own easily.
-- Convert a std_logic_vector to an unsigned integer
function to_uint (a: std_logic_vector) return integer is
alias av: std_logic_vector (1 to a'length) is a;
variable val: integer := 0;
variable b: integer := 1 ;
for i in a'length downto 1 loop
if (av(i) = '1 ') then -- if LSB is '1 ',
val := vat + b; -- add value for current bit position
end if;
b := b*2; -- Shift left 1 bit
end loop;
return val;
end to_uint;
Example from "VHDL Made Easy" by Pellerin and Taylor.
p.s. if you find this post of any use to you then kindly do click on the "helped me" icon. Regards, salam.
3. conv_unsigned vhdl
The following functions are contained in the library arith.vhd. To use them,
place the line “USE ieee.std_logic_arith.ALL” at the beginning of your
VHDL design.
FUNCTION Pass(arg, size) Return
· CONV_INTEGER INTEGER INTEGER
· CONV_INTEGER UNSIGNED INTEGER
· CONV_INTEGER SIGNED INTEGER
· CONV_INTEGER STD_ULOGIC SMALL_INT;
· CONV_UNSIGNED INTEGER, INTEGER UNSIGNED;
· CONV_UNSIGNED UNSIGNED, INTEGER UNSIGNED;
· CONV_UNSIGNED SIGNED, INTEGER UNSIGNED;
· CONV_UNSIGNED STD_ULOGIC, INTEGER UNSIGNED;
· CONV_SIGNED INTEGER, INTEGER SIGNED;
· CONV_SIGNED UNSIGNED, INTEGER SIGNED;
· CONV_SIGNED SIGNED, INTEGER SIGNED;
· CONV_SIGNED STD_ULOGIC, INTEGER SIGNED;
· CONV_STD_LOGIC_VECTOR INTEGER, INTEGER STD_LOGIC_VECTOR
· CONV_STD_LOGIC_VECTOR UNSIGNED, INTEGER STD_LOGIC_VECTOR
· CONV_STD_LOGIC_VECTOR SIGNED, INTEGER STD_LOGIC_VECTOR
· CONV_STD_LOGIC_VECTOR STD_ULOGIC, INTEGER STD_LOGIC_VECTOR
· EXT STD_LOGIC_VECTOR, INTEGER STD_LOGIC_VECTOR;
· SXT STD_LOGIC_VECTOR, INTEGER STD_LOGIC_VECTOR;
The following function is contained in the library unsigned.vhd. To use it, place
the line “USE ieee.std_logic_unsigned.ALL” at the beginning of your VHDL
· CONV_INTEGER(arg: STD_LOGIC_VECTOR) return INTEGER;
The following function is contained in the library signed.vhd. To use it, place
the line “USE ieee.std_logic_signed.ALL” at the beginning of your VHDL
· CONV_INTEGER(arg: STD_LOGIC_VECTOR) return INTEGER;
u can find all info such as the above in this link:
good luck
Advanced Member level 4
Join Date
Jan 2003
53 / 53
conv_signed vhdl
Unfortunately std_logic_arith is obsolute and should no more be used. Instead, numeric_std should exclusively be used.
5. 23rd October 2008, 04:48 #5
Full Member level 1
Join Date
Oct 2008
8 / 8
vhdl conv_signed
ya, use nuemric_std or synopsys lib is good for this conversions
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Dividing by a Binomial
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see your teacher explain things to you on a blackboard in your classroom. This particular lesson includes the teacher's instruction, practice questions as well as end-of-lesson quizzes for practice.
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Undergraduate Catalog
Thermodynamics focuses on the general principles governing the behavior of large ensemble of "things." It answers the basic question, "how do chunks of the universe behave." Because the important
aspect is the large number and not the specific "things," this discipline is universally applicable across all length scales and fields of study. The goal is to provide fundamental preparation for
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induction. Recommended background: working knowledge of the material presented in PH 1110 or PH 1111 and concurrent study of MA 1022. Students may not receive credit for both PH 1120 and PH 1121.
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and resistance, magnetism, and electromagnetic induction. Recommended background: working knowledge of material covered in PH 1111 and concurrent study of MA 1024 (or higher). Students concurrently
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The course provides fundamental preparation for any specialized application of thermodynamics. The material covered includes a general description of large number systems, states, canonical state
variables, state functions, response functions, and equations of state. Focus will be given to the physical meanings of free-energies, enthalpy, chemical potential, and entropy. Connections will be
made to equilibrium states, reversible versus irreversible processes, phases and phase transformation, as well as the arrow of time as applied across disciplines. Recommended background: introductory
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central-force problem, driven harmonic oscillator, an introduction to generalized coordinates, and the Lagrangian and Hamiltonian formulation of mechanics.
Cat. I Introduction to the theory and application of electromagnetic fields, appropriate as a basis for further study in electromagnetism, optics, and solid-state physics. Topics: electric field
produced by charge distributions, electrostatic potential, electrostatic energy, magnetic force and field produced by currents and by magnetic dipoles, introduction to Maxwell?s equations and
electromagnetic waves. Recommended background: introductory electricity and magnetism, vector algebra, integral theorems of vector calculus as covered in MA 2251.
An introduction to the use of optics for transmission and processing of information. The emphasis is on understanding principles underlying practical photonic devices. Topics include lasers, light
emitting diodes, optical fiber communications, fiber lasers and fiber amplifiers, planar optical waveguides, light modulators and photodetectors. Recommended background is PH 1110, PH 1120, PH 1130
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An introduction to the physical principles underlying lasers and their applications. Topics will include the coherent nature of laser light, optical cavities, beam optics, atomic radiation,
conditions for laser oscillation, optical amplifiers (including fiber amplifiers), pulsed lasers (Q switching and mode locking), laser excitation (optical and electrical), and selected laser
applications. Recommended background is PH 1110, PH 1120, PH 1130 and PH 1140 (or their equivalents).
Atomic force microscopes (AFMs) are instruments that allow three-dimensional imaging of surfaces with nanometer resolution and are important enabling tools for nanoscience and technology. The student
who successfully completes this course will understand the functional principles of AFMs, be able to run one, and interpret the data that are collected. Recommended background: PH 1110 and 1120.
Suggested background: PH 1130 and PH 1140.
A selective study of components of the universe (the solar system, stars, nebulae, galaxies) and of cosmology, based on astronomical observations analyzed and interpreted through the application of
physical principles, and organized with the central purpose of presenting the latest understanding of the nature and evolution of the universe. Some topics to be covered include the Big Bang &
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This course introduces the ambient atmospheric and space environments encountered by aerospace vehicles. Topics include: the sun and solar activity; the solar wind; planetary magnetospheres;
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equivalent), and ordinary differential equations (MA 2051 or equivalent).
This course provides an experimental approach to concepts covered in Photonics (PH 2501), Lasers (PH 2502), and Optics (PH 3504). Through a series of individually tailored experiments, students will
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fiber amplifiers, diffraction of light, polarization of light, interferometry. Recommended background: PH 1110/1111, PH 1120/1121. PH 1130, PH 1140, and one or more of the courses PH 2501, PH 2502,
or PH 3504. No prior laboratory background is expected.
Cat. I This course offers experience in experimentation and observation for students of the sciences and others. In a series of subject units, students learn or review the physical principles
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data, and identification of systematic and random errors. Preparation of high-quality experiment reports is also emphasized. Representative experiment subjects are: mechanical motions and vibrations;
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energy measurements. Recommended background: the Introductory Physics course sequence or equivalent. No prior laboratory background beyond that experience is required. Students who have received
credit for PH 2600 or PH 3600 may not receive credit for PH 2651.
Cat. I An introduction to the basic principles of thermodynamics and statistical physics. Topics covered include: basic ideas of probability theory, statistical description of systems of particles,
thermodynamic laws, entropy, microcanonical and canonical ensembles, ideal and real gases, ensembles of weakly interacting spin 1/2 systems. Recommended background: knowledge of quantum mechanics and
thermodynamics at the level of ES 3001.
Cat. I A continuation of PH 2301, this course deals with more advanced subjects in electromagnetism, as well as the study of basic subjects with a more advanced level of mathematical analysis.
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Cat. I This course includes a study of the basic postulates of quantum mechanics, its mathematical language and applications to one-dimensional problems. The course is recommended for physics majors
and other students whose future work will involve the application of quantum mechanics. Topics include wave packets, the uncertainty principle, introduction to operator algebra, application of the
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sequence, including the introduction to the 20th century physics. Suggested background: knowledge (or concurrent study) of linear algebra, Fourier series, and Fourier transforms.
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atom, angular momentum, spin, perturbation theory and examples of the application of quantum mechanics in fields such as atomic and molecular physics, solid state physics, optics, and nuclear
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background: prior knowledge of quantum mechanics at an intermediate level. Suggested background: knowledge of statistical physics is helpful.
This course is intended to acquaint the student with the measurable properties of nuclei and the principles necessary to perform these measurements. The major part of the course will be an
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This course provides an introduction to classical physical optics, in particular interference, diffraction and polarization, and to the elementary theory of lenses. The theory covered will be applied
in the analysis of one or more modern optical instruments. Recommended background: knowledge of introductory electricity and magnetism and of differential equations. Suggested background: PH 2301.
Cat. I A review of the basic principles and introduction to advanced methods of mechanics, emphasizing the relationship between dynamical symmetries and conserved quantities, as well as classical
mechanics as a background to quantum mechanics. Topics include: Lagrangian mechanics and the variational principle, central force motion, theory of small oscillations, Hamiltonian mechanics,
canonical transformations, Hamilton-Jacobi Theory, rigid body motion, and continuous systems. Recommended background: PH 2201 and PH 2202. This is a 14-week course.
Cat. I An introduction to the basic principles of thermodynamics and statistical physics. Topics covered include: basic ideas of probability theory, statistical description of systems of particles,
thermodynamic laws, entropy, microcanonical and canonical ensembles, ideal and real gases, ensembles of weakly interacting spin 1/2 systems. Recommended background: knowledge of quantum mechanics at
the level of PH 3401-3402 and of thermodynamics at the level of ES 3001.
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Algebra 1 Tutors
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Where modal propositions get their truth values from
Posted on: March 16, 2007 - 2:23am
Where modal propositions get their truth values from
A common question asked in Philosophy of Logic is as follows: In virtue of what can a proposition asserting either the possibility, impossibility, or necessity of something be said to be true or
false? A theory of modality is needed to explain it. Some, like David Lewis, claim that all possible worlds exist. However, the metaphysics of this claim is outlandash. Therefore, I have a more
reasonable account.
A possible world is a set of internally consistent propositions. By internally consistent I mean that there is not a contradiction, nor is it possible to prove from those propositions, a
contradiction. Therefore, when one says that some proposition P is possible, they are asserting that there is either a set, T (or one can construct a set, depending upon your philosophy of
mathematics) that P is a member of T, and ~P is disjoined from T. A necessary proposition Q, is such that for any set T, Q is a member of T. The concequence of this, is that every possible world is
infinite, since there are infinitely many true propositions. A contradiction is any set T, which as P as a member unioned with a set T' which has as its member, ~P. This union is strictly
Now, why would I choose to modal a theory of modality within Set Theory? For the simple reason that all of mathematics can be reduced to set theory. Numbers are given set theoretic definitions [the
number 1, for example, is defined as the set of all singletons, the number 2 as the set of all doubles, and so on]. Hence, the success mathematics has with set theory can be extended to an
explanation of where modal logic propositions derive their truth values from.
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
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Posted on: March 17, 2007 - 3:01pm #1
I don't have any answers
I don't have any answers for you, I only have one nit-picky comment.
Chaoslord2004 wrote:
For the simple reason that all of mathematics can be reduced to set theory.
Errrrrr.... not really. Almost all of it can, but set theory can't quite deal with collections "larger" than sets (like proper classes). Unfortunately, such "larger" collections are a natural
conclusion of using set theory, so set theory does a very good job of explaining other things but not entirely itself. Plus, there's category theory, which is basically abstract nonsense.
That said, set theory is a fine base for looking at modal truth values.
Posted on: March 21, 2007 - 10:51pm #2
Yiab wrote: Errrrrr....
Yiab wrote:
Errrrrr.... not really. Almost all of it can, but set theory can't quite deal with collections "larger" than sets (like proper classes). Unfortunately, such "larger" collections are a natural
conclusion of using set theory, so set theory does a very good job of explaining other things but not entirely itself. Plus, there's category theory, which is basically abstract nonsense.
That said, set theory is a fine base for looking at modal truth values.
The set-class distinction is controversal.
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
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Math Help
September 4th 2006, 12:47 PM #1
Junior Member
Oct 2005
I want to prove $\sum_{k=1}^{n}k=\frac{n(n+1)}{2}=<br /> \sum_{k=0}^{n}(-1)^{n+k}*k^2$
So, simply what I chose to do is to first prove
by (mathematical) induction, and then use the same method for
The first works out fine, no problem there at all.
The second part causes problems.
I manage to show P(x) true for x=1 (wow, I am realy amazing)
And then I make my claim:
I then do my calculations and I get:
Witch is not right.
Anyone know or have a clue about where I am off?
Is there another part of my calculations that I should include? Please say so if that is the case, I try to include just the parts where I think I am wrong (everything only if I have no clue).
I want to prove $\sum_{k=1}^{n}k=\frac{n(n+1)}{2}=<br /> \sum_{k=0}^{n}(-1)^{n+k}*k^2$
So, simply what I chose to do is to first prove
by (mathematical) induction, and then use the same method for
The first works out fine, no problem there at all.
The second part causes problems.
I manage to show P(x) true for x=1 (wow, I am realy amazing)
And then I make my claim:
I then do my calculations and I get:
Witch is not right.
Anyone know or have a clue about where I am off?
Is there another part of my calculations that I should include? Please say so if that is the case, I try to include just the parts where I think I am wrong (everything only if I have no clue).
Assume for some integer $m_0$ that:
$<br /> \frac{m_0(m_0+1)}{2}=\sum_{k=0}^{m_0}(-1)^{m_0+k}*k^2<br />$
$<br /> \sum_{k=0}^{m_0+1}(-1)^{m_0+1+k}*k^2$$<br /> =\sum_{k=0}^{m_0}(-1)^{m_0+1+k}*k^2 + (-1)^{2(m_0+1)}(m_0+1)^2<br />$
$<br /> =-\sum_{k=0}^{m_0}(-1)^{m_0+k}*k^2+(m_0+1)^2<br />$
Then by assumption the first term on the right may be replaced by $-\frac{m_0(m_0+1)}{2}$ so:
$<br /> \sum_{k=0}^{m_0+1}(-1)^{m_0+1+k}*k^2$$<br /> =-\frac{m_0(m_0+1)}{2}+(m_0+1)^2<br />$.
Which should simplify to what you are looking for to prove the required equality for $m_0+1$.
Last edited by CaptainBlack; September 4th 2006 at 01:44 PM.
Well then, there is something that I don not understand here:
$<br /> =\sum_{k=0}^{m_0}(-1)^{m_0+1+k}*k^2 + (-1)^{2(m_0+1)}(m_0+1)^2<br />$
and not:
$<br /> =\sum_{k=0}^{m_0}(-1)^{m_0+1}*k^2 + (-1)^{2(m_0+1)}(m_0+1)^2<br />$
Well then, there is something that I don not understand here:
$<br /> =\sum_{k=0}^{m_0}(-1)^{m_0+1+k}*k^2 + (-1)^{2(m_0+1)}(m_0+1)^2<br />$
and not:
$<br /> =\sum_{k=0}^{m_0}(-1)^{m_0+1}*k^2 + (-1)^{2(m_0+1)}(m_0+1)^2<br />$
Because the first of these is the far right term in the original problem
with $n$ replaced by $m_0+1$, that is the next case after the one
assumed to hold.
Oh, sorry a little typo again I mean to say:
$<br /> =\sum_{k=0}^{m_0}(-1)^{m_0+1+k}*k^2 + (-1)^{2(m_0+1)}(m_0+1)^2<br />$
And not:
$<br /> =\sum_{k=0}^{m_0+k}(-1)^{m_0}*k^2 + (-1)^{2(m_0+1)}(m_0+1)^2<br />$
'cuse I did take the m_0+1 term for it self and only got m_0 left...right?
Oh, sorry a little typo again I mean to say:
$<br /> =\sum_{k=0}^{m_0}(-1)^{m_0+1+k}*k^2 + (-1)^{2(m_0+1)}(m_0+1)^2<br />$
And not:
$<br /> =\sum_{k=0}^{m_0+k}(-1)^{m_0}*k^2 + (-1)^{2(m_0+1)}(m_0+1)^2<br />$
'cuse I did take the m_0+1 term for it self and only got m_0 left...right?
It's the same answer as before. The first of these is:
$<br /> \sum_{k=0}^{m_0}(-1)^{m_0+k} k^2<br />$
with $m_0$ replaced by $m_0+1$ throughout, and
the last term taken outside the sum:
$<br /> \sum_{k=0}^{m_0+1}(-1)^{m_0+1+k} k^2=\left[ \sum_{k=0}^{m_0}(-1)^{m_0+1+k} k^2 \right] +$$(-1)^{m_0+1+m_0+1} (m_0+1)^2<br />$
Well, yes off course.
Hmmm, but why do one use n+k?
Isn't there any way to express the same thing with fix numbers, for me it seems like that should be possible, but probably I am wrong.
Thanks a lot CB, you are often of great help.
September 4th 2006, 01:31 PM #2
Grand Panjandrum
Nov 2005
September 4th 2006, 10:15 PM #3
Junior Member
Oct 2005
September 4th 2006, 10:32 PM #4
Grand Panjandrum
Nov 2005
September 4th 2006, 10:42 PM #5
Junior Member
Oct 2005
September 4th 2006, 10:58 PM #6
Grand Panjandrum
Nov 2005
September 4th 2006, 11:26 PM #7
Junior Member
Oct 2005
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On the genus and thickness of graphs
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph
parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
Cited by 33 (0 self)
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph
parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature.
We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with on-line algorithms.
- Graphs Combin , 1998
"... We give a state-of-the-art survey of the thickness of a graph from both a theoretical and a practical point of view. After summarizing the relevant results concerning this topological invariant
of a graph, we deal with practical computation of the thickness. We present some modifications of a ba ..."
Cited by 18 (0 self)
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We give a state-of-the-art survey of the thickness of a graph from both a theoretical and a practical point of view. After summarizing the relevant results concerning this topological invariant of a
graph, we deal with practical computation of the thickness. We present some modifications of a basic heuristic and investigate their usefulness for evaluating the thickness and determining a
decomposition of a graph in planar subgraphs. Key words: Thickness, maximum planar subgraph, branch and cut 1 Introduction In VLSI circuit design, a chip is represented as a hypergraph consisting of
nodes corresponding to macrocells and of hyperedges corresponding to the nets connecting the cells. A chip-designer has to place the macrocells on a printed circuit board (which usually consists of
superimposed layers), according to several designing rules. One of these requirements is to avoid crossings, since crossings lead to undesirable signals. It is therefore desirable to find ways to
handle wi...
- DISCRETE AND COMPUTATIONAL GEOMETRY , 2005
"... Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter
thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restri ..."
Cited by 14 (8 self)
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Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter
thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper
studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/
2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the
maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for
outerthickness, arboricity, and star-arboricity.
- IN: MORE SETS, GRAPHS AND NUMBERS , 2006
"... We survey known results and propose open problems on the biplanar crossing number. We study biplanar crossing numbers of specific families of graphs, in particular, of complete bipartite graphs.
We find a few particular exact values and give general lower and upper bounds for the biplanar crossing n ..."
Cited by 4 (0 self)
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We survey known results and propose open problems on the biplanar crossing number. We study biplanar crossing numbers of specific families of graphs, in particular, of complete bipartite graphs. We
find a few particular exact values and give general lower and upper bounds for the biplanar crossing number. We find the exact biplanar crossing number of K 5;q for every q.
"... This is a survey of studies on topological graph theory developed by Japanese people in the recent two decades and presents a big bibliography including almost all papers written by Japanese
topological graph theorists. ..."
Cited by 1 (0 self)
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This is a survey of studies on topological graph theory developed by Japanese people in the recent two decades and presents a big bibliography including almost all papers written by Japanese
topological graph theorists.
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Center Square, PA Algebra Tutor
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[SOLVED] Prove two definitions of derivative is equivalent
November 28th 2009, 02:06 PM
[SOLVED] Prove two definitions of derivative is equivalent
Hi, everyone! I have a minor problem proving the two following definitions of the derivative are equivalent:
f is differentiable iff $\lim_{x->x_{0}}\frac{f(x)-f(x_{0})}{x-x_{0}}$ exists
f is differentiable iff $\lim_{t->0}\frac{f(x_{0}+t)-f(x_{0})}{t}$ exists.
I know that i can let $t=x - x_{0}$, so that $x=x_{0}+t$ and $\frac{f(x) - f(x_{0})}{x - x_{0}}=\frac{f(x_{0} + t) - f(x_{0})}{t}$, but then how can i explain the limits part, when in first case
i am taking limit as x approaches $x_{0}$ and in second as t approaches 0?
Thanks in advance!
November 28th 2009, 02:27 PM
First, be aware that I not really sure that I understand you difficulty here.
That said, surely it is clear to you that this is true:
$\lim _{x \to x_0 } \left( {x - x_0 } \right) = 0\;\& \,\lim _{t \to 0} t = 0.$
Now tell us what you don't understand.
November 28th 2009, 03:33 PM
Yea, i'm aware of that, but i'm not sure how I can apply that knowledge to my problem, since as far as I am concerned i'm not taking limit of t as t goes to 0, but rather limit of difference
quotient as t goes to 0, where difference quotient need not be t. I guess what I really wanted to know is whether $\lim_{x-x_0\to 0}(Y)=\lim_{x \to x_0}(Y)$ is generally true for any $Y$. Correct
me if i'm totally off the track.
November 28th 2009, 04:03 PM
Hi, everyone! I have a minor problem proving the two following definitions of the derivative are equivalent:
f is differentiable iff $\lim_{x->x_{0}}\frac{f(x)-f(x_{0})}{x-x_{0}}$ exists
f is differentiable iff $\lim_{t->0}\frac{f(x_{0}+t)-f(x_{0})}{t}$ exists.
I know that i can let $t=x - x_{0}$, so that $x=x_{0}+t$ and $\frac{f(x) - f(x_{0})}{x - x_{0}}=\frac{f(x_{0} + t) - f(x_{0})}{t}$, but then how can i explain the limits part, when in first case
i am taking limit as x approaches $x_{0}$ and in second as t approaches 0?
Thanks in advance!
Since you defined $t=x-x_0$ , it follows at once that if $x\rightarrow x_0$ , then $t\rightarrow 0$, and the other way around. This is EXACTLy the reason why both limits above are equivalent.
November 28th 2009, 08:29 PM
Thanks a lot, Tonio!
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Plot a path between 2 points, speed decelerating
November 16th 2008, 05:54 PM #1
Nov 2008
Plot a path between 2 points, speed decelerating
Hi there,
I'm a programmer trying to plot the positions between two points. The number of steps from Point1 to Point2 is variable, as are the x,y positions of the two points.
I can do it if the speed is constant. The function is as follows:
xinc = Beat(x1,x2)-Defeat(x1,x2)
yinc = Beat(y1,y2)-Defeat(y1,y2)
For a = 2 To steps
Next a
That's the program code (PureBasic). Forgive my ignorance but I believe the mathematical formula would be something like:
$Xn = (x2-x1) / steps * n$
$Yn = y2-y1) / steps * n$
assuming that x2 is greater than x1, and y2 is greater than y1.
However, can anyone tell me how to plot the path between two points if the speed is steadily decelerating?
At high school I could possibly have done this by myself but that's years ago and I'm reduced to googling for the formula. Hours later, I still can't find it, so any help would be greatly
Thanks for reading,
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VHDL function to convert an vector datatype to integer
Newbie level 6
Join Date
Apr 2006
0 / 0
VHDL function to convert an vector datatype to integer
Is there a function to convert an vector datatype to integer and vice versa.
Full Member level 2
Join Date
Jun 2004
137 / 137
vhdl conv_integer
Well I don't think there is a ready made function to convert, but you can write down your own easily.
-- Convert a std_logic_vector to an unsigned integer
function to_uint (a: std_logic_vector) return integer is
alias av: std_logic_vector (1 to a'length) is a;
variable val: integer := 0;
variable b: integer := 1 ;
for i in a'length downto 1 loop
if (av(i) = '1 ') then -- if LSB is '1 ',
val := vat + b; -- add value for current bit position
end if;
b := b*2; -- Shift left 1 bit
end loop;
return val;
end to_uint;
Example from "VHDL Made Easy" by Pellerin and Taylor.
p.s. if you find this post of any use to you then kindly do click on the "helped me" icon. Regards, salam.
3. conv_unsigned vhdl
The following functions are contained in the library arith.vhd. To use them,
place the line “USE ieee.std_logic_arith.ALL” at the beginning of your
VHDL design.
FUNCTION Pass(arg, size) Return
· CONV_INTEGER INTEGER INTEGER
· CONV_INTEGER UNSIGNED INTEGER
· CONV_INTEGER SIGNED INTEGER
· CONV_INTEGER STD_ULOGIC SMALL_INT;
· CONV_UNSIGNED INTEGER, INTEGER UNSIGNED;
· CONV_UNSIGNED UNSIGNED, INTEGER UNSIGNED;
· CONV_UNSIGNED SIGNED, INTEGER UNSIGNED;
· CONV_UNSIGNED STD_ULOGIC, INTEGER UNSIGNED;
· CONV_SIGNED INTEGER, INTEGER SIGNED;
· CONV_SIGNED UNSIGNED, INTEGER SIGNED;
· CONV_SIGNED SIGNED, INTEGER SIGNED;
· CONV_SIGNED STD_ULOGIC, INTEGER SIGNED;
· CONV_STD_LOGIC_VECTOR INTEGER, INTEGER STD_LOGIC_VECTOR
· CONV_STD_LOGIC_VECTOR UNSIGNED, INTEGER STD_LOGIC_VECTOR
· CONV_STD_LOGIC_VECTOR SIGNED, INTEGER STD_LOGIC_VECTOR
· CONV_STD_LOGIC_VECTOR STD_ULOGIC, INTEGER STD_LOGIC_VECTOR
· EXT STD_LOGIC_VECTOR, INTEGER STD_LOGIC_VECTOR;
· SXT STD_LOGIC_VECTOR, INTEGER STD_LOGIC_VECTOR;
The following function is contained in the library unsigned.vhd. To use it, place
the line “USE ieee.std_logic_unsigned.ALL” at the beginning of your VHDL
· CONV_INTEGER(arg: STD_LOGIC_VECTOR) return INTEGER;
The following function is contained in the library signed.vhd. To use it, place
the line “USE ieee.std_logic_signed.ALL” at the beginning of your VHDL
· CONV_INTEGER(arg: STD_LOGIC_VECTOR) return INTEGER;
u can find all info such as the above in this link:
good luck
Advanced Member level 4
Join Date
Jan 2003
53 / 53
conv_signed vhdl
Unfortunately std_logic_arith is obsolute and should no more be used. Instead, numeric_std should exclusively be used.
5. 23rd October 2008, 04:48 #5
Full Member level 1
Join Date
Oct 2008
8 / 8
vhdl conv_signed
ya, use nuemric_std or synopsys lib is good for this conversions
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Reciprocal sum of primes diverges... proof? - My Math Forum
December 27th, 2006, 10:01 PM #1
Joined: Nov 2006 Reciprocal sum of primes diverges... proof?
Posts: 20 We'll need the following definitions to solve this problem:
Thanks: 0 definition:
For any number x, Nj(x) is the number of numbers less than or equal to
x whose only prime divisors are in the set of the first j
primes { p1, p2, ..., pj }.
A number is square-free if none of its divisors
is a perfect square (except 1).
(a) Prove that there are exactly 2j square-free numbers whose
only prime divisors are in the set { p1, p2, ..., pj }.
(b) Prove that, for sufficiently large x,
Nj(x)<= 2j * x (Hint: show every number can be written uniquely as the product of a perfect square and a square-free number).
(c) Prove that,
for sufficiently large x, Nj(x) < x. Why does this
mean there must be an infinite number of primes?
(d) For subsequent results, we need a little bit more. Prove that,
for sufficiently large x, Nj(x) < x2 (The proof is
exactly like the one in (c).)
We will use (d) to prove:
theorem: The sum of the reciprocals of the primes
diverge! That is,
1p1 + 1p2 +1p3 +... diverges
[Is it clear that this is even stronger than merely being an infinite set?
For example, the perfect squares are an infinite set, but the sum of the
reciprocals of the perfect squares converges.]
Anyhow to show that the sum of the reciprocals of the primes diverges, it's
enough to show that, for any j:
1/pj+1 + 1/pj+2 + 1/pj+3 + .... > 1/2
(e) Why is it enough to show that the above sum is greater
than 12 ?
(f) Show that for any x, x/pj+1 + x/pj+2 + x/pj+3 + .... >= x - Nj(x).
(Hint: Explain first why x/p is greater than or equal to the number
of numbers less than or equal to x that are divisible by p. What about
x/p + x/q where p and q are relatively prime?)
(g) Use the results from (d) and (f) to complete the proof
that the sums of the reciprocals of the primes diverge.
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Quadrature is the computation of a univariate definite integral. It can refer to either numerical or analytic techniques; one must gather from context which is meant. The term refers to the geometric
origin of integration in determining the area of a plane figure by approximating it with squares.
Cubature refers to higher-dimensional definite integral computation. Likewise, this term refers to the geometric operation of approximating the volume of a solid by means of cubes (and has since been
extended to higher dimensions).
The terms “quadrature” and “cubature” are typically used in numerical analysis to denote the approximation of a definite integral, typically by a suitable weighted sum. Perhaps the simplest
possibility is approximation by a sum of values at equidistant points, i.e. approximate $\int_{0}^{1}f(x)\,dx$ by $\sum_{{k=0}}^{n}f(k/n)/n$. More complicated approximations involve variable weights
and evaluation of the function at points which may not be spaced equidistantly. Some such numerical quadrature methods are Simpson’s rule, the trapezoidal rule, and Gaussian quadrature.
Mathematics Subject Classification
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Groups - Alternating group question
Define $A_n$ as the alternating group. How do I figure out how many 5-cycles $A_5$ has?
Thank you for your post. There are a few more questions concerning this and Sylow theorem that maybe you can help me with. (i) What is the order of every Sylow 5-subgroup of $A_5$ ? (ii) How many
Sylow 5-subgroups does $A_5$ have? For (i) I think the answer is 5 because $60=2^{2}.3.5^{1}$ for (ii) I think using one of the Sylow theorems, that the number of Sylow 5-subgroups is of the form
$5k+1$ and divides the order of $A_5$, so I guess from this there is either 1 or 6 Sylow 5-subgroups in $A_5$, so the answer is 6?? Can anyone verify any of this? (iii) If $P$ is a Sylow p-subgroup
of a finite group $G$, and if $Q$ is a Sylow p-subgroup of a finite group $H$, prove that the direct product $P \times Q$ is a Sylow p-subgroup of the direct product $G \times H$. For (iii) i
attempted to prove it like follows: Take $(p_1,q_1),(p_2,q_2) \in P \times Q$$(p_1,p_2 \in P, q_1,q_2 \in Q)$ $(p_1,q_1)(p_2^{-1},q_2^{-1}) = (p_1p_2^{-1},q_1q_2^{-1}) \in P \times Q$ Is this the
right method? Thanks for any help anyone can provide.
your answers to (i) and (ii) are correct, although you need to explain in (ii) why the number of Sylow 5-subgroups cannot be 1. for that you need to look at your first question about the number of 5
cycles. for (iii) look at the orders: since P and Q are Sylow p-subgroups we have $|P|=p^m, \ |Q|=p^n, \ |G|=p^m r, \ |H|=p^n s,$ where $\gcd(p,r)=\gcd(p,s)=1.$ then $|P \times Q|=p^{m+n}$ and $|G \
times H|=p^{m+n}rs$ and clearly $\gcd(rs,p)=1.$ this completes the proof.
Hi Jason Bourne. If you are allowed to assume that $A_5$ is simple, then the answer is immediately obvious. If there were just 1 Sylow 5-subgroup, this would be a proper nontrivial normal subgroup of
$A_5;$ since $A_5$ is simple, there must therefore be more than 1 Sylow 5-subgroup (indeed more than 1 Sylow subgroup of any order).
Hi Jason Bourne. Are you aware of the result that for each prime $p$ dividing the order of a finite group $G,$ all Sylow $p$-subgroups of $G$ are conjugate to each other? It follows that if $G$ has a
unique Sylow $p$-subgroup, then that Sylow $p$-subgroup is normal in $G.$ Think about it. (Wink)
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Solving Systems of Equation by Substitution and Elimination
The problems with the graphing method are threefold:
• You need an accurate graph.
• Your graph may not be large enough.
• It s hard to estimate the solution if the coordinates are not integers.
In this section, we look at two algebraic methods for finding solutions.
• In both of the methods outline below, there are actually three possible outcomes.
○ You get a single ordered pair as a solution.
In this case, the solution is the ordered pair you find.
○ All variables go away and you get a false statement, such as 0 = 4.
In this case, you have parallel lines, so there is no solution to the system.
○ All variables go away and you get a true statement, such as 0 = 0 or 5 = 5.
In this case, you have the same line, so there are infinitely many solutions.
Procedure: (Substitution Method)
0. Choose a variable and an equation.
1. Solve for the chosen variable in the chosen equation.
2. Substitute the expression you found for the selected variable in the OTHER equation.
3. Solve the resulting equation in one variable.
4. Use the answer you found in 3 to find the value of the other variable.
5. Write your answer as an ordered pair.
x + y = 8
2x - 3y = -9
(3, 5).
Procedure: (Elimination Method)
0. Choose a variable.
1. Multiply one or both equations by whatever is necessary to get the coefficients of the selected variable to be the same, but with opposite signs.
2. Add the equations together. (NOTE: This eliminates the selected variable.)
3. Solve the resulting equation in one variable.
4. Use the answer you found in 3 to find the value of the other variable.
5. Write your answer as an ordered pair.
x + y = 8
3x - y = 0
(2, 6).
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Here's the question you clicked on:
3rd grade math? please help me. show 1500 hours on a clock. the hour hand is shorter.
• one year ago
• one year ago
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Nice picture!!
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So, what time might 15:00 be equal to?
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3 clcok
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exactly! So just draw the hands for the time of 3 o'clock... short hand toward 3, long hand up at 12.
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you can count time 12 hours at a time, like AM and PM, or you can count it from 0:00 up to 23:59. The times larger than 12:00 are in the afternoon, and you just keep counting around to the right
number. So, 18:30 would be like 6:30 PM
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And often, they leave out the " : " (colon) symbol, so 1500 is the same as 15:00 is the same as 3:00 PM... mid afternoon
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oh thank you i thought i was suposed to count from 1 to 1500 and see what time it was
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that's ok... it's not "1500 minutes" It's just a different way of telling time. Sometimes military units use it... "we go to bed at 2200 hours" meaning 10:00 PM.
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Does it make sense now? Given any time larger than 1200, you know it must be after noon, and maybe into the evening... you find the equivalent time on a clock by subtracting 12 hours from the
amount you are given. So 1900 hours is 1900-1200 = 7 o'clock in the evening... 7:00 PM
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yes, thank you!
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Glad to help! Good luck! :)
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is replying to Can someone tell me what button the professor is hitting...
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Bridging the Gap Between Common Core State Standards and Teaching Statistics
How Can We Sort Our Junk? Can You Roll Your Tongue? How Expensive Is Your Name? How Tall Were the Ancestors of Laetoli? What Do Frogs Eat? These are a few of the 20 data analysis and probability
investigations in Bridging the Gap (BTG).
The module's investigations are written as guides for teachers to implement straightforwardly in their K-8 classrooms. Each investigation is based on the four-step statistical process as defined by
the Guidelines for Assessment and Instruction in Statistics Education (GAISE):
1. Formulate a question that can answered by data
2. Design and implement a plan to collect appropriate data
3. Analyze the collected data by graphical and numerical methods
4. Interpret the analysis in the context of the original question
Each investigation contains:
• A descriptive Overview that discusses the big ideas presented in the investigation
• A list of Learning Goals that describe what students will be able to do as a result of completing the investigation
• A reference to the Common Core State Standards for Mathematical Practice, Common Core State Standards Grade Level Content, and NCTM Principles and Standards for School Mathematics
• A list of the Materials needed for the investigation
• A suggested Estimated Time, with the number of days needed to complete the investigation
• Instructional Plan that follows the four-step statistical process as defined by GAISE
• An Example of Interpret the Results report that students need to write for each investigation
• An Assessment with Answers
• Possible Extensions of the material discussed in the investigation
• A list of References pertaining to the investigation
A CD-ROM that contains activity sheets and other material is included for ease of making copies for classroom use.
Graphs are an integral part of data analysis. Those found in BTG include bar graphs, dotplots, stemplots, boxplots, and scatterplots. Statistical measures include those identifying center (mode,
median, mean), spread (range, inter-quartile range, mean absolute deviation), and correlation (quadrant count ratio).
Students who have mastered these investigations will be prepared to pursue those at the high-school level as found in Making Sense of Statistical Studies (MSSS). MSSS is a module of 15 activities
covering the Common Core State Standards in Statistics involving surveys, observational studies, and experiments.
Try it before you buy it. Click on the "Can You Roll Your Tongue?" button for a free investigation.
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Re: random numbers in fortran
"lane straatman" <grumpy196884@xxxxxxxxxxx> wrote in message
Since I want to simulate shuffling a deck of cards, I guess I'll ask
for 52 pseudorandoms at a pop, multiply by 52 and take the floor. I
believe this gives random ints between zero and fifty-one.
It doesn't, there will certainly be duplicates. Here's a simple-minded way
to get what you want:
function scatter(how_many)
integer :: how_many, scatter(how_many), ii, index
real :: numbers(how_many)
call random_number(numbers)
do ii = 1, how_many
index = minloc(numbers, dim=1)
scatter(ii) = index
numbers(index) = 2.0
end do
end function scatter
Mike Metcalf
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Far Rockaway Science Tutor
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West Boxford Math Tutor
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Rank of a matrix
What is the rank of a matrix and how it can be found?What is its uses?Please clarify with examples.
The rank of a matrix is the number of linearly independent rows (or columns) of that matrix. This is usually found by the Gauss elimination method, but other methods exist for more complicated/exotic
circumstances. Matrix rank can be used to characterize some properties of a matrix, or determine the number of solutions to a system of linear equations represented by the matrix (which is the
typical application on first encounters with matrices).
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Marginal revenue and the marginal cost
November 25th 2008, 12:21 PM #1
Junior Member
Nov 2008
Marginal revenue and the marginal cost
The cost function is C(x)= 3x+2 The revenue function is R(X)= 7x-x² x is the number of units produced in thousands and R and C are measured in millions of dollars Cost= Revenue= Thank you.
Marginal revenue and the marginal cost
I'm not exactly sure of the question you're asking, but I'll assume it's "where does the marginal revenue equal the marginal cost?"
The marginal revenue and marginal cost functions are the derivatives of the total revenue and total cost functions. So
MC = 3
MR = 7-2X
And X is solved by setting 3 = 7-2X, since you want to produce where the MC = MR. In this case, that works out to X=2, which, if I understand your note below, implies you would produce 2,000
- Steve J
November 28th 2008, 05:22 PM #2
Junior Member
Nov 2008
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Subnet Q Help Plz - Network+ Exam
hi guys,
im planning to take the exam v soon & was just needed help with the following practise q:
''The plans for a new office requiere 300 network connections, which of the following network sizes should be allocated to the new office?,,
- a: /23
- b: /25
- c: /24
- d: /30
apparently the answer is A but can somebody kindly explain how this is so?
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A Parallelization of Dijkstra's Shortest Path Algorithm
Results 1 - 10 of 23
- In SC ’05: Proceedings of the 2005 ACM/IEEE conference on Supercomputing , 2005
"... Many emerging large-scale data science applications require searching large graphs distributed across multiple memories and processors. This paper presents a distributed breadthfirst search
(BFS) scheme that scales for random graphs with up to three billion vertices and 30 billion edges. Scalability ..."
Cited by 39 (2 self)
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Many emerging large-scale data science applications require searching large graphs distributed across multiple memories and processors. This paper presents a distributed breadthfirst search (BFS)
scheme that scales for random graphs with up to three billion vertices and 30 billion edges. Scalability was tested on IBM BlueGene/L with 32,768 nodes at the Lawrence Livermore National Laboratory.
Scalability was obtained through a series of optimizations, in particular, those that ensure scalable use of memory. We use 2D (edge) partitioning of the graph instead of conventional 1D (vertex)
partitioning to reduce communication overhead. For Poisson random graphs, we show that the expected size of the messages is scalable for both 2D and 1D partitionings. Finally, we have developed
efficient collective communication functions for the 3D torus architecture of BlueGene/L that also take advantage of the structure in the problem. The performance and characteristics of the algorithm
are measured and reported. 1
, 2001
"... This paper addresses the state explosion problem in automata based LTL model checking. To deal with large space requirements we turn to use a distributed approach. All the known methods for
automata based model checking are based on depth first traversal of the state space which is difficult to para ..."
Cited by 29 (12 self)
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This paper addresses the state explosion problem in automata based LTL model checking. To deal with large space requirements we turn to use a distributed approach. All the known methods for automata
based model checking are based on depth first traversal of the state space which is difficult to parallelise as the ordering in which vertices are visited plays an important role. We come up with
entirely different approach which is dependent on locating cycles with negative length in a directed graph with real number length of edges. Our method allows reasonable distribution and the
experimental results confirm its usefulness for distributed model checking.
- in 13th ACMSIAM Symp. on Discrete Algs , 1985
"... Abstract. We present a new scheme for computing shortest paths on real-weighted undirected graphs in the fundamental comparison-addition model. In an efficient preprocessing phase our algorithm
creates a linear-size structure that facilitates single-source shortest path computations in O(m log α) ti ..."
Cited by 12 (3 self)
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Abstract. We present a new scheme for computing shortest paths on real-weighted undirected graphs in the fundamental comparison-addition model. In an efficient preprocessing phase our algorithm
creates a linear-size structure that facilitates single-source shortest path computations in O(m log α) time, where α = α(m, n) is the very slowly growing inverse-Ackermann function, m the number of
edges, and n the number of vertices. As special cases our algorithm implies new bounds on both the all-pairs and single-source shortest paths problems. We solve the all-pairs problem in O(mnlog α(m,
n)) time and, if the ratio between the maximum and minimum edge lengths is bounded by n (log n)O(1) , we can solve the single-source problem in O(m + nlog log n) time. Both these results are
theoretical improvements over Dijkstra’s algorithm, which was the previous best for real weighted undirected graphs. Our algorithm takes the hierarchy-based approach invented by Thorup. Key words.
single-source shortest paths, all-pairs shortest paths, undirected graphs, Dijkstra’s
- In Proc. 11th Annual European Symposium on Algorithms, volume 2832 of LNCS , 2003
"... Abstract. We show how to compute single-source shortest paths in undirected graphs with non-negative edge lengths in O ( p nm/B log n + MST (n, m)) I/Os, where n is the number of vertices, m is
the number of edges, B is the disk block size, and MST (n, m) is the I/O-cost of computing a minimum spann ..."
Cited by 11 (4 self)
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Abstract. We show how to compute single-source shortest paths in undirected graphs with non-negative edge lengths in O ( p nm/B log n + MST (n, m)) I/Os, where n is the number of vertices, m is the
number of edges, B is the disk block size, and MST (n, m) is the I/O-cost of computing a minimum spanning tree. For sparse graphs, the new algorithm performs O((n / √ B) log n) I/Os. This result
removes our previous algorithm’s dependence on the edge lengths in the graph. 1
- Proc. 16th Intl. Par. Distr. Process. Symp. (IPDPS , 2002
"... We study the average-case complexity of the parallel single-source shortest-path (SSSP) problem, assuming arbitrary directed graphs with n nodes, m edges, and independent random edge weights
uniformly distributed in [0; 1]. We provide a new bucket-based parallel SSSP algorithm that runs in T = O(log ..."
Cited by 6 (2 self)
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We study the average-case complexity of the parallel single-source shortest-path (SSSP) problem, assuming arbitrary directed graphs with n nodes, m edges, and independent random edge weights
uniformly distributed in [0; 1]. We provide a new bucket-based parallel SSSP algorithm that runs in T = O(log 2 n min i f2 i L + jV i jg) average-case time using O(n+m+T ) work on a PRAM where L
denotes the maximum shortest-path weight and jV i j is the number of graph vertices with in-degree at least 2 i . All previous algorithms either required more time or more work. The minimum
performance gain is a logarithmic factor improvement; on certain graph classes, accelerations by factors of more than n 0:4 can be achieved. The algorithm allows adaptation to distributed memory
machines, too.
- In Proc. Int’l. Conf. on High Performance Computing (HiPC 2010 , 2010
"... Abstract—Betweenness centrality is a measure based on shortest paths that attempts to quantify the relative importance of nodes in a network. As computation of betweenness centrality becomes
increasingly important in areas such as social network analysis, networks of interest are becoming too large ..."
Cited by 5 (0 self)
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Abstract—Betweenness centrality is a measure based on shortest paths that attempts to quantify the relative importance of nodes in a network. As computation of betweenness centrality becomes
increasingly important in areas such as social network analysis, networks of interest are becoming too large to fit in the memory of a single processing unit, making parallel execution a necessity.
Parallelization over the vertex set of the standard algorithm, with a final reduction of the centrality for each vertex, is straightforward but requires Ω(|V | 2) storage. In this paper we present a
new parallelizable algorithm with low spatial complexity that is based on the best known sequential algorithm. Our algorithm requires O(|V | + |E|) storage and enables efficient parallel execution.
Our algorithm is especially well suited to distributed memory processing because it can be implemented using coarse-grained parallelism. The presented time bounds for parallel execution of our
algorithm on CRCW PRAM and on distributed memory systems both show good asymptotic performance. Experimental results with a distributed memory computer show the practical applicability of our
algorithm. I.
, 2001
"... w\Delta\Theta\Xi\Pi\Sigma\Upsilon\Phi\Omega fffiflffiij`'ae/!"#$%&'()+,-./012345!yA--- ..."
- In , 2010
"... Active messages have proven to be an effective approach for certain communication problems in high performance computing. Many MPI implementations, as well as runtimes for Partitioned Global
Address Space languages, use active messages in their low-level transport layers. However, most active messag ..."
Cited by 4 (2 self)
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Active messages have proven to be an effective approach for certain communication problems in high performance computing. Many MPI implementations, as well as runtimes for Partitioned Global Address
Space languages, use active messages in their low-level transport layers. However, most active message frameworks have low-level programming interfaces that require significant programming effort to
use directly in applications and that also prevent optimization opportunities. In this paper we present AM++, a new user-level library for active messages based on generic programming techniques. Our
library allows message handlers to be run in an explicit loop that can be optimized and vectorized by the compiler and that can also be executed in parallel on multicore architectures. Runtime
optimizations, such as message combining and filtering, are also provided by the library, removing the need to implement that functionality at the application level. Evaluation of AM++ with
distributed-memory graph algorithms shows the usability benefits provided by these library features, as well as their performance advantages.
"... The Parallel Boost Graph Library (Parallel BGL) is a library of graph algorithms and data structures for distributed-memory computation on large graphs. Developed with the Generic Programming
paradigm, the Parallel BGL is highly customizable, supporting various graph data structures, arbitrary verte ..."
Cited by 4 (2 self)
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The Parallel Boost Graph Library (Parallel BGL) is a library of graph algorithms and data structures for distributed-memory computation on large graphs. Developed with the Generic Programming
paradigm, the Parallel BGL is highly customizable, supporting various graph data structures, arbitrary vertex and edge properties, and different communication media. In this paper, we describe the
implementation of two parallel variants of Dijkstra’s single-source shortest paths algorithm in the Parallel BGL. We also provide an experimental evaluation of these implementations using synthetic
and real-world benchmark graphs from the 9 th DIMACS Implementation Challenge. 1
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Topic: Trisection
Replies: 19 Last Post: Feb 11, 2012 9:59 AM
Messages: [ Previous | Next ]
Re: Trisection
Posted: Jul 3, 2002 10:52 AM
I have algebraically checked this construction. It will effectively converge
to the trisection, with a cubic convergence rate. More precisely, if we take A
as the origin and C as the unit point on the X axis, 3*t the angle to trisect,
if h is the difference in abscissa between the first guess and the result
(i.e. the
difference in cosine), and h' the same difference after one iteration, we have:
h' = h^3 / (48*sin(t)^2) + O(h^4)
For the iteration process, it would have been better to first choose point
E on the
circle of center A by B and C, since the construction gives another point
on this circle.
Then D would be the intersection of circle centered in C by E and the
initial circle.
About your question on the imprecision of the practical construction, one may
take the other intersection D' of the circle centered at C and the line BC.
Line ED'
intersects the circle of radius 3 and center A in two points G1 and G2,
two other points E1' and E2' on the unit circle. With the original point
E', these two
points form a nearly equilateral triangle corresponding to the three
solutions of the
trisection problem.
-- Eric
At 16:05 21/06/2002 -0400, John Conway wrote:
>On 21 Jun 2002, Mark Stark wrote:
> > Thanks for the kind words John. I'm afraid this is a case of "have CAD
> > system -will play". It's the result of taking the word impossible as a
> > challenge. The accuracy does vary with the size of the given angle and
> > the quality of the first guess. The smaller the given angle the
> > better. My error calculations were based on a medium sized angle (60
> > degrees) and fair first guess (25 degrees). However, even a 90 deg.
> > angle and a poor guess (45 degrees) yields a first iteration of 30.021
> > deg. and a second iteration within 10E-7.
> It's nice to hear from you. I didn't want to say so in my first
>message, but it's a bit unfortunate that, although the construction
>theoretically has this high accuracy, in practice it's going to be
>weak because you must produce the short line ED to the decidedly
>lone one EF. I expect this defect can be cured in some simple way,
>and hope you'll work on this.
> Here's a question: For a given angle CAB, let D (on the
>straight line CB) and E (on the arc), vary in your manner (so
>that CD = CE). Then what's the envelope of the line DE ?
> The way my thoughts are running is this: if all such lines
>that are tolerably near to the correct one were "understandable",
>then one could use probably this to give an alternative finish to
>the construction.
> The neusis construction I was thinking of is this. Make your
>original circle be a unit one, and draw also the unit circle
>centered at C. Then adjust your ruler, which has two marks X
>and Y one unit apart, so that X lies on this latter circle,
>Y on the straight line CB, and so that it also passes through A.
>Then the ruler will trisect the angle CAB.
> This has the advantage over Archimedes' one (if you don't know
>that, you can find it in "The Book of Numbers", which I wrote
>jointly with Richard Guy), that it trisects the given angle "in
>situ", as it were. It therefore combines very nicely with my
>angle-trisector construction for the regular heptagon (which you
>can also find in The BoN). However, I want to look at that
>again, because for the version in The BoN, the angle to be
>trisected is inconveniently small, and because there may also
>be a nice way to economize by using something twice (once in
>the angle-trisection, and once in the ensuing heptagon construction).
> John Conway
Date Subject Author
6/20/02 Guest
7/3/02 Trisection (and pentasection and septasection etc) Jacques
7/3/02 John Conway
7/3/02 mark
7/3/02 John Conway
7/3/02 Eric Bainville
7/3/02 John Conway
7/15/02 mark
7/16/02 mark
7/16/02 Rouben Rostamian
7/17/02 Eric Bainville
7/17/02 Steve Gray
7/18/02 mark
7/19/02 Eric Bainville
7/21/02 Steve Gray
7/22/02 Rouben Rostamian
7/23/02 mark
7/24/02 mark
7/19/02 Steve Gray
2/11/12 JO 753
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Spring Design Optimization Problem
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This assignment can be completed as a collaborative [DEL:Attach:collaborative50:DEL].[DEL:png assignment. :DEL]Additional guidelines on individual, collaborative, and group assignments are provided
under the [[Main/CourseStandards | Expectations link]].[DEL::DEL]
[INS:Attach:collaborative50.png :INS]This assignment can be completed as a collaborative [INS:assignment:INS].[INS::INS] Additional guidelines on individual, collaborative, and group assignments are
provided under the [[Main/CourseStandards | Expectations link]].
Changed lines 5-6 from:
The specifications and modeling equations for a compression spring create a number of trade-offs that must be considered during design. We wish to determine the spring design that maximizes the force
of a spring at its preload height, [DEL::DEL]'[DEL:'h':DEL]_o_' [DEL:'', :DEL]of 1.0 inches. The spring is to operate an indefinite number of times through a deflection [DEL:o:DEL], of 0.4 inches,
which is an additional deflection from [DEL:ho:DEL]. The stress at the solid height, [DEL:hs:DEL], must be less than [DEL:Sy:DEL] to protect the spring from inadvertent damage.
The specifications and modeling equations for a compression spring create a number of trade-offs that must be considered during design. We wish to determine the spring design that maximizes the force
of a spring at its preload height, [INS:h:INS]'_o_'[INS:,:INS] of 1.0 inches. The spring is to operate an indefinite number of times through a deflection [INS:delta'_o_':INS], of 0.4 inches, which is
an additional deflection from [INS:h'_o_':INS]. The stress at the solid height, [INS:h'_s_':INS], must be less than [INS:S'_y_':INS] to protect the spring from inadvertent damage.
Changed lines 11-30 from:
[DEL:This introductory assignment is designed as a means of demonstrating the optimization capabilities of a number of software packages. Below are tutorials for solving this problem with a number of
software tools:DEL].
[INS:Turn in a report with the following sections:
# Title Page with Summary. The Summary should be short (less than 50 words), and give the main optimization results.
# Procedure: Give a brief description of your model:INS].[INS: You are welcome to refer to the assignment which should be in the Appendix. Also include:
## A table with the analysis variables, design variables, analysis functions and design functions.
# Results: Briefly describe the results of optimization (values). Also include:
## A table showing the optimum values of variables and functions, indicating binding constraints and/or variables at bounds (highlighted)
## A table giving the various starting points which were tried along with the optimal objective values reached from that point.
# Discussion of Results: Briefly discuss the optimum and design space around the optimum. Do you feel this is a global optimum? Also include and briefly discuss:
## A “zoomed out” contour plot showing the design space (both feasible and infeasible) for coil diameter vs. wire diameter, with the feasible region shaded and optimum marked.
## A “zoomed in” contour plot of the design space (mostly feasible space) for coil diameter vs. wire diameter, with the feasible region shaded and optimum marked.
# Appendix:
## Listing of your model with all variables and equations
## Solver output with details of the convergence to the optimal values
Any output from the software is to be integrated into the report (either physically or electronically pasted) as given in the sections above. Tables and figures should all have explanatory captions.
Do not just staple pages of output to your assignment: all raw output is to have notations made on it. For graphs, you are to shade the feasible region and mark the optimum point. For tables of
design values, you are to indicate, with arrows and comments, any variables at bounds, any binding constraints, the objective, etc. (You need to show that you understand the meaning of the output you
have included.):INS]
Added lines 1-11:
[INS:(:title Spring Design Optimization Problem:)
(:keywords nonlinear, optimization, engineering optimization, two-bar optimization, engineering design, interior point, active set, differential, algebraic, modeling language, university course:)
(:description Engineering design of a spring to stay within constraints and meet an optimal criteria. Optimization principles are used to design the system.:)
The specifications and modeling equations for a compression spring create a number of trade-offs that must be considered during design. We wish to determine the spring design that maximizes the force
of a spring at its preload height, ''h'_o_' '', of 1.0 inches. The spring is to operate an indefinite number of times through a deflection o, of 0.4 inches, which is an additional deflection from
ho. The stress at the solid height, hs, must be less than Sy to protect the spring from inadvertent damage.
[[Attach:spring.pdf | Spring Design Assignment]]
This introductory assignment is designed as a means of demonstrating the optimization capabilities of a number of software packages. Below are tutorials for solving this problem with a number of
software tools.:INS]
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.
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Overview Package Class Use Tree Deprecated Index Help
PREV CLASS NEXT CLASS FRAMES NO FRAMES
SUMMARY: NESTED | FIELD | CONSTR | METHOD DETAIL: FIELD | CONSTR | METHOD
Class EdmondsKarpMaximumFlow<V,E>
public final class EdmondsKarpMaximumFlow<V,E>
extends Object
A flow network is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge can not exceed the capacity of the edge (note, that all capacities must
be non-negative). A flow must satisfy the restriction that the amount of flow into a vertex equals the amount of flow out of it, except when it is a source, which "produces" flow, or sink, which
"consumes" flow.
This class computes maximum flow in a network using Edmonds-Karp algorithm. Be careful: for large networks this algorithm may consume significant amount of time (its upper-bound complexity is O(VE^
2), where V - amount of vertices, E - amount of edges in the network).
For more details see Andrew V. Goldberg's Combinatorial Optimization (Lecture Notes).
│ Constructor Summary │
│ EdmondsKarpMaximumFlow(DirectedGraph<V,E> network) │ │
│ Constructs MaximumFlow instance to work with a copy of network. │ │
│ EdmondsKarpMaximumFlow(DirectedGraph<V,E> network, double epsilon) │ │
│ Constructs MaximumFlow instance to work with a copy of network. │ │
│ Method Summary │
│ void │ calculateMaximumFlow(V source, V sink) │
│ │ Sets current source to source, current sink to sink, then calculates maximum flow from source to sink. │
│ V │ getCurrentSink() │
│ │ Returns current sink vertex, or null if there was no calculateMaximumFlow calls. │
│ V │ getCurrentSource() │
│ │ Returns current source vertex, or null if there was no calculateMaximumFlow calls. │
│ Map<E,Double> │ getMaximumFlow() │
│ │ Returns maximum flow, that was calculated during last calculateMaximumFlow call, or null, if there was no calculateMaximumFlow calls. │
│ Double │ getMaximumFlowValue() │
│ │ Returns maximum flow value, that was calculated during last calculateMaximumFlow call, or null, if there was no calculateMaximumFlow calls. │
│ Methods inherited from class java.lang.Object │
│ clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait │
public static final double DEFAULT_EPSILON
Default tolerance.
See Also:
public EdmondsKarpMaximumFlow(DirectedGraph<V,E> network)
Constructs MaximumFlow instance to work with a copy of network. Current source and sink are set to null. If network is weighted, then capacities are weights, otherwise all capacities are equal to
one. Doubles are compared using DEFAULT_EPSILON tolerance.
network - network, where maximum flow will be calculated
public EdmondsKarpMaximumFlow(DirectedGraph<V,E> network,
double epsilon)
Constructs MaximumFlow instance to work with a copy of network. Current source and sink are set to null. If network is weighted, then capacities are weights, otherwise all capacities are equal to
network - network, where maximum flow will be calculated
epsilon - tolerance for comparing doubles
public void calculateMaximumFlow(V source,
V sink)
Sets current source to source, current sink to sink, then calculates maximum flow from source to sink. Note, that source and sink must be vertices of the network passed to the constructor, and
they must be different.
source - source vertex
sink - sink vertex
public Double getMaximumFlowValue()
Returns maximum flow value, that was calculated during last calculateMaximumFlow call, or null, if there was no calculateMaximumFlow calls.
maximum flow value
public Map<E,Double> getMaximumFlow()
Returns maximum flow, that was calculated during last calculateMaximumFlow call, or null, if there was no calculateMaximumFlow calls.
read-only mapping from edges to doubles - flow values
public V getCurrentSource()
Returns current source vertex, or null if there was no calculateMaximumFlow calls.
current source
public V getCurrentSink()
Returns current sink vertex, or null if there was no calculateMaximumFlow calls.
current sink
Overview Package Class Use Tree Deprecated Index Help
PREV CLASS NEXT CLASS FRAMES NO FRAMES
SUMMARY: NESTED | FIELD | CONSTR | METHOD DETAIL: FIELD | CONSTR | METHOD
Copyright © 2013. All rights reserved.
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Zip Code Fractions [Archive] - MapPoint Forums
10-05-2006, 07:33 AM
Hi there,
I am wondering if there is a way to obtain ZIP code fractions that are captured by a radius. I know that you can export all the ZIP codes touched by a radius. But I need the fraction of the ZIP that
is actually inside the radius. Essentially, given a radius, I am looking for an output like:
ZIP Fraction in Radius
------------- -------------
12345 78%
23456 92%
Thanks a lot.
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Simplified and Abstracted Geometry for Forward Dynamics
From GICL Wiki
Geometric simplification has played an important role in the computer graphics field by allowing believable viewing of scenes too complex for timely computation. However, the approaches used in
computer graphics for geometric simplification have as their goal the realistic portrayel of the scene to a viewer, not the similarity between the simplified or abstracted system and the physical
ground truth. They are not then appropriate for simulations of robots in which the geometry plays a role.
Example Problem
Empirical Results
Milling Machine Simplification(PDF)
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Why I hate physics
There is a funny thing about the area of a sphere. If you look at the moon, you see a certain cross-sectional area. But the actual area of the moon is exactly four times what you are looking at. It's
a funny thing, that factor of four.
If you know the area of a sphere, the volume is "trivial", as the mathematicians like to say. From the formula for the volume of a cone, 1/3(base)x(height), you get the volume of the sphere. The
surface of the sphere is simply the "base" of a generalized "cone". Archimedes famously determined the volume of a sphere by a very ingenious and very different argument involving a cone inscribed in
a cylinder. I don't know if he knew about the ratio of 4:1, but that's another pathway to the volume formula.
Of course it works both ways: if you know the volume of a sphere, as Archimedes did, you can back it up to get the formula for area. Again, I don't know if that's what Archimedes did. It would be
nice to find out.
But without solving first for volume, is there an obvious way you can derive that special 4:1 ratio? It turns out you can indeed argue it from geometry without too much difficulty. It starts off
looking a lot like Archimedes' argument: first, you incribe a sphere in a cylinder. And then you take a disc-shaped slice perpendicular to the axis of the cylinder. This is just like Archimedes so
far. But then, to calculate volume, Archimedes inscribes a double-cone inside the cylinder, and it gets pretty intricate. For the surface area, you don't need the cone. You just compare the areas of
the cyldrical section and the spherical section, and it's easy to see they are equal. From this, you immediately get the surface of the sphere.
For a long time, I've had the idea that this ratio had some kind of cosmic connection with physics. I don't remember where I got this idea, but think it might have been a pure accident. See, there's
a funny thing about the apparent size of the sun. The apparent diameter of the sun, as viewed from the earth, is pretty close to one hundredth of a radian. That means if you hold your hand out 50
centimeters in front of your eye, and stick out your pinkie finger, the sun will span about half a centimeter as measured across your fingernail. One percent.
You have to be careful not to compare apples to oranges, or in this case radii to diameters. Your arm is a radius and the sun is a diameter, so you the true ratio is of course 1:200. Since area goes
as the square of the linear dimension, that means the sun occupies one part in forty thousand as compared to...the area of the sky? No, because we don't yet know the area of the dome. We're really
comparing flat circles, which would be the equivalent flat area of the sky. The beauty of the 4:1 business is that we can immediately see that the area of the dome is exactly twice the area of the
corresponding disk; so it follows that the ratio of the sun's area to the total area of the sky is 1:80,000 which is a pretty cool result.
Here's where it gets cosmically weird. The dome we see is exactly one half of the actual "sky", because there is just as much sky on the other half of the world. So the area of the sun to the total
sky becomes 1:160,000 or exactly half of the visible ratio.
The funny thing is that number 160,000 happens to be a perfect fourth power: specifically, it is 20^4. You know it's not that unusual to hit a perfect square. Perfect cubes are not that common.
Perfect fourth powers are pretty rare...obviously, there are only 19 of them smaller than 160,000. But so what?
Fourth powers are not only a bit rare among natural numbers, but even more rare in physical laws. Most laws of physics have squares in them, but there is a law of thermodyanamics that says a black
body radiates heat according to the fourth power of absolute temperature. So if you make something twice as hot, it radiates not twice as much heat, not four times as much heat (square law), but
sixteen times as much.
Now we're going to put it all together. The temperature of the earth is around 300 degrees Kelvin (absolute scale). The earth is therefore radiating heat into the vastness of space according to the
Laws of Thermodynamics. Unless that heat is being replaced by an equivalent source, the earth must therefore be cooling down. The fact that we aren't cooling down tells us how much heat we are
aborbing from outer space.
If outer space consisted of a gigantic dark sphere at a temperature of 300 degrees Kelvin, we would obviously be in thermal equilibrium. We would be radiating heat out to the giant sphere, and it
would be radiating heat back to us. Both bodies would be radiating heat at the same rate. But in fact there is no giant sphere out there, just the endless vacuum. So we are getting nothing back.
Except for this tiny patch of the sky occupied by the sun. Since it is only one part in 160,000 of the total sky, and it is obviously doing the whole job that our hypothetical giant sphere was
otherwise doing, it must be giving off power at a rate 160,000 times as great as the black body of 300 degrees. But in that case, we know how hot the sun is! Knowing that a black body radiates heat
according to the fourth power of absolute temperature, we take the fourth root of 160,000 and find that the sun is exactly 20 times as hot as the earth, or pretty close to 6000 degrees Kelvin.
And that's damn close to the actual temperature of the sun.
Have you ever had your kitchen dimmer switch turned down really low, and then the refrigerator cuts in and the lights go right out? The funny thing is that when the compressor turns off, the lights
don't even come back on again. You've got to turn the dimmer up quite a bit to bring them back.
It works best in older houses where the fridge plug is on the same circuit as the overhead light, but you can often see it quite clearly even when they are on different circuits. I was at a friend's
house the other day and I was showing them how this works, when something very unexpected happened. The fridge cut in and the lights got brighter. I thought we were imagining it but we watched for
several cycles, and it kept happening. Then we tried turning on the microwave, and the lights got dimmer. But the fridge definitely made it get brighter.
I think I figured out what is going on, and the answer is kind of interesting. It's really a two-part question: part 1 is just why is a dimmer switch turned down low so very sensitive to small
changes in line voltage; and part 2, why the anomalous result of the lights getting brighter?
I'm including a simplified schematic of the dimmer circuit for your edification: I'll give you my answer when we return.
(I found the schematic on this very impressive website.)
This is the final installment of my story of how I got fired from UCN back in 2006. My inexplicable refusal to follow the curriculum was the reason they gave for letting me go. But there's an ironic
twist to that story....
That just about covers it except for one small item you might still be wondering about: the mysterious eighth paragraph. I have said already that in seven paragraphs out of eight I am accused of not
following the curriculum. What about the eighth paragraph?
In this paragraph Henning describes my very first day of work, going over how I was shown filing cabinets full of material including worksheets, old tests, blueprints, etc. It happens to be the
longest paragraph of the entire letter, and yet it doesn’t include any particular complaints against me. So what is it doing in my letter of rejection?
The eighth paragraph purports to show that the College provided me with all the resources I needed to do my job: which is to say, they basically shoved me in front of a filing cabinet and said “Knock
yourself out”. But oddly enough, in the long itenerary of resource materials listed by Henning, there is one item conspicuously absent: a copy of the apprenticeship curriculum! I became aware of this
deficiency in November and immediately wrote my supervisor requesting that one be provided. Selwin ignored my request; or to be more precise, he first said that he would get it for me, later said
there were complications (what happened is that Murray actually freaked out when he heard I was asking after the curriculum!), and finally never followed up at all, which was typical behavior on his
part. (Everyone who has ever worked with Selwin knows this is true.) The ultimate irony is that the very curriculum which I am accused of not following is a curriculum which the College wouldn’t give
For the last two weeks I have been re-telling the story about how I was fired as an instructor from the University College of the North. Today's installment is about an incident that did me untold
damage in terms of my reputation, but which ultimately was not even mentioned in the letter which listed the reasons for my dismissal. The reason it didn't make the list is because the College knew
it was a malicious slander from the get-go. The report by independent consultant Joyce Oddidison was subsequently destroyed, but the damage had already been done. Here then is the continuing story...
I suppose in the end it’s all very well for me to write a self-serving letter which makes me out to be the hero and everyone else to be the villains. But what does it really prove? Ultimately, you
can say that I was probably fired because I was a bad teacher. The point is, that’s not what the College chose to say. They said that I wasn’t following the curriculum, and that I was repeatedly
warned and given a chance to comply. That is a lie and a slander which Katherine McNaughton, Selwin Peter, and Murray Oman conspired to establish and maintain, and to which Denise Henning has signed
her name. The truth of my claim can be clearly seen in the patently absurd examples which the College has cited as so-called evidence of my inappropriate teaching.
I wasn’t fired because of the smokestack or the water heater or even the slope question. It’s almost as though I was fired simply because something about me inspires a profound, inexpressible
loathing among certain people; that in an institution which claims to value “diversity” as its highest ideal, I was just too different. And the best excuse they can come up with is the smokestack
issue. In fact, over the course of the year there was a never-ending stream of malicious gossip about me, including at least two claims of sexual harrassment! All of these were investigated by
management in a desparate attempt to throw dirt on my reputation and none of them were found to have any validity. That’s why they’re not brought up in my letter of rejection. But at least one of
these incidents deserves further mention, because everyone on staff at the College heard about it at the time and I would like to set the record straight.
Elvis Balfour was a student of mine who lost his temper in class one day for no sensible reason, jumped over the desks and began shaking me by the throat. I remained calm and put up no physical
resistance. When it was over, I told him to stop acting like a baby and get back in his seat. Subsequently through no action on my part, by the end of the day Elvis had been expelled from the course
by Murray Oman.
What happened next is confusing: Murray has been an instructor for many years and has routinely expelled students in the past without controversy (including the very same Elvis Balfour on a previous
occasion). For unknown reasons, this time the College chose to disregard its own zero-tolerance policy on violence, refused to ratify the expulsion and instead relocated Elvis to The Pas where he was
allowed to finish the course. An outside consultant by the name of Joyce Oddidison was hired by the College to investigate the affair, and she ultimately produced a report that blamed me for
provoking the incident.
Oddly, Ms. Henning chooses not to mention the Elvis Balfour incident in her letter. Why, when she is clearly trying to make the strongest possible case for my dismissal, does she not invoke this
glaring example of misconduct? The answer is simple: there never was any misconduct on my part and Henning knows it. The report by Oddidison is a pack of lies, which she was hired by Selwin Peter to
produce in order to justify his own idiotic decision to keep Balfour in the course.
Most notably, the argument Oddidison uses to condemn my behavior is based on a shameless doctoring of my own written submission. In particular, she places my words, “Stop acting like a big baby...”
....BEFORE the violent outburst rather than after, turning it into a provocation. Perhaps she thought I was lying. If so, she should have said so, and introduced testimony from other witnesses to
back up her conclusion. That’s not what she chose to do: instead, she falsely manipulated my own words to make me guilty by explicit admission!
The College was an active participant in this slander and they know it to be false. The proof is simple: if they believed their own report, they would have included its conclusion in the letter which
gives the reasons for my dismissal. Instead, they say I was fired for measuring the height of a smokestack. What jackasses they are.
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Pseudo Random Distribution | News in the DOTA 2 scene - DotaCinema
Pseudo Random Distribution
For those confused as to what Psuedo Random Distribution means, here is some information taken from an old playDota article. As you know, this system was present in the Warcraft III Engine and has
been brought to Dota 2 via today's patch.
Pseudo Random Distribution
A Pseudo Random Distribution refers to the Warcraft III engine's dynamic probability calculations for certain percentage-based attack modifiers. Rather than using a static percentage, the probability
is first set to a small initial value, then gradually increased with each consecutive attack for which the modifier does not occur. The probability then drops back to the initial value when the
attack modifier does apply. Not only does this system make long strings of successful modifiers unlikely, but it also makes going an entire game without an attack modifier occurring impossible, as
eventually the dynamic probability exceeds 100% and "forces" a modifier on the next attack. The distribution of attack modifiers is therefore not truly random, hence the term Pseudo Random
Distribution. In general, all abilities that are rounded to the nearest 5% in the Warcraft III engine follow this probability distribution.
Pseudo Random Abilities
The list of all currently known base abilities that follow this system of balanced randomness is shown below. Triggered abilities notably do not fall in this list:
• Critical Strike (ex. Phantom Assassin’s Coup de Grace, Crystalys)
• Bash (ex. Faceless Void’s Time Lock. As of its remake, Cranium Basher does notfollow this distribution.)
• Orb of Slow (ex. Maim, Maelstrom)
• Hardened Skin (ex. Stout Shield, Vanguard)
Probability Mechanics
Critical Strike Example:
Consider the case of a 20% chance to Critical Strike. If the game were truly random, then for each attack there would be a 20% chance for the unit to land a Critical Strike, regardless of how many
Critical Strikes had been performed in the past. A simple implementation would therefore be to select a random integer between 1 and 100, and if the integer was less than or equal to 20, then the
engine would cause that attack to be a Critical (which is the usual construct used by triggered percentage-based abilities). While this implementation would, in the long run, average out to 20% of
attacks being Criticals, there is nothing preventing an infinite series of Critical Strikes or, conversely, an absence of Critical Strikes for the entire game, although the chance is rather
insignificant. To prevent the ensuing complaints and balance this random game mechanic, the Warcraft III developers implemented a Pseudo Random Probability Distribution.
Instead of there being a 20% chance to Critical Strike with every attack, the first attack made actually has a 5.57% chance to Critical Strike. If that is not a Critical Strike, then the second
attack has a 11.14% chance to Critical. If that is also not a Critical Strike, then the third has a 16.71% chance to Critical, and so on, adding 5.57% for each consecutive non-Critical. When a
Critical Strike does occur, however, the chance for the next attack resets to 5.57%. In the long run, the number of Critical Strikes divided by the total number of attacks somewhat approximates the
stated 20%, but now it is extremely difficult to have a series of Critical Strikes, and also impossible to go more than 17 attacks without a Critical, because the percentage for the 18th attack
(assuming 17 previous consecutive non-Critical attacks) is 100.26%. In effect, the game causes the number of attacks between Criticals to be skewed towards 1 / 20% = 5, with a maximum limit of 17.
For more detailed information, be sure to check out the whole article here.
Thanks for the explanation :)
Nice info, BTW AWESOME DESIGN OF THE SITE !! Well done.
So it only got added recently to dota2? explains all the weird crit chances i've been getting. Good to see they've finally added it in i suppose.
Does that mean Phantom Assassin's Coupe De Grace is screwed
Is Phantom Lancer illusion affected by this?
@ZeeKz Sure, it's an old article but it explains everything about Pseudo Random Distribution.
inb4 no one plays PA, CK etc and buys Daedalus
Finally we have back the pseudo-random. I'm feeling more "at home" (aka, in the old golden Dota) now.
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math (calculate speed)
Number of results: 256,352
An airplane is heading due south at a speed of 590km/h. If a wind begins blowing from the southwest at a speed of 65.0km/h (average). Calculate magnitude of the plane's velocity, relative to the
ground. Calculate direction of the plane's velocity, relative to the ground. ...
Saturday, September 12, 2009 at 1:15pm by Anne
Olympic champion Justin gaitlin set a new world record for 100.0 m sprint at the last lympics by clocking 9.86 s for the race.Calculate a)His average speed for the race b)Assuming that he accelerated
uniformly for the first 4.86 s after taking off the blocks to reach a maximum...
Monday, September 24, 2007 at 7:05pm by JohnX
Olympic champion Justin gaitlin set a new world record for 100.0 m sprint at the last lympics by clocking 9.86 s for the race.Calculate a)His average speed for the race b)Assuming that he accelerated
uniformly for the first 4.86 s after taking off the blocks to reach a maximum...
Monday, September 24, 2007 at 8:05pm by Sam
A tennis player moves back and forth along the baseline while waiting for her opponent to serve, producing the position-versus-time graph shown in the figure. (The vertical axis is marked in
increments of 1 m and the horizontal axis is marked in increments of 4 s.) (a) Without...
Wednesday, October 3, 2012 at 2:09pm by Anonymous
1. A student is dragging a 5 kg box along the floor. He pulls on the rope with a 25 Newton force. The rope is 37 degrees above horizontal. As the box slides 10 meters, it goes from a speed of 3 m/s
to a speed of 7 m/s. a.) calculate the normal force on the box b.) calculate ...
Wednesday, January 12, 2011 at 9:03pm by Energy
The train slows down uniformly from a speed of 50m/s to a speed of 10m/s in a time of 20sec,During the next 30sec,it accelerates uniformly to a speed of u metres/sec.calculate the retardation from t=
0 to t=20?
Friday, May 3, 2013 at 10:21am by Meggy lep
A motorist travelling at a steady speed of 120km/h covers a section of the highway in 30min. After a speed limit is imposed he finds that when travelling at the maximum speed allowed he takes 20 min.
longer than before to cover the same section. Calculate the speed limit.
Friday, November 30, 2012 at 12:54am by Lindsay
You are driving south on a highway at 24.0 m/s (approximately 54 mi/h) in a snowstorm. When you last stopped, you noticed that the snow was coming down vertically, but it is passing the windows of
the moving car at an angle of 26.6° to the horizontal. Calculate the speed of ...
Sunday, February 5, 2012 at 11:07pm by Jeremy
The distance from TDC (top dead centre) for the pin is given by x when the crank is at angle . Hence : i) Give an equation for x in terms of the other variables (r, L and ). ii) If the crank rod is
rotating at a constant value of ω calculate the speed of ...
Tuesday, May 17, 2011 at 3:16pm by Karl Haxell
A sky diver of mass 80.0 ks jumps from a slow-moving aircraft and reaches a terminal speed of 50.0 m/s. a) What is the acceleration of the sky diver when her speed is 30.0 m/s? What is the drag force
on the diver when her speed is b) 50.0 m/s? c) 30.0 m/s? If you assume drag ...
Thursday, March 1, 2007 at 5:36am by Jean
The speed time graph shows the motion of a car over a period of 40 seconds. a. describe the motion of the car between: 1.) O and A - The car is traveling at constant speed from 0 to 20 m/s. 2.) A and
B - The car is at rest for 20 seconds. 3.) B and C - The car is decelerating ...
Tuesday, February 5, 2013 at 7:45am by RR
The speed time graph shows the motion of a car over a period of 40 seconds. a. describe the motion of the car between: 1.) O and A - The car is traveling at constant speed from 0 to 20 m/s. 2.) A and
B - The car is at rest for 20 seconds. 3.) B and C - The car is decelerating ...
Tuesday, February 5, 2013 at 8:21am by RR
followup question
-- For this question, I know how to find the initial speed. If it asks, "What is the speed of the car just before it lands safely on the other side?" how do i calculate that because it's not the same
as final velocity right? --------------------------------- you got u, the ...
Friday, February 4, 2011 at 4:09pm by Damon
You can calculate frequency from Frequency = (wave speed)/(wavelength) You have provded a time, not a wave speed. What does that time mean? If it is the time for a wave to travel a known distance D,
calculate that wave speed using V = D/T and use it in the formula above.
Sunday, March 9, 2008 at 10:53pm by drwls
A car travels up a hill at a constant speed of 45 km/h and returns down the hill at a constant speed of 70 km/h. Calculate the average speed for the round trip.
Saturday, September 24, 2011 at 2:54pm by Jonathan
A car takes 3 hours to travel 3km at a constant speed on an open road. Calculate the speed of the car
Sunday, September 11, 2011 at 1:46pm by Anonymous
An airplane is heading due south at a speed of 590 km/h. If a wind begins blowing from the southwest at a speed of 65.0 km/h (average). Calculate magnitude of the plane's velocity, relative to the
ground. Calculate direction of the plane's velocity, relative to the ground. ...
Tuesday, September 15, 2009 at 4:18am by physics
A disk rotates about its central axis starting from rest and accelerates with constant angular acceleration. At one time it is rotating at 7 rev/s. 55 revolutions later, its angular speed is 21 rev/
s. (a) Calculate the angular acceleration. (b) Calculate the time required to ...
Sunday, March 11, 2007 at 10:46pm by winterWX
A train travel some distance with a speed of 30km/hr and return with a speed of 45km/hr. Calculate average speed.
Thursday, June 20, 2013 at 3:05am by Anonymous
A train travel some distance with a speed of 30km/hr and return with a speed of 45km/hr. Calculate average speed.
Thursday, June 20, 2013 at 3:05am by Anonymous
A train travel some distance with a speed of 30km/hr and return with a speed of 45km/hr. Calculate average speed.
Thursday, June 20, 2013 at 3:05am by Anonymous
A train travel some distance with a speed of 30km/hr and return with a speed of 45km/hr. Calculate average speed.
Thursday, June 20, 2013 at 3:05am by Anonymous
Science - Waves - Sci. Notation
I know that the formula to calculate the speed of light is c=fλ. F is the frequency, c is the speed, and λ is the wavelength. So how do I calculate the wavelength of a purple light with a frequency
of 750 * 10^12 HZ. Frequency is measured in HZ. The speed is measured...
Thursday, January 31, 2013 at 7:12pm by Jamia
A car moving with an initial speed v collides with a second stationary car that is 50.7 percent as massive. After the collision the first car moves in the same direction as before with a speed that
is 33.2 percent of the original speed. Calculate the final speed of the second ...
Wednesday, March 26, 2014 at 7:48pm by Erin
if co-efficient of friction between tyre and road is 0.5, what is smallest radious at which car turn on a horizontal road when its speed is 30 km/hr? 2.Q calculate angular speed of how hand of a
clock? 3.Q A what angle must cut rack with a bend of 250m banked for safe running ...
Monday, September 27, 2010 at 12:25pm by vishal
A 5.40 Kg package slides 1.47 meters down a long ramp that is inclined at 11.6 degress below the horizontal. The coefficient of kinetic friction between the package and the ramp is = 0.313. Calculate
the work done on the package by friction. Calculate the work done on the ...
Monday, December 3, 2012 at 5:44pm by clariza gonzalez
A vehicle with a mass of 2 tons travels on a horizontal road at a speed of 126km/h.The driver notices a donkey at the side of the road and decides to reduce speed. The brakes are applied and the
vehicle slows down to 72km/h after 5 seconds. Calculate the distance travelled ...
Wednesday, March 6, 2013 at 1:17pm by leslie
The distance from TDC (top dead centre) for the pin is given by x when the crank is at angle . Hence : i) Give an equation for x in terms of the other variables (r, L and ). ii) If the crank rod is
rotating at a constant value of ω calculate the speed of ...
Tuesday, May 17, 2011 at 7:18am by Karl Haxell
The stopping distance d of a car in feet is related to the speed in mph by the equation d(s)=.02s^2+1.1s Calculate the speed when your stopping distance is: 1.) 51ft 2.) 315.12ft
Saturday, February 25, 2012 at 9:24pm by Devin
A car moving with an initial speed v collides with a second stationary car that is 46.0 percent as massive. After the collision the first car moves in the same direction as before with a speed that
is 37.7 percent of the original speed. Calculate the final speed of the second...
Monday, March 24, 2014 at 9:45am by Sami
physics - PLEASE HELP
A softball of mass 0.220 kg that is moving with a speed of 6.5 m/s (in the positive direction) collides head-on and elastically with another ball initially at rest. Afterward it is found that the
incoming ball has bounced backward with a speed of 4.8 m/s. (a) Calculate the ...
Thursday, December 10, 2009 at 6:04pm by Anonymous
In order to convert a tough split in bowling, it is necessary to strike the pin a glancing blow as shown in the figure. Assume that the bowling ball, initially traveling at 15.0 m/s, has four times
the mass of a pin and that the pin goes off at 75° from the original direction ...
Thursday, October 20, 2011 at 11:31pm by kago
A little red wagon with mass 7.0kg moves in a straight line on a frictionless horizontal surface. It has an initial speed of4.00 m/s and then is pushed 3.0m in the direction of the initial velocity
by a force with a magnitude of 10.0 N a.) Use the work-energy theorem to ...
Monday, October 15, 2012 at 1:34pm by Frankie
A little red wagon with mass 7.0kg moves in a straight line on a frictionless horizontal surface. It has an initial speed of4.00 m/s and then is pushed 3.0m in the direction of the initial velocity
by a force with a magnitude of 10.0 N a.) Use the work-energy theorem to ...
Monday, October 15, 2012 at 1:31pm by Frankie
i found the the speed for (a) of when it was launch.. i calculate it from the formula .. v(t)= V+g(t) V=15+9.8(2)=34.6 but how do i calculate it for (b) i tried plugging in 4.2 but i didnt work
Monday, February 7, 2011 at 11:37am by shaknocka
Assuming that the speed of sound at a certain altitude is 330m/s calculate the speed of an airplane that is traveling at match 4.2 ?
Wednesday, June 6, 2012 at 1:52pm by Melissa
A softball of mass 0.200 kg that is moving with a speed of 8.2 m/s collides head-on and elastically with another ball initially at rest. Afterward the incoming softball bounces backward with a speed
of 3.9 m/s. 1. Calculate the velocity of the target ball after the collision. ...
Thursday, November 12, 2009 at 3:37pm by Anonymous
An airplane heads due south with an air speed of 480km/h. Measurements made from the ground indicate that the plane's ground speed is 528km/h at 15 degrees east of south. Calculate the wind speed.
Monday, May 2, 2011 at 4:07pm by Anonymous
Assuming that the speed of sound at a certain altitude is 330m/s calculate the speed of an airplane that is travelling at (A) mach 0.70
Monday, June 4, 2012 at 1:09pm by Melissa
A softball of mass 0.220 kg that is moving with a speed of 4.0 m/s (in the positive direction) collides head-on and elastically with another ball initially at rest. Afterward the incoming softball
bounces backward with a speed of 1.8 m/s. a) Calculate the velocity of the ...
Monday, May 30, 2011 at 12:30pm by Nick
A softball of mass 0.220 kg that is moving with a speed of 4.0 m/s (in the positive direction) collides head-on and elastically with another ball initially at rest. Afterward the incoming softball
bounces backward with a speed of 1.8 m/s. a) Calculate the velocity of the ...
Tuesday, May 31, 2011 at 7:50am by Nick
Distance = Speed/time = 40/(2/3) = ? ?D = Speed/(1/2) Calculate for distance, then speed.
Wednesday, December 5, 2012 at 4:14am by PsyDAG
I know how to get the speed if it accelerated the whole time. How do I calculate the cruising speed? Starting from rest, a car travels 1,350 meters in 1 min. It accelerated at 1 m/s^2 until it
reached its cruising speed. Then it drove the remaining distance at constant ...
Friday, September 7, 2007 at 12:29am by Colin
A cyclist enters a curve of 30 m radius at a speed of 12 m s-1. He applies the brakes and decreases his speed at a constant rate of 0.5 m s-2. Calculate the cyclist’s centripetal (radial) [3 marks]
and tangential [15 marks] accelerations when he is travelling at a speed of 10 ...
Tuesday, February 28, 2012 at 10:27am by alex
The combined mass of a race car and its driver is 600. kilograms. Traveling at a constant speed, the car completes one lap around a circular track of radius 160 meters in 36 seconds. A) Based on the
given information, calculate the speed of the car. B) Calculate the magnitude ...
Sunday, January 10, 2010 at 1:38pm by Nicole
A stone is thrown vertically upward. On its way up it passes point A with speed v, and point B, 7.85 m higher than A, with speed v/2. Calculate (a) the speed v and (b) the maximum height reached by
the stone above point B.
Friday, September 10, 2010 at 11:15am by Antonio
a plane flew from red deer to winnipeg, flying distance of 1260 km. on the return journey, due to a strong head wind, the plane travelled 1200 km in the same time it took to complete the outward
journey. on the outward journey, the plane be able to maintain an average speed ...
Sunday, January 8, 2012 at 11:19pm by Shreya
Calculate his speed going back to the spacecraft after throwing the wrench, using conservation of momentum. Then divide 35.5 m by that speed.
Sunday, December 2, 2012 at 2:19am by drwls
A softball of mass 0.220 kg that is moving with a speed of 5.5 m/s (in the positive direction) collides head-on and elastically with another ball initially at rest. Afterward it is found that the
incoming ball has bounced backward with a speed of 3.9 m/s. (a) Calculate the ...
Sunday, January 10, 2010 at 5:00pm by Anonymous
Physics 4a
A stone is thrown vertically upward. On its way up it passes point A with speed v, and point B, 7.85 m higher than A, with speed v. Calculate (a) the speed v and (b) the maximum height reached by the
stone above point B.
Friday, September 10, 2010 at 10:43am by Antonio
Physics- HELP
How to find average speed with a given initial speed, theta, and delta x? I know average speed= total distance/total time, but I am given an initial speed, an angle and delta x. Mark throws a ball to
Daniel with an initial speed of 20 m/s at an angle of 45 degrees. If they are...
Wednesday, December 1, 2010 at 7:26pm by Amm
math (calculate speed)
m*g*h = 1/2*m*v^2 where m is the mass; g is the acceleration due to gravity; v is the speed. At the top of his dive, the swimmers energy is entirely potential energy = m*g*h; When he enters the pool,
the potential energy is converted to kinetic energy: 1/2*m*v^2 v = = (2*g*h)^0.5
Tuesday, January 8, 2013 at 5:13pm by Jennifer
In order to convert a tough split in bowling, it is necessary to strike the pin a glancing blow as shown in Fig. 9-47. The bowling ball, initially traveling at 13.0 m/s, has four times the mass of a
pin and the pin flies off at 80° from the original direction of the ball. ...
Wednesday, March 5, 2014 at 3:54pm by Anonymous
a motorist goes 78km in 40minutes calculate it speed
Monday, July 4, 2011 at 3:23pm by Anonymous
A car travels up a hill at a constant speed of 32 km/h and returns down the hill at a constant speed of 61 km/h. Calculate the average speed and the answer is not 46.5
Monday, September 19, 2011 at 4:10pm by Ann2793
A car travels up a hill at a constant speed of 45 km/h and returns down the hill at a constant speed of 70 km/h. Calculate the average speed for the round trip.
Thursday, September 22, 2011 at 9:16pm by Jonoathan
A car travels up a hill at a constant speed of 45 km/h and returns down the hill at a constant speed of 70 km/h. Calculate the average speed for the round trip.
Thursday, September 22, 2011 at 9:16pm by Jonoathan
A car travels up a hill at a constant speed of 45 km/h and returns down the hill at a constant speed of 70 km/h. Calculate the average speed for the round trip.
Thursday, September 22, 2011 at 9:16pm by Jonoathan
science waves
You have to know the speed that the waves travel. For sound and light, this speed is well known. It is also easy to measure. Once you know the wave speed V, you can use the relationship (wavelength)x
(frequency) = V With this, you can calculate the frequency from the wavelength...
Sunday, May 18, 2008 at 10:24pm by drwls
a) First you calculate his speed of impact (the speed when he hits the ground). For this you can use the formula: v² = v0² + 2.a.x where: -v is your final speed -v0 is your initial speed (0 m/s in
this case) -a is your acceleration (the gravitational acceleration in this case...
Thursday, December 11, 2008 at 8:04pm by Christiaan
Force F causes displacement d of a 20-Kg object starting from rest along a frictionless path. F= 10 N i + 50 N j + 15 N k and d= 10 m i + 60 m k A) Calculate the work done by the force. B) Calculate
the angle between the vectors F and d. C) Calculate the final speed of the ...
Wednesday, April 22, 2009 at 10:40pm by Sandhya
Car A and B are passing each other in different lanes. Car A has an initial speed of 15m/s and is gaining speed at 1.5m/s^2. Car B has an initial speed of 20m/s and is gaining speed at 2.0m/s^2. Find
when and where the cars will pass one another and their speeds as they are ...
Wednesday, February 20, 2008 at 8:02am by Anonymous
how do you calculate the distance and speed of a motorcycle that is going to jump 4 cars
Wednesday, August 24, 2011 at 8:11pm by linda
A space habitat for a long space voyage consists of two cabins each connected by a cable to a central hub as shown in the figure below. The cabins are set spinning around the hub axis, which is
connected to the rest of the spacecraft to generate artificial gravity. (a) What ...
Monday, November 8, 2010 at 3:24pm by Anonymous
Calculate the magnitude of the linear momentum for the following cases. (a) a proton with mass 1.67 10-27 kg, moving with a speed of 4.00 106 m/s (b) a 16.5 g bullet moving with a speed of 270 m/s
(c) a 70.5 kg sprinter running with a speed of 12.5 m/s (d) the Earth (mass = 5....
Monday, July 7, 2008 at 9:31am by Elisa
A train travel 20km at a uniform speed 60km/1hr and the next 20km at a uniform speed of 80km/1hr calculate its average speed
Tuesday, April 1, 2014 at 8:53am by prabhjot
Chemistry Help
The molecules of a certain gas sample at 367 K have a root-mean-square (rms) speed of 261 m/s. Calculate the most probable speed and the mass of a molecule.
Wednesday, December 5, 2012 at 7:55pm by Anonymous
the speed of light is 3 * 10^ 8m/s. Jupiter is 778 million km from the sun. Calculate the number of minutes it takes for sunlight to reach Jupiter ? A star is 300 lights away from Earth. If the speed
of light is 3 * 10^5 km/s. calculate the disatance from the star to Earth. ...
Monday, November 26, 2007 at 3:03pm by YOYO
applied math - more info
which speed? rotation speed, orbital speed, speed of a falling body?
Tuesday, March 13, 2012 at 3:55am by Anonymous
A car travels up a hill at a constant speed of 33 km/h and returns down the hill at a constant speed of 60 km/h. Calculate the average speed (in km/h) for the round trip.
Sunday, January 16, 2011 at 2:27pm by Ethan
Car travels up hill at a constant speed of 50 km/h and returns down the hill at a constant speed of 65 km/h. Calculate the avg speed for the round trip in km/h.
Thursday, September 22, 2011 at 10:13pm by JP
Calculate upstream rate = d/t. Calculate downstream rate = d/t. subtract rates to obtain speed of the river.
Thursday, October 4, 2007 at 12:10pm by DrBob222
For the first 1 1/2 hours of a 91km journey the average speed was 30km/h.If the average speed for the reminder of the journey was 23km/h, calculate the average speed for the entire journey.
Monday, July 20, 2009 at 9:17pm by Lydia
A car of mass 1200kg starts from rest, accelerates uniformly to a speed of 4.0 meters per second in 2.0 seconds and continues moving at this constant speed in a horizontal straight line for an
additional 10 seconds. the brakes are then applied and the car is brought to rest in...
Tuesday, December 4, 2012 at 5:00am by Ken
The launching speed of a certain projectile is 3.0 times the speed it has at its maximum height. Calculate the elevation angle at launching
Friday, February 8, 2013 at 8:43am by julia
The launching speed of a certain projectile is 9.8 times the speed it has at its maximum height. Calculate the elevation angle at launching.
Thursday, November 14, 2013 at 8:45pm by Vinny
The launching speed of a certain projectile is 9.8 times the speed it has at its maximum height. Calculate the elevation angle at launching.
Thursday, November 14, 2013 at 8:45pm by Vinny
The launching speed of a certain projectile is 9.8 times the speed it has at its maximum height. Calculate the elevation angle at launching.
Thursday, November 14, 2013 at 8:54pm by Vinny
the mean volume folume flow of blood in the arterial system is X m^3 s^-1. the area of the aorata is Y m^2. calculate the speed of flow.
Saturday, March 28, 2009 at 10:10pm by lisa
if the speed of light in air is3.0 by 10 to the 8 power per second calculate speed of light in a glass box
Wednesday, October 20, 2010 at 5:25pm by stela
A spaceship has a speed of 500000 meters per second. Calculate, in scientific notation, the speed in km per hour.
Monday, March 26, 2012 at 8:58am by Cyco
Physics please help!!
The launching speed of a certain projectile is 9.0 times the speed it has at its maximum height. Calculate the elevation angle at launching.
Sunday, February 3, 2013 at 12:42pm by Julia
college math
v = boat speed speed downstream = (v + 6) speed upstream = (v-6) time = distance/speed 160/(v+6) = 96/(v-6) 10/(v+6) = 6/(v-6) 5/(v+6) = 3/(v-6) 3(v+6) = 5(v-6) 3 v + 18 = 5 v - 30 2 v = 48 v = 24
Friday, March 9, 2012 at 4:57pm by Damon
Using Heisenberg's Principle, calculate for the following positions. a) a 1.54-mg mosquito moving at a speed of 1.82 m/s if the speed is known to within ± 0.01 m/s b) a proton moving at a speed of
(4.67 ± 0.02) 104 m/s When I solved (a) I got 1.88e-30 and (b) I got 3.41e-5 but...
Wednesday, March 17, 2010 at 7:48pm by Anonymous
A 245 g particle is released from rest at point A inside a smooth hemispherical bowl of radius 35.0 cm, as shown in Figure 5-21. (a) Calculate the gravitational potential energy at A relative to B. 1
. (b) Calculate the particle's kinetic energy at B. 2 (c) Calculate the ...
Thursday, October 20, 2011 at 2:15pm by Brooks
(a) a person in a wheelchair approaches a 0.50-m-high ramp with a speed of 3.5 m/sec. calculate the final speed of the wheelchair at the top of the ramp if the person coasts up and friction is
negligible. (b) what initial speed would be necessary to be 1.0 m.sec? i need this ...
Friday, May 4, 2012 at 11:28pm by sara
a stone is dtropped from the top of 40m heigh calculate its speed after 2 seconds .Also find the speed ith the stone strikes the ground.
Thursday, September 15, 2011 at 3:43am by Anonymous
The rumble is a sound, and sound travels at a speed of 343 m/s in air. The answer that was given is more than twice the speed of sound. Since we already know the speed of sound, we can only calculate
the time required to travel 10km. VT = 10,000 m 343T = 10,000
Tuesday, May 22, 2012 at 12:50am by Henry
A soccer ball iskicked with an intial horizontal speed of 6 m/s and initial vertical speed of 3.5 m/s. Assume that the projection and landing height are the same, and neglect air resistance.
Calculate the following a) the ball's projection speed and angle b)the ball's ...
Thursday, November 7, 2013 at 9:46pm by Crystal
A stone thrown horizontally from a height of 7.8 m hits the ground at a distance of 12.0 m. Calculate the speed of the ball as it hits the ground. I was able to find the initial speed of the ball,
which is 9.52 m/s. However, I'm unsure about whether I need to use this number ...
Saturday, November 3, 2007 at 4:24pm by Lindsay
A bicyclist travels in a circle of radius 35.0 m at a constant speed of 7.00 m/s. The bicycle-rider mass is 72.0 kg. Calculate the magnitude of the force of friction on the bicycle from the road.
Calculate the magnitude of the net force on the bicycle from the road.
Monday, February 12, 2007 at 3:40am by COFFEE
The launching speed of a certain projectile is 6.4 times the speed it has at its maximum height. Calculate the elevation angle at launching. I have noo idea how to start this one. Please help!
Wednesday, November 7, 2007 at 9:09pm by Lindsay
calculate the range of the projectile using the initial speed and angle. use 9.8m/s. Initial speed 18m/s and the is 75 degrees. I came up with 8.29 is that right
Sunday, January 23, 2011 at 12:29pm by Anonymous
physics - SHM
Calculate the speed of the pulse from the following: y(x,t) = 2/((x - 3t)^2 + 1) Well the speed of the pulse is given by: y(x,t) = f (x - vt) for a pulse traveling to the right and y(x,t) = f (x +
vt) for a pulse traveling to the left but in this case the function is (x - 3t)^...
Saturday, April 14, 2007 at 5:47pm by COFFEE
Which formula can be used to calculate the Speed of sound which is about 540 miles per hour in minutes
Sunday, October 16, 2011 at 6:07pm by John vick
A small mass attached to a spring oscillates with simple harmonic motion with amplitude of 70mm,taking 13 seconds to make 40 complete oscillations.Calculate the following showing steps; a.Its angular
ferquency b.Its maximum speed c.Its maximum acceleration d.Its speed when ...
Thursday, April 5, 2012 at 3:33am by Nkamo
the speed of a cyclist reduces uniformly from 2.5 m/s to 1.0 m/s in 12 s. (a) calculate the deceleration of the cyclist. (b) calculate the distance travelled by the cyclist in this time
Sunday, October 31, 2010 at 7:55pm by valerie
I've tried this problem again and again using conservation of momentum and energy but I am not having any luck: A ball of mass 0.300kg that is moving with a speed of 8.0m/s collides head-on and
elastically with another ball initially at rest. Immediately after the collision, ...
Friday, October 11, 2013 at 12:06am by Alison
Clocking Tectonic Plates
Explain how you could calculate the average speed of the Pacific Plate over the past five million years, then calculate the value? I tried searching the internet for this, I came up with nothing, and
I just dont get this question.
Thursday, January 13, 2011 at 6:50pm by Anonymous
A woman throws a cricket ball upwards into the air. The ball leaves the womans hand at 1.8m above the ground at speed of 9.7. It rises and then falls back to the ground. Considering the energies
involved calculate the speed at the point it hits the ground. accerlation due to ...
Wednesday, December 15, 2010 at 8:17am by joleen
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infinity-Chern-Simons theory -- examples
This is a sub-entry of infinity-Chern-Simons theory. See there for context.
1d Chern-Simons functionals
2d Chern-Simons functionals
Poisson $\sigma$-model
3d Chern-Simons functionals
Ordinary Chern-Simons theory
Let $\mathfrak{g}$ be a semisimple Lie algebra. For the following computations, choose a basis $\{t^a\}$ of $\mathfrak{g}^*$ and let $\{r^a\}$ denote the corresponding degree-shifted basis of $\
Notice that in terms of this the differential of the Chevalley-Eilenberg algebra is
$d_{CE(\mathfrak{g})} : t^a \mapsto -\frac{1}{2}C^a{}_{b c}t^b \wedge t^c$
and that of the Weil algebra
$d_{W(\mathfrak{g})} : t^a \mapsto -\frac{1}{2}C^a{}_{b c}t^b \wedge t^c + r^a$
$d_{W(\mathfrak{g})} : r^a \mapsto -C^a{}_{b c} t^b \wedge r^c \,.$
Let $P_{a b} r^a \wedge r^b \in W(\mathfrak{g})$ be the Killing form invariant polynomial. This being invariant
$d_{W(\mathfrak{g})} P_{a b} r^a \wedge r^b = 2 P_{a b} C^{a}{}_{d e} t^d \wedge r^e \wedge r^b = 0$
is equivalent to the fact that the coefficients
$C_{a b c} := P_{a a'}C^{a'}{}_{b c}$
are skew-symmetric in $a$ and $b$, and therefore skew in all three indices.
A Chern-Simons element for the Killing form invariant polynomial $\langle -, - \rangle = P(-,-)$ is
\begin{aligned} cs &= P_{a b} t^a \wedge (d_{W(\mathfrak{g})} t^b) + \frac{1}{3} P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c \\ & = P_{a b} t^a \wedge r^b - \frac{1}{6} P_{a a'}C^{a'}{}_{b c} t^
a \wedge t^b \wedge t^c \end{aligned} \,.
In particular the Killing form $\langle -,-\rangle$ is in transgression with the degree 3-cocycle
$\mu = -\frac{1}{6}\langle -,[-,-]\rangle \,.$
We compute
\begin{aligned} d_{W(\mathfrak{g})} \left( P_{a b} t^a \wedge r^b + \frac{1}{2}P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c \right) & = P_{a b} r^a \wedge r^b \\ & -\frac{1}{2} P_{a b} C^a{}_{d
e} t^d \wedge t^e \wedge r^b \\ & + P_{a b} C^b{}_{d e} t^a \wedge t^d \wedge r^e \\ & - \frac{3}{6} P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge r^c \\ & = P_{a b} r^a \wedge r^b \\ & + \frac{1}{2}
C_{a b c} t^a \wedge t^b \wedge r^c \\ & - \frac{1}{2} C_{a b c} t^a \wedge t^b \wedge r^c \\ & = P_{a b } r^a \wedge r^b \end{aligned} \,.
Under a Lie algebra-valued form
$\Omega^\bullet(X) \stackrel{}{\leftarrow} W(\mathfrak{g}) : A$
this Chern-Simons element is sent to
$cs(A) = P_{a b} A^a \wedge d A^b + \frac{1}{3} C_{a b c} A^a \wedge A^b \wedge A^c \,.$
If $\mathfrak{g}$ is a matrix Lie algebra then the Killing form is the trace and this is equivalently
$cs(A) = tr(A \wedge d A) + \frac{2}{3} tr(A \wedge A \wedge A) \,.$
This is a traditional incarnation of the standard Chern-Simons form in degree 3.
For $\Sigma$ a 3-dimensional smooth manifold the corresponding action functional $S_{CS} : \Omega^1(\Sigma, \mathfrak{g}) \to \mathbb{R}$
$S_{CS} : A \mapsto \int_\Sigma cs(A)$
is the standard action functional of Chern-Simons theory.
Covariant phase space
The covariant phase space of ordinary Chern-Simons theory is the space of those Lie algebra valued form $A$ whose curvature 2-form $F_A$ vanishes
$P = \{A \in \Omega^1(\Sigma, \mathfrak{g}) | F_A = 0\} \,.$
The presymplectic structure on this space is
$\omega : (\delta A_1, \delta A_2) \mapsto \int_{\partial \Sigma} \langle \delta A_1, \delta A_2 \rangle \,.$
This is a special case of prop. \ref{TheEquationsOfMotion}, prop. \ref{PresymplecticStructure} in view of corollary \ref{CovariantPhaseSpaceForBinaryNonDegenerateInvariantPolynomial}, using that the
Killing form is a binary and non-degenerate invariant polynomial.
Obstruction theory
The circle n-bundle with connection given by ordinary Chern-Simons theory is known as the Chern-Simons circle 3-bundle . The non-triviality of its underlying class is the obstruction to lifting the
$G$-principal bundle to a string structure. The non-triviality of its connection is the obstruction to having a differential string structure. In general it defines a twisted differential string
Courant $\sigma$-model
4d Chern-Simons functionals
BF-theory and topological Yang-Mills theory
Let $\mathfrak{g} = (\mathfrak{g}_2 \stackrel{\partial}{\to} \mathfrak{g})_1$ be a strict Lie 2-algebra, an L-∞ algebra concentrated in the lowesr two degrees and with the only nontrivial bracket
being the binary one.
We have
1. every invariant polynomial $\langle -\rangle_{\mathfrak{g}_1} \in inv(\mathfrak{g}_1)$ on $\mathfrak{g}_1$ is a Chern-Simons element on $\mathfrak{g}$, exhibiting a transgression to a trivial
∞-Lie algebra cocycle;
2. for $\mathfrak{g}_1$ a semisimple Lie algebra and $\langle - \rangle_{\mathfrak{g}_1}$ the Killing form, the corresponding Chern-Simons action functional on L-∞ algebra valued forms
$\Omega^\bullet(X) \stackrel{(A,B)}{\leftarrow} W(\mathfrak{g}_2 \to \mathfrak{g}_1) \stackrel{(\langle - \rangle_{\mathfrak{g}_1}, d_W \langle - \rangle_{\mathfrak{g}_1} )}{\leftarrow} W(b^{n-1}
is the sum of the action functionals of topological Yang-Mills theory with BF-theory with cosmological constant (in the sense of gravity as a BF-theory):
$CS_{\langle-\rangle_{\mathfrak{g}_1}}(A,B) = \langle F_A \wedge F_A\rangle_{\mathfrak{g}_1} - 2\langle F_A \wedge \partial B\rangle_{\mathfrak{g}_1} + 2\langle \partial B \wedge \partial B\
rangle_{\mathfrak{g}_1} \,,$
where $F_A$ is the ordinary curvature 2-form of $A$.
This is from (SSSI).
For $\{t_a\}$ a basis of $\mathfrak{g}_1$ and $\{b_i\}$ a basis of $\mathfrak{g}_2$ we have
$d_{W(\mathfrak{g})} : \mathbf{d} t^a \mapsto d_{W(\mathfrak{g}_1)} + \partial^a{}_i \mathbf{d} b^i \,.$
Therefore with $\langle -\rangle_{\mathfrak{g}_1} = P_{a_1 \cdots a_n} \mathbf{d} r^{a_1} \wedge \cdots \mathbf{d} t^{a_n}$ we have
$d_{W(\mathfrak{g})} \langle - \rangle_{\mathfrak{g}_1} = n P_{a_1 \cdots a_n}\partial^{a_1}{}_i \mathbf{d} b^{i} \wedge \cdots \mathbf{d} t^{a_n} \,.$
The right hand is a polynomial in the shifted generators of $W(\mathfrak{g})$, and hence an invariant polynomial on $\mathfrak{g}$. Therefore $\langle - \rangle_{\mathfrak{g}_1}$ is a Chern-Simons
element for it.
Now for $(A,B) \in \Omega^1(U \times \Delta^k, \mathfrak{g})$ an ∞-Lie algebra-valued form, we have that the 2-form curvature is
$F_{(A,B)}^1 = F_A - \partial B \,.$
\begin{aligned} CS_{\langle -\rangle_{\mathfrak{g}_1}}(A,B) & = \langle F_{(A,B)}^1\rangle_{\mathfrak{g}_1} \\ & = \langle F_A \wedge F_A\rangle_{\mathfrak{g}_1} - 2\langle F_A \wedge \partial B\
rangle_{\mathfrak{g}_1} + 2\langle \partial B \wedge \partial B\rangle_{\mathfrak{g}_1} \end{aligned} \,.
7d Chern-Simons functionals
7d $String$-Chern-Simons theory
In heterotic string theory Chern-Simons circle 3-bundles appear as the ordinary differential cohomology-incarnation of magnetic charge of NS-fivebranes that twists the Kalb-Ramond field as described
by the Green-Schwarz mechanism.
At least when this twist vanishes there is expected to be an electric-magnetic dual description with a Chern-Simons circle 7-bundle.
Let $\langle -,-,-,-\rangle$ be the canonical quaternaty invariant polynomial on the special orthogonal Lie algebra $\mathfrak{so}$. This lifts directly also to an invariant polynomial on the string
Lie 2-algebra $\mathfrak{string} := \mathfrak{so}_\mu$.
Obstruction theory
The circle n-bundle with connection given by 7-dimensional Chern-Simons theory is the Chern-Simons circle 7-bundle . The non-triviality of its underlying class is the obstruction to lifting the
string 2-group-principal 2-bundle to a fivebrane structure. The non-triviality of its 7-connection is the obstruction to having a differential fivebrane structure. In general it defines a twisted
differential fivebrane structure.
Higher dimensional abelian Chern-Simons theory
The line Lie n-algebra carries a canonical invariant polynomial. The $\infty$-Chern-Simons theory associated with this data is in the literature known as abelian higher dimensional Chern-Simons
Also every higher-degree invariant polynomial on an ordinary Lie algebra gives rise to a higher dimensional Chern-Simons action functional. An concrete example of this is considered (below).
$\infty$-Dijkgraaf-Witten theory
We consider the case where the target space object $\mathbf{B}G$ is a discrete ∞-groupoid.
$\mathbf{B}G := Disc B G$
with $B G$ the delooping of an ∞-group $G \in$Grpd $\simeq$Top.
As we discuss below, $\infty$-Chern-Simons theory for this setup subsumes and generalizes Dijkgraaf-Witten theory (and the Yetter model in next higher dimension). Therefore we speak of $\infty$
-Dijkgraaf-Witten theory.
By the $(\Pi \dashv Disc \dashv \Gamma)$-adjoint triple of the ambient cohesive (∞,1)-topos and usinf that $Disc$ is a full and faithful (∞,1)-functor we have
\begin{aligned} \mathbf{\Pi} \mathbf{B}G & \simeq Disc \Pi Disc B G \\ & \simeq Disc B G \\ & \simeq \mathbf{B}G \end{aligned}
and therefore, using the $(\Pi \dashv Disc)$-zig-zag identity, the constant path inclusion
$\mathbf{B}G \to \mathbf{\Pi} \mathbf{B}G$
is an equivalence. Therefore the intrinsic de Rham cohomology of $\mathbf{B}G$ is trivial
\begin{aligned} \mathbf{H}_{dR}(\mathbf{B}G, \mathbf{B}^n U(1)) & \simeq \mathbf{H}(\mathbf{\Pi}(\mathbf{B}G), \mathbf{B}^n U(1)) \prod_{\mathbf{H}(\mathbf{B}G, \mathbf{B}^n U(1))} * \\ & \simeq * \
and so the intrinsic universal curvature class
$curv : \mathbf{H}(\mathbf{B}G, \mathbf{B}^n U(1)) \to \mathbf{H}_{dR}(\mathbf{B}G, \mathbf{B}^n U(1))$
is trivial.
3d Dijkgraaf-Witten theory
Dijkgraaf-Witten theory is the analog of Chern-Simons theory for discrete structure groups. We show that this becomes a precise statement in $Smooth \infty Grpd$: the Dijkgraaf-Witten action
functional is that induced from applying the $\infty$-Chern-Simons homomorphism to a characteristic class of the form $Disc B G \to \mathbf{B}^3 U(1)$, for $Disc : \infty Grpd \to Smooth \infty Grpd$
the canonical embedding of discrete ∞-groupoids into all smooth ∞-groupoids.
Let $G \in Grp \to \infty Grpd \stackrel{Disc}{\to} Smooth \infty Grpd$ be a discrete group regarded as an ∞-group object in discrete ∞-groupoids and hence as a smooth ∞-groupoid with discrete smooth
cohesion. Write $B G = K(G,1) \in \infty Grpd$ for its delooping in ∞Grpd and $\mathbf{B}G = Disc B G$ for its delooping in Smooth∞Grpd.
We also write $\Gamma \mathbf{B}^n U(1) \simeq K(U(1), n)$. Notice that this is different from $B^n U(1) \simeq \Pi \mathbf{B}U(1)$, reflecting the fact that $U(1)$ has non-discrete smooth structure.
For $G$ a discrete group, morphisms $\mathbf{B}G \to \mathbf{B}^n U(1)$ correspond precisely to cocycles in the ordinary group cohomology of $G$ with coefficients in the discrete group underlying the
circle group
$\pi_0 Smooth\infty Grpd(\mathbf{B}G, \mathbf{B}^n U(1)) \simeq H^n_{Grp}(G,U(1)) \,.$
By the $(Disc \dashv \Gamma)$-adjunction we have
$Smooth\infty Grpd(\mathbf{B}G, \mathbf{B}^n U(1)) \simeq \infty Grpd(B G, K(U(1),n)) \,.$
For $G$ discrete
• the intrinsic de Rham cohomology of $\mathbf{B}G$ is trivial
$Smooth \infty Grpd(\mathbf{B}G, \mathbf{\flat}_{dR}\mathbf{B}^n U((1)) \simeq * ;$
• all $G$-principal bundles have a unique flat connection
$Smooth\infty Grpd(X, \mathbf{B}G) \simeq Smooth\infty Grpd(\Pi(X), \mathbf{B}G) \,.$
By the $(Disc \dashv \Gamma)$-adjunction and using that $\Gamma \circ \mathbf{\flat}_{dR} K \simeq *$ for all $K$.
It follows that for $G$ discrete
• any characteristic class $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ is a group cocycle;
• the $\infty$-Chern-Weil homomorphism coincides with postcomposition with this class
$\mathbf{H}(\Sigma, \mathbf{B}G) \to \mathbf{H}(\Sigma, \mathbf{B}^n U(1)) \,.$
For $G$ discrete and $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^3 U(1)$ any group 3-cocycle, the $\infty$-Chern-Simons theory action functional on a 3-dimensional manifold $\Sigma$
$Smooth\infty Grpd(\mathbf{\Pi}(\Sigma), \mathbf{B}G) \to U(1)$
is the action functional of Dijkgraaf-Witten theory.
By proposition \ref{IntrinsicIntegrationTheorem} the morphism is given by evaluation of the pullback of the cocycle $\alpha : B G \to B^3 U(1)$ along a given $abla : \Pi(\Sigma) \to B G$, on the
fundamental homology class of $\Sigma$. This is the definition of the Dijkgraaf-Witten action (for instance equation (1.2) in FreedQuinn).
Obstruction theory
The flat Dijkgraaf-Witten circle 3-bundle on $\Sigma$ is the obstruction to lifting the $G$-principal bundle to a $\hat G$-principal 2-bundle, where $\hat G$ is the discrete 2-group classified by the
group 3-cocycle.
4d Yetter model
Closed string field theory
For the moment see closed string field theory .
AKSZ theory
We consider symplectic Lie n-algebroids $\mathfrak{P}$ equipped with a canonical invariant polynomial and show that the ∞-Chern-Simons action functional associated to this data is the action
functionalof the AKSZ theory sigma model with target space $\mathfrak{P}$.
This is taken from (FRS11). See there for more details.
This means that
• on each coordinate chart $U \to X$ of the base manifold $X$ of $\mathfrak{P}$, there is a basis $\{x^a\}$ for $CE(\mathfrak{a}|_U)$ such that
$\omega = \omega_{a b} \mathbf{d}x^a \wedge \mathbf{d}x^b$
with $\{\omega_{a b} \in \mathbb{R} \hookrightarrow C^\infty(X)\}$ and $deg(x^a) + deg(x^b) = n$;
• the coefficient matrix $\{\omega_{a b}\}$ has an inverse;
• we have
$d_{\mathrm{W}(\mathfrak{P})} \omega = d_{\mathrm{CE}(\mathfrak{P})} \omega + \mathbf{d} \omega = 0 \,.$
This $\infty$-Lie theoretic structure is essentially what in the literature is mostly considered in terms of symplectic dg-geometry :
We may think of an L-infinity-algebroid $\mathfrak{a}$ as a graded manifold $X$ whose global function ring is the graded algebra underlying the Chevalley-Eilenberg algebra
$C^\infty(X) := \mathrm{CE}(\mathfrak{a})$
and which is equipped with a vector field $v_X$ of grade 1 whose graded Lie bracket with itself vanishes $[v_X, v_X] = 0$, given, as a derivation, by the
differential on the Chevalley-Eilenberg algebra:
$v_X := d_{\mathrm{CE}(\mathfrak{a})} : \mathrm{CE}(\mathfrak{a}) \to \mathrm{CE}(\mathfrak{a}) \,.$
The pair $(X,v)$ is a differential graded manifold . In this perspective the graded algebra underlying the Weil algebra of $\mathfrak{a}$ is the de Rham complex of $X$
$\Omega^\bullet(X) := \mathrm{W}(\mathfrak{a}) \,,$
but the de Rham differential is just $\mathbf{d}$, not the full differential $d_{\mathrm{W}(\mathfrak{a})} = \mathbf{d} + d_{\mathrm{CE}(\mathfrak{a})}$ on the Weil algebra. The latter is thus a
twisted de Rham differential on $X$.
From this perspective all standard constructions of Cartan calculus usefully apply to $L_\infty$-algebroids. Notably for $v$ any vector field on $X$ there is the contraction derivation
$\iota_v : \Omega^\bullet(X) \to \Omega^{\bullet -1}(X)$
and hence the Lie derivative
$\mathcal{L}_v := [\mathbf{d}, \iota_{v}] : \Omega^\bullet(X) \to \Omega^\bullet(X) \,.$
So in the above notation we have in particular
$d_{\mathrm{W}(\mathfrak{a})} = \mathbf{d} + \mathcal{L}_{v_X} : \mathrm{W}(\mathfrak{a}) \to \mathrm{W}(\mathfrak{a}) \,.$
For $X$ a dg-manifold, let $\epsilon \in \Gamma(T X)$ be the vector field which over any coordinate patch $U \to X$ is given by the formula
$\epsilon|_U = \sum_a \mathrm{deg}(x^a) x^a \frac{\partial}{\partial x^a} \,,$
where $\{x^a\}$ is a basis of generators and $\mathrm{deg}(x^a)$ the degree of a generator.
We write
$N := [\mathbf{d}, \iota_\epsilon]$
for the Lie derivative of this vector field. The grade of a homogeneous element $\alpha$ in $\Omega^\bullet(X)$ is the unique natural number $n \in \mathbb{N}$ with
$\mathcal{L}_\epsilon \alpha = N \alpha = n \alpha \,.$
• This implies that for $x^i$ an element of grade $n$ on $U$, the 1-form $\mathbf{d}x^i$ is also of grade $n$. This is why we speak of grade (as in “graded manifold”) instead of degree here.
• The above is indeed well-defined: on overlaps of patches the $\{x^a\}$ of positive degree/grade transform by a degreewise linear transformation, which manifestly preserves $\sum_a \mathrm{deg}(x^
a) x^a \frac{\partial}{\partial x^a}$. Notice that the ordinary (degree-0 coordinates) do not appear in this formula. And indeed the vector field locally defined by $\sum_a x^a \frac{\partial}{\
partial x^a}$ (thus including the coordinates of grade 0) does not in general exist globally.
The existence of $\epsilon$ implies the following useful statement, which is a trivial variant of what in grade 0 would be the standard Poincare lemma.
On a graded manifold every closed differential form $\omega$ of positive grade $n$ is exact: the form
$\lambda := \frac{1}{n} \iota_\epsilon \omega$
$\mathbf{d}\lambda = \omega \,.$
Using this differential geometric language we can now capture something very close to def. 1 in more traditional symplectic geometry terms.
A symplectic dg-manifold of grade $n \in \mathbb{N}$ is a dg-manifold $(X,v)$ equipped with 2-form $\omega \in \Omega^2(X)$ which is
• \item non-degenerate;
• closed;
as usual for symplectic forms, and in addition
• of grade $n$;
• $v$-invariant: $\mathcal{L}_v \omega = 0$.
Example. It follows that a symplectic dg-manifold of grade 0 is the same as an ordinary symplectic manifold. In the following we are mostly interested in the case of positive grade.
The function algebra of a symplectic dg-manifold $(X,\omega)$ of grade $n$ is naturally equipped with a Poisson bracket
$\{-,-\} : C^\infty(X)\otimes C^\infty(X) \to C^\infty(X)$
which decreases grade by $n$. On a local coordinate patch this is given by
$\{f,g\} = \frac{\partial f}{\partial x^a} \omega^{a b} \frac{\partial g}{\partial x^b} \,,$
where $\{\omega^{a b}\}$ is the inverse matrix to $\{\omega_{a b}\}$.
For $f \in C^\infty(X)$ and $v \in \Gamma(T X)$ we say that f$is a Hamiltonian for$v_ or equivalently that _v$is the [[nLab:Hamiltonian vector field]] of$f$ if
$\mathbf{d}f = \iota_v \omega \,.$
The dg-manifold itself is identified with an $L_\infty$-algebroid as in observation 6. For $\omega \in \Omega^2(X)$ a symplectic form, the conditions $\mathbf{d} \omega = 0$ and $\mathcal{L}_v \omega
= 0$ imply $(\mathbf{d}+ \mathcal{L}v)\omega = 0$ and hence that under the identification $\Omega^\bullet(X) \simeq \mathrm{W}(\mathfrak{a})$ this is an invariant polynomial on $\mathfrak{a}$.
It remains to observe that the $L_\infty$-algebroid $\mathfrak{a}$ is in fact a Lie $n$-algebroid. This is implied by the fact that $\omega$ is of grade $n$ and non-degenerate: the former condition
implies that it has no components in elements of grade $gt n$ and the latter then implies that all such elements vanish.
Let $(\mathfrak{P},\omega)$ be a symplectic Lie $n$-algebroid for positive $n$ in the image of the embedding of prop. 10. Then it carries the canonical $L_\infty$-algebroid cocycle
$\pi := \frac{1}{n+1} \iota_\epsilon \iota_v \omega \in \mathrm{CE}(\mathfrak{P})$
which moreover is the Hamiltonian, according to def. 9, of $d_{\mathrm{CE}(\mathfrak{P})}$.
The required condition $\mathbf{d}\pi = \iota_v \omega$ from def. 9 holds by observation 7.
Our central observation now is the following.
The cocycle $\pi$ from prop. 11 is in transgression with the invariant polynomial $n \omega$. A Chern-Simons element witnessing the transgression according to def. \ref{TransgressionAndCSElements} is
$\mathrm{cs} = \iota_\epsilon \omega + \pi \,.$
It is clear that $i^* \mathrm{cs} = \pi$. So it remains to check that $d_{\mathrm{W}(\mathfrak{P})} \mathrm{cs} = n\omega$. Notice that
$[d_{\mathrm{CE}(\mathfrak{P})}, \iota_\epsilon] = [\mathcal{L}_v, \iota_\epsilon] = \iota_{[v,\epsilon]} = - \iota_{v}$
by Cartan calculus. Using this we compute the first summand in $d_{\mathrm{W}(\mathfrak{P})} ( \iota_{\epsilon} \omega + \pi )$:
\begin{aligned} d_{\mathrm{W}(\mathfrak{P})} \iota_{\epsilon} \omega & = ( \mathbf{d} + d_{\mathrm{CE}(\mathfrak{P})} ) \iota_\epsilon \omega \\ &= n \omega + [d_{\mathrm{CE}(\mathfrak{P})}, \iota_\
epsilon] \omega \\ &= n\omega - \iota_v \omega \\ & = n \omega - \mathbf{d}\pi \end{aligned} \,.
The second summand is simply
$d_{\mathrm{W}(\mathfrak{P})} \pi = \mathbf{d}\pi$
since $\pi$ is a cocycle.
For $(\mathfrak{P}. \omega)$ a symplectic Lie $n$-algebroid coming from a symplectic dg-manifold by prop. 10, the higher Chern-Simons action functional associated with its canonical Chern-Simons
element $\mathrm{cs}$ from prop. 12 is the AKSZ Lagrangean:
$L_{\mathrm{AKSZ}} = \mathrm{cs} \,.$
We work in local coordinates $\{x^a\}$ where
$\omega = \frac{1}{2}\omega_{a b} \mathbf{d}x^a \wedge \mathbf{d}x^b$
and the Chern-Simons element is
$\mathrm{cs} = \sum_a \omega_{a b} \mathrm{deg}(x^a) x^a \wedge \mathbf{d}x^b + \pi \,.$
We want to substitute here $\mathbf{d} = d_{\mathrm{W}}- d_{\mathrm{CE}}$. Notice that in coordinates the equation
$\mathbf{d}\pi = \iota_v \omega$
\begin{aligned} \mathbf{d}x^a \frac{\partial \pi}{\partial x^a} & = \omega_{a b} v^a \wedge \mathbf{d} x^b \\ & = \omega_{a b} \mathbf{d}x^a \wedge v^b \end{aligned} \,.
\begin{aligned} \sum_a \omega_{a b} \mathrm{deg}(x^a) x^a \wedge d_{\mathrm{CE}} x^b & = \sum_a \omega_{a b} \mathrm{deg}(x^a) x^a \wedge v^b \\ & = \sum_a \mathrm{deg}(x^a)x^a \frac{\partial \pi}{\
partial x^a} \\ &= (n+1) \pi \end{aligned} \,.
$\mathrm{cs} = \sum_{a b} \mathrm{deg}(x^a) \,\omega_{a b} x^a \wedge \mathbf{d}x^b - n \pi \,.$
This means that for $\Sigma$ an $(n+1)$-dimensional manifold and
$\Omega^\bullet(\Sigma) \leftarrow \mathrm{W}(\mathfrak{P}) : X$
a $\mathfrak{P}$-valued differential form on $\Sigma$ we have
\begin{aligned} \mathrm{cs}(X) &= \sum_{a,b} \mathrm{deg}(x^a)\,\omega_{a b} X^a \wedge d_{\mathrm{dR}} X^b - n \Pi(X) \end{aligned} \,.
This is indeed $L_{\mathrm{AKSZ}}(X)$.
Remark The AKSZ $\sigma$-model action functional interpretation of $\infty$-Chern-Weil functionals for binary invariant polynomials on $L_\infty$-algebroids from prop. 13 gives rise to the following
dictionary of concepts\
$\array{ Chern-Weil theory && quantum field theory \\ \\ cocycle & \pi & Hamiltonian \\ \\ transgression element & cs & Lagrangean \\ \\ curvature characteristic & \omega & symplectic structure }
Covariant phase space
The covariant phase space of AKSZ theory with target $(\mathfrak{P}, \omega)$ is the space of those ∞-Lie algebroid-valued forms $A$ whose curvature $(n+1)$-form $F_A$ vanishes
$P = \{A \in \Omega^1(\Sigma, \mathfrak{g}) | F_A = 0\} \,.$
The presymplectic structure on this space is
$\omega : (\delta A_1, \delta A_2) \mapsto \int_{\partial \Sigma} \omega(\delta A_1, \delta A_2 ) \,.$
This is a special case of prop. \ref{TheEquationsOfMotion}, prop. \ref{PresymplecticStructure} in view of corollary \ref{CovariantPhaseSpaceForBinaryNonDegenerateInvariantPolynomial}, using that, by
definition of symplectic Lie n-algebroid, $\omega$ is a binary and non-degenerate invariant polynomial.
$n=0$ – The topological particle
For $X$ a smooth manifold we may regard its cotangent bundle $\mathfrak{a} = T^* X$ as a Lie 0-algebroid and the canonical 2-form $\omega \in W(\mathfrak{a}) = \Omega^\bullet(X)$ as a binary
invariant polynomial in degree 2.
The Chern-Simons element is the canonical 1-form $\alpha$ which in local coordinates is $\alpha = p_i d q^i$.
The corresponding action functional on the line
$\int_{\mathbb{R}} \gamma^* (p_i\, d q^i)$
is the familiar term for the action functional of the particle (missing the kinetic term, which makes it “topological”).
$n=1$ – The Poisson $\sigma$-model
Let $(X, \{-,-\})$ be a Poisson manifold. Over a Darboux chart the corresponding Poisson Lie algebroid has coordinates $\{x^i\}$ of degree 0 and $\partial_i$ of degree 1. We have
$d_{\mathrm{W}} x^i = -\pi^{i j}\mathbf{\partial}_j + \mathbf{d}x^i$
where $\pi^{i j} := \{x^i , x^j\}$ and
$\omega = \mathbf{d}x^i \wedge \mathbf{d}\partial_i \,.$
The Hamiltonian cocycle from prop. 11 is
\begin{aligned} \pi &= \iota_v \iota_\epsilon \omega \\ &= \iota_v \partial_i \wedge \mathbf{d}x^i \\ & = \partial_i \wedge [\iota_v,\mathbf{d}]x^i \\ &= -\partial_i \wedge [\mathbf{d},\iota_v]x^i \\
&= + \partial_i \pi^{ij}\partial_j \end{aligned}
and the Chern-Simons element from prop. 12 is
\begin{aligned} \mathrm{cs} &= \iota_\epsilon \omega + \pi \\ &= \partial_i \wedge \mathbf{d}x^i + \pi^{ij}\partial_i \partial_j \end{aligned} \,.
In terms of $d_{\mathrm{W}}$ instead of $\mathbf{d}$ this is
\begin{aligned} \cdots & = \partial_i \wedge (d_{\mathrm{W}} - d_{\mathrm{CE}}) x^i + \pi^{ij}\partial_i \partial_j \\ &= \partial_i \wedge \mathbf{d}x^i + 2 \pi^{ij}\partial_i \partial_j \end
So for
$\Omega^\bullet(\Sigma) \leftarrow \mathrm{W}(\mathfrak{P}) : (X,\eta)$
a Poisson-Lie algebroid valued differential form – which in components is a function $\phi: \Sigma \to X$ and a 1-form $\eta \in \Omega^1(\Sigma, \phi^* T^* X)$ – the corresponding Chern-Simons form
$\mathrm{cs}(X,\eta) = \langle d_{\mathrm{dR}}X \wedge \eta \rangle + 2 \pi(\eta \wedge \eta) \,.$
This is the Lagrangean of the Poisson $\sigma$-model \cite{CattaneoFelder}.
$n=2$ – Ordinary Chern-Simons theory
We show how the ordinary Chern-Simons form arises from this perspective. So let $\mathfrak{a} = \mathfrak{g}$ be a semisimple Lie algebra and $\omega := \langle -,-\rangle\in \mathrm{W}(\mathfrak{g})
$ its Killing form invariant polynomial. For $\{t^a\}$ a dual basis for $\mathfrak{g}$ we have
$d_{\mathrm{W}} t^a = - \frac{1}{2}C^a{}_{b c} t^a \wedge t^b + \mathbf{d}t^a$
where $C^a{}_{b c} := t^a([t_b,t_c])$ and
$\omega = \frac{1}{2} P_{a b} \mathbf{d}t^a \wedge \mathbf{d}t^b \,,$
where $P_{ab} := \langle t_a, t_b \rangle$. The Hamiltonian cocycle $\pi$ from prop. 11 is
\begin{aligned} \pi & = \frac{1}{2+1}\iota_\epsilon \iota_v \omega \\ & = \frac{1}{3} \iota_v \iota_\epsilon \omega \\ & = \frac{1}{3}\iota_v P_{a b} t^a \wedge \mathbf{d}t^b \\ & = \frac{1}{3} P_{a
b} t^a \wedge [\iota_v,\mathbf{d}]t^b \\ & = -\frac{1}{3} P_{a b} t^a \wedge [\mathbf{d}, \iota_v]t^b \\ &= -\frac{1}{3} P_{a b} t^a \wedge (-\frac{1}{2})C^b{}_{d e} t^d \wedge t^e \\ & = +\frac{1}
{6} C_{abc}t^a \wedge t^b \wedge t^c \end{aligned} \,.
Therefore in this case the Chern-Simons element from def. 12 becomes
\begin{aligned} \mathrm{cs} & = \iota_\epsilon \omega + \pi \\ & = P_{a b} t^a \wedge \mathbf{d}t^b + \frac{1}{6} C_{abc} t^a \wedge t^b \wedge t^c \end{aligned} \,.
This is indeed the familiar standard choice of Chern-Simons element on a Lie algebra. Notice that evaluated on a $\mathfrak{g}$-valued form
$\Omega^\bullet(\Sigma) \leftarrow \mathrm{W}(\mathfrak{g}) : A$
this is
$\mathrm{cs}(A) = \langle A \wedge F_A\rangle + \frac{1}{6}\langle A \wedge [A \wedge A]\rangle \,.$
If $\mathfrak{g}$ is a matrix Lie algebra then the Killing form is proportional to the trace of the matrix product: $\langle t_a,t_b\rangle = \mathrm{tr}(t_a t_b)$. In this case we have
\begin{aligned} \langle A \wedge [A \wedge A]\rangle &= A^a \wedge A^b \wedge A^c \,\mathrm{tr}(t_a (t_b t_c - t_c t_b)) \\ &= 2 A^a \wedge A^b \wedge A^c \,\mathrm{tr}(t_a t_b t_c ) \\ &= 2 \,\
mathrm{tr}(A \wedge A \wedge A) \end{aligned}
and hence
$\mathrm{cs}(A) = \mathrm{tr}(A \wedge F_A) + \frac{1}{3}\,\mathrm{tr}(A \wedge A \wedge A) \,.$
Often this is written in terms of the de Rham differential 2-form $d_{\mathrm{dR}} A$ instead of the curvature 2-form $F_A := d_{\mathrm{dR}} A + \frac{1}{2}[A \wedge A]$. Since the former is the
image under $A$ of $d_{\mathrm{W}(\mathfrak{g})}$ we can alternatively write
\begin{aligned} \mathrm{cs} &= P_{a b} t^a \wedge (d_{\mathrm{W}} - d_{\mathrm{CE}})t^b + \frac{1}{6} C_{abc} t^a \wedge t^b \wedge t^c \\ &= P_{a b} t^a \wedge d_{\mathrm{W}} t^b - P_{ab} t^a \wedge
(-\frac{1}{2})C^b{}_{cd}t^c \wedge t^d + \frac{1}{6} C_{abc} t^a \wedge t^b \wedge t^c \\ & = P_{ab} t^a \wedge d_{\mathrm{W}}t^b + (\frac{1}{2}+\frac{1}{6}) C_{abc} t^a \wedge t^b \wedge t^c \\ & =
P_{ab} t^a \wedge d_{\mathrm{W}}t^b + \frac{2}{3}C_{abc} t^a \wedge t^b \wedge t^c \end{aligned} \,.
$\mathrm{cs}(A) = \langle A \wedge d_{\mathrm{dR}} A\rangle + \frac{2}{3} \langle A \wedge [A \wedge A]\rangle \,.$
$n=2$ – The Courant $\sigma$-model
The notion of Chern-Simons elements for $L_\infty$-algebras and the associated $\imnfty$-Chern-Simons Lagrangians is due to
The induced construction of the ∞-Chern-Weil homomorphism with special attention to the Chern-Simons circle 3-bundle and the Chern-Simons circle 7-bundle is in
In the general context of cohesive (∞,1)-toposes $\infty$-Chern-Simons theory is discussed in section 4.3 of
The case of the AKSZ sigma-model is discussed in
Discussion of symplectic Lie n-algebroids is in
• Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds PhD thesis (arXiv)
On the structure of graded symplectic supermanifolds and Courant algebroids (arXiv)
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proportional relationship between variables
September 21st 2010, 02:44 PM #1
Sep 2008
proportional relationship between variables
Does a proportional relationship have to start at (0,0)? I know that if they are continuous then a proportional relationship contains the point zero. What about a discrete relationship? Can a
discrete relationship be proportional?
It's a little hard to understand what you are talking about but apparently you are asking if a the graph of a "proportional relationship" (y= ax for direct proportion) contains the point (0, 0).
The answer is yes, if "0" is in the domain but that is not always true.
I am assuming here that you mean "direct" proportion. The graph of an "inverse" proportion (y= a/x) never contains (0, 0).
It's a little hard to understand what you are talking about but apparently you are asking if a the graph of a "proportional relationship" (y= ax for direct proportion) contains the point (0, 0).
The answer is yes, if "0" is in the domain but that is not always true.
I am assuming here that you mean "direct" proportion. The graph of an "inverse" proportion (y= a/x) never contains (0, 0).
ok so, if I have a beaker and put 200ml water in it and place a marble in the beaker to determine its volume, the realationship between total volume is a direct proportion? because it causes the
water to rise by a constant "K" each time, and the graph will start at (0,200).
Yes, the graph is linear. I would call that a "linear" relation, not a "proportional" relation because, while x= 0 (no marble) is in the domain, (0, 0) is not on the graph.
September 22nd 2010, 02:09 AM #2
MHF Contributor
Apr 2005
September 22nd 2010, 12:32 PM #3
Sep 2008
September 23rd 2010, 03:26 AM #4
MHF Contributor
Apr 2005
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Quantity System
The best part in making the Quantity System, is that I am re-discovering mathematics with it. It gives me an immense pleasure to learn mathematics in a practical way and to try and make corrections
on two levels, Programming and Mathematics.
This post will be a usage scenario for the quantity system. The scenario will be applicable for the latest source code (which can be downloaded from http://quantitysystem.codeplex.com/SourceControl/
list/changesets labeled by version [Qs 1.2.5])
In this post I will explain the mathematical foundation with my language/realization (so I am not tied to the formal way of books)
Coordinates Introduction
In studying geometry, we always encounter (x,y,z) coordinate system. This system simply is a way to describe the location of point.
And if we are talking with Vectors, then {x y z} is an arrow coming from {0 0 0} to {x y z} and we are calling it Vector.
Using {x[1] x[2] x[3]} labeling instead of {x y z} then we have indexes 1,2, and 3 usually referred as R^3 Space.
What does that mean ??
R^2: 2 dimensional coordinate system (2D Space)
R^3: 3 dimensional coordinate system (3D Space)
R^4: 4 dimensional coordinate system (4D Space) (yes it exists somehow refer to Hypersphere coordinate system http://en.wikipedia.org/wiki/N-sphere )
Note: Using ‘R’ refer to the numbering system type we are using which is Real Numbers.
So in R^n the point/vector can be described by {x[1] x[2] x[3] x[4] .. x[n]} and to generalize the point location we end into defining x[i]
where i: 1,2, 3, .., n for number of coordinates
Cartesian Coordinates
They are simply {x y} or {x y z}, R^2 or R^3 which we all know and studied during school. (an important note here is that the angle between the unit vectors is 90 degree or Orthogonal)
Curvilinear Coordinates
In this type of coordinates the unit vectors that represent the coordinate system is not a straight forward lines, but on contrary they are making curves in some way.
When dealing with curvilinear coordinates every coordinate has a tangent vector (e) and orthogonal vector (E)
Without going into a lot of details these extra vectors are the foundation of covariant and contravariant vectors
The curse of Curvilinear coordinates is that the location of point can be represented by two vectors covariant or contravariant.
We label the two types of vectors by this
Contravariant vector: x^i superscript index or simple {x^1 x^2 x^3}
Covariant vector: x[i] subscript index {x[1] x[2] x[3]}
Best to be seen by the following figure
Note: we are obliged to use two types of vectors specially when we are beginning to use geometric properties of geometric objects (for example the line length in curvilinear coordinates is not the
same as Cartesian coordinates, think about line over spherical object like big line on earth) line length = sqrt(x*x + y*y + z*z) however in curvilinear length = sqrt(x[1]*x^1 + x[2]*x^2 +x[3]*x^3)
or simply length = sqrt(x[i]*x^i) Einstein summation notation.
Coordinates Transformation
We can transform the point from Cylindrical to Cartesian (here I mean changing the representation of point/vector from r,q,z to x,y,z )
x = r cos(q)
y = r sin(q)
z = z
a vice versa transformation exist but I will not write it here
for simplicity let us write that any {r q z} vector we can transform it by multiplying it with {r*cos(q) r*sin(q) z}
so for any two coordinate systems there is a transformation
Note: in mathematical books they are referring to Jacobian calculation to make sure that such transformation exist
How can we make this simple with quantity system?
• Define a vector that contains a symbolic quantities
• Get the function from this vector
• Now you can get x, y, z vector from r q z vector by the function fc
• Also using named parameters (so you don’t get ambiguous about parameters order)
• And for fun you can pass Symbolic parameters (remember that quantity system blend the usage between numbers and symbols)
Note: Quantity system permits you to make calculations on symbols as if they were first citizen types in Qs to define symbol you can precede dollar sign before it like $x $y $myvariable and to define
complex expression you can use ${expression} like ${x+y*r} or ${sqrt(sin(u*v^2))} please refer to the separate symbolic variable engine that also made by me in http://symbolicalgebra.codeplex.com
Covariant and Contravariant vectors Dilemma
A question is raised here, how can I get covariant and contravariant components of vector with the guide of Transformation Equations and existing vector?
This dilemma is solved by the Metric Tensor. The metric tensor is a matrix like object that when multiplied by vector gets the corresponding vector
There are also two types of metric tensor (covariant metric tensor and contravariant metric tensor)
To calculate the metric tensor we get back to the definition of base vector
In the figure above the x^1,x^2,x^3 represent a curvilinear coordinate system and g[1],g[2],g[3] represent covariant base vectors (the base vectors has been defined by the differentiation of the
location point transformation of curvilinear coordinate system)
Let us take the cylindrical transformation to Cartesian
x = r cos(q)
y = r sin(q)
z = z
g[1] = g[1x]*i[1]+g[1y]*i[2]+g[1z]*i[3]
to get these components we differentiate each term of transformation vector with r
g[1x] = diff(r*cos(q), r)
g[1y] = diff(r*sin(q), r)
g[1z] = diff(z, r)
then we get the other
g[2] also is the differentiation of all components with q
g[3] is differentiating with z
Now to make the metric tensor we will use the dot product of vectors to get this 2^nd order tensor which looks like matrix
g[1].g[1] g[1].g[2] g[1].g[3]
g[2].g[1] g[2].g[2] g[2].g[3]
g[3].g[1] g[3].g[2] g[3].g[3]
I will not dive into details about characteristics of metric tensor but there are common rules about them
1. contravariant * covariant metric tensors = Identity
2. Covariant Metric Tensor * Covariant Vector = Contravariant Vector
3. Contravariant Metric Tensor * Contravariant Vector = Covariant Vector.
So by getting the metric tensor for the curvilinear coordinate system (or any higher spaces if we want) then we can get the covariant and contravariant components of vector.
Metric Tensor with Quantity System
As my reader can see, you have to know many skills to get the metric tensor for specific coordinate system
• Transformation equations
• Differentiations
• Dot Product
• Matrix Algebra (for matrix inverse)
Without another mumbling let’s use the precious quantity system for this operation.
Continuing on the last transformation for cylindrical coordinates
• Get g[1]
• Get g[2]
• Get g[3 ]
• Get the covariant metric tensor
You can notice that if you do Trigonometric simplification to the covariant metric tensor of cylindrical coordinates would be
Which is the right deduction for it (I didn’t implement simplification in symbolic algebra yet bear with me please J)
Calculating contravariant vector
• Get a function from the metric tensor
• Suppose that we have vector with r=3 theta=2 and z=10 as a covariant vector
• Then to get contravariant vector we simply do the following
• Quantity system make it easy to calculate the metric tensor
• Quantity system make it easy to get covariant and contravariant vectors
• I am very biased J
J to the quantity system (I can’t help it actually)
Of course I am not claiming that my implementation is 100% error free (and you will see mistakes while playing with these calculations) however I am trying to say that the idea of blending symbolic
calculations beside numerical calculations and getting the result easily and intuitively should enhance the global or macro view of mathematics and may increase the researcher ability to distinguish
between these concepts and testing it easily while he is studying.
I also wish that I can reach a stable state that makes these calculations really reachable and easy for all of those interested beside being easy in using as an embedded language inside C#
applications and .NET framework.
Thank you all for reading this so far.
Ahmed Sadek
Today I am going to talk about the types that the Quantity System contains in its runtime
I should make a distinction between two labels now:
• Quantity System Framework: is the library responsible about Quantities and their units. It forms the required functionality that holds the necessary classes for embedding units and quantities in
your application.
• Quantity System: is the runtime that is built over the framework, that is evolving with time and its main concern is a new vision to apply the mathematics as a unit aware expressions.
I admit that I am learning mathematics again while programming this marvelous library, I learnt the compiler theory, parsing techniques, object oriented, and how can I apply it to physical
I will try also to shorten my posts and increasing them to indicate the features that I’ve implemented so far. Also you (Dear reader) may encounter a lot of my thinking during the writings in this
However let’s avoid my philosophy view of what I am doing and dive directly into the architecture.
Qs Types
There is a complex type that called QsScalar which serve the basic calculation type in Qs.
The architecture of types that is calculated is illustrated on the next diagram
Figure 1: Logical Relation between Qs Scalar and Tensor
The scalar in quantity is a single quantity that holds a defined set of fields beside the unit of that field.
The field is simply a Ring (Mathematical Point of View) or (Entity if I am talking from a programmer point of view)
Integer Numbers are a field, Real Numbers are a field, Rational Numbers are a field, Irrational Numbers are fields also.
But the concept of field goes beyond these samples also, because you can find Functions as Field.
The following diagram will illustrate the Scalar Object supported fields in Quantity System
Figure 2: Supported Mathematical Fields in QsScalar Type
Real Number
Rational Number
Which contains {Numerator, Denominator}
Complex Number
Which contains 2 components {Real Value, Imaginary Value}
Which contains 4 compnents {Real, i, j, k}
Symbolic variables that help in symbolic calculations
Referring to the function as a single quantity that can be added or subtracted
As I promised that’s all for now J
In the next posts, I will talk about the rest functionalities of Quantity System. So be tuned J
you may wondering why is this long long version number
actually I don’t want to reach 1.2 soon, thats why I slow down in my releases counting
especially that I am still in ALPHA release
the new release is featuring the new symbolic quantity concept which makes you do calculations with symbolics like mathematica and maxima BUT with my point of view of units
to declare a symbolic quantity you precede it with dollar sign ‘$’ and any number of charachters
$x+$y is a valid expression for x plus y
lets have some fun
make a rank two matrix
h = [$x<in> $y<ft>; 4<fm> 3<mm>]
ofcourse <fm> is Femto Meter ;)
get the determinant
Qs> |h|
Area: 3*x-4.8E-11*y <in.mm>
What about 3 ranked matrix
Qs> m = [$x<in> $y<ft> $z<pm>; 4<fm> 3<mm> 2<yd>; $u<m> $v<ft> $w<rod>]
Qs> |m|
Volume: 3*x*w-110.836363636364*x*v+4363.63636363636*y*u-4.8E-11*y*w+9.54426151276544E-24*z*v-2.34848954546393E-11*z*u <in.mm.rod>
This release also feature a tensor support
the tensor syntax will use ‘<|’ ‘|>’ which I don’t have a names for them now
as I understand (because I am not sure) I was reading the global relativity theory of Einstien (and I repeat I am not sure if I got it right) that Tensor is the ability to transfer your point of
view from local co-ordinates into another reference co-ordinates
so that when you look into a matrix for example you see it as a square or rectangle
and to go into z-direction you have to use a tensor view like a cube (this is the 3rd order tensor)
however to go into more reference like 4th order tensor you need some sort of recursive representation for this problem
I found that I can use some sort of recursive magic in syntax
for tensor of matrix resemblance you can use the same matrix syntax
T2 = <| 3 4; 8 9 |>
go into 3rd order tensor
T3 = <| 3 4; 8 9 | 8 7; 3 2|>
go into 4th order
T4 = <| <| 3 4; 8 9 | 8 7; 3 2|> | <| 3 4; 8 9 | 8 7; 3 2|> |>
T4 = <| T3 | T3 |>
and yes in storing this in memory I use a lot of inner objects (remember that I didn’t think about performance yet )
etc the 5th and 6th orders to the degree you want
BUT :(
all sources I read is only dealing with tensor of 2nd order
Needless to say how I get frustrated to understand what the heck is the tensor it really is (but it exists).
covariant, and contra variant vectors and tensors (some help needed here)
Another confusing thing (made my head spin)
If you make a vector, don’t safely consider it a first order tensor, also tensor of first order is NOT a vector
(I don’t know the validity of previous statement)
how is this differ in quantity system
make two vectors
v1 = {3 4 6}
v2 = {9 8 3}
multipy them tensorial ‘(*)
Qs> v1 (*) v2
27 <1> 24 <1> 9 <1>
36 <1> 32 <1> 12 <1>
54 <1> 48 <1> 18 <1>
Great isn’t it
However what about tensor from the first order
Qs> tv1 = <|3 4 6|>
QsTensor: 1st Order
3 <1> 4 <1> 6 <1>
Qs> tv2 = <|9 8 3|>
QsTensor: 1st Order
9 <1> 8 <1> 3 <1>
Qs> tv1*tv2
QsTensor: 2nd Order
27 <1> 24 <1> 9 <1>
36 <1> 32 <1> 12 <1>
54 <1> 48 <1> 18 <1>
The difference is that ordinary tensor multiplication is different than the tensorial product of mathematical types other then the tensor.
The first one you have to use ‘(*)’ explicitly and the result was matrix
The second one you only used ‘*’ for multiplication and the result was a tensor from the 2nd order (and it called dyadic product)
another headache product called (kronecker product) for matrices
Try this
Qs> fm = [1 2; 3 4]
1 <1> 2 <1>
3 <1> 4 <1>
Qs> sm = [0 5; 6 7]
0 <1> 5 <1>
6 <1> 7 <1>
Qs> fm (*) sm
0 <1> 5 <1> 0 <1> 10 <1>
6 <1> 7 <1> 12 <1> 14 <1>
0 <1> 15 <1> 0 <1> 20 <1>
18 <1> 21 <1> 24 <1> 28 <1>
the result haven’t changed
Note: you may try two regular multiplication between 2nd order tensors but you will get an exception (because I didn’t implement it yet unless I understand)
About understanding all of these I didn’t imagine that I will go into all of this details (so I really walk into it as it appears to me)
I realized that (vectors, marices, and tensors) are another types of quantities
thats why their becomes a must if I would say.
what else ??
yep I tried to speed up things so I tweaked my parser and made a lot of improvements (but as an inner feeling something is not right, the speed is not satisfactory)
that was a long post as usual :)
good to write again :) and see you safe and sound later :)
I have started Symbolic Algebra Project in CodePlex
WHY ??
Because I am a Silly Guy WHO really like to reinvent the WHEEL :D
I think this project will be the same concept of GiNaC
however being .NET natively should make my hear rest in peace this way
I am going to add derivatives also so wait for it to be released
(MAY be AfTeR long Time :) :) )
I am really sorry
I didn’t notice that I haven’t posted how did I solve the dilemma of Torque and Work Problem
however the solution was there in my discussion of quantity System http://quantitysystem.codeplex.com/Thread/View.aspx?ThreadId=23672
and also on this post of my personal blog http://blog.lostparticles.net/?p=28
Problem Core: Length
Simply I defined TWO Length Types
Normal Length (NL)
This is the normal length that we refer to it in our daily life
Radial Length (RL)
This is the radius length that have an origin point in center of circle
Work: Force * Normal Length
Torque: Force * Radial Length
Angle: Normal Length / Radial Length
In quantity system I made the L dimension as NL+RL
I made it explicitly a quantity that is NOT dimensionless
and THIS SOLVED ALL my problems of this Problem
I won’t argue much let us test :)
Torque * Angle = Work
F*RL * (NL/RL) = F * NL <== see what I mean
Torque * Angular Speed = Power
F*RL * (NL/RL*T) = F*NL/T <== where T is the time.
so you may wonder about Angle and Solid Angle
in SI they are all dimensionless but in Quantity System YOU CAN’T consider them like this
(By the way I’ve break the checking for these two quantities to be summable with dimensionless numbers – just to keep the fundamentals as it is although I am not convinced and I may remove it in
future but damn it I need support from any physics guy)
ANGLE : NL/RL
Solid Angle: NL^2/RL^2 because its area over area
Frequency = 1/T
Angular Velocity = (NL/RL)/T
do you remember the conversion between RPM to frequency
RPM (Revolution Per Minute) is a Angle / Time
lets test the law
Omega = 2*pi*frequency
pi: radian value which is NL/RL
Omega = (NL/RL) * 1/T = (NL/RL)/T which angular velocity
PI value
PI is ratio of any circle‘s circumference to its diameter WHICH MEANS (NL/RL)
this is the same value as the ratio of a circle’s area to the square of its radius WHICH MEANS (NL^2/RL^2)
which corresponds to radian unit and stradian unit for angle and solid angle quantities.
Reynolds Number
I am not holding back ( Reynolds number is a dimensionless number that measure inertial forces to viscous forces)
WHY we always differentiating between Flow in Pipes and Flow on Flat plate
let’s see with normal length in quantity system
Qs> rho = 3[Density]
Density: 3 <kg/m^3>
Qs> v=0.5<m/s>
Speed: 0.5 <m/s>
Qs> l=4<m>
Length: 4 m
Qs> mue = 2<Pa.s>
Viscosity: 2 <Pa.s>
Qs> rho*v*l/mue
DimensionlessQuantity: 3 <kg/m.s^2.Pa>
it shows it is a dimensionless quantity
ok let us force my theory about flow in pipes
the pipe have a diameter and Reynolds number is calculated by rho*v*DIAMETER/viscosity
so let me add d as a diameter and solve again
Qs> d=0.5<m!>
RadiusLength: 0.5 m
Qs> rho*v*d/mue
DerivedQuantity: 0.375 <kg/m.s^2.Pa>
NOTE: Adding ‘!’ after the length unit will mark the quantity as Radial Length quantity.
ERROR it shouldn’t be DerivedQuantity at all it should be Dimensionlesss
so there is another term that should be fixed. Do you know which term ???
lets try the velocity with vr=0.5<m!/s>
Qs> vr=0.5<m!/s>
DerivedQuantity: 0.5 <m/s>
Qs> rho*vr*d/mue
DerivedQuantity: 0.375 <kg/m.s^2.Pa>
ok let us try the density
Qs> rhor = 3<kg/m!^3>
DerivedQuantity: 3 <kg/m^3>
Qs> rhor*v*d/mue
SolidAngle: 0.375 <kg/m.s^2.Pa>
CAN YOU SEE THE SOLID ANGLE Quantity
do you remember the above argue about Solid Angle is dimensionless number
can I pretend now that reynolds number for pipes is CORRECT ???
the new density which is kg/m!^3
This is driving me nuts
Pump Affinity Laws
Also pump equations led to the same SolidAngle Quantity not DimensionLess one
I’ve said what I’ve discovered till now about this problem I hope that may be someday someone explain to me or convince me that Angle is really a Dimensionless number after all of this and that my
assumption about Torque and Work is Wrong.
I want to catch up with what I’ve implemented so far.
This is not what’s new, but rather a glimpse on what I’ve done and still remembering it
I really need to organize my thoughts about what is happening in this project
the scattering thoughts are frightening me :S
Tensor Parsing:
I want to add Tensor badly in Qs because I wasn’t able to understand it I had to include it
however I’ve Added Tensor parsing
Qs> <|4 3;3 2| 4 5; 5 3 |>
Yes this is a 3rd rank tensor or you may see it as a two matrices in the Z-Axis
what about 4th Tensor (You may ask) ??!! Bear with me I am still thinking of a natural syntax to add it.
also I would like to remind my self that Stress tensor is a 2nd rank tensor (which looks like matrix but with special operations)
using <|…|> syntax you can define also zero rank tensor up to 3rd rank tensor (what will I do with this?? I don’t KNOW)
yes I included the text at last (I was struggling against putting text in Qs, but it was necessary :S or let me say I couldn’t come up with another strategy.
Qs> ml = “c d e f g a b c>”;
Qs> Music:Play(ml + ” ” ml); #Music:Play is an extension function that play midi and the music library is 100% implemented by me (I mean sequencer part and music DSL as it contains eastern tones
also – another story indeed)
and yes you can add text to text, also imagine you can add text to number
Qs> 4 + “44″
DimensionlessQuantity: 48 <1>
but the opposite is will not act like this (because I am casting to the left type always – which is the first one in fact)
What is meant by adding text is to differentiate between sending text to functions and sending other types.
so Windows:MessageBox function now accept Text
Windows:MessageBox(“Hello There”) will result into messagebox ofcourse (just try it)
to escape quotation mark use back slash “\”hello\””
I like it :) (do you ?!? :):))
Function Extensions
With C# or any .Net language frankly {put static class with static functions under Qs.Modules namespace}
the class name becomes namespace in Qs.
- I’ve enhanced the function binding so you can write your C# native function with native numerical type and I will cast it to you without you notice it ( no more QsValue as a parameter type),
however this will add other processing lag (but who cares :D everything is cached after this)
also I can map vector to array of numbers so use ff(int[] ia) declaration :) it will work
Why I was enhancing the extension functions {simply because I am planning to include OpenGL } asking me why :O :O :O ??
silly, every numerical system has a 3d device to render to it.
but frankly I am thinking of visualizing Scalars, vectors, and tensors into 3d conceptual drawings
however OpenGL is a very far idea
don’t forget the Graph:Plot(xvector, y1vector) it really draws and I consumed a free opensource library to do it.
may be a freakin plot but hey it works and I can visualize some quick samples indeed.
Function Named Arguments
Quantity System Engine can consume different declaration to the function with the same name
however you must change the parameter names
that’s why you can call the function with named arguments (shhsh argument==parameter so I use it interchangeably)
all the following are correct declarations
f(x) = x
f(y) = y^1.5
f(z) = z^2
f(er) = er/4
f(g) = g+9/g^2
Note: the first function name declared is the (default function) the one you call it without specify named arguments
to call any of these functions use the ‘=’ equal sign
Function Variables
Now you can treat the functions as normal variables because they are inherited from QsValue (internal structure of Qs)
so that when declaring
u(x) = x^2
v(x) = x/2
w(x) = x*9
r = @u+@v+@w # <== this sum all functions and generate new function in r
r(x) = x^2+x/2+x*9
and don’t worry about parameters, as they are merged for you with any number of them
all basic operations supported (simply I merge the function declaration text with the desired operator :))
HOWEVER this is going to change after I put the symbolic algebra library :) (another library I am making) :( :(
What about named argument functions??!! (clever question)
imagine you have a x-component speed in u function
u(x) = x
u(y) = y/2
u(z) = z^2.1
you can write
u = @u(x) + @u(y) +@u(z)
and yes
Qs> u(3,4,5) #IT WILL Work
because you may not remember but I encode the functions in the memory based on their arguments number also
so u(x,y,z) now is the default function of u in 3 arguments ;) (impressive for me I admit)
my beloved feature.
sequence now can return a function series
for example
e[n](x) ..> x^n/n!
e[1](1) gives 1
however calling the sequence without specify argument (IN argument enables sequence)
e[1] gives _(x) = x^1/1!
and you can assign it to a function also
let me make it fast
exp = e[0++20]
_(x) = (x^0/0!) + (x^1/1!) + (x^2/2!) + (x^3/3!) + (x^4/4!) + (x^5/5!) + (x^6/6!) + (x^7/7!) + (x^8/8!) + (x^9/9!) + (x^10/10!) + (x^11/11!) + (x^12/12!) + (x^13/13!) + (x^14/14!) + (x^15/15!) + (x^
16/16!) + (x^17/17!) + (x^18/18!) + (x^19/19!) + (x^20/20!)
which when you can call your exp normally
Shifting Operators
I’ve added ‘>>‘ right shift as C family and ‘<<‘ left shift
simply the operators is shifting vectors and matrices
not implemented yet for tensors
and also you will find these operators starting of Qs 1.1.9.94 release Changeset 39690
I hope I listed things that I remembered for now
and chaao :D
In Quantity System Calculator (DLR) I’ve implemented a new way of conditional expression that can be written continuously and fits into one line.
I called it ‘when-otherwise’ expression
‘[true result] when [test] otherwise [false result] when …’
let us define a simple function with it
f(x) = x when x< 10 otherwise x^2 when x<20 otherwise x^5
I know that I am going against the famous ‘if-then’ statement, however ‘if-then’ statement is an imperative syntax, and I needed a functional statement that fits in one line.
when I was thinking about this new syntax I also didn’t ignore the question syntax of C,C++, C#
‘Test?True Part, False Part’ , but also this syntax is somehow complex to use, and give me a headache actually.
thats why I invented it this way. You can see that the result part precede the condition part and the syntax is truly easy to remember (hard to type) but easy to extend.
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