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Distinctive property of the primes 17 and 19?
up vote 12 down vote favorite
Consider the question whether it is true that a prime number $p$ divides $1^1+2^2+3^3+....+(p-1)^{p-1}$ if and only if $p \in \{17,19\}$.
For the obvious heuristic reasons, for large $n$ one would expect there to be roughly $\ln(\ln(n))$ such primes $p < n$, however it seems that presently no examples other than 17 and 19 are known.
Is there a more efficient way of looking for examples than the brute force method of testing the primes one-by-one?
nt.number-theory open-problem prime-numbers
11 Do it yourself. – Fernando Muro May 27 '13 at 7:17
6 See the FAQ's section on open problems. Also, what evidence is there that this is true? If you pick a random number mod each prime, this should be $0$ infinitely often, but the number of times
you pick $0$ among the first $n$ primes will grow as $\log \log n$. So, if you see two examples among the first $1000$ or $10^6$, this is still not much evidence that there is anything special
about those two examples, and that there aren't infinitely many examples. – Douglas Zare May 27 '13 at 8:09
15 I can't do it myself. Please note that after being used to reading problems phrased in this manner in most books, it is perfectly understandable for students (especially foreign) to state
questions as starting with "prove that". – Gjergji Zaimi May 27 '13 at 8:14
8 Gjerji, if it's an open problem then asking us to prove is seems a bit too much. If it's not an open problem, I'd like to see some more motivation behind this question. – Asaf Karagila May 27 '13
at 8:17
I was interested in the determinant (mod $p$) and the trace (mod $p$) of a matrix defined as follows :- For each prime $p$, define a $(p-1)*(p-1)$ integer matrix $M^{p}$ as $M^{p}_{ij} \equiv i^
6 {j}$ (mod $p$), $0<i<p, 0<j<p$. I was surprised to see that the trace was 0 (mod $p$) only for 17 and 19 in the first 3000 primes and thus I thought of posting this question. (I know that 3000 is
very small number and one should not make any blind conjecture based on only 3000 evidences). – Mihir Sheth May 29 '13 at 5:53
show 12 more comments
2 Answers
active oldest votes
exp(exp(3)) is about 5e8 and testing the primes up to that is probably feasible with a few days of computer abuse. I tested up to 1e5 in 6 minutes with a trivial, single-threaded Haskell
up vote 1 script (no more p's found). I don't see any particular reason to think there are no more p's though. It would surprise me if searching didn't turn up another p.
down vote
3 Testing up to $N$ takes time proportional to $N^2$ (the sum mod $p$ is a sum of $p$ terms, each of which takes about $\log p$ operations mod $p$, and there are $N/\log N$ terms to try.
So going from $10^5$ to $5 \cdot 10^8$ would multiply the computing time not by $500$ but by about a quarter-million, unless there's some clever speedup I'm not seeing to a smaller
power than $N^2$. – Noam D. Elkies May 28 '13 at 3:15
One passably clever speedup would memoize mod p the results of $m^m$ and $n^n$ to aid in calculating $mn^{mn}$, but I don't see it reducing the exponent on the runtime. Gerhard "Has Not
Thought It Through" Paseman, 2013.05.27 – Gerhard Paseman May 28 '13 at 6:27
add comment
This should be a comment, but I can't post them, sorry. To follow up the above: Prof. Elkies is of course right about the $O(n^2)\,$ complexity of the basic algorithm, but it should be
possible to get large constant-factor speedups with better implementation:
• First of all there's probably 10x-50x available just by rewriting the program in C using the naive algorithm, and running on a multi-core processor, depending on available hardware.
Let's say 25x on an 8-core AMD processor.
• There might be another 2x(?) using Montgomery's representation (from cryptography) to get rid of most of the integer division operations in the modular exponentials. Actually maybe
up vote 0 a lot more than 2x.
down vote
• Finally, combining the above (I'm less sure of this) it might be possible to run on a graphics accelerator giving 100x or more.
Assuming 50x (perhaps using two or three computers) that would give 250000*6 minutes / 1440 minutes/day = about 3 weeks, which is above my guess of a "few days" but I think still
feasible if someone was really interested. I'm surprised by how many upvotes this thread got.
2 See also arxiv.org/abs/1209.3436 for ideas about possible mathematical speedups. – François Brunault May 30 '13 at 8:31
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Introduction to Linear Regression Analysis
Introduction to Linear Regression Analysis is a textbook intended for an upper-level undergraduate or introductory graduate course in regression analysis. It assumes students have taken an
introductory statistics courses and are familiar with concepts such as hypothesis testing, confidence intervals, and the normal, t, F and chi-squared distributions. It also requires some knowledge of
matrix algebra and access to a computer and statistical software such as SAS or MINITAB.
Introduction to Linear Regression Analysis is organized in two parts. The first ten chapters are the nucleus of the book and cover topics which would typically be included in any regression course,
beginning with an introduction to the basic concepts of linear regression and model building. Topics spill over from one chapter to another, and it is assumed that all ten chapters will be covered in
a basic regression course. In contrast, each of last five chapters are self-contained and focus on a specific topic, such as robust regression, which may be included or omitted depending on the
specific focus of the class. The final chapter includes brief introductions to some more recent topics in regression, including bootstrapping, classification and regression trees, and neural
Introduction to Linear Regression Analysis was developed from course notes for a regression analysis course for students in engineering, chemistry, physical science, statistics, mathematics and
management. Most of the examples pertain to those fields, making it less attractive as a text in the social sciences. It is also more mathematical than the typical statistics texts used in social
science departments. The strongest merits of Introduction to Linear Regression Analysis are the clarity of exposition, including many illustrations and solved examples, and the integration of
computer use into the instruction. Instructions for executing the different types of analyses in SAS and MINITAB are included, as are annotated output from those programs. An appendix also provides a
basic introduction to SAS. Introduction to Linear Regression Analysis is also an excellent reference and could also serve as a self-teaching text for anyone with a basic level of statistical
The webpage for Introduction to Linear Regression Analysis includes electronic versions of the data sets used in the book. An instructor's manual also includes electronic copies of the data sets as
well as solutions to all exercises and suggested examination problems, and the student solutions guide includes solutions to selected exercises.
The authors of Introduction to Linear Regression Analysis have varied backgrounds. Douglas C. Montgomery is a Professor of Industrial & Management Systems Engineering at Arizona State University; his
research interests focus on engineering applications of statistics and operational research methods. Elizabeth A. Peck is a Logistics Modeling Specialist at the Coca-Cola Company in Atlanta, GA. G.
Geoffrey Vining is Professor and Head of the Statistics Department at Virginia Polytechnic Institute; his research.
Sarah Boslaugh, PhD, MPH, is a Performance Analyst for BJH HealthCare in Saint Louis, Missouri. She published An Intermediate Guide to SPSS Programming with Sage in 2005 and is currently editing The
Encyclopedia of Epidemiology for Sage (forthcoming, 2007) and writing Secondary Data Sources for Public Health (forthcoming, 2007) for Cambridge University Press. | {"url":"http://www.maa.org/publications/maa-reviews/introduction-to-linear-regression-analysis","timestamp":"2014-04-17T19:20:07Z","content_type":null,"content_length":"98909","record_id":"<urn:uuid:1d75080e-d2eb-425b-a6da-510ef9a815cb>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00311-ip-10-147-4-33.ec2.internal.warc.gz"} |
RE: st: Survival analysis and control variables
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RE: st: Survival analysis and control variables
From "D.W.Richards" <d.w.richards@open.ac.uk>
To "'statalist@hsphsun2.harvard.edu'" <statalist@hsphsun2.harvard.edu>
Subject RE: st: Survival analysis and control variables
Date Wed, 4 May 2011 13:04:16 +0100
Hi Martin,
Thanks for the feedback. I am trying control for wealth in relation the survival time of losses, not only survival time in general. In this case, does this equation seem logical?
h(t)= ho(t) exp ( LossB1 + lossxageB2 + ageB3 + lossxwealthB4 +wealthB5)
Also do you know if it's possible to get a statistic like an "r-square" to test the difference between the above model and model below:
h(t)= ho(t) exp ( LossB1 + lossxageB2 + ageB3)
Thanks, for the info about intervening v confounding variables. I was probably getting a little off track and you brought me back to my focus. The variable I am hoping to control for through proxies such as wealth and age is "investor sophistication". I am interested in whether some psychological variables (e.g. emotion regulation) have an influence after controlling for this variable in my analysis.
-----Original Message-----
From: maarten buis [mailto:maartenlbuis@googlemail.com]
Sent: 04 May 2011 12:32
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: Survival analysis and control variables
--- Wed, May 4, 2011 at 12:15 PM, D.W.Richards wrote:
> My research involves survival analysis to analyse whether a stock is held for longer
> if it is a gain or a loss. For each stock traded by an investor, I create a time varying
> variable which indicates whether it trades at a loss or not on each day it is held. I
> then interact demographic variables like age, gender, with the loss variable to
> understand whether the demographic variables reduce the tendency to hold stocks
> at a loss. For example, age of the investor holding a stock decreases the survival
> time of stocks at a loss. The equation in a Cox model is:
> h(t)= ho(t) exp ( LossB1 + lossxageB2 + ageB3)
> where ageB3 is a control variable. I am most interested in B1 and B2 to assess
> whether they influence holding of losses.
> My next step which I am stuck on is adding further control variables. For example,
> age decreases the selling of losses and investor wealth also decreases the selling
> of losses. But age is positively correlated with wealth. I want to discover the effect
> of age after controlling for wealth.
> How do I go about doing this?
Just add wealth to your model. So estimate:
h(t)= ho(t) exp ( LossB1 + lossxageB2 + ageB3 + wealthB4)
The real question is: what does age mean nett of wealth and other
control variables? I
find it hard to believe that age has a direct effect. I would think
that age primarily works
through a set of indirect effect, you mentioned wealth, others are
position in the labor
market, risk aversion and time preference. If you start controlling
for these you will just
end up with random noise. Remember, we only want to control for confounding
variables but _not_ for intervening variables.
Hope this helps,
Maarten L. Buis
Institut fuer Soziologie
Universitaet Tuebingen
Wilhelmstrasse 36
72074 Tuebingen
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At A Pressure Of 1 Atm, The Heat Of Vaporization ... | Chegg.com
At a pressure of 1 atm, the heat of vaporization of water is 22,260,000 J/kg; this is the heat required to convert 1 kg of water at 100 celcius into water vapor at the same temperature. Given that
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Two way analysis of variance models can be fitted to data using the R Commander GUI. The general approach is similar to fitting the other types of model in R Commander described in previous posts.
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Source code files accompanying article are located on MacTech CD-ROM or source code disks.
You will have noticed the change in the column s title: a recent reader survey has shown that Forth and Basic are the two languages that our readers would most like to see less of in MacTutor. That s
a shame, but we re responsive (at least we try)
So, from now on my monthly column will have a wider scope. As you might have seen, I have very often used Forth as a vehicle to explain general concepts of Macintosh programming. Since many
subscribers don t seem to be happy with Forth - in fact, people often have asked about things that had been explained in a Forth column, but they just hadn t read. The column about the Notification
Manager in V5#6 is a good example: at the bottom of page 42, one of the ideas to be written was explained as: An example that places a Notification Manager request in low System memory, and
starts a Timer routine ; this is just the example that was in the Forth Forum that covered the Notification Manager. Anyway, I ll try to use other vehicles to convey the message from now on. Such as
assembly, or maybe even C. Mach2 Forth still is a very good assembly-language development system because of its interactivity. Of course, you cannot create complicated macros, or structures, and have
to resort to Forth code for those purposes. I ll still inform you about interesting things I come across on the Forth scene, but this won t be an exclusive Forth column anymore. Emphasis will be on
two things: basic system-level things such as drivers, trap patches, INITs, network, new managers as they come up; and - on the other side of the spectrum - object-oriented programming in C++.
Apple will Real Soon Now release a C++ under MPW, and we ll hopefully have a pre-release by the time you read this. C++ is a very interesting language, much more than a simple extension of C;
reading Stroustrup s book I got this feeling of yes, that s how one should have done it in the first place that I had 15 years ago when all I knew was Algol 60, and came across the description of
Algol 68. Sadly enough, Algol 68 never really caught on; hopefully, C++ will. The C++ column will start with the next issue; including program examples if we get the pre-release soon enough, just
dry swimming if not.
This month, we ll talk once more about one of my favorite subjects, number crunching, speed (or the lack of it), and intelligence in compilers (or the lack of it).
Since I am doing more and more everyday computation (mostly Fortran) on the MacII, I m obviously interested in a good optimizing compiler. Now, a standard trick that every decent compiler should have
in its repertoire is the elimination of constant expressions from loops, or assignment of array elements that are not dependent on the loop index to intermediate variables.
Imagine my surprise when I found out that (in Language Systems Fortran 1.2.1) I could speed up a loop that looked like this:
do k=1,n3 do i=1,n1 c(i,k) = 0x0 do j=1,n2 c(i,k) = c(i,k)+a(i,j)*b(j,k) end do end do end do
do k=1,n3 do i=1,n1 sum = 0x0 do j=1,n2 sum = sum+a(i,j)*b(j,k) end do c(i,k) = sum end do end do
Now, in undergraduate programming classes, years ago, we were actually taught to look for constant arithmetical or index expressions in loops and put them outside if possible. Today, almost everybody
assumes that the compiler is smart enough to take care of that; incorrectly, as you see. To see how good the compilers available under MPW can do, I wrote a Fortran program (listing 1) that calls
several versions of this matrix multiplication loop, written in Fortran (Lang. Sys. 1.2.1), Pascal (Apple 3.0 beta), and C (Apple 3.0 beta). Surprise: none of the compilers was good enough to move
the indexing outside of the loop. The following table gives the results (Mac IIx):
A difference of more than a factor of 2 between the slowest Fortran and the fastest Pascal code. Apple Pascal lived up to its good reputation here, but even that could be improved a lot by
eliminating the constant index expression.
Surprised, I ran the Fortran benchmark on a Microvax II, and found that even there some speed could be gained by hand-optimizing the code:
However, the difference between the machine-optimized and the hand-optimized version is not quite as big as for the MPW languages (15% for the VAX vs. 27-30% for MPW). If you compile the VAX code
without optimization, you get a bigger difference (23%):
Therefore, take-home lesson one: don t take compiler optimizations for granted.
Benchmarks have been run on lots of different machines, using lots of different compilers. I was interested in how the code generated by the MPW compilers actually differed. A job for Nosy, and the
results are shown in the last listing. I ve only printed the innermost loops. Don t be overwhelmed by the pile of assembly code, just note some important details.
First, for the loop optimization examples discussed here, there seems to be no tradeoff between code length and speed. On the contrary, the fastest code is also the shortest. On the other hand, there
are some obvious pieces of code which are clearly redundant. The most blatant example is the Fortran-generated code at the end of the listing, where an index expression is recalculated that was
actually in register A1 all the time! 14 extra lines of machine code on each pass through the loop will add up to quite some extra time lost. Another point is that Language System obviously has no
great trust in the quality of the 68000/20/30, otherwise how can one explain that they repeat the EXT.L D2 instruction each time it occurs? To make sure it works at least once?
Language Systems Fortran makes other funny assumptions about the machine, for instance it seems to think there are only two floating point registers in the 68881, FP0 and FP7. I have looked at some
code which had great potential for optimization by using enough FP registers. Language Systems is, however, known for its responsiveness towards customers, so I hope we won t have to wait too long
until a well-optimized Fortran shows up.
Both Pascal and C like juggling floating point registers. Why generate (like Apple s C):
when a simple FADD FP0,FP7 would suffice? Eliminates two floating point instructions per loop. Pascal does
would give the same result. One floating point instruction per loop eliminated. The timing difference between the Pascal and C routines is partly because of the one extra floating point instruction.
Last remark: I haven t seen the Absoft MPW 3.0 Fortran yet. If anyone from Absoft is reading this, I d like an evaluation copy to run the same analysis (since you claim in your ads you have such a
great optimizer). If I get enough other languages collected together, we ll have a follow-up on this article.
The MacHack is over (thanks, Aimée, Carol, and all the others, for organizing such a good meeting), and I ll tell you some of my impressions in the next column. Otherwise, we ll start with an
introduction to C++; I hope the compiler will arrive here in time.
Listing 1: Matrix multiplication benchmark !!S Main program matrix c c Main program in Language Systems Fortran c c Some line breaks in the Fortran program are due to c editing. c implicit none
integer i,j,ticks1,ticks2 extended a(50,50), b(50,50), c(50,50) extended time1,time2 integer ticks type *, Matrix multiplication benchmark type *, ------------------------------ type * type *, This
program compares the number crunching power type *, of some of the popular MPW compilers. type *, Written under MPW 3.0 by J. Langowski / MacTutor 1989' type * type *, Setting up 50x50 matrices...
ticks1 = ticks() do i=1,50 do j=1,50 a(i,j) = (i-1) + j-1 b(j,i) = a(i,j) end do end do ticks2 = ticks() time1 = (ticks2-ticks1)/60. type * write (6, (f8.4, seconds for setting up matrices ) )
time1 ticks1 = ticks() call mat_mult_for3(c,50,a,50,b,50,50,50,50) ticks2 = ticks() time1 = (ticks2-ticks1)/60. type * write (6, (f8.4, seconds for multiplying matrices , * using FORTRAN
routine, opt=3' ) ) time1 type *, c(25,25) = ,c(25,25) ticks1 = ticks() call mat_mult_for(c,50,a,50,b,50,50,50,50) ticks2 = ticks() time1 = (ticks2-ticks1)/60. type * write (6, (f8.4, seconds for
multiplying matrices , * using FORTRAN routine, opt=0' ) ) time1 type *, c(25,25) = ,c(25,25) ticks1 = ticks() call mat_mult_for1(c,50,a,50,b,50,50,50,50) ticks2 = ticks() time1 =
(ticks2-ticks1)/60. type * write (6, (f8.4, seconds for multiplying matrices , * using FORTRAN routine, hand-optimized ) ) time1 type *, c(25,25) = ,c(25,25) ticks1 = ticks() call
mat_mult_for0(c,50,a,50,b,50,50,50,50) ticks2 = ticks() time1 = (ticks2-ticks1)/60. type * write (6, (f8.4, seconds for multiplying matrices , * using FORTRAN routine, constant dimensions ) )
time1 type *, c(25,25) = ,c(25,25) ticks1 = ticks() call mat_mul_pas(c,%val(50),a,%val(50),b,%val(50),%val(50),%val(50),%val(50)) ticks2 = ticks() time1 = (ticks2-ticks1)/60. type * write (6, (f8.4,
seconds for multiplying matrices , * using PASCAL routine ) ) time1 type *, c(25,25) = ,c(25,25) ticks1 = ticks() call mat_mul_pas_opt(c,%val(50),a,%val(50),b,%val(50),%val(50),%val(50),%val
(50)) ticks2 = ticks() time1 = (ticks2-ticks1)/60. type * write (6, (f8.4, seconds for multiplying matrices , * using PASCAL routine, hand-optimized ) ) time1 type *, c(25,25) = ,c(25,25)
ticks1 = ticks() call mat_mul_c(c,%val(50),a,%val(50),b,%val(50),%val(50),%val(50),%val(50)) ticks2 = ticks() time1 = (ticks2-ticks1)/60. type * write (6, (f8.4, seconds for multiplying matrices ,
* using C routine ) ) time1 type *, c(25,25) = ,c(25,25) ticks1 = ticks() call mat_mul_c_opt(c,%val(50),a,%val(50),b,%val(50),%val(50),%val(50),%val(50)) ticks2 = ticks() time1 = (ticks2-ticks1)
/60. type * write (6, (f8.4, seconds for multiplying matrices , * using C routine, hand-optimized ) ) time1 type *, c(25,25) = ,c(25,25) end !!S Main subroutine mat_mult_for3
(c,nc,a,na,b,nb,n1,n2,n3) c sets c=a.b c na,nb,nc are first dimensions c n1 n2 n3 are problem dimensions c c is n1xn3 c a n1 n2 c b n2 n3 implicit none integer na,nb,nc,n1,n2,n3 integer*2 i,j,k
extended c(nc,n3),a(na,n2),b(nb,n3) do k=1,n3 do i=1,n1 c(i,k) = 0x0 do j=1,n2 c(i,k) = c(i,k)+a(i,j)*b(j,k) end do end do end do return end !!S Main subroutine mat_mult_for1(c,nc,a,na,b,nb,n1,n2,n3)
c csame as before, invariant matrix element eliminated from loop c implicit none integer na,nb,nc,n1,n2,n3 integer*2 i,j,k extended c(nc,n3),a(na,n2),b(nb,n3) extended sum do k=1,n3 do i=1,n1 sum =
0x0 do j=1,n2 sum = sum+a(i,j)*b(j,k) end do c(i,k) = sum end do end do return end !!S Main subroutine mat_mult_for0(c,nc,a,na,b,nb,n1,n2,n3) c csame as before, with constant dimensions c implicit
none integer na,nb,nc,n1,n2,n3 integer*2 i,j,k extended c(50,50),a(50,50),b(50,50) extended sum do k=1,n3 do i=1,n1 sum = 0x0 do j=1,n2 sum = sum+a(i,j)*b(j,k) end do c(i,k) = sum end do end do
return end !!S Main integer function ticks ticks = long(362) return end
Listing 2 : non-optimized Fortran routine !!S Main subroutine mat_mult_for(c,nc,a,na,b,nb,n1,n2,n3) c cduplicate of mat_mul_for3 for compiling without optimization c implicit none integer
na,nb,nc,n1,n2,n3 integer*2 i,j,k extended c(nc,n3),a(na,n2),b(nb,n3) do k=1,n3 do i=1,n1 c(i,k) = 0x0 do j=1,n2 c(i,k) = c(i,k)+a(i,j)*b(j,k) end do end do end do return end
Listing 3 : Pascal routine {$S Main} {$R-} unit matmul; interface type matrix = array [1..50,1..50] of extended; procedure mat_mul_pas (var c : matrix; nc : longint; var a : matrix; na : longint; var
b : matrix; nb : longint; n1,n2,n3:longint); procedure mat_mul_pas_opt (var c : matrix; nc : longint; var a : matrix; na : longint; var b : matrix; nb : longint; n1,n2,n3:longint); implementation
procedure mat_mul_pas; var i,j,k:integer; begin for k:=1 to n3 do for i:=1 to n1 do begin c[i,k] := 0; for j:=1 to n2 do c[i,k] := c[i,k]+a[i,j]*b[j,k]; end; end; procedure mat_mul_pas_opt; var
i,j,k:integer; sum:extended; begin for k:=1 to n3 do for i:=1 to n1 do begin sum := 0; for j:=1 to n2 do sum := sum+a[i,j]*b[j,k]; c[i,k] := sum; end; end; end.
Listing 4 : C routine pascal void mat_mul_c (extended c[50][], long nc, extended a[50][], long na, extended b[50][], long nb, long n1, long n2, long n3) { int i,j,k; for ( k=1 ; k <= n3; k++ ) for (
i=1 ; i <= n1 ; i++ ) { c[i][k] = 0.0; for ( j=1 ; j <= n2 ; j++ ) c[i][k] = c[i][k]+a[i][j]*b[j][k]; } } pascal void mat_mul_c_opt (extended c[50][], long nc, extended a[50][], long na, extended b
[50][], long nb, long n1, long n2, long n3) { register int i,j,k; register extended sum; for ( k=1 ; k <= n3; k++ ) for ( i=1 ; i <= n1 ; i++ ) { sum = 0.0; for ( j=1 ; j <= n2 ; j++ ) sum = sum+a[i]
[j]*b[j][k]; c[i][k] = sum; } }
Listing 5 : inner loops compared, Nosy-disassembled pascal, optimized lan_3 MOVEA.L param2(A6),A0 MOVE D6,D0 MULS #$258,D0 MOVE D5,D1 MULS #12,D1 ADD D1,D0 MOVEA.Lparam3(A6),A1 MOVE D5,D1 MULS #
$258,D1 MOVE D7,D2 MULS #12,D2 ADD D2,D1 LEA -$264(A0),A0 FMOVE.X0(A0,D0.W),FP0 LEA -$264(A1),A0 FMUL.X 0(A0,D1.W),FP0 FADD FP7,FP0 ; could use FMOVE FP0,FP7 ; FADD FP0,FP7 here ADDQ #1,D5 BVS.S
lan_5 lan_4 CMP.W van_1(A6),D5 BLE lan_3 c, optimized lar_1 MOVE.LD7,D0 MOVE.L D0,D1 MULU #12,D0 SWAP D1 MULU #12,D1 SWAP D1 CLR D1 ADD.L D1,D0 ADD.L D6,D0 MOVE.L D5,D1 MOVE.L D1,D2 MULU #12,D1 SWAP
D2 MULU #12,D2 SWAP D2 CLR D2 ADD.L D2,D1 ADD.L D7,D1 FMOVE.X 0(A3,D0.L),FP0 FMUL.X 0(A4,D1.L),FP0 FMOVE FP7,FP1 FADD FP0,FP1 FMOVE FP1,FP7 ADDQ.L #1,D7 lar_2 CMP.L D7,D4 BGE lar_1 pascal, plain
lam_3 MOVEA.L param2(A6),A0 MOVE D6,D0 MULS #$258,D0 MOVE D5,D1 MULS #12,D1 ADD D1,D0 MOVEA.L param3(A6),A1 MOVE D5,D1 MULS #$258,D1 MOVE D7,D2 MULS #12,D2 ADD D2,D1 LEA -$264(A0),A0 FMOVE.X 0
(A0,D0.W),FP0 LEA -$264(A1),A0 FMUL.X 0(A0,D1.W),FP0 MOVE D6,D0 MULS #$258,D0 MOVE D7,D1 MULS #12,D1 ADD D1,D0 LEA -$264(A4),A0 FADD.X 0(A0,D0.W),FP0 MOVE D6,D0 MULS #$258,D0 MOVE D7,D1 MULS #12,D1
ADD D1,D0 LEA -$264(A4),A0 FMOVE.X FP0,0(A0,D0.W) ADDQ #1,D5 BVS.S lam_5 lam_4 CMP.W vam_1(A6),D5 BLE lam_3 c, plain lao_3 MOVE.LD5,D0 MOVE.L D0,D1 MULU #12,D0 SWAP D1 MULU #12,D1 SWAP D1 CLR D1
ADD.L D1,D0 ADD.L D6,D0 MOVE.L D7,D1 MOVE.L D1,D2 MULU #12,D1 SWAP D2 MULU #12,D2 SWAP D2 CLR D2 ADD.L D2,D1 ADD.L D6,D1 MOVEA.L param3(A6),A0 MOVE.L D5,D2 MOVE.L D2,D3 MULU #12,D2 SWAP D3 MULU #
12,D3 SWAP D3 CLR D3 ADD.L D3,D2 ADD.L D7,D2 FMOVE.X 0(A4,D1.L),FP0 FMUL.X 0(A0,D2.L),FP0 FADD.X 0(A3,D0.L),FP0 MOVE.L D5,D0 MOVE.L D0,D1 MULU #12,D0 SWAP D1 MULU #12,D1 SWAP D1 CLR D1 ADD.L D1,D0
ADD.L D6,D0 FMOVE.X FP0,0(A3,D0.L) ADDQ.L #1,D7 lao_4 CMP.L D7,D4 BGE lao_3 Fortran, optimized lah_3 MOVE-172(A6),D2 EXT.L D2 EXT.L D2 SUB.L -142(A6),D2 MULS.L #12,D2 MOVE.L D2,D0 MOVE -170(A6),D2
EXT.L D2 EXT.L D2 SUB.L -130(A6),D2 MULS.L -134(A6),D2 ADD.L D0,D2 MOVEA.L 32(A6),A0 ADDA.L D2,A0 FMOVE.X (A0),FP7 MOVE -170(A6),D2 EXT.L D2 EXT.L D2 SUB.L -118(A6),D2 MULS.L #12,D2 MOVE.L D2,D1 MOVE
-168(A6),D2 EXT.L D2 EXT.L D2 SUB.L -106(A6),D2 MULS.L -110(A6),D2 ADD.L D1,D2 MOVEA.L 24(A6),A1 ADDA.L D2,A1 FMUL.X (A1),FP7 FADD.X -94(A6),FP7 FMOVE.X FP7,-94(A6) ADDQ #1,-170(A6) SUBQ.L #1,D5 BGT
lah_3 Fortran, plain lae_3 MOVE-164(A6),D2 EXT.L D2 EXT.L D2 SUB.L -134(A6),D2 MULS.L #12,D2 MOVE.L D2,D0 MOVE -162(A6),D2 EXT.L D2 EXT.L D2 SUB.L -122(A6),D2 MULS.L -126(A6),D2 ADD.L D0,D2 MOVEA.L
32(A6),A0 ADDA.L D2,A0 FMOVE.X (A0),FP7 ; get a(i,k) MOVE -162(A6),D2 EXT.L D2 EXT.L D2 SUB.L -110(A6),D2 MULS.L #12,D2 MOVE.L D2,D1 MOVE -160(A6),D2 EXT.L D2 EXT.L D2 SUB.L -98(A6),D2 MULS.L -102
(A6),D2 ADD.L D1,D2 MOVEA.L 24(A6),A1 ADDA.L D2,A1 FMUL.X (A1),FP7 ; multiply by b(i,k) MOVE -164(A6),D2 EXT.L D2 EXT.L D2 SUB.L -158(A6),D2 MULS.L #12,D2 MOVE.L D2,D1 MOVE -160(A6),D2 EXT.L D2 EXT.L
D2 SUB.L -146(A6),D2 MULS.L -150(A6),D2 ADD.L D1,D2 MOVEA.L 40(A6),A1 ADDA.L D2,A1 FADD.X (A1),FP7 ; add c(i,k) MOVE -164(A6),D2; this EXT.L D2; whole EXT.L D2; stuff SUB.L -158(A6),D2; is MULS.L #
12,D2 ; MOVE.L D2,D1 ; R MOVE -160(A6),D2; E EXT.L D2; D EXT.L D2; U SUB.L -146(A6),D2; N MULS.L -150(A6),D2; D ADD.L D1,D2 ; A MOVEA.L 40(A6),A1; N ADDA.L D2,A1 ; T !!!! FMOVE.X FP7,(A1) ; put back
c(i,k) ADDQ #1,-162(A6) SUBQ.L #1,D5 BGT lae_3
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he Birthday Paradox
I have a couple problems with the code below and was hoping you could take a look at it.
First problem:
Run-Time Check Failure #2 - Stack around the variable 'Set' was corrupted.
I get this when I select menu 2 to check the birthday paradox.
Second problem:
When the results are displayed, my matches are too high. I should be getting 50%, maybe a little more, out of a set of 1000. I am getting more matches between 600 and 700 out of a 1000. So my
"paradox" is too high.
Third problem:
For the life of me, I cannot discover why my Average formula isn't working in the DisplayBirthdayParadoxResults function. It always comes out to 0% when it is displayed.
#include <iostream>
#include <ctime>
#include <iomanip>
using namespace std;
void DisplayMenu();
void MenuChoice(char Choice);
void VerifyBirthdayParadox();
void DisplayBirthdaySet();
void GenerateBirthdaySet(int BirthdaySet[]);
void SelectionSort(int List[], int Last);
void DisplayBirthdayParadoxResults(int Matches);
const int SAMPLE_SIZE = 23;
const int NUMBER_OF_SETS = 1000;
const int DASH_LENGTH = 60;
int main()
char Choice;
Choice = '\0';
while(Choice != 'e' && Choice != 'E');
void DisplayMenu()
cout<<"1) Explain birthday paradox\n";
cout<<"2) Check birthday paradox by generating 1000 sets of birthdays\n";
cout<<"3) Display one set of 23 birthdays\n";
cout<<"E) Exit\n\n";
void MenuChoice(char Choice)
case '1':
cout<<"If 23 people are chosen at random, then the chances are more than 50%\n"
<<"that at least two will have the same birthday!\n\n";
case '2':
case '3':
cout<<"Menu 3\n\n";
case 'E':
case 'e':
cout<<"Invalid selection\n\n";
void VerifyBirthdayParadox()
int Set[SAMPLE_SIZE];
int NumberOfMatches = 0;
for(int k = 0; k <= NUMBER_OF_SETS; k++)
for(int j = 0; j <= SAMPLE_SIZE; j++)
if(Set[j] == Set[j - 1])
void GenerateBirthdaySet(int BirthdaySet[])
for(int i = 0; i < SAMPLE_SIZE; i++)
BirthdaySet[i] = (rand() % 365) + 1;
SelectionSort(BirthdaySet, SAMPLE_SIZE);
void DisplayBirthdayParadoxResults(int Matches)
int Average = (Matches / NUMBER_OF_SETS) * 100;
char DashLine[DASH_LENGTH];
cout<<"Generating "<<NUMBER_OF_SETS<<" sets of "<<SAMPLE_SIZE
<<" birthdays and checking for matches...\n\n";
cout<<"Results : "<<Matches<<" out of "<<NUMBER_OF_SETS<<" ("
<<"%) of the sets contained matching birthdays\n";
for(int i = 0; i < DASH_LENGTH; ++i)
DashLine[i] = '=';
Thanks ahead of time for any help you can bring me to this. | {"url":"http://www.dreamincode.net/forums/topic/293986-c-finding-the-birthday-paradox/","timestamp":"2014-04-19T15:25:08Z","content_type":null,"content_length":"148750","record_id":"<urn:uuid:5118c76b-b647-401d-8ad5-0972f7dbec92>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00208-ip-10-147-4-33.ec2.internal.warc.gz"} |
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hey can you see if I got this right 2. –6 + 3x = –9 (1 point)–1 –6 –5 –3 I think it is -5
• one year ago
• one year ago
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is ur question? 6+3x=-9?
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4. –3x + 6 = –9 (1 point)–3 5 3 1 5. Paul rented a car for $129.00 plus $0.25 per mile. The total bill at the end of his trip was $216.50. Use the equation 129 + 0.25x = 216.50 to find the number
of miles he drove. (1 point)1,382 miles 350 miles 864 miles 607 miles I think it is -3 and 607
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ur answer -5 is right if ur question is 6+3x=-9
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it is how about the other ones
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what is ur answer for 4th ques?
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for number four it is 5 and for the last one it is 350
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ya 4th is right too:)
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5th is too right:) very good:)
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Perfect set property for projective hierarchy
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.
Is there any paper discussing the consistency strength (or possible equivalents, maybe large cardinals) of just assuming the perfect set property for certain levels of the
projective hierarchy?
up vote 7 down vote
favorite set-theory descriptive-set-theory lo.logic
add comment
Is there any paper discussing the consistency strength (or possible equivalents, maybe large cardinals) of just assuming the perfect set property for certain levels of the projective hierarchy?
Analytic sets have the perfect set property, provable in, say, ZF+DC. This goes back to Suslin, and is discussed in Kanamori's book "The higher infinite" (Around section 12).
Large cardinals imply that projective sets have the perfect set property, but the cardinals needed (Woodin cardinals) are more than those needed to establish mere consistency--an
inaccessible suffices, of course, by Solovay's result: If $\kappa$ is inaccessible, and ${\mathbb P}$ is the Levy collapse that makes countable all cardinals below $\kappa$, then in the
forcing extension by ${\mathbb P}$, all sets of reals in $L({\mathbb R})$ have the classical regularity properties (Lebesgue measurability, the property of Baire, and the perfect set
property). (Solovay's result is stronger than I've stated, this is discussed in detail in sections 10 and 11 of Kanamori's book.)
In fact, for sets of reals in $L({\mathbb R})$, an inaccessible is required as well. This goes back to Specker, the point being that if $\aleph_1\le{\mathfrak c}$, then there is a set of
reals without the perfect set property, so $\omega_1^{L[r]}<\omega_1$ for all reals $r$, and it follows that $\omega_1$ is inaccessible in all $L[r]$.
For (boldface) $\Sigma^1_2$ sets, sharps for reals imply the result and already an inaccessible is needed in consistency strength: Solovay showed that if $A$ is $\Sigma^1_2$ in a
up vote 12 parameter $r$, then either $A\in L[r]$ or else $A$ contains a perfect set. If $A$ is the standard well-ordering of ${\mathbb R}\cap L[r]$, then $A$ is $\Sigma^1_2$ in $r$. But then, if $
down vote \omega_1^{L[r]}=\omega_1$, we have that $A$ does not have a perfect subset and is uncountable (This can be unearthed from Section 14 of Kanamori's book). It follows that already at the $
accepted \Sigma^1_2$ level we need $\omega_1$ to be inaccessible in all $L[r]$.
In summary: An inaccessible is precisely what is needed.
Bartoszynski and Judah' book "Set theory. On the structure of the real line" discusses extensively the effect of Martin's axiom (and large cardinals) on the regularity properties of the
projective sets, but it focuses on the Baire property and Lebesgue measurability. David Milovich asked a closely related question not long ago, and you may want to look at the answer
However, I suspect the real question is what is the consistency strength of, say, "the $\Sigma^1_5$ sets have the perfect set property but the $\Sigma^1_6$ do not." I think this is wide
open in general. It is also open with Lebesue measurability or Baire property in place of the perfect set property, as far as I know.
add comment
Analytic sets have the perfect set property, provable in, say, ZF+DC. This goes back to Suslin, and is discussed in Kanamori's book "The higher infinite" (Around section 12).
Large cardinals imply that projective sets have the perfect set property, but the cardinals needed (Woodin cardinals) are more than those needed to establish mere consistency--an inaccessible
suffices, of course, by Solovay's result: If $\kappa$ is inaccessible, and ${\mathbb P}$ is the Levy collapse that makes countable all cardinals below $\kappa$, then in the forcing extension by ${\
mathbb P}$, all sets of reals in $L({\mathbb R})$ have the classical regularity properties (Lebesgue measurability, the property of Baire, and the perfect set property). (Solovay's result is stronger
than I've stated, this is discussed in detail in sections 10 and 11 of Kanamori's book.)
In fact, for sets of reals in $L({\mathbb R})$, an inaccessible is required as well. This goes back to Specker, the point being that if $\aleph_1\le{\mathfrak c}$, then there is a set of reals
without the perfect set property, so $\omega_1^{L[r]}<\omega_1$ for all reals $r$, and it follows that $\omega_1$ is inaccessible in all $L[r]$.
For (boldface) $\Sigma^1_2$ sets, sharps for reals imply the result and already an inaccessible is needed in consistency strength: Solovay showed that if $A$ is $\Sigma^1_2$ in a parameter $r$, then
either $A\in L[r]$ or else $A$ contains a perfect set. If $A$ is the standard well-ordering of ${\mathbb R}\cap L[r]$, then $A$ is $\Sigma^1_2$ in $r$. But then, if $\omega_1^{L[r]}=\omega_1$, we
have that $A$ does not have a perfect subset and is uncountable (This can be unearthed from Section 14 of Kanamori's book). It follows that already at the $\Sigma^1_2$ level we need $\omega_1$ to be
inaccessible in all $L[r]$.
Bartoszynski and Judah' book "Set theory. On the structure of the real line" discusses extensively the effect of Martin's axiom (and large cardinals) on the regularity properties of the projective
sets, but it focuses on the Baire property and Lebesgue measurability. David Milovich asked a closely related question not long ago, and you may want to look at the answer there.
However, I suspect the real question is what is the consistency strength of, say, "the $\Sigma^1_5$ sets have the perfect set property but the $\Sigma^1_6$ do not." I think this is wide open in
general. It is also open with Lebesue measurability or Baire property in place of the perfect set property, as far as I know.
The classical argument Andres mentions can be taken one step further to show that if every uncountable $\Pi_1^1$ set contains a perfect subset then $\aleph_1$ is inaccessible in $L$.
up vote 10 This follows from the same fact for $\Sigma^1_2$-sets since, by the uniformization theorem, every $\Sigma^1_2$-set is the injective projection of a $\Pi^1_1$-set.
down vote
add comment
The classical argument Andres mentions can be taken one step further to show that if every uncountable $\Pi_1^1$ set contains a perfect subset then $\aleph_1$ is inaccessible in $L$. This follows
from the same fact for $\Sigma^1_2$-sets since, by the uniformization theorem, every $\Sigma^1_2$-set is the injective projection of a $\Pi^1_1$-set. | {"url":"http://mathoverflow.net/questions/72252/perfect-set-property-for-projective-hierarchy/72531","timestamp":"2014-04-19T22:15:27Z","content_type":null,"content_length":"61067","record_id":"<urn:uuid:1b4b4095-69ff-4eae-b336-131ee11c6cce>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00391-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Strength of Nonstandard Analysis
Nonstandard analysis has its roots, or, more properly, its foreshadowing, in the formulation of the infinitesimal calculus, independently by Leibniz and Newton in the 17^th century. Newton,
performing a famous act of prudence, took great care to phrase his published arguments about fluxions and fluents (and other such calculus fauna) in purely geometric or physical terms. Was he trying
to avoid the inherent difficulties he knew would be raised by the appearance of infinitesimals on the scene in this new way of doing mathematics?
Newton was right to be prudent, of course, much as Leibniz was perhaps less circumspect in this regard, for infinitesimals indeed caused controversy from the moment of their first appearance; one
need but recall Bishop Berkley’s allusion to “ghosts of departed quantities.” Scholars were immediately cognizant of the fact that the admittedly brilliantly successful new methods of Newton and
Leibniz were based on notions that were in dire need of deep investigation, echoing nothing less than the concerns expressed already by the ancient Greeks in this connection. Zeno’s paradoxes come to
mind right away, of course, and one is reminded of Euclid’s bizarre and unsatisfying phrasing of the notions of point and line in the Elements. Already at that early stage of history the dangers
presented by the infinitely large and infinitely small were recognized and taken note of.
Still, even before the foundations were properly lain, the mathematics was right: Euclidean geometry was right, and so was the infinitesimal calculus. And it is irresistible to note that even in much
more recent (and, presumably, rigorous) times the state of affairs is not dissimilar: in quantum mechanics Dirac’s delta function entered the scene well before Laurent Schwartz defined distributions.
In any case, the presence of infinities and infinitesimals in Mathematics had already been a bête noir throughout history when, in the face of the incomparable successes of 18^th century analysis,
this state of affairs grew nigh on intolerable. In the beautifully written foreword to the book under review, W. A. J. Luxemburg quotes Lagrange, who, in 1774, about a century after the birth of
calculus, asked for “[a] clear and precise theory of what is known as ‘Infinity’ in Mathematics.” Lagrange was at the time the Head of the Mathematics section of the Berlin Academy of Sciences and an
explicit challenge was issued in the form of a prize contest to give “an explanation why it is that so many correct theorems have been deduced from [the] contradictory assumption [of the existence of
an ‘infinite magnitude’].” The award was fifty ducats.
Luxemburg claims that the Berlin Academy’s contest was not properly settled until the appearance of Abraham Robinson’s famous paper, “Non-standard analysis,” in 1961, one hundred seventy five years
after the contest’s 1786 deadline. The fifty ducats did not go unclaimed, however: the Swiss mathematician Simon l’Hullier won with an essay titled “The infinite is the abyss in which our thoughts
are engulfed.” But the prize committee noted explicitly that the pressing if subtle mathematical question of why so much real and correct mathematics had emerged, and was emerging still, from such
ill-defined foundations had gone unaddressed in each of the essays submitted for the prize.
However, in due course, in the immensely fertile 19^th century, classical analysis did of course receive a fully rigorous and gloriously elegant foundation. Says Luxemburg, “The construction of the
real number system (linear continuum) by Cantor and Dedekind in 1872 and the Weierstrass ε–δ techniques gradually replaced the use of infinitesimals,” and we are on very familiar ground: this is the
standard analysis we learn in school today, and it has been this way for on the order of a century. This raises the obvious question of why we should concern ourselves with non-standard analysis at
this stage of mathematical history.
The question could — or should — be asked of Abraham Robinson himself, and Luxemburg provides the answer in connection with work by Thoralf Skolem concerning models of Peano arithmetic possessing
“infinitely large numbers:”
Robinson, rereading Skolem’s paper, wondered what systems would emerge if he would apply Skolem’s method to the axiom system of the real numbers. In doing so, Robinson immediately realized that…
in particular, the set of infinitesimals lacking a least upper bound was an external set [meaning that the set was not realizable by a formula in the according formal language].”
So, in a certain sense, nonstandard analysis is “natural.”
We admittedly discover in Robinson’s nonstandard analysis a framework for calculus (and analysis) in a form that might particularly please Leibniz and, possibly to a lesser extent, Newton. But, being
so deeply steeped in mathematical logic, nonstandard analysis has come to suffer the same status vis à vis mainstream mathematics as does logic itself. Luxemburg quotes Augustus De Morgan: “We know
that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact sciences are mathematics and logic; the mathematical sect puts out the logical eye; the logical sect
puts out the mathematical eye; each believing it can see better with one eye than two.” Luxemburg observes that “[w]e owe Abraham Robinson a great deal for having taught us the use of both eyes… This
book shows clearly that we have learned our lesson well.”
And this brings us, at last, to The Strength of Nonstandard Analysis , edited by Imme van den Berg and Vítor Neves, and suffice it to say right off that it is an interesting and valuable book, or,
more properly, collection of articles. One is immediately struck by the introductory article by the prominent model theorist, H. Jerome Keisler, (also) entitled “The strength of nonstandard
analysis,” and the short note, “The virtue of simplicity,” by Edward Nelson, who, as Luxemburg notes in the foreword, gets credit for having given a full axiom system for Robinson’s “non-standard
methodology” in 1977.
Keisler’s paper, which exemplifies hard-core mathematical logic with a vengeance, is concerned with a framework for “reverse mathematics,” a subject originating in the 1970s with Harvey M. Friedman
and Steve G. Simpson, about which Kazuyuki Tanaka says (MR1770738) that “the ultimate goal… is to answer the question: Which sets of logical principles are necessary and sufficient to prove the
theorems of ordinary mathematics?”
Nelson strikes an entirely different note in his contribution, as is illustrated by his comment to the effect that “[a] prime example of unnecessary complication is, in my opinion, Kolmogorov’s
foundational work on probability expressed in terms of Cantor’s set theory and Lebesgue’s measure theory.” Nelson goes on to provide nonstandard analytical proofs of some core theorems in the field,
including Radon-Nikodym.
The two articles just mentioned belong to the section, “Foundations,” the first of five, with the other four being, in order, “Number theory,” “Statistics, probability, and measure,” “Differential
systems and equations,” and “Infinitesimals and education.” The articles in these sections are all of interest, with some obviously being more eclectic (or even austere) than others. Here is a random
sampling of topics touched on: the Erdös-Turan conjecture (that if {a(n)}⊂ N and ∑a(n)^-1 < ∞, then {a(n)} contains arbitrarily long arithmetic progressions), quantum Bernoulli experiments and
quantum stochastic processes, the Navier-Stokes equations, and (more on) Radon-Nikodym. The Strength of Nonstandard Analysis ends with two thought-provoking articles by Keith Stroyan and Richard
O’Donovan, respectively, concerning the possible use of infinitesimals and nonstandard analysis in the teaching of calculus and pre-university analysis.
Although the reader should be cautioned that the level of difficulty in the articles varies considerably, the book contains interesting material for almost everyone and is well worth looking at. But
a good deal of preparation may be indicated for the reader, especially as regards model theory, for example. (The last two articles may be something of an exception.)
Nonstandard analysis is an intrinsically fascinating subject deserving a lot more airplay. The Strength of Nonstandard Analysis is a serious contribution to the cause.
Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles, CA. | {"url":"http://www.maa.org/publications/maa-reviews/the-strength-of-nonstandard-analysis?device=desktop","timestamp":"2014-04-19T01:21:05Z","content_type":null,"content_length":"103788","record_id":"<urn:uuid:d4fdd841-c530-4fe4-8496-382d2e2652e0>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00287-ip-10-147-4-33.ec2.internal.warc.gz"} |
Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 324.04005
Autor: Erdös, Paul; Galvin, Fred; Hajnal, András
Title: On set-systems having large chromatic number and not containing prescribed subsystems. (In English)
Source: Infinite finite Sets, Colloq. Honour Paul Erdös, Keszthely 1973, Colloq. Math. Soc. Janos Bolyai 10, 425-513 (1975).
Review: [For the entire collection see Zbl 293.00009.]
This lengthy paper is devoted to a variety of problems concerning the chromatic number of set systems. A ``set system'' is defined to be a family of S of sets each of at least two elements. The
``chromatic number'' of S is the smallest cardinal \kappa for which there is a partition of length \kappa of \cup S, \cup S = \cup {P[\nu]; \nu < \kappa }, such that A \not\subseteq P[\nu] whenever A
in S, \nu < \kappa. The family S is said to be an (n,i)-system if it is a family of n-tuples every two of which have at most i elements in common. In an earlier paper [Cambridge Summer School Math.
Logic, Cambridge 1971, Lecture Notes Math. 337, 531-538 (1973; Zbl 289.04002)], P. Erdös, A. Hajnal and B. Rothchild have shown that if 1 \leq i \leq n < \omega \leq \kappa then there is an (n,i)
-system with chromatic number > \kappa, and that the cardinality of such a system must be large. One of the aims of the present paper is to determine just how large. The following are proved: (a)
Assume mi+2 \leq n < \aleph[0]. Let S be a system of n-tuples with |S| = \aleph[\alpha+m] such that every \aleph[\alpha+1]-size subset of S has at most i elements in its intersection. Then S has
chromatic number at most \aleph[\alpha]. (b) Assume 2 \leq n < mi+2 < \aleph[0] and that G.C.H. holds. Then there is an (n,i)-system of cardinality \aleph[\alpha+m] and chromatic number > \aleph[\
alpha]. The G.C.H. is needed in (b), for it is shown that if MA[\kappa] holds then every (3,1)-system of cardinality \kappa has chromatic number at {u in P |u ~ x, u ~ y, u ~ z }. Bose has proved
that |tr(x,y,z)| = q+1. The triple (x,y,z), x \not ~ , y \not ~ z, z \not ~ x, is said to be regular provided each point collinear with at least three points of tr(x,y,z) is actually collinear with
all points of tr(x,y,z). If for a point x each triple (x,y,z), with x \not ~ y, y \not ~ z, z \not ~ x, is regular, x is said to be regular. The following theorems are proved. a) If the 4-gonal
configuration S with parameters r = q^2+1 and k = q+1, where q is even and q > 2, possesses a regular point, then S is isomorphic to a 4-gonal configuration T(0) (i.e. a 4-gonal configuration arising
from an ovoid 0 of PG(3,q)) of J. Tits. b) If each point of the 4-gonal configuration S with parameters r = q^2+1 and k = q+1 (q > 2) is regular, then S is isomorphic to the 4-gonal configuration Q
(5,q) arising from a non-singular hyperquadric Q of index 2 of PG(5,q). c) Suppose that the 4-gonal configuration S = (P,B,I), with parameters r = q^2+1 and k = q+1 (q > 1), has a 4-gonal
subconfiguration S' = (P' ,B' ,I), with parameters k' = r' = k, for which the following condition is satisfied: if x,y,z in P' with x \not ~ y, y \not ~ z, z \not ~ x, then the triple (x,y,z) is
regular and moreover sp(x,y,z) \subset P'. Then we have (i) S has an involution with fixes P' pointwise (ii) S' is isomorphic to the 4-gonal configuration Q(4,q) arising from a non-singular
hyperquadric in PG(4,q). Remark: Recently the author has proved a) for q odd.
Reviewer: P.Erdös
Classif.: * 04A20 Combinatorial set theory
04A10 Ordinal and cardinal numbers; generalizations
05C15 Chromatic theory of graphs and maps
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag
│Books │Problems │Set Theory │Combinatorics │Extremal Probl/Ramsey Th. │
│Graph Theory │Add.Number Theory│Mult.Number Theory│Analysis │Geometry │
│Probabability│Personalia │About Paul Erdös │Publication Year│Home Page │ | {"url":"http://www.emis.de/classics/Erdos/cit/32404005.htm","timestamp":"2014-04-20T00:58:39Z","content_type":null,"content_length":"8730","record_id":"<urn:uuid:d5064b99-92d6-477d-a37a-79585aec683c>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00208-ip-10-147-4-33.ec2.internal.warc.gz"} |
How to identify primes in quadratic fields
August 3rd 2010, 05:13 AM #1
Junior Member
Jul 2010
How to identify primes in quadratic fields
I'm aware that if p is an integer in a Q field, and if its norm is prime, then p is a prime in Q. However, what if I myself need to confirm whether or not a number is prime in a specific field?
Is there anyway I can identify/confirm that a number is prime when its properties are exclusive of this theorem?
I'm not quite sure what you're asking exactly, but a prime number in a quadratic field will always have a norm that is prime or the prime squared.
So does that mean ALL integers with a norm that is either a prime or a prime squared, is a prime in Q, or does your statement only work in one direction?
August 3rd 2010, 07:58 AM #2
August 5th 2010, 02:38 AM #3
Junior Member
Jul 2010 | {"url":"http://mathhelpforum.com/number-theory/152678-how-identify-primes-quadratic-fields.html","timestamp":"2014-04-16T19:38:46Z","content_type":null,"content_length":"34668","record_id":"<urn:uuid:d7f6cfb2-96ab-49ad-a580-cbd8c78cb968>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00189-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Students will learn about NASA's Radiation Belt Storm Probes (RBSP), Earth's van Allen Radiation Belts, and space weather through reading a NASA press release and viewing a NASA eClips video segment.
Then students will use simple linear functions to... (View More) examine the scale of the radiation belts and the strength of Earth's magnetic field. This activity is part of the Space Math
multimedia modules that integrate NASA press releases, NASA archival video, and mathematics problems targeted at specific math standards commonly encountered in middle school textbooks. The modules
cover specific math topics at multiple levels of difficulty with real-world data and use the 5E instructional sequence. (View Less)
This is an activity about the Signal-to-Noise Ratio. Learners will engage with a hands-on activity and an online interactive to understand the terms signal and noise as they relate to spacecraft
communication; quantify noise using a given dataset;... (View More) and calculate the signal-to-noise ratio. The activity also includes a pencil-and-paper component that addresses relevant topics,
such as proportions and ratios. Includes teacher background information, student data sheets, answer guide, extensions and adaptions. (View Less)
This is an activity about spacecraft radio communications. Learners will explore spacecraft radio communications concepts, including the speed of light and the time-delay for signals sent to and from
spacecraft. Learners measure the time it takes... (View More) for a radio signal to travel to a spacecraft using the speed of light, demonstrate the delay in radio communication signals to and from a
spacecraft, and devise unique solutions to the radio-signal-delay problem. In an extension, learners are asked to calculate the distance the spacecraft traveled. All NASA spacecraft missions have a
telecommunications system and use radio waves to transmit signals. The context for this activity is sending a command to the New Horizons spacecraft telling it to take a picture of Pluto. Includes
teacher background, adaptations, and student data sheets. (View Less)
This lesson is a case study of atmospheric ozone levels developed from observations over Thule, Greenland in 2002. Students will download a composite graph of this stratospheric ozone data taken from
two different sources: the SAGE III satellite and... (View More) an ozone sensor on a weather balloon. Instructions for downloading the graph to either a computer or a graphing calculator (the TI-84
Silver Plus is recommended) are included in the lesson. Students will then compare and analyze the two data sets shown on the graph. Emphasis is placed on the applicability of using the weather
balloon data to validate the satellite data. This lesson uses student- and citizen science-friendly microsets of authentic NASA Earth system science data from the MY NASA DATA project. It also
includes related links, extensions, and an online glossary. (View Less)
This article describes an approach designed to decrease math anxiety and teach students about the use of mathematical symbols simplifying radicals. A deck of cards is used in a demonstration, and a
problem set using real life examples to master the... (View More) use of radicals is included. This resource is from PUMAS - Practical Uses of Math and Science - a collection of brief examples
created by scientists and engineers showing how math and science topics taught in K-12 classes have real world applications. (View Less)
In this activity, students construct base-two slide rules that add and subtract base-2 exponents (log distances), in order to multiply and divide corresponding powers of two. Students use these slide
rules to generate both log and antilog equations,... (View More) learning to translate one in terms of the other. This is activity C1 in the "Far Out Math" educator's guide. Lessons in the guide
include activities in which students measure,compare quantities as orders of magnitude, become familiar with scientific notation, and develop an understanding of exponents and logarithms using
examples from NASA's GLAST mission. These are skills needed to understand the very large and very small quantities characteristic of astronomical observations. Note: In 2008, GLAST was renamed Fermi,
for the physicist Enrico Fermi. (View Less)
This is a mathematical lesson utilizing algebra to investigate Earth's magnetosphere. Learners will solve algebraic distance equations that will show how the distance to the Earth's magnetopause
depends on the incoming solar wind pressure. This is... (View More) the twentieth and final activity in the Exploring the Earth's Magnetic Field: An IMAGE Satellite Guide to the Magnetosphere
educators guide. (View Less) | {"url":"http://nasawavelength.org/resource-search?facetSort=1&topicsSubjects=Mathematics%3AAlgebra&learningTime=45+to+60+minutes","timestamp":"2014-04-16T20:34:45Z","content_type":null,"content_length":"56769","record_id":"<urn:uuid:70a71ddb-f387-4cb0-9d28-0b06bbd63841>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00438-ip-10-147-4-33.ec2.internal.warc.gz"} |
Scalar Triple Product Assignment Help - Transtutors
Scalar Triple Product Assignment Help
It is defined for three vectors ^[]in that order as the scalar ([]×[]).[]×[].
i.e. V = magnitude of []×[].[]×[].
The value of scalar triple product depends on the cyclic order of the vectors and is independent of the position of the dot and cross. These may be interchange at pleasure. However and anti-cyclic
permutation of the vectors changes the value of triple product in sign but not a magnitude.
Properties of Scalar Triple Product:
* If are given as [^]
* []×[].[]×[].[]×[].
* [], [], [].[].
Solved Example 1:
Show that ^.
Key concept:
In such type of questions first we reduce each term as a product of two or three vector, by substituting a group of vectors by a single vector and then solve.
Email Based Homework Assignment Help in Scalar Triple Product
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Rearranging a Polynomial
Date: 02/23/2002 at 11:07:29
From: Amanda
Subject: Pythagorean Theorem
How do you do this question?
2n^2 + y^2 = 66^2
Thank you so much!
Date: 02/23/2002 at 12:10:26
From: Doctor Schwa
Subject: Re: Pythagorean Theorem
Hi Mandy,
my first idea is to rearrange a bit:
2n^2 = 66^2 - y^2,
and then
2n^2 = (66-y)(66+y).
Then, since the left side is even, the right side is even too, so y
must be even, call it 2r:
2n^2 = (66-2r)(66+2r) = 4(33-r)(33+r).
n^2 = 2(33-r)(33+r).
Now, since the right side is even, the left side must be even too, so
n must be even, call it 2j:
4j^2 = 2(33-r)(33+r)
2j^2 = (33-r)(33+r).
Now, since the left side is even, the right side is even too, so r
must be odd, call it 2s+1:
2j^2 = (33-2s-1)(33+2s+1)
2j^2 = (32-2s)(34+2s) = 4(16-s)(18+s), so
j^2 = 2(16-s)(18+s)
By now the problem is getting small enough that you could try all the
possible values of s (0 through 16, with 0 and 16 giving you the silly
solution n = 0, y = 66 to your original problem), or you could keep
applying the same idea (now j has to be even, so call j = 2k, and
substitute yet again...)
- Doctor Schwa, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/56441.html","timestamp":"2014-04-20T01:15:50Z","content_type":null,"content_length":"6064","record_id":"<urn:uuid:6f30a3f0-415b-4a60-8d34-5fb7218d1d21>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00015-ip-10-147-4-33.ec2.internal.warc.gz"} |
Antonio Gulli's coding playground
I was looking for a library implementing
Range Trees
Interval Trees
, two less known but very useful data structures for retrieving overlapping segments and points. Those data structures are used in databases and in many other contexts.
Well, what I found is a larger golden mine. The Computational Geometry Algorithms Library (
) is a software library that aims to provide easy access to efficient and reliable algorithms in
computational geometry
. While primarily written in C++, Python bindings are also available.
table of content
shows an impressive amount of already cooked algorithms and data structures, ranging from basic
2D and 3D Geometry Kernel,
Convex Hulls and Delaunay Triangulations,
Voronoi Diagrams
I found the Search Structures part quite interesting with implementations of
2D Range and Neighbor Search
Interval Skip List
Spatial Searching
Neighbor Searching
Range Searching
k-d tree
Range and Segment Trees
Intersecting Sequences of dD Boxes
The library has additional tools for
Linear and Quadratic Programming Solver
Spatial Sorting
and support for third party software such as the G
libraries Qt, Geomview, and the Boost Graph Library. (Gosh all these are together?)
A huge amount of software all using Template C++ with an approach similar to Boost.
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I'm constructing an arm for my robot and I want to make sure I got the idea for servo torque right. I've read some other posts about servo torque and its helped a little, but I still want to know a
few things:
Suppose I have a servo that has specs of 47 ounces per inch. Suppose I mount this servo for the elbow joint. Now, if the forearm was 9 inches long, this servo would be able to...see that's where I
get stuck in my calculations. I understand that as the arm gets longer, it takes more power for the servo to lift it. I want the servo to easily lift a 9 inch forearm made of sheet polyethylene
plastic (two sheets cut to similar dimensions and bracketed together to be, say two inches wide). If I use a 47 oz per inch servo, and at a beam of 1 inch it lifts an object of 94 oz, than at 9
inches it would lift the arm if the arm weighed...uh, okay. What's the mathematical equation to calculate the torque of a servo that's measured by oz per inch? Is the torque just halved for each
additional inch? Or is it 1/3rd for three inches? That's what I'd like to know, just so I make sure I buy a powerful enough servo.
Can someone please explain this for me, I'm really confused. Thanks. | {"url":"http://www.societyofrobots.com/robotforum/index.php?topic=2063.0","timestamp":"2014-04-18T11:24:10Z","content_type":null,"content_length":"42602","record_id":"<urn:uuid:03052d64-6bf7-4317-b1d1-a4c5f4d301be>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00034-ip-10-147-4-33.ec2.internal.warc.gz"} |
Energy dependence of transverse mass spectra of kaons produced in p+p and p+pbar interactions : a compilation (2003)
The data on mT spectra of K0S K+ and K- mesons produced in all inelastic p + p and p + pbar interactions in the energy range sqrt(s)NN=4.7-1800GeV are compiled and analyzed. The spectra are
parameterized by a single exponential function, dN/(m_T*dm_T)=C exp(-m_T/T), and the inverse slope parameter T is the main object of study. The T parameter is found to be similar for K0S, K+ and
K- mesons. It increases monotonically with collision energy from T~30MeV at sqrt(s)NN=4.7GeV to T~220MeV at sqrt(s)NN=1800GeV. The T parameter measured in p+p and p+pbar interactions is
significantly lower than the corresponding parameter obtained for central Pb+Pb collisions at all studied energies. Also the shape of the energy dependence of T is different for central Pb+Pb
collisions and p+p(pbar) interactions. | {"url":"http://publikationen.stub.uni-frankfurt.de/solrsearch/index/search/searchtype/authorsearch/author/%22Marek+Gazdzicki%22/start/0/rows/10/yearfq/2003/author_facetfq/Benjamin+Lungwitz","timestamp":"2014-04-17T16:28:40Z","content_type":null,"content_length":"17144","record_id":"<urn:uuid:1bbd4c82-4b23-416c-8abb-568e249c0212>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00621-ip-10-147-4-33.ec2.internal.warc.gz"} |
Famous mathematical quotes
up vote 74 down vote favorite
Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?
Standard community wiki rules apply: one quote per post.
soft-question big-list ho.history-overview
16 Why did a question with so much positive feedback get closed? – Romeo Nov 28 '10 at 23:21
11 Diminishing marginal utility. – Qiaochu Yuan Jan 31 '11 at 2:46
2 Closing this solved what problem? – Matt Brin Jan 18 '12 at 18:35
@Matt: standards for what kind of questions people want on MO have changed over time, and keeping this question opens gives a false impression to new users of what kind of questions we want on
2 MO. It's less confusing to close it. This happens on other SE sites as well; many of the most popular questions on StackOverflow, for example, are also closed. There's also the more practical
issue that if it's open people keep adding answers and, again, the marginal utility of each additional answer is decreasing. – Qiaochu Yuan Nov 12 '13 at 3:03
show 2 more comments
closed as no longer relevant by Scott Morrison♦ Apr 28 '10 at 17:51
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97 Answers
active oldest votes
I heard this one while taking a differential geometry class in Mexico City. I love it.
"Groups, as men, will be known by their actions".
up vote 87 down vote
-Guillermo Moreno.
add comment
We often hear that mathematics consists mainly of "proving theorems." Is a writer's job mainly that of "writing sentences?" - Gian-Carlo Rota
up vote 80 down vote
20 Depends on the writer? – Harry Gindi Nov 30 '09 at 13:47
7 Similarly, depends on the mathematician.... – Suvrit Feb 1 '11 at 18:04
add comment
“The difference between mathematicians and physicists is that after physicists prove a big result they think it is fantastic but after mathematicians prove a big result they think it is
up vote 80 trivial.” Lucien Szpiro during Algebra 1 lecture.
down vote
7 This reminds me of Surely You're Joking, Mr. Feynman!, in which the physics grad students at Princeton propose the theorem that mathematicians only prove trivial theorems. – Pete L.
Clark Jan 8 '10 at 6:27
12 I once heard Henry McKean say of mathematical models in physics that "First, they’re pretty disgraceful. Second, they work extremely well...One of the faults of mathematicians is:
when physicists give them an equation, they take it absolutely seriously." (I wrote this down on the spot.) – Steve Huntsman Apr 23 '10 at 15:48
add comment
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." --- John von Neumann. (From a 1947 ACM keynote, recalled by
up vote 78 down Alt in this 1972 CACM article.)
show 1 more comment
"Mathematics is the art of giving the same name to different things." Henri Poincaré.
up vote 76 down (This was in response to "Poetry is the art of giving different names to the same thing.")
1 This idea is in his "Science and Method" with an extensive discussion, and he does not mention anything about this being a response to something about poetry. – timur Nov 28
'10 at 4:01
add comment
"The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to
take such childish steps..."
up vote 71 down
vote -Alexander Grothendieck, writing to Ronald Brown
4 Grothendieck was French, and I suppose the correct translation would be "digit" not "cipher". – Tom Ellis Feb 1 '11 at 9:39
13 As far as I know, Grothendieck is still French. – José Figueroa-O'Farrill Nov 7 '11 at 15:23
13 As far as I know, Grothendieck does not hold any nationality. But I may well be wrong. – Jonathan Chiche Oct 19 '12 at 8:52
add comment
Le but de cette thèse est de munir son auteur du titre de Docteur.
Beginning of A. Douady's thesis. Quoted by Michèle AUdin in her Conseils aux auteurs de textes mathématiques.
up vote 68 down
vote In a less barbarous language: The purpose of this thesis is to obtain the degree of Doctor for its author.
12 "The goal of this thesis is to furnish its author with the title of Doctor" is perhaps closer still. A. Douady was indeed a rather interesting person! [He was my co-supervisor,
along with John Hubbard]. – Jacques Carette Mar 26 '10 at 2:30
5 the sentence ends with : "and (some set) with (some structure)", playing on the two meanings of "munir" = furnish – Feldmann Denis Aug 24 '11 at 3:04
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Oh, he seems like an okay person, except for being a little strange in some ways. All day he sits at his desk and scribbles, scribbles, scribbles. Then, at the end of the day, he takes
up vote 67 the sheets of paper he's scribbled on, scrunches them all up, and throws them in the trash can. --J. von Neumann's housekeeper, describing her employer.
down vote
32 The weatherman at Kitty Hawk thought Wilbur Wright wasted a lot of time watching gulls fly. And Darwin's housekeeper thought he would get more work done if he didn't keep staring at
the ants in an anthill. – SixWingedSeraph Nov 30 '09 at 1:22
3 That's fantastic! – Tom Leinster Nov 30 '09 at 18:27
add comment
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona
up vote 61
down vote
27 Obviously there is something wrong with ZF for proving their equivalence. I blame infinity. – Andrew Critch Nov 30 '09 at 1:36
3 Zorn's Lemma seems the most intuitive out of the three, but well-ordering isn't so counter-intuitive, since all it comes down to is being able to well-order a set with cardinality
strictly larger than the natural numbers. Thinking about well-ordering the reals gives a false impression of the difficulty, since the well-ordering only has to do with the underlying
set. – Harry Gindi Nov 30 '09 at 6:25
2 @Harry: I can't really agree with either statement. Naively, if we could truly "see" a well-ordering of a set of continuum cardinality, then intuitively we should be able to compare
it to $\aleph_1$ and "see" whether it is larger. – Todd Trimble♦ Jul 3 '11 at 12:07
add comment
It's hard to beat John Stembridge's page of quotes. My single favorite one on this page: "If I have not seen as far as others, it is because there were giants standing on my
up vote 55 down shoulders." - Hal Abelson.
18 Along those lines, R. W. Hamming said "Mathematicians stand on each other’s shoulders while computer scientists stand on each other’s toes." – John D. Cook Nov 30 '09 at 3:02
11 My version of this quote, after working my way backwards through a series of papers, each relying on the previous ones: If I can't see a darn thing it's because I stand on the
shoulders of giants... – Ehud Friedgut Dec 2 '09 at 5:49
7 What was Gell-Mann's version? Something like, "If I have seen farther than others, it's because I'm surrounded by pygmies?" – Todd Trimble♦ Jan 31 '11 at 1:15
add comment
You know, for a mathematician, he did not have enough imagination. But he has become a poet and now he is fine.
up vote 55 D. Hilbert, talking about an ex-student. I'd love to remember where I got this from!
down vote
You might have gotten it from the Monthly in 1993, or a mailing list or blog that derives from that source. Google Books traces it back to a book about Ernst Cassirer published in
2 1949. Since it is quoted there as Cassirer's verbal story, it's less clear what Hilbert himself said. archive.org/stream/philosophyoferns033109mbp/… – Greg Kuperberg Nov 29 '09 at
show 3 more comments
Algebra is the offer made by the devil to the mathematician...All you need to do, is give me your soul: give up geometry --Michael Atiyah
up vote 54 down vote
8 A quote that obviously inspired the soul theorem. en.wikipedia.org/wiki/Soul_theorem – Greg Kuperberg Dec 18 '09 at 7:24
2 @Ronnie, maybe there is no such duality in mathematics, but there is duality in our perception. – Anton Petrunin Aug 9 '12 at 23:35
show 1 more comment
"The art of doing mathematics is finding that special case that contains all the germs of generality." -- David Hilbert
up vote 53 down vote
add comment
Grothendieck comparing two approaches, with the metaphor of opening a nut: the hammer and chisel approach, striking repeatedly until the nut opens, or just letting the nut open naturally by
immersing it in some soft liquid and let time pass:
"I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply
water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand
up vote pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or
51 down hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the
vote resistant substance."
Grothendieck, of course, always pioneered this approach, and considered for example that Jean-Pierre Serre was a master of the "hammer and chisel" approach, but always solving problems in a
very elegant way.
16 A friend who I won't name at the moment once told me this, paraphrased: An excellent problem-solver might not always be a great mathematician, while a bad problem-solver can still be an
okay mathematician. On the other hand, a good Grothendieck is a great mathematician, while a bad Grothendieck is really terrible! – Greg Kuperberg Dec 18 '09 at 7:20
12 Greg, the symmetry in that statement would be nicer if you replaced "problem-solver" with "Erdos." – Qiaochu Yuan Dec 27 '09 at 8:11
11 That's a good suggestion, but I can't edit comments. So let me just rephrase the aphorism: "An excellent Erdos might reach certain limits as a mathematician, while a bad Erdos can still
be an okay mathematician. On the other hand, a good Grothendieck can be a truly great mathematician, while a bad Grothendieck is really terrible." – Greg Kuperberg Jan 19 '10 at 19:36
add comment
"Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of
up vote 49 down mathematics. In one respect this last point is accurate." - David Mumford
24 So you guys are trying to take over the rest of mathematics? And I thought I was just paranoid. – Michael Lugo Nov 30 '09 at 0:33
8 A professor of mine recently said something along the lines of, "Liking algebraic geometry is a clich\'e". – Harry Gindi Nov 30 '09 at 6:30
add comment
In mathematics you don't understand things. You just get used to them.
--John von Neumann, reply to a physicist at Los Alamos who had said "I don't understand the method of characteristics."
up vote 48 down ---- footnote on page 226 of Gary Zukav, The Dancing Wu Li Masters: An Overview of the New Physics, Rider, London, 1990.
(taken from Warren Dicks' Home Page)
6 I don't know how much I agree with that. It may be true about physics, but not math. – Harry Gindi Nov 30 '09 at 6:20
20 I feel that it's true of both mathematics and physics, but when talking about mathematics it's a much deeper statement. – Dan Piponi Dec 2 '09 at 19:37
8 I may have a surprise for you... "Understanding"="getting used to"! :) – efq Dec 4 '09 at 21:20
6 I totally disagree with this quote ... – Martin Brandenburg Feb 2 '10 at 15:45
1 Different understandings of the quote lead to different styles of math-study. People who agree are doing 'potential infinity' mathematics, while people who disagree are doing
'actual infinity' mathematics. – Sunni Mar 26 '10 at 3:56
show 4 more comments
"God exists since mathematics is consistent, and the Devil exists since we cannot prove it."- André Weil
up vote 46 down
1 So, mathematics is consistent? I think we can more or less rely on dividing by three, for the rest we shouldn't be that sure... – Jose Brox Dec 8 '09 at 15:42
5 @Jose: I don't know if you were referring to this paper indirectly, but you should check out math.dartmouth.edu/~doyle/docs/three/three.pdf, where it is proven that division by
three is possible. – Steven Gubkin Apr 23 '10 at 12:42
add comment
"The shortest path between two truths in the real domain passes through the complex domain." -- Jacques Hadamard
up vote 44
down vote
Here's a simple example. What is the radius of convergence for the power series of 1/(x^2 + 1) centered at 0? Looking only at the real line there's no apparent reason for the radius to
6 be only 1. But in the complex plane, you can see that the radius is 1 because that's the distance from the center to the singularity at i. Another example would be using contour
integration to compute integrals over the real line. – John D. Cook Mar 26 '10 at 18:35
show 1 more comment
"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." - Weyl
up vote 42
down vote
3 I never understood this quote, I mean, there's nasty algebra and nice algebra, so isn't that statement a bit strong? – Harry Gindi Nov 30 '09 at 6:28
4 I actually think it refers to something Mephistophelian about algebra, as if such heights of abstraction are meant not for mortals, or such symbolic calculation lacks intuitive
"soul". (That would be a tendentious way of putting it!) – Todd Trimble♦ Jan 31 '11 at 1:39
2 Oh, and I swear that I wrote that before seeing Anton Petrunin's contribution! – Todd Trimble♦ Jan 31 '11 at 1:42
show 1 more comment
"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions", F. Klein (from Reed & Simon:
up vote 41 down Methods of modern mathematical physics)
add comment
Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you
have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you
don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow
up vote undertaking, with sophisticated techniques.
41 down
vote Gian-Carlo Rota, in an interview with David Sharp.
3 Here is an online copy: fas.org/sgp/othergov/doe/lanl/pubs/00326965.pdf – Jose Capco Nov 30 '09 at 11:11
6 Combinatorics is discrete functional analysis in my world view, while functional analysis is applied combinatorics. – Bill Johnson Mar 21 '10 at 19:28
5 Bill, would you mind elaborating? As someone not particularly familiar with either field, I can imagine that by combinatorics being "discrete functional analysis" you mean e.g.
generating function methods, or perhaps the general ambition of associating some sort of linear operator to combinatorial objects (e.g. adjacency matrix). But what do you mean by
functional analysis being applied combinatorics. – Erik Davis Apr 15 '10 at 1:01
31 It takes balls to do combinatorics. – Todd Trimble♦ Jan 31 '11 at 1:45
show 4 more comments
"The question you raise, "how can such a formulation lead to computations?" doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit,
elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out
up vote 40 would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered." - Grothendieck
down vote
show 3 more comments
" Last time, I asked: "What does mathematics mean to you?" And some people answered: "The manipulation of numbers, the manipulation of structures." And if I had asked what music means
up vote 39 down to you, would you have answered: "The manipulation of notes?" "- Serge Lang
add comment
Not famous yet, maybe from now on!
At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of
up vote 39 down digits which terminate.
Terence Tao
2 Yikes,need to email Terry and compliment him on that one! That's pretty cool. – Andrew L Apr 28 '10 at 0:35
add comment
"Wir müssen wissen, wir werden wissen." - Hilbert.
up vote 37 down vote Translation: We must know, we will know.
show 1 more comment
"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a
truly marvellous proof of this, which this margin is too narrow to contain."
up vote 36
down vote Pierre de Fermat
show 2 more comments
"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies
between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."
--Stefan Banach
up vote 35
down vote "Good mathematicians see analogies between theorems or theories. The very best ones see analogies between analogies."
--Stanislaw M. Ulam quoting Stefan Banach
13 So, category theorists? – Qiaochu Yuan Nov 29 '09 at 22:55
25 @Qiaochu: Higher category theorists! – Harrison Brown Nov 29 '09 at 22:59
5 Who can see analogies between analogies between analogies? – timur Nov 28 '10 at 4:28
add comment
The introduction of numbers as coordinates is an act of violence.
up vote 35 down vote Hermann Weyl, Philosophy of Mathematics and Natural Science.
1 Cannot agree more. – Sam Derbyshire Dec 1 '09 at 5:22
add comment
"The price of metaphor is eternal vigilance." Norbert Wiener.
up vote 33 down vote
show 1 more comment
"It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out."
up vote 32 down vote -Emil Artin, Geometric Algebra
add comment
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The halo phenomenon after laser eye surgery
Laser eye surgery has become increasingly popular to help those with shortsightedness. Many have underwent this procedure and are able to see without glasses. But one of the side effects of
corrective eye surgery is the occurence of seeing halos around lights at night time. This makes it difficult for one to see clearly at night.
The follow picture illustrates how a halo around a light at night time would appear to a person with eye surgery
A normal person without eye surgery, looking at the same light would see a a much clearer view.
To understand this phenomena, we'll first look at how the eye works and then a shortsighted person's eye, before they undergo any surgery.
Diagram of eye
The eye has a transparent, outer surface at the front called the cornea. This is where light enters the eye. The cornea is spherical in shape, allowing it to refract light.
The iris acts like an aperature, narrowing or widening its circular opening, controlling the amount of light that enters the lens. The lens acts as an optical lens that can change its curvature to
refract light rays by different amounts, so an in-focus image forms at the retinal surface. The image at the retinal surface is interpreted by the brain to create the image one sees.
The remaining interior of the eye is filled with aqueous fluid, having a composition close to water.
Diagram of normal eye focusing light
As light rays hit the cornea, they refract and bend towards the center. This is because of Snell's law, so light rays going from the air to the cornea, refract towards the surface normal, because
light travels slower in the cornea and has a higher index of refraction that air. When the rays reach the lens, they are refracted again and focus on the retinal surface at the back of the eye. Light
rays from a distance object are focused to one point that lies on the retina, creating a sharp image.
To mathematically model the eye as a system of optical lenses, we'll use a linear approximation involving matrices.
Tracing light rays using matrices
Each ray starts at an initial vertical plane and has a height y on the plane, and angle q between the ray and lens axis. For further simplification, we replace the direction q with T = nq , where n
is the current index of refraction. Thus (y,T) is a ray starting at the vertical plane.
A linear matrix can be used to describe the transformation of a ray (y, T), as it refracts at a surface or travels (translates) through a medium.
In the refraction matrix, n = current index of refraction, R = radius of curvature of the lens, with a positive curvature being convex to the left. In the translation matrix, s = horizontal distance
the ray travels, and n = current index of refraction.
We apply our matrices as follows. Our new ray (y*,T*) is computed by applying it to one of the above matrices.
So the ray (y,T) = (0.4, 0) in the previous picture, has the index of refraction in air n = 1.0 and the radius of curvature of the first lens surface R = 1. The resulting refracted ray is
So the path of ray so far look like:
If we continue with the computation, we get the following picture, showing where the ray eventually will focus.
The myopic eye can't focus light from far away, (parallel light rays (y,0) ), but can focus light from close objects. Since the lens in the eye cannot change its curvature enough to focus light rays
at the retina, we'll look at changing the curvature of the cornea, which is done in refractive eye surgery.
The 4 matrices describing the refraction at the outer and inner surface of the cornea and translation of rays through to the lens are shown below. M is the transfer matrix describing the cornea.
The index of refraction of the cornea is 1.37 and its outer radius of curavature, R[1] = 7.8mm. Its inner radius of curvature is smaller than its outer curvature and the corneal thickness is 0.5mm.
Given this information, we can compute the values of P[1], P[2] and S, in the matrix and the final matrix for the cornea.
The corneal lens matrix M, will be represented by the matrix to help simplify the notation.
To understand how a myopic eye focusses light, we'll look at conjugate planes. The light at a point on an object infront of the eye focusses at some point on the retina, so these two points define
the conjugate planes for each other. Two planes are conjugate if light from a point on one plane focusses on some other point on the other plane.
Rays at conjugate planes in myopic eye
To see why the focus in a myopic eye is infront of the retina, we must look at the lens system for conjugate planes. The resulting transfer matrix for this conjugate plane system is
In terms of matrices, incoming light rays have a height y = 0, and a resulting height y* = 0. So in terms of the resulting transfer matrix.
Ay + BT = y* = 0
BT = 0 so B = 0 because y* is independent of T , since changing T has no effect on y*.
Now we solve B=0 from the conjugate plane matrix and derive a formula for the focus in terms of the object's distance s.
Substituing in for a,b,c and d the values of the corneal lens system, the focal equation becomes:
We can see that as you increase s, the distance between the object and the eye, the denominator increases quicker than the numerator, so f, the focal distance decreases. So as a short-sighted person
attempt to view objects further away, the image will focus a some point infront of the eye, creating an out-of-focus image on the retina
Myopic eye viewing distance object
So how does flattening the cornea help one to focus light from far away so they are in-focus at the retina? Lets look at the matrix describing the focus of the corneal lens.
Since we have parallel rays from infinity, T = 0 and the resulting rays focus at some point where y* = 0. So in terms of the transfer matrix for the system
Ay + BT = y* = 0
Ay = 0, so A=0
Solving for A = 0, the upper left term in the transfer matrix, we get the following focal equation for the corneal lenses.
If we wish to make f, the focal length larger, so that the image will focus at the retina, we can try to change P[1], by changing R[1], the radius of curvature for the outer surface of the cornea. As
we flatten the cornea, R[1] increases making P[1] decrease.
SP[1] and SP[2] are small terms since S is minute in value, so we can ignore these terms when considering the effect of changing P[1]. As P[1] decreases, the denominator becomes less and less
negative. Since the numerator is dominately negative, f increases positively as P[1] decreases. This is why laser eye surgery flattens the cornea to obtain a greater radius of curvature in the outer
cornea. The amount by which you flatten the cornea depends upon the required curvature needed to focus light properly at the retina.
This is what the cornea look like after surgery: The central cornea is flatter than the peripheral corneal, so light focusses correctly at the retina.
Myopic eye after eye surgery with cornea flattened
Now we look at how this relates to halos. The iris acts like an aperature that restricts the amount of light that reaches the lens and eventually the retina. If the eye is receiving alot of light,
the iris constricts itself to reduce the amount of light entering. If it's dark, then the iris will open larger to allow more light to enter the eye. Halos around lights are seen when it is dark,
because the iris open wider, allowing light from the peripheral area of cornea to reach the lens. Light from these regions will focus infront of the retina, since the eye was original myopic.
To determine what one actually sees, we'll look at where light rays passing through the peripheral regions of the cornea that end up hitting the retina.
Light rays refracting at peripheral cornea at night with enlarged iris opening
We can see these rays focus at the old focus of the myopic eye, so when they reach the retina, the rays become out-of-focus. The two bands of light rays that reach the retina create the fuzzy, halo
of light, as depicted at the beginning of our discussion.
Combining all the rays, refracting through the flattened cornea and the peripheral cornea, one sees the following:
Halo phenomenom in laser corrected eye
There is a sharp focus of the image of the light source at the retina, so one sees the light unblurred, but the two extra bands of rays, surround the sharp image of the light, blurring the clarity of
the light.
Krachmer, Jay H. MD, ed., Basic Science, Refraction, and Pathology (Elsevier Science, 2002).
Azar, Dimitri T., Excimer Laser Photorefractive Keratectomy (Williams & Wilkins 1997).
Seiler, T, Beckmann, W, and Maloney, RK "Effective spherical aberration of the cornea as a quantitative descriptor in corneal opography," Journal or Cataract & Refractive Surgery Vol 19, 1993,
(Suppl): 155-165.
Yuan-Chieh Lee, Fung-Rong Hu and I. -Jong Wang "Quality of vision after laser in situ keratomileusis: Influence of dioptric correction and pupil size on visual function," Journal of Cataract &
Refractive Surgery Vol 29, Issue 4, April 2003, 769-777.
Birgit Lackner et al., "Glare and halo phenomena after laser in situ keratomileusis", Journal of Cataract & Refractive Surgery Vol 29, Issue 3, March 2003, 444-450 .
Author: Lyndon Hiew
Created: April 28, 2003 | {"url":"http://www.math.ubc.ca/~cass/courses/m309-03a/m309-projects/hiew/lasik.html","timestamp":"2014-04-20T03:25:14Z","content_type":null,"content_length":"12411","record_id":"<urn:uuid:64e80db7-3f16-4fe9-b988-36c6b6b04c20>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00548-ip-10-147-4-33.ec2.internal.warc.gz"} |
double integration questions help please
September 9th 2010, 07:03 AM #1
Jul 2010
double integration questions help please
Hi everyone, can anybody please show me how to solve these two double integration questions please?
Here they are:
(a) $\int_0^1 \int_0^{\arccos(x)} \cos(\sin y)} \, dy \,dx<br />$
(b) $\int_0^4 \int_{\sqrt{2}}^{3} \int_0^y z\cos (y^6) \, dx \,dy \,dz$
For the first one is painful,I cannot even start,but I guess maybe I need to change the order,am I right?
Second one is a bit horrible, I cannot continue after I integrated with respect to x, it was getting really messy.
Please show me what I should do,thanks a lot.
are you sure that you write that correctly ?
$\cos ({\sin {y}})$
For the first problem, change the order of integration. Look at the region over which you're integrating and carefully look at the new limits. The result will be much easier to work with.
For the second integral, are you sure it isn't
$\displaystyle{\int_{0}^{4} \int_{\sqrt{2}}^{3} \int_{0}^{y}z\cos^{6}(y) \, dx \,dy \,dz?}$
The one you wrote is nasty and has incomplete gamma functions everywhere.
Yes, friend, I'm sure I wrote the right thing. I know this one is very nasty because of that $y^6$, that's why I found it is so hard. Can you please help me with that? Thanks a lot.
With regard to the second integral, here's what Mathematica gives:
$\displaystyle{\int_{0}^{4}\int_{\sqrt{2}}^{3}\int_ {0}^{y}z\,\cos(y^{6})\,dx\,dy\,dz=}$
$\displaystyle{-\frac{2}{3}(-1)^{1/6}\left(-\Gamma\left(\frac{1}{3},-8i\right)+(-1)^{2/3}\Gamma\left(\frac{1}{3},8i\right)+\Gamma\left(\f rac{1}{3},-729i\right)-(-1)^{2/3}\Gamma\left(\frac{1}
Here the $\Gamma$ is the upper incomplete Gamma function defined by
The numerical value of this result is approximately $-0.326074.$
I might be able to explain portions of the journey to get you there.
Let's take your original integral. You can immediately do the z integration. You have the following:
$\displaystyle{\int_{0}^{4}\int_{\sqrt{2}}^{3}\int_ {0}^{y}z\,\cos(y^{6})\,dx\,dy\,dz=\left(\frac{z^{2 }}{2}\right)\Bigg|_{0}^{4}\int_{\sqrt{2}}^{3}\int_ {0}^{y}\cos(y^{6})\,dx\,dy=8\int_{\sqrt
{2}}^{3}\in t_{0}^{y}\cos(y^{6})\,dx\,dy.}$
Next, we perform the $x$ integral to obtain
$\displaystyle{8\int_{\sqrt{2}}^{3}y\,\cos(y^{6})\, dy.}$
From here, I'd probably do the substitution $u=y^{2},$ from which we get $du=2y\,dy.$ Hence, the integral becomes
This function actually has an antiderivative in terms of the gamma function. I'm not sure I can explain it to you. Perhaps Opalg could weigh in on that one. Once you have that antiderivative, all
you do is plug in the limits, and you have your result.
September 9th 2010, 07:15 AM #2
September 9th 2010, 07:27 AM #3
September 9th 2010, 07:42 AM #4
September 9th 2010, 05:38 PM #5
Jul 2010
September 9th 2010, 05:40 PM #6
Jul 2010
September 10th 2010, 02:11 AM #7 | {"url":"http://mathhelpforum.com/calculus/155654-double-integration-questions-help-please.html","timestamp":"2014-04-21T06:43:29Z","content_type":null,"content_length":"55138","record_id":"<urn:uuid:c4c4470b-3be5-42fe-b1a8-c45fb66c97aa>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00511-ip-10-147-4-33.ec2.internal.warc.gz"} |
Möbius at Fermilab
Fermilab's Wilson Hall. Courtesy of Fermilab.
Soaring into the sky like a medieval cathedral, the twin towers of the structure known as Wilson Hall dominate the flat countryside surrounding the Fermi National Accelerator Laboratory (Fermilab) in
Batavia, Ill.
Named for physicist and accelerator builder Robert Rathbun Wilson (1914-2000), the building celebrates Wilson's vision and skill, not only as a scientist but also as an artist. Made leader in 1967 of
the effort to build a world-class proton synchrotron and founding director of Fermilab, Wilson played a central role in designing the entire laboratory. He was also widely recognized as an
accomplished sculptor. Wilson died on Jan. 16 at the age of 85.
The Fermilab grounds feature several of Wilson's sculptures (see http://www.fnal.gov/projects/history/sculpture.html). One of the most striking examples is a tubular form 8 feet in diameter, which
sits in the middle of a circular pool atop the Norman F. Ramsey Auditorium in front of Wilson Hall. Having spent part of his youth on a Wyoming cattle ranch, Wilson was no stranger to the blacksmith
shop, and he welded together the sculpture from 3 by 5-inch pieces of stainless steel.
"Möbius Strip" by Robert R. Wilson. Courtesy of Fermilab.
Wilson called the sculpture "Möbius Strip," but it isn't the usual rectangular strip with its ends taped together after giving one end a 180-degree twist (see Möbius and His Band, July 8, 2000). It's
a three-dimensional version of this "one-sided" surface. The tube's cross section is essentially an equilateral triangle, and this triangle is rotated through 120 degrees along the tube.
Wilson used the same Möbius form as the basis for a gleaming bronze sculpture now installed near an entrance to the Science Center at Harvard University in Cambridge, Mass.
It's possible to create a variety of three-dimensional variants of the Möbius band. Thickening a standard Möbius strip so that its edge becomes as wide as the strip's side produces a
three-dimensional object with a square cross section. The resulting form has two edges and two faces (see Möbius in the Playground, May 22, 1999).
"Topological III" by Robert R. Wilson.
You can construct a model of this surface from a length of foam rubber with a square cross section. After rotating one end through 180 degrees, you simply fasten the ends together to form the twisted
loop. If instead you rotate one end through 90 degrees before linking the ends, you wind up with a twisted loop with just one edge and one face!
You can do the same thing with a length of foam rubber having a triangular cross section to get the shape on which Wilson based his intriguing Möbius sculptures.
Wilson is just one of many artists who have sculpted Möbius strips. One of the first to do so was Max Bill, who lives in Zumikon, Switzerland. Starting in the 1930s, he created a variety of "endless
ribbons." You can see two examples crafted out of metal at http://www.cpm.informatics.bangor.ac.uk/sculpmath/pagesm/mb.html and a remarkable one carved from granite at http://www.bluffton.edu/
Bill has been a strong advocate of using mathematics as a framework for art. "I am convinced it is possible to evolve a new form of art in which the artist's work could be founded to quite a
substantial degree on a mathematical line of approach to its content," he argued in a 1949 essay.
English sculptor John Robinson has also explored various mathematical themes in his series of "symbolic" sculptures. Robinson's "Immortality" bronze, for instance, is a Möbius band in the form of a
trefoil knot (see http://www.cpm.informatics.bangor.ac.uk/sculpture/pages/bangor.html and http://www.bradshaw-foundation.com/jr/1immorta.htm). Another sculpture, named "Journey," is at Macquarie
University in Sydney, Australia (see http://www.cpm.informatics.bangor.ac.uk/sculpture/pages/3journey.html). This tall, angular form, made of polished stainless steel, is based on the so-called Brehm
model of the Möbius strip (see http://www.cpm.informatics.bangor.ac.uk/sculpmath/pagesm/brehm.html).
In 1980, Robinson independently discovered and created several versions of the same three-dimensional Möbius form that had inspired Wilson.
"I prepared 100 equilateral triangles, each with a hole in the middle, and threaded them onto a ring," Robinson recounts. Following the faces around the loop, he saw that he had a surface with three
edges and three bands. "By changing one of the edges to meet another, I found I was left with only one edge and one band," he says. "It was a magical moment when I realized that I had created
Polished bronze versions of "Eternity" can be seen in Canberra, Australia, and Aspen, Colorado (http://www.cpm.informatics.bangor.ac.uk/sculpture/pages/1eternit.html).
Visitors to Washington, D.C., can find several Möbius forms in very public places–if they know where to look or understand what they're seeing.
"Infinity" by José de Rivera.
A Möbius strip in the form of an 8-foot-high, stainless-steel sculpture rests atop a tall pedestal in front of the National Museum of American History. Designed by José de Rivera, who titled the
piece "Infinity," this swooping form has been revolving majestically since 1967.
Across the Mall, a more complicated Möbius form stands guard at the entrance to the National Air and Space Museum. Created by sculptor Charles O. Perry of Norwalk, Conn., it is called "Continuum."
Möbius sculptures created by Charles O. Perry: "Continuum" (left) and "Calligraphic Möbius" (right).
Several miles away in a plaza in front of the U.S. Patent and Trademark Office in Arlington, Va., stands another Möbius strip–a giant, calligraphic loop of twisted, red-painted steel–also created by
Perry. A companion piece, a short distance away, presents an entirely different perspective on this intriguing topological shape.
The amazingly different ways in which artists can depict a Möbius surface are nice reminders that a topological form retains its essential character–in this case, its one-sidedness–no matter how much
the figure is deformed, just as long as it isn't punctured or torn. This quality offers a vast playground for creative reconfiguration of an intriguing shape.
Charles Perry (right) helping with the installation of "Helix Möbius Mace" in Crystal City, Arlington, Va.
Perry's monumental sculptures can be found in many cities around the world. An astonishingly large proportion of them involve the Möbius strip in some way. If there's a Perry sculpture in your
neighborhood or you happen to encounter one when you're traveling, take a close look at it and try to decipher its geometric message.
"My pieces are ordered unto themselves, as if space had a set of rules similar to the counterpoint of music," Perry remarks, perhaps a little cryptically. "At times, the works will have a single
generating order which will grow to completion and stop, while others will have overlays of mutating effects. Most, however, are given intertwining themes which are meant to be played upon by
sunlight. Many have some form of transparency to them. All have a sequential movement."
You'll find Perry sculptures in San Francisco, El Segundo, Costa Mesa, and Beverly Hills, Calif.; Sydney, Perth, and Ringwood, Australia; Dallas and Midland, Texas; Singapore; Fairfield, Westport,
Greenwich, and Stamford, Conn.; Falls Church and Virginia Beach, Va.; Minneapolis, Minn.; Knoxville, Tenn.; Tampa, Jacksonville, and Maitland, Fla.; Atlanta, Ga.; Hanover, N.H.; Charlotte, N.C.;
Tokyo and Kokubo, Japan; Bloomington, Ind.; and elsewhere. See some examples at http://www.cs.berkeley.edu/~sequin/SCULPTS/PERRY/ and http://www.charlesperry.com/.
Who would have thought that looking for Möbius could turn into a remarkably engrossing, wide-ranging mathematical treasure hunt?
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Assumptions on a commutative C*-algebra to get a nice C(X) - space
up vote 2 down vote favorite
I have the following question,
Is it possible to get somehow a compact Hausdorff space $X$ which is second-countable from a unital commutative C*-algebra. If it is possible, what should we assume for our C*-algebra.
Gelfand-Naimark theorem gives us $C(X)$, where $X$ is a compact Hausdorff space, but I'm asking how to get it with the second axiom of countability. Thank you in advance for any answers or hints.
oa.operator-algebras gn.general-topology
add comment
1 Answer
active oldest votes
The space $X$ is second countable if and only if $C(X)$ is separable for the norm. This is proved, for example, as Theorem 2.4 of the little article "Notes on the Separability of C*
up vote 4 down algebras" by Chun-Yen Chou. Actually, the short proof given there works also for locally compact Hausdorff spaces and therefore non-unital $C^{*}$-algebras.
vote accepted
1 It is also a standard exercise in text books. – Bill Johnson Aug 30 '12 at 0:31
You're right Bill. This was probably best left as a long comment, should the OP have liked to move the question to math.stackexchange. – Jon Bannon Aug 30 '12 at 0:54
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Indefinite Integrals
You can think of integration by parts as a way to undo the product rule. While integration by substitution lets us find antiderivatives of functions that came from the chain rule, integration by
parts lets us find antiderivatives of functions that came from the product rule.
Integration by parts requires learning and applying the integration-by-parts formula. Here's the formula, written in both Leibniz and Lagrange notation:
Why the Formula is True
It's time to be skeptical. Why should
be true? Just because we wrote down an equation doesn't mean it's valid.
Sample Problem
• What should be the derivative of x?
• What should be the derivative of x?
• What's the derivative of uv?
• uv'. If we take the derivative of any function in that family, we get uv'. So it makes sense for the derivative of uv'. In symbols,
• vu'. If we take the derivative of any one of those functions, we get vu', so the derivative of vu'. In symbols,
• This is a straightforward application of the product rule. The derivative of uv is
uv' + u'v.
Let's return to the question of why the equation
should be valid. Each side of this equation describes a family of functions. If each side of the equation describes the same family of functions, then the equation is valid.
describes the family of functions with derivative uv'.
The expression
describes the family of functions with derivative
Since uv' and vu' are the same, when we simplify we get
u'v + uv' – vu' = u'v.
This means the expression
also describes the family of functions with derivative u'v.
Since the expressions
describe the same family of functions, the equation
is valid.
How to Use the Formula
Hopefully, you're now convinced that
To use this formula we have to
• pick u and v',
• figure out what u ' and v are, and
• stick everything in the formula and simplify.
If we're using the notation
then we pick u and dv, figure out du and v, then apply the formula. | {"url":"http://www.shmoop.com/indefinite-integrals/indefinite-integral-parts.html","timestamp":"2014-04-17T10:13:24Z","content_type":null,"content_length":"31169","record_id":"<urn:uuid:66ab16b7-4070-41f2-8a60-d64578d8ab59>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00654-ip-10-147-4-33.ec2.internal.warc.gz"} |
PhET Teacher Ideas & Activities: Motion and Moving Man Simulation Homework
published by the Physics Education Technology Project
written by Katherine K. Perkins and Carl E. Wieman
This is a set of homework questions developed for use with the PhET simulation "The Moving Man". It assesses student understanding of position, velocity, and acceleration graphs. It was designed for
implementation in cooperative learning groups. Most of the questions are in multiple-choice format, with a few requiring short written responses. This resource has been used in large-enrollment
classes, but is also appropriate for algebra-based high school physics courses.
Please note that this resource requires at least version 1.4, Java WebStart of Java.
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Classical Mechanics - Instructional Material
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= Acceleration = Curriculum support
- Lower Undergraduate
= Position & Displacement = Problem/Problem Set
= Velocity - Assessment Material
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Additional information is available.
acceleration, assessment, graph, graph interpretation, homework, kinematics, position, problem set, question, velocity
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Metadata instance created November 16, 2008 by Caroline Hall
Record Updated:
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Other Collections:
AAAS Benchmark Alignments (2008 Version)
4. The Physical Setting
4F. Motion
• 6-8: 4F/M3a. An unbalanced force acting on an object changes its speed or direction of motion, or both.
• 9-12: 4F/H1. The change in motion (direction or speed) of an object is proportional to the applied force and inversely proportional to the mass.
• 9-12: 4F/H8. Any object maintains a constant speed and direction of motion unless an unbalanced outside force acts on it.
9. The Mathematical World
9B. Symbolic Relationships
• 6-8: 9B/M3. Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: increase or decrease steadily,
increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease, increase or decrease in steps,
or do something different from any of these.
• 9-12: 9B/H1b. Sometimes the rate of change of something depends on how much there is of something else (as the rate of change of speed is proportional to the amount of force acting).
11. Common Themes
11B. Models
• 6-8: 11B/M4. Simulations are often useful in modeling events and processes.
• 6-8: 11B/M6. A model can sometimes be used to get ideas about how the thing being modeled actually works, but there is no guarantee that these ideas are correct if they are based on the model
• 9-12: 11B/H3. The usefulness of a model can be tested by comparing its predictions to actual observations in the real world. But a close match does not necessarily mean that other models would
not work equally well or better.
Common Core State Standards for Mathematics Alignments
Expressions and Equations (6-8)
Represent and analyze quantitative relationships between dependent and independent variables. (6)
• 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent
variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate
these to the equation.
Functions (8)
Define, evaluate, and compare functions. (8)
• 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Use functions to model relationships between quantities. (8)
• 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph
that exhibits the qualitative features of a function that has been described verbally.
High School — Functions (9-12)
Linear, Quadratic, and Exponential Models^? (9-12)
• F-LE.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
• F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
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<a href="http://www.compadre.org/portal/items/detail.cfm?ID=8304">Perkins, Katherine, and Carl Wieman. PhET Teacher Ideas & Activities: Motion and Moving Man Simulation Homework. Boulder: Physics
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K. Perkins and C. Wieman, (Physics Education Technology Project, Boulder, 2008), WWW Document, (http://phet.colorado.edu/en/contributions/view/3081).
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Perkins, Katherine, and Carl Wieman. PhET Teacher Ideas & Activities: Motion and Moving Man Simulation Homework. Boulder: Physics Education Technology Project, January 14, 2008. http://
phet.colorado.edu/en/contributions/view/3081 (accessed 17 April 2014).
Perkins, Katherine, and Carl Wieman. PhET Teacher Ideas & Activities: Motion and Moving Man Simulation Homework. Boulder: Physics Education Technology Project, 2008. 14 Jan. 2008. 17 Apr. 2014 <http:
@misc{ Author = "Katherine Perkins and Carl Wieman", Title = {PhET Teacher Ideas & Activities: Motion and Moving Man Simulation Homework}, Publisher = {Physics Education Technology Project}, Volume =
{2014}, Number = {17 April 2014}, Month = {January 14, 2008}, Year = {2008} }
%A Katherine Perkins
%A Carl Wieman
%T PhET Teacher Ideas & Activities: Motion and Moving Man Simulation Homework
%D January 14, 2008
%I Physics Education Technology Project
%C Boulder
%U http://phet.colorado.edu/en/contributions/view/3081
%O text/html
%0 Electronic Source
%A Perkins, Katherine
%A Wieman, Carl
%D January 14, 2008
%T PhET Teacher Ideas & Activities: Motion and Moving Man Simulation Homework
%I Physics Education Technology Project
%V 2014
%N 17 April 2014
%8 January 14, 2008
%9 text/html
%U http://phet.colorado.edu/en/contributions/view/3081
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PhET Teacher Ideas & Activities: Motion and Moving Man Simulation Homework:
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I got -28 for the answer but didn't know if it was right
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?? Is there a value for x?
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no just to simplify.
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its 3 over x + 13 over x-11 and I am just supposed to simplify.
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given \[\frac{ 3 }{ x } + \frac{13}{x-11}\] we can multiply any term by 1, right? We want a common denominator, so try multiplying each term by a version of 1... \[\frac{x-11}{x-11}\] and \[\frac
{x}{x}\] respectively to see what happens.
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I did that and after everything i got -28
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If you don't have a value for x your answer should still include x. When you multiplied (x-11)/(x-11) and x/x what did you end up with?
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3x-33 over x^2-11x + 13x over x^2-11x
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would that be it or could i simplify more.
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Perfect. Now that you have a common denominator, you can combine your two fractions into one, combining like terms in your numerator: [(3x-33)+13x]/(x^2-11x)
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oh okay, so in the end it would be 16x-33 over x^2-11x?
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exactly! That's as simple as you can get. :) Good work
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College Physics Problem Capacitor / Inductor
A 10.0 Microfarad Capacitor is charged to 170 Micro Coulombs then connected across the ends of a 4.00mH inductor.
(a) Find the maximum current in the inductor.
(b) At the instant the current in the inductor is maximum. How much charge is on the capacitor parallel plates?
(c) Find the maximum potential across the capacitor.
(d) At the instant the potential across the capacitor is maximum. What is the current in the inductor?
(e) Find the maximum energy stored in the inductor.
(f) At the instance the energy stored in the inductor is maximum. what is the current in the circuit?
I am really stuck on calculating this physics problem, especially part a. Could someone please give me some assistance in calculating this Physics Problem. I really appreciate your help. Thanks
Hi aamartineng17, Welcome to Physics Forums.
These kinds of problems are often easier if you think in terms of conservation laws. Here there are two energy storage devices (a Capacitor and an Inductor) which are going to trade energy back and
forth. The result is oscillations of current and voltage.
When the circuit begins operation, only the capacitor has any energy stored since it has an initial voltage; There's no current flowing so the inductor has no stored energy.
What do you know about energy for these devices? Do you have formulas? | {"url":"http://www.physicsforums.com/showthread.php?p=3797052","timestamp":"2014-04-19T22:42:23Z","content_type":null,"content_length":"32634","record_id":"<urn:uuid:4536d9e8-42bb-45e0-aa00-492af68d7b07>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00259-ip-10-147-4-33.ec2.internal.warc.gz"} |
the encyclopedic entry of operational semantics
computer science
operational semantics
is a way to give meaning to
computer programs
in a mathematically rigorous way. Other approaches to providing a
formal semantics of programming languages
axiomatic semantics
denotational semantics
The operational semantics for a programming language describes how a valid program is interpreted as sequences of computational steps. These sequences then are the meaning of the program. In the
context of functional programs, the final step in a terminating sequence returns the value of the program. (In general there can be many computation sequences and many return values for a single
program, because the program could be nondeterministic.)
For the first time, the concept of operational semantics was used in defining the semantics of Algol 68. The following statement is a quote from the revised ALGOL 68 report:
The meaning of a program in the strict language is explained in terms of a hypothetical computer which performs the set of actions which constitute the elaboration of that program. (Algol68,
Section 2)
The first use of the word "operational semantics" in its present meaning is attributed to Dana Scott (Plotkin04b). What follows is a quote from Scott's seminal paper on formal semantics, in which he
mentions the "operational" aspects of semantics.
It is all very well to aim for a more ‘abstract’ and a ‘cleaner’ approach to semantics, but if the plan is to be any good, the operational aspects cannot be completely ignored. (Scott70)
Perhaps the first formal incarnation of operational semantics was that of the lambda calculus. Abstract machines in the tradition of the SECD machine are also closely related.
Structural Operational Semantics (SOS)
Structural Operational Semantics (SOS) was introduced by Gordon Plotkin in (Plotkin04a) as a logical means to defining operational semantics. The basic idea behind SOS is to define the behavior of a
program in terms of the behavior of its parts, thus providing a structural, i.e., syntax oriented and inductive, view on operational semantics. An SOS specification defines the behavior of a program
in terms of a (set of) transition relation(s). SOS specifications take the form of a set of inference rules which define the valid transitions of a composite piece of syntax in terms of the
transitions of its components.
For a simple example, we consider part of the semantics of a simple programming language; proper illustrations are given in Plotkin04a and Hennessy90, and other textbooks. Let $C_1, C_2$ range over
programs of the language, and let $s$ range over states (e.g. functions from memory locations to values). If we have expressions (ranged over by $E$), values ($V$) and locations ($L$), then a memory
update command would have semantics $frac\left\{langle E,srangle Rightarrow V\right\}\left\{langle L:=E,,,sranglelongrightarrow \left(suplus \left(Lmapsto V\right)\right)\right\}$
Informally, the rule says that "if the expression $E$ in state $s$ reduces to value $V$, then the program $L:=E$ will update the state $s$ with the assignment $L=V$".
The semantics of sequencing can be given by the following rules $frac\left\{langle C_1,srangle longrightarrow langle C_1",s\text{'}rangle\right\} \left\{langle C_1;C_2 ,,sranglelongrightarrow langle
C_1\text{'};C_2,, s\text{'}rangle\right\} quad frac\left\{langle C_1,srangle longrightarrow s\text{'}\right\} \left\{langle C_1;C_2 ,,sranglelongrightarrow langle C_2, s\text{'}rangle\right\} quad
frac\left\{\right\} \left\{langle mathbf\left\{skip\right\} ,sranglelongrightarrow s\right\}$
Informally, the first rule says that, if the program $C_1$ in state $s$ can reduce to the program $C_1"$ with state $s"$, then the program $C_1;C_2$ in state $s$ will reduce to the program $C_1";C_2$
. The second rule says that if program $C_1$ in state $s$ finishes in state $s"$, then the program $C_1;C_2$ in state $s$ will reduce to the program $C_2$ in state $s"$. The semantics is structural,
because the meaning of the sequential program $C_1;C_2$, is defined by the meaning of $C_1$ and the meaning of $C_2$.
If we also have Boolean expressions over the state, ranged over by $B$, then we can define the semantics of the while command: $frac\left\{langle B,srangle Rightarrow mathbf\left\{true\right\}\right
\}\left\{langlemathbf\left\{while\right\} B mathbf\left\{ do \right\} C,sranglelongrightarrow langle C;mathbf\left\{while\right\} B mathbf\left\{do\right\} C,srangle\right\} quad frac\left\{langle
B,srangle Rightarrow mathbf\left\{false\right\}\right\}\left\{langlemathbf\left\{while\right\} B mathbf\left\{ do \right\} C,sranglelongrightarrow s\right\}$
Such a definition allows formal analysis of the behavior of programs, permitting the study of relations between programs. Important relations include simulation preorders and bisimulation. These are
especially useful in the context of concurrency theory.
Thanks to its intuitive look and easy to follow structure, SOS has gained great popularity and has become a de facto standard in defining operational semantics. As a sign of success, the original
report (so-called Aarhus report) on SOS (Plotkin04a) has attracted some 900 citations according to the CiteSeer , making it one of the most cited technical reports in Computer Science.
See also | {"url":"http://www.reference.com/browse/operational+semantics","timestamp":"2014-04-16T23:15:56Z","content_type":null,"content_length":"87362","record_id":"<urn:uuid:8f2e05d1-13ed-491e-8c28-8c2dcdafe95e>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00144-ip-10-147-4-33.ec2.internal.warc.gz"} |
APL Symbols Meaning
This page is a list of of APL symbol's meanings.
Name Notation Meaning
Roll ? One integer selected randomly from the first B integers
Ceiling ⌈ Least integer greater than or equal to B
Floor ⌊ Greatest integer less than or equal to B
Shape ⍴ Number of components in each dimension of B
Not ∼ Logical: ∼1 is 0, ∼0 is 1
Absolute value ∣ Magnitude of B
Index generator ⍳ Vector of the first B integers
Exponential ⋆ e to the B power
Negation − Changes sign of B
Identity + No change to B
Signum × ¯1 if B<0; 0 if B=0; 1 if B>0
Reciprocal ÷ 1 divided by B
Ravel , Reshapes B into a vector
Matrix inverse ⌹ Inverse of matrix B
Pi times ○ Multiply by π
Logarithm ⍟ Natural logarithm of B
Reversal ⌽ Reverse elements of B along last axis
Reversal ⊖ Reverse elements of B along first axis
Grade up ⍋ Indices of B which will arrange B in ascending order
Grade down ⍒ Indices of B which will arrange B in descending order
Execute ⍎ Execute an APL expression
Monadic format ⍕ A character representation of B
Monadic transpose ⍉ Reverse the axes of B
Factorial ! Product of integers 1 to B
Name Notation Meaning
Add + Sum of A and B
Subtract − A minus B
Multiply × A multiplied by B
Divide ÷ A divided by B
Exponentiation ⋆ A raised to the B power
Circle ○ Trigonometric functions of B selected by A. A=1: sin(B) A=2: cos(B) A=3: tan(B)
Deal ? A distinct integers selected randomly from the first B integers
Membership ∈ 1 for elements of A present in B; 0 where not.
Maximum ⌈ The greater value of A or B
Minimum ⌊ The smaller value of A or B
Reshape ⍴ Array of shape A with data B
Take ↑ Select the first (or last) A elements of B according to ×A
Drop ↓ Remove the first (or last) A elements of B according to ×A
Decode ⊥ Value of a polynomial whose coefficients are B at A
Encode ⊤ Base-A representation of the value of B
Residue ∣ B modulo A
Catenation , Elements of B appended to the elements of A
Expansion \ Insert zeros (or blanks) in B corresponding to zeros in A
Compression / Select elements in B corresponding to ones in A
Index of ⍳ The location (index) of B in A; 1+⌈/⍳⍴A if not found
Matrix divide ⌹ Solution to system of linear equations Ax = B
Rotation ⌽ The elements of B are rotated A positions
Rotation ⊖ The elements of B are rotated A positions along the first axis
Logarithm ⍟ Logarithm of B to base A
Dyadic format ⍕ Format B into a character matrix according to A
General transpose ⍉ The axes of B are ordered by A
Combinations ! Number of combinations of B taken A at a time
Less than < Comparison: 1 if true, 0 if false
Less than or equal ≤ Comparison: 1 if true, 0 if false
Equal = Comparison: 1 if true, 0 if false
Greater than or equal ≥ Comparison: 1 if true, 0 if false
Greater than > Comparison: 1 if true, 0 if false
Not equal ≠ Comparison: 1 if true, 0 if false
Or ∨ Logic: 0 if A and B are 0; 1 otherwise
And ∧ Logic: 1 if A and B are 1; 0 otherwise
Nor ⍱ Logic: 1 if both A and B are 0; otherwise 0
Nand ⍲ Logic: 0 if both A and B are 1; otherwise 1 | {"url":"http://xahlee.info/comp/APL_symbol_meaning.html","timestamp":"2014-04-20T00:58:30Z","content_type":null,"content_length":"15851","record_id":"<urn:uuid:adb3e568-a450-4f35-a7d2-aa0f25cfdc4e>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00514-ip-10-147-4-33.ec2.internal.warc.gz"} |
Patent application title: SYMMETRIC KEY CRYPTOGRAPHIC METHOD AND APPARATUS FOR INFORMATION ENCRYPTION AND DECRYPTION
Inventors: Pal Bela Domosi (Debrecen, HU)
IPC8 Class: AH04L900FI
USPC Class: 380255
Class name: Cryptography communication system using cryptography
Publication date: 2009-04-09
Patent application number: 20090092251
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For information encryption and decryption the apparatus uses the same hardware keys, which have the transition matrix of a key-automaton with no output signal and with an initial state and a final
state burnt in. To each character in the character set of the plaintext message there is one or more final states of the key-automaton assigned. During encryption the message is read in sequentially
character by character and the key automaton assigns to each character a random character string, whose length is adjustable length within a given length range. The process is the following: for each
character in the message the apparatus generates a character string of adjustable length and with no initial- and end markers, which takes the apparatus from its current state into the final state
that corresponds to the subsequent character of the message. The apparatus creates the encrypted message by linking these character strings together. The encrypted message can be decrypted using the
same apparatus.
An apparatus for encrypting and decrypting information, which has an encryption and/or decryption unit with plaintext and ciphertext input and output and a further input with an encryption and/or
decryption key automaton attached, wherein to each element of the character set of the plaintext there is one or more final states of the key automaton assigned and each final state is assigned to
one and only one element of the character set for every state pair, whose first element is the initial state or a final state, its second element is any final state, there are several, preferably
sup.128 or even more distinct input signal strings such that the second element of the state pair is the same as the last element of the state string which state string is assigned by the extended
transition function to the pair consisting of the first element of the state pair and the given input string and none of the intermediate elements of the state string is a final state.
The cryptographic apparatus in claim 15, wherein the key automaton is a Rabin-Scott key automaton, a deterministic automaton, a non-deterministic automaton, a weighted automaton, a cellular
automaton, a two-way automaton, an automata network, a linearly limited/bounded automaton, a stack automaton, a tree automaton, a Turing machine, a transducer or some combination of these.
The cryptographic apparatus in claim 15, wherein the input and output of the cryptographic apparatus is attached to the processing unit through a line driver.
The cryptographic apparatus in claim 15, wherein a further input of the processing unit has a two-way connection with a key-automaton and data memory through a data bus.
The cryptographic apparatus of claim 15, wherein there is a program memory attached to the program input of the processing unit with a two-way connection and a further input of the processing unit
has a two-way connection with a random number source preferably with a random number generator.
The cryptographic apparatus of claim 15, wherein the processing unit is a microcontroller, preferably an ATMegal
28. 21.
The cryptographic apparatus of claim 15, wherein the key automaton is burnt into a 4 kilobyte flash memory integrated into a microcontroller ATMegal28; the program memory is 128 kilobyte PROM module
type integrated into ATMegal28; the data memory is 4 kilobyte MCU SMT type integrated into ATMegal28; the random number generator is PROTEGO R300 SMT type; the line driver (1,5) is RS232 with PC type
The cryptographic apparatus of claim 15, wherein the character set of the plaintext and the ciphertext is fixed, preferably a binary signal string of length 1, 2, 4 or 8 bits, the states of the
key-automaton are also fixed, preferably binary signal strings of length 8 or 9 bits, the initial state of the key automaton is of numerical value 0, which is at the same time the final state
assigned to the character of numeric value 0, for each character the state which is assigned to it as a final state has the same numeric value as that of the character.
Cryptographic method executable with the apparatus of claim 15, for encrypting and decrypting information, wherein in encryption mode the apparatus reads in the plaintext character by character,
generates a randomly chosen character string of length falling into a given, adjustable length interval for each character in the plaintext taking the key automaton from its state (from the initial
state first) without any intermediate final state into a final state assigned to the read plaintext character and the ciphertext consists of these character strings linked together.
The cryptographic method of claim 23, wherein in decryption mode the apparatus reads in the ciphertext character by character taking the key automaton from its state (from the initial state first)
into the state which is associated by the transition function with the state of the key automaton and the read ciphertext character, and decryption is accomplished by linking together the input
characters associated with the occurring final states, which provides the message in its original form.
The cryptographic method of claim 23, wherein the encrypted message is sent to several recipients simultaneously when used in broadcasting.
The cryptographic method of claim 23, wherein the plaintext message may include a digital signature, as a prefix of the plaintext, as a common secret key for authentication and identification, the
actual message, the description of a subsequent key automaton for the forthcoming messages, for the sake of increased security.
The cryptographic method of claim 26, wherein the description of the subsequent key automaton is the same as the current one.
The cryptographic method of claim 26, wherein the digital signature or the description of the subsequent key automaton or both are omitted.
PRIOR ART [0001]
The aim of a coding device is to make message transfer from a sender to one or more recipients secure, so that only the intended recipients can decrypt the message. A coding device converts the
original message into an encrypted message. Encryption means the manipulation or transformation of the message with the help of encryption key or keys. The recipient decrypts the message by reversing
the manipulation or transformation process, that is, the encrypted message is converted, with the help of a secret decryption key or keys, into the original message. Since the secret keys are known
only to the sender and the recipient, the message transfer is secure. A cryptographic device will mean a device which can, given the encryption unit and the encryption key, convert the message into
an encrypted message and conversely, given the decryption unit and the decryption key, can convert the encrypted message into the original one. Among cryptographic devices we distinguish between:
symmetric key cryptographic devices and public key (or asymmetric) cryptographic devices, which may use a variety of algorithms, that is, they use both symmetric and asymmetric encryption. There is a
huge number of such devices starting with Enigma, developed between the two world wars, then various decoders, access points and hardware keys equipped with the smallest microcontrollers as well as
smart cards. They are used in message encryption, in proving the integrity of a message, genuineness confirmation, in realising secure network communication, in data protection, in limiting broadcast
pickup, in electronic bank transfers and in shopping etc. The baud rate is very important in many applications as well as security. For example coding and decoding in digital television broadcasting,
internet encryption, or encryption of network communication. All these are applications, where the large number of people interested (in the case of television broadcasting this number may reach or
even increase a hundred million) enhances the danger of hacking efforts. Furthermore, many hackers publish the broken code on the internet, so it spreads fast. This unfortunate fact causes the most
important and still unsolved problem for broadcasting companies of these days. Similar devices used earlier and today (e.g. Rainbow iKey 2032 USB-key, Avaya Wireless PC card, USRobotics Wireless
Turbo Access Point & Router), if they were symmetric key (e.g. DES, TripleDES keys), proved to be easy to hack, if they were asymmetric key (e.g. RSA, ElGamal keys) have a long encoding and decoding
time, which is a drawback in real-time applications.
The cryptographic apparatus described in the present patent is symmetric key, where the incoming encrypted message is decrypted using the encrypting and decrypting apparatus and the encryption and
decryption keys, and encrypts the plaintext message using the encrypting and decrypting apparatus and the encryption and decryption keys. The technical novelty of the apparatus in the patent, when
compared with the devices above, is that if there is the transition matrix of a suitable finite automaton without outputs as a key automaton burnt into the apparatus, and encryption and decryption is
performed in the described way, then micro-sized technical realization is relatively easy (microcontrollers, smart cards etc.) and, in spite of the use of symmetric key, security is immensely
increased without any effects on the speed.
The theory of (abstract) automata, which was initially conceived to model computers and other discrete physical devices, has become an intensely researched field. According to the assumptions of the
theory an (abstract) automaton functioning in a discrete time scale has, at every point in time, a well-defined (inner) state, while it receives input signals from outside. The automaton reacts an
output signal for each input signal, and this output signal is uniquely determined by its current inner state and the received input signal, and then the input signal takes the automaton into a new
state, which is also determined by its current state and the input signal, and this new state will be its current state at the next point in time. A special case of this (abstract) automaton
type--which is often called Mealy machine--is the automaton without outputs, where either the current state of the automaton is considered to be the output signal or there is supposed to be no output
signal at all. An introduction to the theory of automata recommended is `Automata Theory` by M. Simon (World Scientific Press, Singapore, 1999) or `Introduction to Automata Theory` by J. E. Hopcroft,
R. Motwani and J. D. Ullman (second edition, Addison-Wesley Series in Computer Science, Addison-Wesley Co., Reading, Mass., 2001). Due to development in automata theory, research in cellular automata
started about 30 years ago and has been intensive ever since and it has an important role in modelling discrete physical systems of multiple components. The components of a cellular automaton are
atomic automata with no output signal, called cells, whose input signals are the (inner) states of their neighbouring cells, where the neighbourhood of a cell is determined by a neighbourhood
topology. A survey of cellular automata is provided by e.g. M. Garzon: `Models of Massive Parallelism. Analysis of Cellular automata and Neural Networks` (Texts in Theoretical Computer Science. An
EATCS Series. Springer-Verlag, Berlin, 1995).
Automata theory provides a natural basis for designing cryptosystems and several such systems have been designed. Some of them are based on Mealy automata or their generalisation, while others are
based on cellular automata.
Almost all cryptosystems can be modelled with Mealy machines (as sequential machines) or generalised sequential machines. During encryption the system first receives a preliminary input, which
contains the encryption key automaton (and sometimes a secondary encryption key as well). Following this key the input contains the plaintext stream, as a string of input signals for the automaton,
and starting the automaton from a given state, the encrypted message can indeed be generated as an output signal string initiated by the input signal string. Decryption is performed in a likewise
manner: the decryption key is substituted by the encryption key so that plaintext and ciphertext messages change roles. This method has several variants.
Rayward-Smith systems [V. J. Rayward-Smith: Mealy machines as coding devices. In: H. J. Beker and F. C. Piper, eds., Cryptography and Coding, Claredon Press, Oxford, 1989.] use ordinary Mealy
machines. Gysin systems [M. Gysin: One-key cryptosystem based on a finite non-linear automaton. In: E. Dawson and J. Golic, eds., Proc. Int. Conf. Proceedings of the Cryptography: Policy and
Algorithms, CPAC'95, Brisbane, Queensland, Australia, Jul. 3-5, 1995. Lecture Notes in Computer Science 1029 Springer-Verlag, Berlin, 1995, 165-163.] use special Mealy encryption and decryption
automata, where encryption and decryption are realised in several repeated simple steps, called turns. Similarly to Rayward-Smith systems and Gysin systems, Tao systems [R. Tao: On finite automaton
one-key cryptosystems. In: R. Anderson, ed., Proc. 1
Fast Software Encryption Workshop--FSE'93. Proceedings of the Security Workshop held in Cambridge, Cambridge, UK, Dec. 9-11, 1993, LNCS 809, Springer-Verlag, Berlin, 1994, 135-148.] are based on
sequential Maely machines.
The vulnerability of these systems is due to the well-known fact that automaton mappings are length and prefix preserving. Knowing a great number of encrypted messages the system can be attacked by
brute force search. Based on this fact Wichmann developed a system of cryptoanalysis [P. Wichmann: Cryptoanalysis of a modified rotor machine. In: J.-J. Quisquarter, J. Vandewalle, eds., Proc. Conf.
Advances in Cryptology--EUROCRYPT'89, Workshop on the Theory and Applications of Cryptographic Techniques, Houthalen, Belgium, Apr. 10-13, 1989, LNCS 434, Springer-Verlag, Berlin, 1990, 395-402.].
Atanasiu systems [A. Atanasiu: A class of coders based on gsm. In. Acta Informatica, 29 (1992), 779-791.] are similar to Rayward-Schmith systems, with the considerable difference, that they use
generalised sequential machines. It is also well-known, though, that mappings induced by generalised sequential machines are prefix preserving. Thus Atanasiu systems have the same security problems
as systems based on Mealy machines.
Tao-Chen public key FAPKC systems [R. Tao and S. Chen: Finite automata public key cryptosystem and digital signature. In: Computer Acta 8 (1985), 401-409 (in Chinese).] use the mathematical
conjecture that for delayed, weakly invertible automata it is difficult to find the reverse automaton. Unfortunately these systems can also be defeated [F. Bao and Y. Igarashi: Break finite automata
public key cryptosystems. In: Zoltan Fulop, Ferenc Gecseg, eds., Proc. 22
Int. Coll. on Automata Languages and Programming--ICALP'95, Szeged, Hungary, Jul. 10-14, 1995, LNC 944, Springer-Verlag, Berlin, 1995, 147-158.] So as to prevent attacks there was a refinement of
this system developed, called FAPKC-3 [R. Tao, S. Chen and X. Chen: FAPKC3: a new finite automata public key cryptosystem. Publ.: Technical report No. ISCAS-LCS-95-05, Laboratory for Computer
Science, Institute of Chinese Academy of Sciences, Beijing, 1995.]. Meskaten discusses two methods to break this cryptographic system [T. Meskaten: On finite automaton public key cryptosystems.
Publ.: TUCS Technical Report No. 408, Turku Centre for Computer Science, Turku, 2001, 1-42.].
All the well-known cryptosystems based on Mealy machines have proved to be vulnerable, which is a serious drawback in applications.
Almost from the very beginning of researches of cellular automata, there have been serious attempts at cryptographic applications. Cryptosystems of this kind usually use the plaintext as the initial
state of the cellular automaton, the encryption key being the transition rules for the cells. The state reached after a given number of steps provides the encrypted message. Decryption is performed
in a similar manner, starting from the encrypted message as a state, the initial state is determined which is then the decrypted message.
The Wolfram system [S. Wolfram: Cryptography with Cellular Automata. In: C. W. Hugh, ed., Proc. Conf. Advances in Cryptology--CRYPTO'85, Santa Barbara, Calif., USA, Aug. 18-22, 1985, LNCS 218,
Springer-Verlag, Berlin, 1986, 429-432.] use one dimensional cellular automata, which produce a random bit string. This bit string is added to the plaintext stream bit by bit. Meier and Staffelbach
[W. Meier and O. Staffelbach: Analysis of pseudo-random sequences generated by cellular automata. In: D. W. Davies, ed., Proc. Conf. Advances in Cryptology--EUROCRYPT'91, Workshop on the Theory and
Application of Cryptographic Techniques, Brighton, UK, Apr. 8-11, 1991, LNCS 547 Springer-Verlag, Berlin, 1991, 186-199.] gave a cryptoanalysis of Wolfram system. Furthermore, although extensive
statistical tests have been conducted to analyse pseudo-random numbers [S. Wolfram: Random sequence generation by cellular automata. In: Adv. Applied Math. 7 (1986), 123-169.], there is no
mathematical proof available. The same applies to the system patented by Bianco and Red [U.S. Pat. No. 5,048,086], as it is known [H. B. Lin: Elementary Symbolic Dynamics and Chaos in Dissipative
Systems. Publ.: World Scientific, Singapore, 1989.], that is the generated pseudo-random bit stream will not be random for most choices of parameter. Habutsu at al. [T. Habutsu, Y. Nishio, I. Sasase,
S. Mori: A Secret Key Cryptosystem by Iterating a Chaotic Map. In: D. W. Davies, ed., Proc. Conf. Advances in Cryptology--EUROCRYPT'91, Workshop on the Theory and Application of Cryptographic
Techniques, Brighton, UK, Apr. 8-11, 1991, LNCS 547 Springer-Verlag, Berlin, 1991, 127-140 .] recommend a cryptosystem based on the iteration of chaotic maps. Two methods, described by Biham [E.
Biham: Cryptoanalysis of the chaotic map cryptosystem suggested at EUROCRYPT'91. In: D. W. Davies, ed., Proc. Conf. Advances in Cryptology--EUROCRYPT'91, Workshop on the Theory and Application of
Cryptographic Techniques, Brighton, UK, Apr. 8-11, 1991, LNCS 547 Springer-Verlag, Berlin, 1991, 532-534.] can be used to defeat these systems. Recently, further efforts have been taken by Sen at al.
[S. Sen, C. Shaw, D. R. Chowdhuri, N. Ganguly, P. P. Chaudhuri: Cellular automata based cryptosystem (CAC). In: R. Deng, S. Qing, F. Bao, J. Zhou eds., Proc. 4
Int. Conf. on Information and Data Security--ICIS 2002, Dec. 9-12, 2002, Singapore LNCS 2513, Springer-Verlag, 2002, 303-314.] to create cellular cryptosystems (CACs) based on cellular automata,
where affine cellular automata are combined with non affine transformations. Bao showed F. Bao: Cryptoanalysis of partially known cellular automata. In: a) R. Safavi-Naini, J. Seberry, eds., Proc. 8
Australian Conf. on Inf. Security and Privacy--ACISP'03, Wollongong, Australia, Jul. 9-11, 2003, LNCS 2727, Springer-Verlag, Berlin 2003, 416-427, b) IEEE Trans. on Computers, 53 (2004), 1493-1497.]
that these systems are not secure either. Bao also stated [F. Bao: Cryptoanalysis of partially known cellular automata. In: IEEE Trans. on Computers, 53 (2004), 1493-1497.] that minor changes are not
enough to make this system secure.
There are other cellular automaton based public key cryptosystems that are secure, but they are not practical. Guan's public key cryptosystem [P. Guan: Cellular automaton public key cryptosystem. In:
Complex Systems, 1 (1987), 51-56.] is secure, but it requires a very careful choice of the cellular automata used. Guan's method involves solving systems of polynomial equations during encryption,
hence its practical realization leads to slow performance. The strong security of Kari's public key cryptosystem [J. Kari: Cryptosystems based on reversible cellular automata. Publ: University of
Turku, Finland, April, 1992, preprint.] is based on the well-known result that for cellular automata of at least two dimensions, the question whether they have an inverse or not cannot be answered
[J. Kari: Reversibility of 2D Cellular Automata is Undecidable. In: Physica D, Vol 45 (1990), 379-385.]. This fundamental result highlights a considerable problem. It is very difficult or if not
impossible to build a reverse automaton for a given cellular automaton. This fact sets considerably limits for the applicability of reversible cellular automata in cryptography.
These difficulties are overcome by Gutowitz's cellular automaton based, symmetric encoding system [Gutowitz, U.S. Pat. No. 5,365,589]. More precisely, Gutowitz systems are highly secure in spite of
their use of a random number generator. This high level of security is partly due to the fact that unlike other cellular automata systems using random number generator, the random number generator in
Gutowitz systems can be entirely independent from the structure of the key-automaton used. At the same time--unlike Guan systems--the simplicity of its components makes Gutowitz systems fast.
Finally, when compared to Kari systems, it is easy to generate the suitable key-automata for Gutowitz systems. Besides all this, like most cellular automaton based cryptosystems Gutowitz systems are
essentially coded block ciphers, where encoding is performed with a string of cellular automata. This is why--no matter whether the functioning control of the automaton string is sequential or
parallel--serious technical difficulties are faced in micro-sized technical realization of Gutowitz systems. Apart from the technical difficulties, using Gutowitz systems is not economical in
situations when the specific nature of the process requires bit-by-bit or character-by-character encoding/decoding. This is so, because Gutowitz systems are of coded block ciphers.
The best-known cellular automaton based cryptosystems all share the common problem of serious realization difficulties: some systems are easy to defeat, the technical realization of others result in
slow performance, and still others exhibit difficulties in the choice of the key-automaton. A further common drawback of cellular automaton based cryptosystems is (just like Gutowitz systems) that
their micro-sized technical realization poses serious difficulties and they are not always economical either.
The subject matter of the present invention is a cryptographic apparatus with a Rabin-Scott automaton as key for encoding and decoding information. Unlike the above described cryptosystems the key is
neither a Mealy machine nor a cellular automaton.
The present invention is similar to Mealy machine based cryptosystems in that encoding and decoding is performed using a key automaton. Unlike Mealy machine based or generalised sequential machine
based cryptosystems, whether certain mappings preserve length and prefix or other mappings preserve prefix or not has no significance here, so these properties cannot be exploited for hacking. The
properties of weakly reversible automata do not affect the invention either, so this invention cannot be hacked with methods used for defeating FAPKC and FAPKC-3 cryptosystems either.
The present invention is similar to cellular based cryptosystems in that the key-automaton is an automaton without outputs, like the cells in a cellular automaton. Furthermore, the encoded message
here is created in a somewhat similar manner (via manipulations on states) to the functioning of cellular automata, i.e. using the changes in the state of the key automaton. This present invention is
particularly similar to Gutowitz cellular based cryptosystem, in the sense that there is also a random number generator, which is independent from the key. Apart from having all the advantages of
Gutowitz systems, the present invention does not pose any serious difficulties for micro-sized technological realization.
The present invention does not share the problems the abovementioned cryptosystems have. Although the invention uses a random number generator, it can take random number generators which are proved
to be random indeed, or it can use any radioactive or other physical random number sources. Therefore, the recipient of the message does not need to know how this randomness is realised in the
specific message which is generated during the encryption of the original dataflow. The plaintext message consists of such fixed length blocks (with no initial or end marker), where each block
represents a character (of one byte or half a byte or quarter of a byte or one bit length). To each plaintext message there are several (in theory, there are infinitely many) corresponding encoded
messages. Ciphertext messages constitute of blocks (with no initial or end marker) of varying length, where each block stands for a character string. (The plaintext message and the encoded message
have the same number of blocks.) Since there are no initial or end markers in the encoded message, the blocks cannot be identified without the key-automaton. So, without the key automaton, even the
length of the message is difficult to estimate, since block lengths and the number of blocks are not public for hackers. Due to the large number of possible key-automata, brute force attacks do not
lead to success either.
Thus incorporating the advantages of known cryptosystems the invention makes hacking impossible without knowing the key, and as it also has a simple structure encoding and decoding is fast. Also due
to its simple structure its micro-technological realization (microcontrollers, smart cards, etc.) is not difficult.
SUMMARY OF THE INVENTION [0021]
The essence of the invented apparatus is
to each element of the character set of the plaintext there is one or more final states of the key automaton assigned and each final state is assigned to one and only one element of the character set
for every state pair, whose first element is the initial state or a final state, its second element is any final state, there are arbitrarily many distinct input signal strings such that
the last element of state string, assigned to the first element of the state pair and the given input signal string by the generalised transition function, is the same as the second element of the
state pair
and none of the intermediate elements of the state string is a final state.
The essence of the invented method is
in encryption mode the key automaton
reads in the plaintext character by character starting from the initial state
generates a randomly chosen character string of length falling into a given, adjustable length interval for each character in the plaintext
the ciphertext consists of these character strings linked together
in decryption mode the key automaton
reads in the ciphertext character by character starting from the initial state
and decryption is accomplished by linking together the plaintext characters associated with the occurring final states, which provides the message in its original form.
During decryption the key automaton
starting from the initial state reads in the ciphertext character by character
and decryption is accomplished by linking together the input signals associated with the occurring final states, which provides the message in its original form.
DEFINITIONS [0037]
An initial automaton with an initial state and final states and no outputs (which is also referred to as Rabin-Scott automaton in the literature for short) is an algebraic structure which consists of
two non-empty sets and a function, and where one element and a non-empty subset of one of the sets is distinguished. One of the two non-empty sets is called the state set and the other is the input
set. The function which maps the product of the state set and the input set into the state set is usually called transition function. The elements of the state set are the states and the elements of
the input set are the input signals. A transition function can thus also be given as the function that associates a state with each pair, whose first element is a state and its second element is an
input signal. The distinguished element of the state set is the initial state; the distinguished subset is the set of final states. The elements of the set of final states are the final states. It
may happen that the initial state is a final state as well (this is not excluded).
Initial automata with an initial state and final states and with no outputs will henceforth be called automata for short.
In our description we assume that the state set, the input set (as well as the set of final states) is finite. We shall also assume that both the state set and the input set are ordered sets, so we
shall use the terms zeroth, first, second . . . and last elements. (For technical reasons counting starts with zero instead of one.) For finite state sets and input sets the transition functions are
also represented as matrices, which are called transition matrices. The transition matrix has as many rows as the number of input signals and as many columns as the number of states the automaton
has. An element k in the i-th (starting from zero) row and j-th (starting from zero) column of the transition matrix is the number (starting from zero) of the state which by the transition function
associates with the pair of the j-th state and the i-th input signal. We say about the element k of the i-th row and j-th column of the transition matrix that the i-th input signal takes the
automaton from the j-th state into the k-th state. (We also say that the automaton goes from its j-th state into its k-th state under the effect of the i-th input signal.)
A finite list of the elements of the state set is called a state string; a finite list of elements of the input set is called an input signal string. (Input signal strings and state strings of length
one that is of one element are not excluded.) Strings of binary elements that is of 0 and 1 are also called bit strings. The commas between the elements either in state strings or in input signal
strings will be omitted (as is customary). If a state string a
. . . a
has at least three elements, the states a
and a
, . . . , and as a
-1 are also called intermediate states. A sequence of one or two elements has no intermediate states. The transition function of the automaton is extended in the usual way, so that to each state and
input signal the extended transition function assigns a state string as follows:
Let a be a state and x
. . . x
be an input signal string (where x
, and x
, . . . , x
are input signals). Let a
be the state into which the input signal x
takes the automaton from state a, then a
be the state into which the automaton is taken by the input signal x
, and then a
the state where the automaton is taken by the input signal x
. . . , and a
be the state into which the automaton is taken from the state a
-1 by the input signal x
. Then the extended transition function assigns, by definition, the state string a
. . . a
to the pair consisting of state a and the input signal string x
. . . x
. Then we also say that the input signal string x
. . . x
takes the automaton from state a into state a
through the state sequence a
. . . a
Finally, if an automaton has the features discussed above in the summary of the invention, then it is a key-automaton. The key-automaton accomplishes encryption by generating a random character
string with no initial or end marker for each character in the plaintext and then linking them together provides the ciphertext. We shall assume that these random character strings have a minimal
block length, which is the minimal block length and a maximal length that is the maximal block length.
DETAILED DESCRIPTION OF THE INVENTION [0043]
FIG. 1 shows an example cryptographic apparatus, whose input is linked to the data input (2) of the processing unit (3) through an input line driver (1). The data output (4) of the processing unit
(3) is linked to the output of the cryptographic apparatus through an output line driver (5). A further input of the processing unit (3) has a two-way connection with the key automaton (7) and data
memory (8) through data bus (6). 3 The processing unit has a direct two-way connection with program memory (9). A further input of the processing unit has a two-way connection with a random number
source, preferably a random number generator (10).
If encoding and decoding is accomplished by two separate devices, an input of the processing unit in the encoding device needs to be in a two-way connection with a random number source, preferably a
random number generator (10). A cryptographic device used solely for decoding does not need a random number generator (10).
The invented cryptographic apparatus is capable of encoding and decoding as well, and then decoding is accomplished in the opposite direction in the device. The input of the device reads in the
plaintext when encoding and the ciphertext when decoding. The output is the ciphertext when encoding and the plaintext in its original form, when decoding, respectively.
One or more final states of the key-automaton (7) are assigned to each element in the character set of the plaintext, and each final state is assigned to only one element of the character set. To
each state pair, where the first element is the initial or a final state, the second element is any final state, there can be found a great number of distinct input signal strings, so that the last
element in the state sequence, assigned by the generalised transition function to the first element in the state pair and the input signal string, is the same as the second element in the state pair,
while none of the other elements in the state string is a final state.
The key-automaton (7) in the invention is preferably realised by a Rabin-Scott key automaton, but other key automaton (7), like deterministic automaton, non-deterministic automaton, weighted
automaton, cellular automaton, two-way automaton, automaton networks, linearly limited automaton, stack automaton, tree automaton, Turing machines, transducers, two or more taped automaton, two or
more taped transducers, or some combination of these are also possible.
The blocks in the cryptographic device shown in the example are realised by the following products:
The key-automaton (7) is installed (burnt) in a 4 kilobyte flash memory integrated into ATMegal28; the program memory (9) is 128 kilobyte PROM module integrated into ATMegal28; the 8 data memory (8)
is a 4 kilobyte MCU SRAM integrated into ATMegal28; the random number generator (10) is PROTEGO R300 and the line drivers (1,5) are RS232 with PC links.
The cryptographic device in the invention accomplishes encoding and decoding in the following way:
The device functions as an encoding device when in encoding mode and as a decoding device when in decoding mode. The transition matrix in the key-automaton (7) is realised as a continuous row array
and the minimal and maximal block lengths of the encrypted message is burnt in the flash memory. Working data, while processing, is stored in the data memory (8). It also stores the current state of
the key-automaton (current state), the state of the key-automaton currently to be reached (state to be reached) as well as a working array for storing character strings (where the character set,
depending on the actual technical realisation, can be of two, four or eight bits), and whose length is also the maximal block length of the encrypted message, and a pointer, pointing to the element
under processing in the working array. The program memory (9) has the program of the encrypting and decrypting algorithm burnt in, which is interpreted by the processing unit (3).
Steps of Encryption:
1. Providing space in the data memory (8) for the current state, the state to be reached, the work array and the pointer, and the current state takes the value zero.
2. The plaintext is fed into the processing unit (3) from the input line driver (1) in the form of a character stream.
3. The next character is read in (first character first).
4. The state to be reached in the data memory (8) will be of the same numeric value as that of the incoming character.
5. The pointer in the data memory (8) takes value zero.
6. The following steps are repeated as long as the current state and the state to be reached in the data memory become the same.
If the value of the pointer is greater than the maximal block length of the encrypted message minus one, then quit the cycle and go to step 5.
Generate a random character with the random number generator (10).
The generated random character will be the element of the work array (enumeration starting with zero) to which the pointer points.
The value of the pointer increases by one.
The new current state of the transition matrix in the key-automaton will be that element, whose row number is (starting from zero) the numeric value of the generated random character, its column
number (starting from zero) is the numeric value of the current state.
If the numeric value of the new current state is the same as the numeric value of the character, but the new current state is different from the state to be reached, quit the cycle and go to step 5.
Generate a random bit (zero or one) based on the random number generated by the random number generator (10).
Check the validity of the following properties:
The value of the generated random bit is one.
The new current state and the state to be reached are different.
The value of the pointer is at least the value of the minimal block length of the encrypted message minus one.
The value of the pointer is at most the value of the maximal block length of the encrypted message minus one.
There is a character that takes the key automaton from the current state into the state to be reached.
If all the properties above hold, accomplish the following steps
Generate a random character based on the random number generated by the random number generator (10) which takes the key-automaton (7) from its current state into the state to be reached.
Write the generated random character such as the element of the work array to which the pointer points.
The value of the pointer increases by one.
The actual state becomes the state to be reached.
7. If the value of the pointer is less than then minimal block length of the encrypted message minus one go to step 5.
8. The processing unit (3) sends the content of the work array (sequentially), where the length is determined by the value of the pointer, as output data (4) to the output line driver (5), and then
the pointer takes the value zero.
9. The current state in the data memory becomes the state to be reached after step 4.
10. If there are more input characters, then functioning continues from step 3. If there are no more input characters, encryption is finished.
Steps of Decryption:
1. Providing space in the data memory (8) for the current state, the state to be reached, the work array and the pointer, and the current state takes the value zero.
2. The encrypted message is fed into the processing unit (3) through the input line driver (1) as a character stream.
3. The next character is read in (first character first).
4. The value of the new current state in the data memory (8) will be the element of the transition matrix of the key-automaton (7) whose row number (starting from zero) is the numeric value of the
character read in step 3, and its column number (starting from zero) is the numeric value of the current state.
5. If the numeric value of the current state is the same as the value of a character, then this character is written to the output line driver (5).
6. If there are more incoming characters, go to step 3. If there are no more incoming characters, decryption is finished.
During encryption the plaintext is read in character by character starting from the initial state of the key-automaton (7), and to each character, a random character string of adjustable length in a
given length interval is assigned, and by linking these character strings together we get the encrypted message. These random character strings will take the key automaton from its current state into
the final state assigned to the character read in, so that none of intermediate states are final states. Before reading in the first character the current state of the key automaton in the encryption
device is the initial state, and then before reading in the next character it is the final state assigned to the previous character. During decryption the encrypted message is read in character by
character starting from the initial state of the automaton, and by linking together the input signals assigned to the final states found we get the original message.
A Possible Realization for Binary Message Streams:
Micro-technological realization is easiest for binary message streams. The input and output characters of the device(s) are binary, in this case, and the random number generator (10) is also binary,
that is, it is used for generating a single random bit only.
Other Possible Ways of Realization
The above described realization of the invention--including the version for binary message streams--assigns to each character the state of the key-automaton (7) which has the same numeric value as
the numeric value of this character. There are other possible ways of realization--to increase security--when there can be more than one final states for each character. In such a realization the
flash memory of the key-automaton (7) has to store (in the form of a linked list for example) which final states are assigned to which character. Encryption changes only in step 4. from the
realization described above, so that the random number generator (10) or another random number source determines which of the possible final states associated with the character should be the current
state to be reached. Decryption changes in step 5. only, so that whenever the current state is a final state, the character to which the current state corresponds as a final state is written to the
output line driver (5).
SHORT DESCRIPTION OF THE FIGURE [0093]
FIG. 1. The block diagram of the invention shows the structure of the two functional (encryption and decryption) device and the connections between its components.
EXAMPLE [0094]
Let the transition matrix of the key-automaton (where the initial state is 0) be the following:
-US-00001 4 1 3 0 3 0 4 1 2 2
For simplicity suppose that the (binary) character set of the plaintext is {0,1}. Let us assign to the character 0 the final state 0, and to the character 1 the final state 1. Let L
denote, for any pair i,j from {0,1,2,3,4}, the set of all input signal strings that take the automaton from its state i to state j, so that none of the intermediate states is a final state.
Suppose that the device is in encryption mode, so the encryption program is stored in the program memory (9). There is space allocated in the data memory (8) to the current state of the key-automaton
(current state), to the state to be reached by the key-automaton (state to be reached), to a work array, the size of which (in bits) is also the maximal block length of the encrypted message, and a
pointer pointing to the element of the array under processing. Suppose that the size of the work array is 6 bits. In other words, suppose that the maximal block length of the encrypted message is 6
bits. Let the minimal block length of the encrypted message be 1 bit.
Let us consider the hexadecimal 4F4B ASCII code of the word OK. Converting this 4F4B code into binary code we get the string 0100111101001011. Suppose, that this is our plaintext. Also suppose that
this is the message the invention will encrypt in encryption mode. The apparatus will show, on 5 the output line driver random signal strings of at least 1 and at most 6 bits length for each of the
values 0,1,0,0,1,1,1,1,0,1,0,0,1,0,1,1 arriving sequentially into the input line driver (1), of the message 0100111101001011.
More precisely, when used as an encryption device, the reading in of the first bit from the input line driver (1) generates a random bit string (of at least 1 at most 6 bits length) and writes it on
5 the output line driver, which takes the key automaton from its initial state (i.e. state 0) to the final state (i.e. 0) assigned to the value of that first bit (i.e. 0) so that none of the
intermediate states is a final state (i.e. neither 0, nor 1). More precisely, the device when receiving the signal 0 from the input line driver (1), it will write a random element (of at least 1 and
at most 6 bits length) of L
to the output line driver (5).
Then the current state will be the previous state to be reached (i.e. state 0) and the next bit read in from 1 the input line driver will cause the generation of a random bit string (of at least 1
and at most 6 bits length) and writes it on the output line driver (5), which takes the key automaton from its current state (i.e. state 0) to the final state (i.e. 1) assigned to the value of the
bit read in (i.e. 1) so that none of the intermediate states is a final state (i.e. either 1 or 0). More precisely, the device triggered by the signal 1 coming from the input line driver (1) will
write a random element (of length at least 1 and at most 6 bits) of L
to the output line driver (5).
The device works similarly for the third, fourth, . . . , sixteenth bit read in, when after having a random element (of length at least 1 and at most 6 bits) of L
and of L
from the input line driver (1)) it will generate a random element (of length at least 1 and at most 6 bits) of the sets L
, L
, L
, L
, L
, L
, L
, L
, L
, L
, L
, L
, L
, L
Suppose, for instance, that these random bit strings (of length at least 1 and at most 6 bits) are the following:
-US-00002 1 (is in L
, 0) 0011 (is in L
, 1) 100 (is in L
, 0) 1 (is in L
, 0) 011 (is in L
, 1) 0 (is in L
, 1) 0 (is in L
, 1) 1011 (is in L
, 1) 100 (is in L
, 0) 011 (is in L
, 1) 1100 (is in L
, 0) 1 (is in L
, 0) 0011 (is in L
, 1) 10100 (is in L
, 0) 001011 (is in L
, 1) 0 (is in L
, 1)
In this case the bit strings 1, 0011, 100, 1, 011, 0, 0, 1011, 100, 011, 1100, 1, 0011, 10100, 001011, 0 appearing on the output line driver (5) will be linked together to give the bit string
100111001011001011100011110010011101000010110, which is a random encrypted message for the message 0100111101001011.
Let us consider the bit string 100111001011001011100011110010011101000010110 as an encrypted message and suppose that the device is in decryption mode, so the decryption software is stored in the
program memory (9). Also suppose that this is the message the device will decrypt in decryption mode.
For the values 1,0,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,1,1,1,0,1,0- ,0,0,0, 1,0,1,1,0 of the bit string 100111001011001011100011110010011101000010110 arriving bit by bit at the
input at the input line driver (1) it will provide a suitable bit string for output line driver (5). Reading in the encrypted message bit by bit the device checks what state sequence the device goes
through starting from its initial state (i.e. from state 0) due to the incoming bits as input signals. The final states (states 0 or 1) occurring during the checking process are written to the output
line driver (5). (The occurrence of states 2, 3, 4 are not considered.) These values will correspondingly be 0,1,0,0,1,1,1,1,0,1,0,0,1,0,1,1. Linking these binary signals we get the bit string
0100111101001011, which is exactly the decrypted plaintext. Converting this bit string into hexadecimal form we get the hexadecimal character string 4F4B, which considered as an ASCII code we get the
word OK.
Number of Key-automata
It is a subject of further research to give a better estimate for the number of key automata with a given number of input signals and a given number of states, and to find methods which can choose
random elements from a wider class of key automata. We shall see that using the method described (which uses a relatively limited class of key automata for choosing randomly element) the number of
key automata to choose from is huge.
Suppose, for simplicity, that the character set of the original message coincides with the input set {0, . . . , m-1} of the key automaton, and let {0, . . . , n-1} be the set of states, where m is
at least 2 and n is at least two times m. Let 0 be the initial state and assign the state k as the final state to each element k in the character set of the original message. (It is not a problem
that the initial state is a final state as well.) Let us create the transition matrix of the automaton as follows:
Step 1: for every k in the set {0, 1, . . . , m-1} let us give a random integer u between 0 and m-1, and an integer v between m and n-1, and write v in the u-th row and k-th column of the matrix
(numbering of the rows and columns begins with 0);
Step 2: for every k in {m, . . . , 2m-1} give a random integer u between 0 and m-1 and write (k-m) into the u-th row and k-th column of the matrix;
Step 3: for each element k in {m, . . . , n-2)} give a random integer u between 0 and m-1 and write (k+1) into the u-th row and k-th column of the matrix;
Step 4: give a random integer u between 0 and m-1 and write m into the u-th row and (n-1)-st column of the matrix;
Step 5: choose a random number between 0 and n-1 for the other elements of the matrix.
By Step 1. there is a state j in {m, . . . , n-1} and an input signal for every state i in {0, 1, . . . , m-1} that takes the automaton from state i into state j. By Step 2., there is a state i in
{0, 1, . . . , m-1} and an input signal for every state j in {m, m+1, . . . , 2m-1} which takes the automaton from state j into state i.
By steps 3. and 4. a suitable input signal will take the automaton from any state in {m, . . . , n-2} into the subsequent state, and a suitable signal will take it from state n-1 into state m. Thus
for any state pair i,j in {m, . . . , n-1} there can be found arbitrarily many input signal strings that take the automaton from state i into state j, so that none of the intermediate states is in
{0, 1, . . . , m-1}.
By Step 1., for each state i in {0, 1, . . . , m-1} there is a state j in {m, . . . , n-1} and an input signal which take the automaton from state i into state j. By Step 2., for each state j in {m,
m+1, . . . , 2m-1} there is a state i in {0, . . . , m-1} and an input signal that take the automaton from state j into state i. Then by Steps 3. and 4. for every state pair i,j in {0, 1, . . . ,
m-1} there can be found any number of input signal strings, which take the automaton from state i into state j, so that none of the intermediate states is in {0, 1, . . . , m-1}. Since 0 is not the
initial state alone, but the final state assigned to the input signal 0 as well, it means that the automaton generated is a key automaton indeed.
Suppose first, that m=2. Then before Step 5. there are n-2 empty entries in the matrix. Each of these entries can filled in n different ways in Step 5., which gives us n
-2 possibilities. Furthermore it is obvious that if we change two elements in the columns of the transition matrix, we get another key automaton, which multiplies the number of possibilities by 2
. If n=256, say, then the number of all the possible key automata is 256
, that is 2
Let m now be greater than 2. It is obvious that before Step 5. there is at least one empty entry in each column of the transition matrix, and it is also easy to see that they can be filled in n ways.
Then in Step 5. we have at least n
ways to fill the rest of the entries in the matrix. So, if n=256 there are at least 256
, that is the number of possible key automata is more than 2
Authentication, Identification, Changing Keys
Authentication and identification are very important issues in computer applications. The person or the device we want to communicate with could be in the other room or on an other continent; neither
the usual visual nor the acoustic traces help us here. So the aim of this invention is also to prove the person's or device's identity, including digital signatures as well--to assure the
authenticity and identity of the message. Attackers, of course, do not have access to the key automaton used for encryption and decryption. However if relatively short messages are also allowed, the
attacker can send a message to the recipient, which by mere chance will be an encrypted message. So it is better if the original message also has a secret key for authentication and identification.
So for the proposed invention a typical message to be encrypted should consist of the following parts:
a digital signature as a common key for authentication and identification
the message itself
and, to advance security, the key automaton to be used for encrypting and decrypting the next message.
The digital signature for authentication may be omitted if it is not needed. Furthermore, usually, there is no need for a new key automaton for each interaction. So the description of the next key
automaton may not be included, or if it is there, it can be the same as the current automaton.
Advantages of the Invention
The encrypted message in the invented device can represent any number of various plaintext messages, which makes breaking attempts, no matter how sophisticated and powerful background they use,
practically impossible; it can be used as often as needed without any further key sharing and due to the simplicity of the operations used it is fast and the micro-technological realization is also
1) Resistance to code breaking and interference. The resistance of this invention to code breaking and interference is also due to the difficulty of a ciphertext only attack or a known plaintext
attack as well. The key automaton is chosen randomly from a large set of elements. The implementation can be a key automaton with 256 states and 128 input signals, where the character set of the
message can be coincident with the input set of the key automaton. As we have seen, there are more than 256
possible key automata to be randomly generated. So this gives more than 2
options for choosing the key automaton. It is obviously impossible to break such a system using brute force. Furthermore it is enough to have a one bit mistake either in the key or in the encrypted
message to cook the message. (It is not the aim of the invention to treat bit loss or bit changes due to transfer problems. This problem can be solved with the well-known protocol family TCP/IP.)
2) Each plaintext can be associated with several ciphertexts. The security of this invention is also due to the property that with a given key every message can have several encrypted versions.
Suppose, for instance that the key automaton has 256 states, it has 128 input signals and final states, and let the minimal block length of the encrypted message be 10. In this case for any state the
expected value of the number of input signals that take this state into an intermediate state is 128. Then the expected value of the number of input signal strings that take this state into a final
state so that none of the intermediate states is a final state is 128
. So the expected value of the number of encrypted character strings that can be associated with a character of the plaintext is above 2
. In this case (taking a page of typed text to be of 3 kilobytes storage capacity) the expected value of the number of encrypted messages for one page of typed text is 3×2
. This means that even if the attacker can identify a plaintext--ciphertext pair, this information cannot be used to identify another encryption of the same message.
3) One character corresponds to several encrypted character strings. Following the previous line of argumentation it is expected after every 3.8×10
(which is more than 2
) pages to have the same encrypted character string for a given character. (We note that during computer software simulation we used a minimal block length of 10 characters and a maximal block length
of 65536 characters for the ciphertext which meant that the ciphertext was about 14 times as long as the plaintext.) So code breaking based on the study of the frequency of repetitions is not
successful either.
4) The length of the ciphertext (in a statistical sense) depends on the length of the plaintext. However choosing longer minimal and maximal block lengths for shorter messages and shorter minimal and
maximal block lengths for longer messages, hackers cannot estimate even the length of the message from the ciphertext, although there are no dummy characters used.
5) Fast processing. The simplicity of the key automaton allows for fast processing. Testing software simulations of this invention were implemented using a computer programme written in C++. The
implementation was tested on a conventional computer of 1.1 GHz frequency (IBM X40). The test results show that for a big key automaton (265 input signals and 512 states) and with big block lengths
(10 bytes minimum and 65536 maximum) 120 kb/s as encryption speed and 210 kb/s as decryption speed, relative to the length of the plaintext. Choosing suitable hardware devices (microcontrollers,
smart cards etc.) in the technical realisation can considerably improve the processing speed.
6) Broadcasting applicability. Test results show that the invention is efficient in broadcasting applications where the encryption device of the transmitter is relatively fast (at least twice as
fast) compared to the speed of the receiver's decryption device.
Patent applications in class COMMUNICATION SYSTEM USING CRYPTOGRAPHY
Patent applications in all subclasses COMMUNICATION SYSTEM USING CRYPTOGRAPHY
User Contributions:
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Mplus Discussion >> Basic questions concerning LGM
Basic questions concerning LGM
Tiina Ojanen posted on Tuesday, October 18, 2005 - 2:18 am
I'm conducting LGM with Mplus using three contsructs (each with 3 measurement points) as growth curves 1. social goals (multiple indicator curve), 2., withdrawal behavior (multiple indicator cureve)
and social status, and interested in conducting regression among these constructs. I was wondering the following set of questions:
1) When wanting simply to indicate, that one construct (A) predicts increases in another construct (B) over time, which again predicts increases in another contruct (c), is it O.K. simply to predict
the trend of B with a level of A, and the trend of C with the level of B?
Sometimes reseacrhes use the level of A to predict the level of B, and the trend of A to predict the trend of B, and so on. If I'm not interested in predicting change with a change, is the former
approach suitable?
2) When setting linear trends as growht factors, is it common to stick to the original 0, 1, 2, loadings, even though there is a singificant negative mean for the trend, suggesting that there is a
decrease rather than increase over time?
If so, then the results would be merely interpreted taking into consideration a decreasing rather than increasing trend? For instance, when wanting so show that the higher the level of goals, the
higher the increases in withdrawal over time, there is a significant, negative estimate when predicting the (decreasing) trend of withdrawal with the level (intercept) of goals. Does this seem right?
Given that the loading of 0 defines the level of a given construct, it would seem that if the trend was replaced with loadings of 2,1,0, the interpretation of the level would change?
3) How excatly are multigroup differences for gender conducted in the LGM framework? Are each of the estimated latent constructs, along with the estimated regression paths, tested individually by
estimating them freely and constrained to be equal among genders (i.e., setting the variace of the given parameters equal accross the groups with a (1) marking in the syntax)?
4) I have defined a model including the three growth curves mentioned at the beginning, and have significant regression coefficients among the level of A and the trend of B, and the level of B and
the trend of C., and almost O.K. model fit. However, due to negative variance of two of T1 and T3 withdrawal (latent) indicators, these were set to zero, after which the CFI dropped to .86. Given
that RMSEA coefficient is O.K., this seems unfortunate. Is there something that I could do about this? For instance, have I forgot to fix some correlations os something, or should I do something for
the starting values of the withdrawal curve? I have newer defined starting values before.
5) If a level of a trend is negative (esimate of the intercept mean is significantly negative), is this O.K., if the variable in question (measured in three time points) can have negative values
(there is a growing trend in this construct with loadings 0, 1, 2)? Does this indicate that the average level of that contsruct is below zero?
Thank you so much in advance,
bmuthen posted on Tuesday, October 18, 2005 - 7:05 pm
Here are quick answers that should be viewed critically since I don't know enough about your research situation and don't want to get into general consulting.
1) Certainly sounds reasonable. As long as the intercepts are allowed to covary.
2) All of that is correct and it is perfectly fine to stay with 0, 1, 2, ... even with a negative trend.
3) Yes, you put say (1) at the growth factor related parameters that you are interested in testing gender invariance for - such parameters are typically means and regression slopes.
4) One common reason for a misspecified multiprocess model is correlated residuals across processes - look at your Modification indices. Starting values shouldn't matter here.
5) Say that you have observed outcome values for an individual at the 3 time points of -1, -2, -3 and have a negative mean for the slope factor of -1. With time scores of 0, 1, 2 this still adds up
fine since this models says that the outcome mean declines by 1 each time point (the mean change is the time score times the slope mean).
Charles Saunders posted on Tuesday, November 15, 2005 - 1:44 pm
In an unconditional growth model that is set up in logs (i.e. ln(Y) = a + b(ln(Time)) - how do I interpret b and a? In OLS b would give the % change per % change in x - but how do I interpret that in
terms of time? Thanks.
bmuthen posted on Tuesday, November 15, 2005 - 5:53 pm
As ln(Time) changes one unit, ln(Y) changes b units. Note then that ln(Time) changing one unit corresponds to different time changes for different time levels since ln(X) is nonlinear in X. So you
have to consider specific time intervals in the interpretation.
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Manipulatives Make Abstract Math Concepts Concrete
Manipulatives Make Abstract Math Concepts Concrete
Using math lessons that include manipulatives can help cement learning.
If a picture is worth a thousand words, a hand-held object is worth a million. Using manipulatives in mathematics can help bring conceptual ideas alive for students. Yet, because mathematics is
traditionally taught more abstractly, relying on textbooks, paper, and pencil, teachers in middle school and high school are often reluctant to use manipulatives. However, manipulatives can help
relieve student boredom, spark imaginations, and promote cooperative learning at any age level.
Things always get exciting when the props come out in a classroom, and just about anything can be used. Pennies, beads, sea-shells, M&Ms, calculators, rulers, dominos, dice, spinners, playing cards,
buttons, and even pebbles are just some objects that allow students to make mathematics more concrete. Such manipulatives can easily be used when focusing on estimation techniques, fractions,
measurement, ratios, probability, and properties of whole numbers, for example.
There are also many companies that make more specialized manipulatives. Tangrams, pattern blocks, geoboards, algebra tiles, Cuisenaire Rods, geometric solids, mini cubes, square tiles, and base ten
sets are just some of the manipulatives available for purchase. Many of these manipulatives are familiar to elementary math teachers, but they can also be utilized by middle school and high school
teachers by creating more complex challenges and experiments for students.
Manipulatives may also encourage practical applications of mathematical concepts, and increase the retention of information. Rather than assigning problem sets every day for homework, teachers
could instead ask students to design a probability simulation using a manipulative. Additionally, manipulatives often allow for multicultural and historical connections to mathematics. For instance,
tangrams have Chinese roots. Students might also study the evolution of the abacus through various cultures and number systems.
While it is true using manipulatives requires teacher to spend more time planning, setting up, and cleaning up, their use typically proves to be worth it because it provides a engaging, kinesthetic
change for students.
Students strategize how to vary different numbers of tiles to form shapes, and then find minimum and maximum perimeters. Students examine area with either square tiles or geoboards.
Students analyze patterns of exponential models by collecting data using paper folding and M&Ms. They use graphing calculators to create scatter plots that lead to equations for the exponential
models. They then apply this model to predict the future population changes in African rhinos.
Students listen as the teacher identifies and defines four math properties. They use manipulatives (small and large rocks) to experience non-traditional number systems. They use the "rocks and
pebbles" to demonstrate the properties.
Students use paper cups and colored chips to observe properties of operations with real numbers. As a class, students brainstorm using manipulatives to demonstrate the associative, commutative,
distributive identity and inverse properties.
Geoboard Exploration of the Pythagorean Relationship
Students create a right triangle on a geoboard or on dot paper. They construct a square on each side of the triangle, and complete a table and determine a pattern. Students use the Pythagorean
Theorem to derive the distance formula. | {"url":"http://www.lessonplanet.com/article/math/manipulatives-make-abstract-math-concepts-concrete","timestamp":"2014-04-19T14:49:15Z","content_type":null,"content_length":"41772","record_id":"<urn:uuid:f47ceb07-ef44-4522-8fd7-264923752438>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00039-ip-10-147-4-33.ec2.internal.warc.gz"} |
RP 6: Throwing a fooball, Part II | Science | WIRED
• By Rhett Allain
• 12.27.09 |
• 8:00 am |
In part I of this post, I talked about the basics of projectile motion with no air resistance. Also in that post, I showed that (without air resistance) the angle to throw a ball for maximum range is
45 degrees. When throwing a football, there is some air resistance this means that 45 degree is not necessarily the angle for the greatest range. Well, can’t I just do the same thing as before? It
turns out that it is a significantly different problem when air resistance is added. Without air resistance, the acceleration was constant. Not so now, my friend.
The problem is that air resistance depends on the velocity of the object. Search your feelings, you know this to be true. When you are driving (or riding) in a car and you stick your hand out the
window, you can feel the air pushing against your hand. The faster the car moves, the greater this force. The air resistance force depends on:
• Velocity of the object. The typical model used for objects like a football would depend on the direction and the square of the magnitude of the velocity.
• The density of air.
• The cross sectional area of the object. Compare putting an open hand out the car window to a closed fist out the car window.
• Some air drag coefficient. Imagine a cone and a flat disk, both with the same radius (and thus same cross sectional area). These two objects would have different air resistances due to the shape,
this is the coefficient of drag (also called other things I am sure).
So, since the air force depends on the velocity, it will not be a constant acceleration. Kinematic equations won’t really work. To easily solve this problem, I will use numerical methods. The basic
idea in numerical calculations is to break the problem into a whole bunch of little steps. During these small steps, the velocity does not change much so that I can “pretend” like the acceleration is
constant. Here is a diagram of the forces on the ball while in the air.
Before I go any further, I would like to say that there has been some “stuff” done on throwing a football before – and they probably do a better job than this post. Here are a few references
(especially with more detailed discussion about the coefficient of drag for a spinning football):
And now for some assumptions:
• I hereby assume that the air resistance is proportional to the square of the magnitude of the velocity of the object.
• The orientation of the football is such that the coefficient of drag is constant. This may not actually be true. Imagine if the ball were thrown and spinning with the axis parallel to the ground.
If the axis stayed parallel to the ground, for part of the motion the direction of motion would not be along the axis. Get it?
• Ignore aerodynamic lift effects.
• Mass of the ball is .42 kg.
• The density of air is 1.2 kg/m^3.
• The coefficient of drag for the football is 0.05 to 0.14
• Typical initial speed of a thrown football is around 20 m/s.
And finally, here is the recipie for my numerical calculation (in vpython of course):
• Set up initial conditions
• Set the angle of the throw
• Calculate the new position assuming a constant velocity.
• Calculate the new momentum (and thus velocity) assuming a constant force.
• Calculate the force (it changes when the velocity changes)
• Increase the time.
• Keep doing the above until the ball gets back to y=0 m.
• Change the angle and do all the above again.
The answer
First, I ran the program with an initial velocity of 20 m/s. Here is the data:
At 35 degrees, this gives a distance of 23 meters (25 yards). This doesn’t seem right. I know a quarterback can throw farther than that. What if I change the coefficient to 0.05? Then the greatest
angle is closer to 40 degrees and it goes 28 meters. Still seems low (think Doug Flutie). What about with no air resistance? Then it goes 41 meters (at 45 degrees). So, here is the Doug Flutie throw.
From the video, it looks like he threw the ball from the 36ish yard line to about the 2 yard line. This would be 62 yards (56.7 meters). I am going to assume a coefficient of 0.07 (randomly). So,
what initial speed will get this far? If I put in an initial velocity of 33 m/s, the ball will go 55.7 meters at an angle of 35 degrees.
Really the thing that amazes me is that someone (not me) can throw a ball that far and essentially get it where they want it. Even if they are only sometimes successful, it is still amazing. How is
it that humans can throw things somewhat accurately? We obviously do not do projectile motion calculations in our head – or maybe we do? | {"url":"http://www.wired.com/2009/12/rp-6-throwing-a-fooball-part-ii/","timestamp":"2014-04-19T02:36:25Z","content_type":null,"content_length":"106900","record_id":"<urn:uuid:a5ee426a-d016-42ef-a61e-5fdf06b12185>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00623-ip-10-147-4-33.ec2.internal.warc.gz"} |
[FOM] Fwd: Paul Cohen was wrong
[FOM] Fwd: Paul Cohen was wrong
Brian White white at math.stanford.edu
Sun Sep 11 19:08:52 EDT 2011
On Sep 10, 2011, at 12:43 PM, Monroe Eskew wrote:
> I know the existence of \aleph_1 can be logically separated from the
> existence of P(\omega). But in his quote, he says that cardinals are
> built "from ideas deriving from the replacement axiom." I disagree.
Actually Cohen does not say that (in the passage you quote).
But he does certainly suggest it.
The phrase "ideas deriving from the replacement axiom"
is rather vague, and I have no idea what he meant by it.
Do you?
In your original message ("Paul Cohen was wrong"), you wrote:
>>> But his argument is self-defeating! Because in fact, \aleph_1 cannot
>>> be built without the powerset axiom. The collection HC of
>>> hereditarily countable sets satisfies ZFC minus powerset.
Do you think Cohen wasn't very well aware of that?
> . But what is the difference in character
> between your A1 and A2 below and the following?
> B1: There is a set that contains all subsets of omega.
> B2: There is a set that contains all subsets of P(omega).
The difference is that B1 is more powerful than A1: B1 implies
A1 but not vice versa. In a sense, it is much more powerful than
A1 because even A1 plus A2 plus A3 etc do not imply B1.
> The A's and B's are both just special cases of powerset.
> On Fri, Sep 9, 2011 at 9:53 PM, Brian White <white at math.stanford.edu> wrote:
>> I think you've misunderstood what Cohen is asserting.
>> He doesn't say you can get aleph_1, aleph_2, etc using
>> ZFC minus the power set axiom.
>> Rather, he says you can get them with something weaker than
>> the power set axiom, in particular with higher axioms of
>> infinity.
>> He doesn't say what those axioms
>> are, but for example the following would do:
>> A1: there is an ordinal that contains all countable ordinals.
>> (Of course there is then a smallest such ordinal, namely aleph_1).
>> A2: there is an ordinal that contains all ordinals equipotent to aleph_1
>> etc.
>> On Sep 8, 2011, at 7:49 PM, Monroe Eskew wrote:
>>> Consider the following quote from Paul Cohen's book, Set Theory and
>>> the Continuum Hypothesis:
>>> <<"A point of view which the author [Cohen] feels may eventually come
>>> to be accepted is that CH is obviously false. The main reason one
>>> accepts the axiom of infinity is probably that we feel it absurd to
>>> think that the process of adding only one set at a time can exhaust
>>> the entire universe. Similarly with the higher axioms of infinity. Now
>>> \aleph_1 is the cardinality of the set of countable ordinals, and this
>>> is merely a special and the simplest way of generating a higher
>>> cardinal. The set C [the continuum] is, in contrast, generated by a
>>> totally new and more powerful principle, namely the power set axiom.
>>> It is unreasonable to expect that any description of a larger cardinal
>>> which attempts to build up that cardinal from ideas deriving from the
>>> replacement axiom can ever reach C.
>>> Thus C is greater than \aleph_n, \aleph_\omega, \aleph_a where a =
>>> \aleph_\omega, etc. This point of view regards C as an incredibly rich
>>> set given to us by one bold new axiom, which can never be approached
>>> by any piecemeal process of construction. Perhaps later generations
>>> will see the problem more clearly and express themselves more
>>> eloquently.">>
>>> But his argument is self-defeating! Because in fact, \aleph_1 cannot
>>> be built without the powerset axiom. The collection HC of
>>> hereditarily countable sets satisfies ZFC minus powerset. The bold,
>>> powerful principle which allows C to exist is no different from that
>>> which allows \aleph_1 to exist.
>>> _______________________________________________
>>> FOM mailing list
>>> FOM at cs.nyu.edu
>>> http://www.cs.nyu.edu/mailman/listinfo/fom
>> _______________________________________________
>> FOM mailing list
>> FOM at cs.nyu.edu
>> http://www.cs.nyu.edu/mailman/listinfo/fom
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
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RE: st: Computing medcouple
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RE: st: Computing medcouple
From "Lachenbruch, Peter" <Peter.Lachenbruch@oregonstate.edu>
To <statalist@hsphsun2.harvard.edu>
Subject RE: st: Computing medcouple
Date Thu, 13 Aug 2009 08:15:38 -0700
A few years ago I tried to develop a measure of skewness based on
"Skewness" = [(P75-P50)-(P50-P25)]/[P75-p25]
And also similar ones based on P90 and P10. I did fairly extensive
simulations and found that the P90, P10 based ones did a bit better in
expressing skewness. In addition, the distribution of this was not
nicely behaved, but by log-transforming you would get a statistic that
looked very nicely normal.
In doing this I learned of the l-moments articles and was delighted that
Nick had already written a routine for this.
Peter A. Lachenbruch
Department of Public Health
Oregon State University
Corvallis, OR 97330
Phone: 541-737-3832
FAX: 541-737-4001
-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Nick Cox
Sent: Wednesday, August 12, 2009 12:34 PM
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: Computing medcouple
This sounds a little similar in spirit to using L-moments to calculate a
skewness measure. The latter approach arguably has two features: it is
systematic and it is already implemented in Stata through -lmoments-
from SSC.
As far as medcouple is concerned, you could compute it exactly or by
sampling. I've no code to offer.
My prejudice here is that for most problems you would be better off
either transforming the data or using a graph form that discarded less
of the information than a box plot does. Otherwise put, if the data are
very skew you usually need to see more detail about the tails than a
boxplot provides.
Ronnie Babigumira <rb.glists@gmail.com>
Vandervieren and Hubert (2004) present what they call a robust measure
of skewness using the medcouple
Given a distribution F, medcouple (MC(F)) is defined as
MC(F) = med h(xi,xj) given xi<med<xj
- h = (xj-m_F)-(m_F-xi)
- m_F is the median of F
I would like to compute MC but dont know how to even start. Any pointers
will be much appreciated.
Vanderviere, E. & Huber, M. (2004). An adjusted boxplot for skewed
distributions. In J. Antoch
(Ed.), COMPSTAT2004 Symposium: proceedings in computational statistics
(pp. 1933-1940). Heidelberg,
Germany: Physica-Verlag.
The paper can be downloaded here
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American Mathematical Society
AMS Presidents: A Timeline
AMS Presidents: A Timeline home page
7. Thomas Scott Fiske (1865-1944)
President, 1903-1904
Ph.D. Columbia University, New York, New York, 1888
Fiske founded the American Mathematical Society in 1888 to foster comradeship and communication among mathematicians. He had visited England where he witnessed a vibrant mathematical community, and
upon his return home he called for the establishment of such a society in the United States: "It is proposed by some recent students of the graduate school of Columbia College to establish a
mathematical society for the purpose of preserving, supplementing, and utilizing the results of their mathematical studies." It was Fiske's strong support that saw the Colloquium lecture series
begin in 1896--and continue to this day.
For more information:
• A Semicentennial History of the American Mathematical Society, 1888--1938, by Raymond Clare Archibald (AMS, 1938), which contains CV, honors, bibliography, biographical notes and sources.
• The MacTutor History of Mathematics Archive
• Mathematics Genealogy Project
• Columbia University Archival Collections
• A Century of Mathematics in America, Part I, Edited by Peter Duren (American Mathematical Society, 1988), "Thomas S. Fiske and Charles S. Peirce," by C. Eisele, p.41
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Spring Geometry Tutor
my name is Kevin, and I can tutor on a variety of subjects. I am a research scientist and Yale educated with a post graduate degree. I have tutored in the past.
85 Subjects: including geometry, English, Spanish, reading
...If what I am saying isn’t getting across, I change the approach. It is important to me to find out exactly where the student is at in understanding, so that time is not wasted going over
material the student is already proficient in, and also to ensure that critical points a student may be strug...
14 Subjects: including geometry, algebra 1, algebra 2, economics
I stress the learning of problem-solving skills above and beyond just learning facts. If you are willing to try and are open to new approaches, I can help you develop skills that can last a
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49 Subjects: including geometry, reading, English, writing | {"url":"http://www.purplemath.com/spring_tx_geometry_tutors.php","timestamp":"2014-04-18T19:01:34Z","content_type":null,"content_length":"23502","record_id":"<urn:uuid:3413c6df-d93c-4ace-bff3-2ca9561ea47e>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00499-ip-10-147-4-33.ec2.internal.warc.gz"} |
Peck, Kelley / Unit 4: Factors, Fractions, & Exponents; Pythagorean Theorem
Simplifying fractions (numbers only, no variables) practice that gets more difficult the higher you score.
Play as a solo player (team) or against a friend (on the same computer). Questions are about factors, multiples, and GCF/LCM. Numbers only, no variables.
Practice PFAC with trees (prime factorization). Numbers only, no variables.
Review your knowledge of divisibility rules in this game.
Test your knowledge of identifying prime numbers at three different levels.
Increasing levels of difficulty matching equivalent fractions. Numbers only, no variables.
4 minutes of review!
Review problems with explanations and plenty of examples. Then 6 practice problems at the end with answers & explanations.
on simplifying and exponents. You decide how many and the type of problem. Also create an answer key!
Type in your answer and submit to move to the next problem.
Choose from a variety of practice activities with exponents, variables, and multiplying and dividing terms.
Given the number, convert into scientific notation. | {"url":"http://www.tesd.k12.pa.us/Page/7252","timestamp":"2014-04-21T09:36:03Z","content_type":null,"content_length":"85846","record_id":"<urn:uuid:40b179a1-334a-41a9-9fa8-863d7f366e25>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00118-ip-10-147-4-33.ec2.internal.warc.gz"} |
Overview of Geometric Algebra in Physics
Oersted Medal Lecture 2002:
Reforming the Mathematical Language of Physics
In: D. Hestenes, Am. J. Phys. 71 (2), February 2003, pp. 104--121.
The connection between physics teaching and research at its deepest level can be illuminated by Physics Education Research (PER). For students and scientists alike, what they know and learn about
physics is profoundly shaped by the conceptual tools at their command. Physicists employ a miscellaneous assortment of mathematical tools in ways that contribute to a fragmentation of knowledge.
We can do better! Research on the design and use of mathematical systems provides a guide for designing a unified mathematical language for the whole of physics that facilitates learning and
enhances physical insight. This has produced a comprehensive language called Geometric Algebra, which I introduce with emphasis on how it integrates and simplifies classical, relativistic and
quantum physics. Introducing research-based reform into a conservative physics curriculum is a challenge for the emerging PER community. Join the fun!
Oersted Medal Lecture (short version) in Power Point form: ppt version
A sequel to the Oersted paper:
Spacetime Physics with Geometric Algebra
In: D. Hestenes, Am. J. Phys. 71 (7), July 2003, pp. 691--714.
This is an introduction to spacetime algebra (STA) as a unified mathematical language for physics. STA simplifies, extends and integrates the mathematical methods of classical, relativistic and
quantum physics while elucidating geometric structure of the theory. For example, STA provides a single, matrix-free spinor method for rotational dynamics with applications from classical rigid
body mechanics to relativistic quantum theory -- thus significantly reducing the mathematical and conceptual barriers between classical and quantum mechanics. The entire physics curriculum can be
unified and simplified by adopting STA as the standard mathematical language. This would enable early infusion of spacetime physics and give it the prominent place it deserves in the curriculum.
A third paper in the Oersted sequence:
A new gauge theory of gravity on flat spacetime has recently been developed by Lasenby, Doran, and Gull. Einstein's principles of equivalence and general relativity are replaced by gauge
principles asserting, respectively, local rotation and global displacement gauge invariance. A new unitary formulation of Einstein's tensor leads to resolution of long-standing problems with
energymomentum conservation in general relativity. Geometric calculus provides many simplifications and fresh insights in theoretical formulation and physical applications of the theory.
Alternative versions of the energymomentum complex in general relativity are given compact new formulations with spacetime algebra. A new unitary form for Einstein's equation greatly simplifies
the derivation and analysis of gravitational superpotentials. Interpretation of Einstein's equations as a gauge field theory on flat spacetime is shown to resolve ambiguities in energymomentum
conservation laws and reveal intriguing new relations between superpotential, gauge connection and spin angular momentum with rich new possibilities for physical interpretation.
Geometric Calculus is developed for curved-space treatments of General Relativity and comparison with the flat-space gauge theory approach by Lasenby, Doran and Gull. Einstein's Principle of
Equivalence is generalized to a gauge principle that provides the foundation for a new formulation of General Relativity as a Gauge Theory of Gravity on a curved spacetime manifold. Geometric
Calculus provides mathematical tools that streamline the formulation and simplify calculations. The formalism automatically includes spinors so the Dirac equation is incorporated in a
geometrically natural way.
Gauge Gravity and Electroweak Theory
In H. Kleinert, R. T. Jantzen & R. Ruffini (Eds.), Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity (World Scientific, Singapore, 2008) pp. 629-647.
Reformulation of the Dirac equation in terms of the real Spacetime Algebra (STA) reveals hidden geometric structure, including a geometric role for the unit imaginary as generator of rotations in
a spacelike plane. The STA and the real Dirac equation play essential roles in a new Gauge Theory Gravity (GTG) version of General Relativity (GR). Besides clarifying the conceptual foundations
of GR and facilitating complex computations, GTG opens up new possibilities for a unified gauge theory of gravity and quantum mechanics, including spacetime geometry of electroweak interactions.
The Weinberg-Salam model fits perfectly into this geometric framework, and a promising variant that replaces chiral states with Majorana states is formulated to incorporate zitterbewegung in
electron states.
We present a complete formulation of the 2D and 3D crystallographic space groups in the conformal geometric algebra of Euclidean space. This enables a simple new representation of translational
and orthogonal symmetries in a multiplicative group of versors. The generators of each group are constructed directly from a basis of lattice vectors that define its crystal class. A new system
of space group symbols enables one to unambiguously write down all generators of a given space group directly from its symbol.
Hunting for Snarks in Quantum Mechanics.
D. Hestenes, in P. Goggins and C-Y Chan (eds.), Bayesian Inference and Maximum Entropy Methods in Science and Engineering (American Institute of Physics, 2009). (17 pages)
A long-standing debate over the interpretation of quantum mechanics has centered on the meaning of Schroedinger’s wave function ψ for an electron. Broadly speaking, there are two major opposing
schools. On the one side, the Copenhagen school (led by Bohr, Heisenberg and Pauli) holds that ψ provides a complete description of a single electron state; hence the probability interpretation
of ψψ* expresses an irreducible uncertainty in electron behavior that is intrinsic in nature. On the other side, the realist school (led by Einstein, de Broglie, Bohm and Jaynes) holds that ψ
represents a statistical ensemble of possible electron states; hence it is an incomplete description of a single electron state. I contend that the debaters have overlooked crucial facts about
the electron revealed by Dirac theory. In particular, analysis of electron zitterbewegung (first noticed by Schroedinger) opens a window to particle substructure in quantum mechanics that
explains the physical significance of the complex phase factor in ψ. This led to a testable model for particle substructure with surprising support by recent experimental evidence. If the
explanation is upheld by further research, it will resolve the debate in favor of the realist school. I give details. The perils of research on the foundations of quantum mechanics have been
foreseen by Lewis Carroll in The Hunting of the Snark!
In a previous conference honouring Hermann Grassmann’s profound intellectual contributions (Schubring 1996), I cast him as a central figure in the historical development of a universal geometric
calculus for mathematics and physics (Hestenes 1996). Sixteen years later I am here to report that impressive new applications in this tradition are rapidly developing in computer science and
robotics as well as physics and mathematics. Especially noteworthy is the emergence of Conformal Geometric Algebra as an ideal tool for computational geometry, as it fulfils at last one of
Grassmann’s grandest goals and confirms the prescience of his mathematical insight. Geometric Calculus has finally reached sufficient maturity to serve as a comprehensive geometric language for
the whole community of scientists, mathematicians and engineers. Moreover, its simplicity recommends it as a tool for reforming high school mathematics and physics, as Grassmann had envisioned.
Last modified on 7 May 2010. | {"url":"http://geocalc.clas.asu.edu/html/Overview.html","timestamp":"2014-04-17T12:51:53Z","content_type":null,"content_length":"13550","record_id":"<urn:uuid:819957ea-762a-458c-b379-755adecd9708>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00602-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts by
Total # Posts: 115
can you help me without the pic? please?..
A thin uniform rod has mass M = 0.5 kg and length L= 0.49 m. It has a pivot at one end and is at rest on a compressed spring as shown in (A). The rod is released from an angle θ1= 55.0o, and moves
through its horizontal position at (B) and up to (C) where it stops with ...
I really appreciate all your help, if I said anything that made it sounds disrespectful I am deeply apologizing.. Thank you very much for all your help.
the first one I know, can you help me with the others?..
M, a solid cylinder (M=2.35 kg, R=0.111 m) pivots on a thin, fixed, frictionless bearing. A string wrapped around the cylinder pulls downward with a force F which equals the weight of a 0.750 kg
mass, i.e., F = 7.357 N. 1)Calculate the angular acceleration of the cylinder. The...
I like your answer and explain . step by step, now I understand how to do that.thank you, Jai
If x=(2/y^2)+2,what is the value of y in terms of x ? please answer and explain
a. 9 b. 12 c.102 d. 46,080 e. 138,240
Jennifer is setting the dinner table. She needs to set the table for 4 people, and each person needs to have an identical place setting to the other 3. She has 6 red place mats,6 green place mats,
and 8 blue place mats as well as 4 maroon napkins, 3 orange napkins,5 beige napk...
If #(a,b,c,d)means you subract the absolute value of d from the sum of a,b,and c,then #(2,5,6,-10)+#(1,2,3,7)= a. 2 b. 4 c. 16 d. 22 e. 36 please answer and explain
If m#= m/m^2-m ,what is the value of 6#-[(-5)#]? a. 1/30 b. 1/20 c. 1/4 d. 11/30 e. 9/20 please answer and explain
A relay with n laps was run by 4 people. The first runner ran half the laps. The second runner ran half the amount of laps the first runner ran. The third runner ran half the amount of laps the
second runner ran. The remaining laps were completed by the fourth runner. How many...
Scott, Mike, and Holly shared an order of French fries. Scott ate x fries. Mike ate one more than twice the amount of fries that Scott ate. Holly ate 5 fewer fries than Scott did. If there were 6
fries left, how many fries were in the order? a. x-5 b. x c. 2x+`1 d. 4x-4 e. 4x+...
There are x students in Mrs. Schwartz's third-grade class. One less than half the students have blue eyes. Six fewer children have green eyes than have blue eyes. The rest of the students have brown
eyes. How many students have brown eyes? a. 7 b. 8 c. (x/2)-7 d. x-7 e. x-...
The notation ##(a,b,c)produces a number that is a less than the product of b and c raised to the a power. When ##(2,5,x)=23, what is the value of x? a. 0 b. 1/2 c. 1 d. 2 e. 5 please answer and
The number a is a number less than -3. What is the range of possibl values of 1/a^2 ? a. 1/a^2 <-9 b. 1/a^2 <-3 c. -1/9<1/a^2<1/9 d. 0<1/a^2<1/9 e. 1/9<1/a^2<1/3 please answer and explain
Which of the following is equivalent to the inequality 7>-3x>-12? a. -7<x<12 b. 7/3 <x<4 c. -7/3<x<4 d. -4<x<-7/3 e. -4<x<7 please answer and explain
math help
3ab=6 ab =2 a =2/b answer b
math help
If 3ab=6, what is the value of a in terms of b? a. 2 b. 2/b c. 2b^2 d. 2b e. 2b^2 My answer is d, but not sure, anyone can tell me right or wrong? 3ab=6 3a=6b a=2b
A bag contains x marbles. Half of the marbles are blue. Four less than the number of blue marbles are red. The remaining marbles are purple. What is the probability of choosing a purple marble at
random? a. 2 b. 4 c. 1/2-(x/2-4)/x d. x/2-4 e. x/2 please answer and explain
so, the teacher give me the wrong answer? my answer is right
Y is divided by 6, the remainder is 4, so I think 6x6+4=40 , so y=40 40-6/3 =34/3 remainder=1
When y is divided by 6, the remainder is 4. What will be the remainder if y-6 is divided by 3 ? a. 0 b. 1 c. 2 d. 3 e. 4 my answer is b. 1 , but teacher give me the answer is d. 3 , I don't
understand, anyone can explain
Stacey is x years old. Ali is four years less than three times Stacey's age. Carli is 10 years more than twice Ali's age. What is the age difference between Stacey and Carli? a. 2x+6 b. 3x-4 c. 3x+6
d. 5x+2 e. 6x+2 please answer and explain
d, right
If n is an integer, which of the following must be odd? a. 3n-5 b. 3n+4 c. 4n+10 d. 4n-5 e. 5n+7 please answer and explain
math help
If x people working together make a total of y dollars after an hour of work, how much money will z people make if they work 4 hours at the same rate per person ? a. 4xyz b. xy + 4z c. 4yz/x d. 4xy/z
e. xyz/4 please answer and explain
wrong, the answer is a
The answer is d, right ?
Which set of numbers has a greatest common factor of 1 ? a. 2,3,and 6 b. 2,200, and 400 c. 3,9,and 27 d. 22,33,and 99 e. 100,500, and 2,500 please answer and explain
I understand so, #2 how to multiple of both numbers
#1 how to divide both mumbers
1. What is the greatest common factor of: 2x2x5x7x7 2x5x5x7x7 a. 2x5 b. 2x5x7 c. 2x5x7x7 d. 2x2x5x5x7x7 e. 2x2x2x5x5x5x7x7x7x7 2. What is the least common multiple of: 3x3x5x5x5x7 2x3x5x5x7x7 a. 2 b.
5x5x7 c. 3x3x5x5x5x7 d. 2x3x3x5x5x5x7x7 e. 2x3x3x3x5x5x5x5x5x7x7x7 Please ans...
math help
thank you
math help
5x+10y +15z-15x+10y-5z+5x-10y-5z / (5x+10y+15z)-(15x+10y-5z)+(5x-10y-5z) a. 1/3 b. 1 c. x+2y+z / x-2y+3z d. x-2y-z / x+2y-3z e. x-2y-z / x+2y+z my answer is none of above, is - 1/3, I didn't know I
done it wrong or the question is not correct
math help
(2np.3r/4r)-(2np-3n)=? a. -1/2np+3n b. -1/2np-3n c. -np/2r+3n d. 1/2np+3n e. 1/2np-3n Please answer and explan
(x^2 + x) / 4(x-3x)
(x^2 + x) / x-3x .4=? mean x 4
(x^2 + x) / x-3x (4) = ? Please answer and explain
(x^2 + x) / x-3x . 4 =
(3x / x) ( 2x / 6) =
thank you, Reiny
= 1/2 (4x +12x +12x^2) = 1/2 ( 16x + 12x^2 ) = 8x + 6x^2 is it correct answer ?
last question 1/2[4x+6x(2+2x)]= ?
thank you, Reiny
y+y^2+(y+y)^2= ? a. y^2 + 5y b. 2y^2 + y c. 3y^2 + y d. 5y^2 + y e. 4y^4 + y^2 + y please answer and explain
yes,4xy(1/(2y) + 3/(2x) ) thank you
4xy(1/2y+3/2x)= ? a. 2x + 6y b. 2x + 12y c. 4x + 6y d. 4x + 12y e. 4x - 12y please answer and explain
A television set listed at $160 is offered at a 12 1/2% discount during a store wide sale. If an additional 3% is allowed on the net price for payment in cash, how much can Josh save by buying this
set during the sale for cash ?
math help
A gear with 20 teeth revolving at 200 revolutions per minute is meshed with a second gear turning at 250 revolutions per minute. How many teeth does this gear have ?
Two boys weighing 60 pounds and 80 pounds balance a seesaw. How many feet from the fulcrum must the heavier boy sit if the lighter boy is 8 feet from the fulcrum ?
my answer is A
How much money is saved by buying an article priced at $80 with a 40% discount,rather than buying an article marked at $90 with discounts of 35% and 10% ? A. $ 4.65 B. $1.50 C. $10.50 D. $3.15 E.
$4.25 not just answer
oh, I get it now, thank you
the decimal answer is 0.1666666
The enrollment of Kennedy High School dropped from 1200 to 1000 over a three-year period. What the percent of decrease during this time? A. 20 B 16 2/3 C. 25 D. 200 E. 2 My answer is 1200-1000=200
200/1200= 1/6 ( some wrong )
A pulley revolving at 200 revolutions per minute has a diameter of 15 inches. It is belted to a second pulley which revolves at 150 revolutions per minute. Find the diameter, in inches, of the second
pulley. not just answer, please
so, the answer is 50?
my answer is 1/2 (same ) But the answer they give me is A. 66 2/3 B. 75 C. 50 D. 33 1/3 E. 25
At Baker High, 3 out of every 4 graduates go on to college . Of these 2 out of every 3 graduate from college. What percent of students graduating from Baker High will graduate from college ?
152*0.625=95 95-50=45
A baseball team won 50 of the first 92 games played in a season. If the season consists of 152 games, how many more game must the team win to finish the season winning 62 1/2% of games played?
math help
thank you
math help
I think 1+3+5+7+9+11+13+15+17+19+21+23+25+27=196 answer:196/14=14 but I am not sure
math help
Find the average of the first 14 positive odd integers.
math help
thank you, I understand now.
math help
The average of 2P, 3Q, and another number is S. Represent the third number in terms of P,Q,and S. A. S-2P-3Q B. S-2P+3Q C. 3S-2P+3Q D. 3S-2P-3Q E. S+2P-3Q not just answer,please
A gear with 60 teeth is meshed to a gear with 40 teeth. If the larger gear revolves at 20 revolutions per minute, how many revolutions does the smaller gear make in a minute? not just an answer
please (A)13 1/3 (B)3 (C)300 (D)120 (E)30
There are fewer than 33 eggs in a basket. If they are counted out 6 at a time, there are 4 left over. If they are counted out 5 at a time, there are 3 left. How many eggs are in the basket? not just
an answer
oh made a mistake it is 44
The raisins in a bag weigh 9/16 of pound. If the raisins are divided into 6 equal shares, how much will 4 shares weigh? not just the answer
math help
How many different two- digit numbers can be formed using each of the digits 3, 5, and 7 only once ?
math help
I know if four digit using the digit 3,5,7,and 9 , the answer is 4*3*2*1=24, but if three digit using the digit 3, 5,7, and 9, the answer is 3*2*1=6?
math help
If digits cannot be repeated,how many three digit numbers can be made using the digits 3, 5, 7 and 9 ?
math help
if the digits 1,3,5.7,9, what is the different
math help
How many 5-digit odd numbers can be made using the digits 3, 5, 6, 7 and 9, if digits cannot be repeated?
math help
If m/n = 5/6, then what is 3m + 2n ? (A) 0 (B) 2 (C) 7 (D) 10 (E) it cannot be determined from the information given not just answer
math help
answer B
math help
math help
In an audience of adults and children, 1/6 of the audience are young boys and 1/3 are young girls. What percentage of the audience are children ? (A) 66 2/3 (B) 50 (C) 40 (D) 37 1/2 (E) 33 1/3 not
just answer
math help
i understand now, thank you
math help
i still not understand what is 2 feet from 6 ft to 24 ft.
math help
11, 16.5, 22
math help
i mean E
math help
A Certain type of board is sold only in length of multiples of 2 feet from 6 ft to 24 ft. A builder needs a large quantity of this type of board in 5 1/2 foot lengths for minimum waste. The lengths
in feet to be ordered should be (A) 6 (B) 12 (C) 16 (D) 18 (E) 22 (F) 24 not ju...
math help
I understand now, thank you
math help
why 1.38*2.5 what is 2.5 comefrom
math help
d. $0.45
math help
The total saving in purchasing 30 13-cent lollipops for a class party at a reduced rate of $1.38 per dozen is: (A) $0.35 (B) $0.38 (C) $0.40 (D) $0.45 (E) $0.50 (F) $0.55 not just answer
The price of mangoes ranges from 25 cents to 75 cents a piece. What is the greatest number of mangoes that can be bought with $3 ?
x exceeds y by the same amount that 7 exceeds 2. If y is 16, what is the value of x ? (A) 18 (B) 19 (C) 20 (D) 21 (E) 23 [ not only answer ]
math help
S is the sum of n integers and A is their average. Which of the following statements is correct? (A) S=A+n (B) S=nA (C) A=nS (D) N=AS (E) None of these
x is a positive integer and x-5 is negative. what is the greatest possibel value of x ?
Muscle are made up of muscle functions called____ ____. Its two words... I was thinking myosin and actin , however, not sure if i AM CORRECT.
What is the verbal expression of 2(n-6)?
Two blocks are connected by strings and hung from the ceiling as shown. Tension T1 is 187 N and tension T2 is 78 N. Find the mass of each block.
17. Burger King's Drive-Through: Suppose that cars arrive at Burger King's drive-through at the rate of 20 cars every hour between 12:00 noon and 1:00 PM. A random sample of 40 one-hour time periods
between 12:00 noon and 1:00 P.M. is selected and has 22.1 as the mean ...
how many girls are there girls drink 4 cups e/a of water 7 cups are consume by the boys e/a and coach drinks 9 cups of water with a total of 43 cups consume how many girls are in the team
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Mathematics Test
The CAAP Mathematics Test is a 35-item, 40-minute test designed to measure students' proficiency in mathematical reasoning. The test assesses students' proficiency in solving mathematical problems
encountered in many postsecondary curricula. It emphasizes quantitative reasoning rather than the memorization of formulas.
The content areas tested include prealgebra; elementary, intermediate, and advanced algebra; coordinate geometry; and trigonometry. Descriptions of the content areas and the approximate proportions
of items in each are provided below.
• Prealgebra. Items in this category involve operations with whole numbers, decimals, and fractions; order concepts; percentages; averages; exponents; scientific notation; and similar concepts.
• Elementary Algebra. Items in this category involve basic operations with polynomials, setting up equations, and substituting values into algebraic expressions. They may also require the solution
of linear equations in one variable and other related topics.
• Intermediate Algebra. Items in this category assess students' understanding of exponents, rational expressions, and systems of linear equations. Other concepts such as the quadratic formula and
absolute value inequalities may also be tested.
• Coordinate Geometry. Knowledge and skills assessed in this category may include graphing in the standard coordinate plane or the real number line, graphing conics, linear equations in two
variables, graphing systems of equations, and similar types of skills.
• College Algebra. Items in this category are based on advanced algebra concepts including rational exponents, exponential and logarithmic functions, complex numbers, matrices, inverses of
functions, and domains and ranges.
• trigonometry. Items in this category include concepts such as right triangle trigonometry, graphs of trigonometric functions, basic trigonometric identities, and trigonometric equations and
Three scores are reported for the CAAP Mathematics Test:
1. a total test score based on all 35 items
2. a subscore in Basic Algebra based on 17 items
3. a subscore in College Algebra based on the remaining 18 items
The Basic Algebra subscore is composed of test questions from the prealgebra, elementary algebra, intermediate algebra, and coordinate geometry content areas. The College Algebra subscore is composed
of test questions from the College Algebra and trigonometry content areas.
Content Specifications Summary for the CAAP Mathematics Test
Content Category Proportion of Test Number of Items
Basic Algebra .49 17
Prealgebra .09 .14 3 5
Elementary Algebra .09 .14 3 5
Intermediate Algebra .09 .11 3 4
Coordinate Geometry .14 .18 5 6
College Algebra .51 18
College Algebra .40 14
trigonometry .11 4
Total 1.00 35 | {"url":"http://www.act.org/caap/test/math.html","timestamp":"2014-04-17T02:10:47Z","content_type":null,"content_length":"41518","record_id":"<urn:uuid:bffa8545-0ed2-4eb9-8182-e4a0ddf6f905>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00231-ip-10-147-4-33.ec2.internal.warc.gz"} |
Results 1 - 10 of 55
- International Journal of Computational Geometry and Applications , 1997
"... We introduce the notion of expansiveness to characterize a family of robot configuration spaces whose connectivity can be effectively captured by a roadmap of randomly-sampled milestones. The
analysis of expansive configuration spaces has inspired us to develop a new randomized planning algorithm. T ..."
Cited by 210 (37 self)
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We introduce the notion of expansiveness to characterize a family of robot configuration spaces whose connectivity can be effectively captured by a roadmap of randomly-sampled milestones. The
analysis of expansive configuration spaces has inspired us to develop a new randomized planning algorithm. This algorithm tries to sample only the portion of the configuration space that is relevant
to the current query, avoiding the cost of precomputing a roadmap for the entire configuration space. Thus, it is well-suited for problems where a single query is submitted for a given environment.
The algorithm has been implemented and successfully applied to complex assembly maintainability problems from the automotive industry.
, 2000
"... We present a randomized motion planner for robots that must avoid moving obstacles and achieve a specified goal under kinematic and dynamic constraints. The planner samples the robot's statetime
space by picking control inputs at random and integrating the equations of motion. The result is a roa ..."
Cited by 190 (12 self)
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We present a randomized motion planner for robots that must avoid moving obstacles and achieve a specified goal under kinematic and dynamic constraints. The planner samples the robot's statetime
space by picking control inputs at random and integrating the equations of motion. The result is a roadmap of sampled statetime points, called milestones, connected by short admissible trajectories.
The planner does not precompute the roadmap as traditional probabilistic roadmap planners do; instead, for each planning query, it generates a new roadmap to find a trajectory between an initial and
a goal statetime point. We prove in this paper that the probability that the planner fails to find such a trajectory when one exists quickly goes to 0, as the number of milestones grows. The planner
has been implemented and tested successfully in both simulated and real environments. In the latter case, a vision module estimates obstacle motions just before planning starts; the planner is then
allocated a small, fixed amount of time to compute a trajectory. If a change in the obstacle motion is detected while the robot executes the planned trajectory, the planner recomputes a trajectory on
the fly. 1
, 1998
"... ... This paper provides foundations for understanding the effect of passages on the connectedness of probabilistic roadmaps. It also proposes a new random sampling scheme for finding such
passages. An initial roadmap is built in a "dilated" free space allowing some penetration distance of the robot ..."
Cited by 167 (35 self)
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... This paper provides foundations for understanding the effect of passages on the connectedness of probabilistic roadmaps. It also proposes a new random sampling scheme for finding such passages.
An initial roadmap is built in a "dilated" free space allowing some penetration distance of the robot into the obstacles. This roadmap is then modified by resampling around the links that do not lie
in the true free space. Experiments show that this strategy allows relatively small roadmaps to reliably capture the free space connectivity
- In Proc. IEEE Int. Conf. Robot. Autom. (ICRA , 1999
"... Probabilistic roadmap planning methods have been shown to perform well in a number of practical situations, but their performance degrades when paths are required to pass through narrow passages
in the free space. We propose a new method of sampling the configuration space in which randomly generate ..."
Cited by 142 (32 self)
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Probabilistic roadmap planning methods have been shown to perform well in a number of practical situations, but their performance degrades when paths are required to pass through narrow passages in
the free space. We propose a new method of sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space. We give
algorithms that perform this retraction while avoiding explicit computation of the medial axis, and we show that sampling and retracting in this manner increases the number of nodes found in small
volume corridors in a way that is independent of the volume of the corridor and depends only on the characteristics of the obstacles bounding it. Theoretical and experimental results are given to
show that this improves performance on problems requiring traversal of narrow passages. 1
, 1998
"... We provide an analysis of a recent path planning method which uses probabilistic roadmaps. This method has proven very successful in practice, but the theoretical un- derstanding of its
performance is still limited. Assuming that a path 7 exists between two configurations a and b of the robot, we ..."
Cited by 94 (17 self)
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We provide an analysis of a recent path planning method which uses probabilistic roadmaps. This method has proven very successful in practice, but the theoretical un- derstanding of its performance
is still limited. Assuming that a path 7 exists between two configurations a and b of the robot, we study the dependence of the failure probability to connect a and b on (i) the length of 7, (ii) the
distance function of 7 from the obstacles, and (iii) the number of nodes N of the probabilistic roadmap constructed. Importantly, our results do not depend strongly on local irregularities of the
configuration space, as was the case with previous analysis. These results are illustrated with a simple but illuminat- ing example. In this example, we provide estimates for N, the principal
parameter of the method, in order to achieve failure probability within prescribed bounds. We also compare, through this example, the different approaches to the analysis of the planning method.
, 2001
"... This paper describes a nev probabilistic roadmap (PRM) path planner that is: (1) single-query instead of pre-computing a roadmap covering the entire free space, it uses the tvo input query
configurations as seeds to explore as little space as possible; (2) hi-directional it explores the robotis free ..."
Cited by 84 (4 self)
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This paper describes a nev probabilistic roadmap (PRM) path planner that is: (1) single-query instead of pre-computing a roadmap covering the entire free space, it uses the tvo input query
configurations as seeds to explore as little space as possible; (2) hi-directional it explores the robotis free space by concur- rently building a roadmap made of tvo trees rooted at the query
configurations; (3) adaptive it makes longer steps in opened areas of the free space and shorter steps in cluttered areas; and (4) lazy in checking collision it delays collision tests along the edges
of the roadmap until they are absolutely needed. Experimental results shov that this combination of techniques drastically reduces planning times, making it possible to handle difficult problems,
including multi-robot problems in geometrically complex environments.
- INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH , 1996
"... Several randomizod path planners have been proposed during the last few years. Their attractiveness stems from their applicability to virtually any type of robots, and their empirically observed
success. In this paper we attempt to present a unifying view of these planners and to theoretically expla ..."
Cited by 82 (26 self)
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Several randomizod path planners have been proposed during the last few years. Their attractiveness stems from their applicability to virtually any type of robots, and their empirically observed
success. In this paper we attempt to present a unifying view of these planners and to theoretically explain their success. First, we introduce a general planning scheme that consists of randomly
sampling the robot' s configuration space. We then describe two previously developed planners as instances of planners based on this scheme, but applying very different sampling strategies. These
planners are probabilistically complete: if a path exists, they will find one with high probability, if we let them run long enough. Next, for one of the planners, we analyze the relation between the
probability of failure and the running time. Under assumptions characterizing the "goodness" of the robot's free space, we show that the running time only grows as the absolute value of the logarithm
of the probability of failure that we are willing to tolerate. We also show that it increases at a reasonable rate as the space goodness degrades. In the last section we suggest directions for future
, 1999
"... We present a randomized approach to path planning for articulated robots that have closed kinematic chains. The approach extends the probabilistic roadmap technique which has previously been
applied to rigid and elastic objects, and articulated robots without closed chains. Our work provides a frame ..."
Cited by 70 (4 self)
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We present a randomized approach to path planning for articulated robots that have closed kinematic chains. The approach extends the probabilistic roadmap technique which has previously been applied
to rigid and elastic objects, and articulated robots without closed chains. Our work provides a framework for path planning problems that must satisfy closure constraints in addition to standard
collision constraints. This expands the power of the probabilistic roadmap technique to include a variety of problems such as manipulation planning using two open-chain manipulators that
cooperatively grasp an object, forming a system with a closed chain, and planning for reconfigurable robots where the robot links may be rearranged in a loop to ease manipulation or locomotion. We
generate the vertices in our probabilistic roadmap by sampling random con gurations that ignore kinematic closure, and by performing randomized gradient descent to force satisfaction of the closure
constraints. We generate...
, 1999
"... Most computational models of protein-ligand interactions consider only the energetics of the final bound state of the complex and do not examine the dynamics of the ligand as it enters the
binding site. We have developed a novel approach to study the dynamics of protein-ligand interactions base ..."
Cited by 66 (17 self)
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Most computational models of protein-ligand interactions consider only the energetics of the final bound state of the complex and do not examine the dynamics of the ligand as it enters the binding
site. We have developed a novel approach to study the dynamics of protein-ligand interactions based on motion planning algorithms from the field of robotics. Our algorithm uses electrostatic and van
der Waals potentials to compute the most energetically favorable path between any given initial and goal ligand configurations. We use probabilistic motion planning to sample the distribution of
possible paths to a given goal configuration and compute an energy-based "difficulty weight" for each path. By statistically averaging this weight over several randomly generated starting
configurations, we compute the relative difficulty of entering and leaving a given binding configuration. This approach yields details of the energy contours around the binding site and can be used
to cha...
- INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH , 2002
"... This paper describes the foundations and algorithms of a new probabilistic roadmap (PRM) planner that is: single-query -- instead of pre-computing a roadmap covering the entire free space, it
uses the two input query configurations to explore as little space as possible; bi-directional -- it explo ..."
Cited by 65 (16 self)
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This paper describes the foundations and algorithms of a new probabilistic roadmap (PRM) planner that is: single-query -- instead of pre-computing a roadmap covering the entire free space, it uses
the two input query configurations to explore as little space as possible; bi-directional -- it explores the robot's free space by building a roadmap made of two trees rooted at the query
configurations; and lazy in checking collisions -- it delays collision tests along the edges of the roadmap until they are absolutely needed. Several observations motivated this strategy: (1) PRM
planners spend a large fraction of their time testing connections for collision; (2) most connections in a roadmap are not on the final path; (3) the collision test for a connection is most expensive
when there is no collision; and (4) any short connection between two collision-free configurations has high prior probability of being collision-free. The strengths of single-query and bi-directional
sampling techniques, and those of delayed collision checking reinforce each other. Experimental results | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=35912","timestamp":"2014-04-23T10:54:49Z","content_type":null,"content_length":"40373","record_id":"<urn:uuid:476d57a6-115a-4df5-9cc2-69bf4e45aba4>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00033-ip-10-147-4-33.ec2.internal.warc.gz"} |
Chinese Ten
In Chinese this game is known as Jiăn Hóng Diăn (pick up red spots) (揀紅點 or in simplified characters 拣红点). It is very similar to the Thai game Red Frog Black Frog (Gob Dum Gob Dang -
กบดำกบแดง). The playing process whereby a card is flipped from the face-down deck after each play from a player's hand is characteristic of Oriental fishing games.
The description below was originally based with permission on the account by Jen Minute (Bullock) formerly available on the page http://members.xoom.com/_XMCM/niferbee/frames/Cards/chinese.htm which
unfortunately disappeared around the end of 2001, along with the whole of her 'House of Cards' site. Additional information was contributed by David Halber, Stephen Coleman, Florent Barraco and
Kantaphon Tongmee.
Players, Cards and Deal
The game is played with a standard 52-card pack. The number of cards dealt to players varies depending on how many are playing. For a game with two players, each player gets twelve cards. For a game
with three players, each player gets eight cards. For a game with four players, each player gets six cards. It is easy to remember how many get dealt to each player by simply dividing 24 by the
number of people who are playing (i.e. 24 divided by 2 = 12, each player gets 12 cards).
After all the cards have been dealt, four cards are turned over face up from the remaining deck and placed in a layout like this:
X X
D (X = card; D = deck)
X X
The layout isn't that important, if you want your own setup, go right on ahead... (Just as long as you begin the game four cards out and the remaining deck on the table.)
The object of the play is to capture cards from the layout. A numeral card from Ace (1) to Nine captures another numeral card if their values add up to exactly ten - so for example Ace captures Nine,
Nine captures Ace, Three captures Seven, a Five captures another Five, and so on. Tens and picture cards can only capture another card of the same rank: Ten captures Ten, Queen captures Queen, and so
on. Suits don't matter for the purpose of capturing.
At your turn you play one card from your hand. If it captures a card from the layout, you take both cards and place them face down in front of you in your pile of won cards; if not you leave it face
up as a new layout card. Whether you captured or not, you next flip the top card of the face down deck face up. If this captures a card from the layout you take both cards; if not it remains in the
layout. After you have played one card and flipped one card, whether you captured or not, the turn passes to the next player.
Note that each card played or turned up can only capture one card from the layout. If the initial layout is 3-3-6-Q and you play a Seven, you can only capture one of the Threes from the layout (3+7=
10). If you play an Ace you do not capture anything (you cannot use 1+3+6=10 to capture two cards at once).
Note also that although the layout always begins with four cards, during the game there may often be more or fewer than four cards in the layout.
For another example, look at the following initial layout with the Jack of hearts, Ace of hearts, Six of clubs, and Two of hearts.
To make a move on a turn (with this layout), a player would need to play (out of his hand) a Jack (on the Jack), a Nine (on the Ace), a Four (on the Six), or an Eight (on the Two). Suppose he plays a
Jack. He takes the pair of Jacks and stacks them face down in front of him (for later evaluation). He next flips over the top card from the deck and places it face up on the table. Suppose it's the
Six of spades. This does not capture anything. It is now the next player's turn and the new layout is:
If, after capturing the Jack, the player had flipped a Four, capturing the Six from the table, then only two layout cards would remain for the beginning of the next player's turn:
If the rules are followed correctly, then you should finish with no cards in any player's hand, no cards in the face-down deck and no cards in the layout - on the final play, the last card from the
deck will always capture the last card from the layout.
Special cases
If the initial layout contains three kings, three queens, three jacks, three tens or three fives, then the fourth card of that rank captures all three matching cards.
If the initial layout consists of four kings, four queens, four jacks, four tens or four fives, then the dealer automatically captures the four of a kind.
These two rules are necessary to avoid having cards in the layout that can never be captured.
At the end of the game, each player should have a stack of cards next to him. When there are two players, only the red cards have a value; when there are three players, the Ace of spades also scores;
when there are four players the red cards and both black Aces score. Black cards other than Aces are always worthless. The point values of the cards are as follows:
│ Cards │ Point Value │
│ red 2 - 8 │ face value │
│ red 9 - King │ 10 points each │
│ red Ace │ 20 points each │
│ Ace of spades │ 30 points when there are three or four players - otherwise zero │
│ Ace of clubs │ 40 points when there are four players - otherwise zero │
The tie score is
• 105 points in a two-player game
• 80 points in a three-player game
• 70 points in a four-player game.
The difference of your score from the "tie score" is the amount you win or lose by. If your score is exactly the tie score you have neither won nor lost.
Example of scoring:
Let's suppose a person had the following red cards in a two-player game:
K, Q, 9, 6, 7, 3, 4, 5, 5, and an Ace of Hearts.
The points would add up as follows:
10 + 10 + 10 + 6 + 7 + 3 + 4 + 5 + 5 + 20 = 80.
This player has lost by 25 points (105 - 80 = 25), and the other player should have 130 points, winning by 25 (because 130 - 105 = 25).
Taiwan variant
Stephen Coleman describes a variation played in Taiwan. This is for four players. Deal and play are clockwise and the turn to deal passed to the left after each round (i.e. when all the cards have
been played and scored). The dealer shuffles and the player to dealer's right cuts the deck and may look at the cut card, which will become the bottom card of the deck. The dealer plays first.
In this version the Ace of clubs is worth nothing; the Ace of spades is the only black card worth points (30). Red aces are 20 points, red K-9 are 10 points, red 8-2 are face value as above. The
tie-score (or 'base') is 60 points. It is often played as a gambling game: Losers pay into the pot $1 (Taiwan dollar) for each point less than 60, and winners collect from the pot $1 for each point
over 60.
Red 5 Bonus. If a player manages to capture both red 5's in a single play, then the other players must each give him 10 of their points at the end of the round (by giving him one of their captured
10-point cards). [The strategy to avoid this, if you have no red 5's in your hand, is to lay down a black 5 as soon as possible, hopefully forcing another player to play his red 5 on a black one,
preventing the bonus.]
Florent Barraco describes a similar version for 2 or 4 players in which the red aces are worth only 10 points rather than 20, like the K-9, red 8-2 are face value and the ace of spades is 30. This
gives a total of 220 points, so the tie-score is 110 with 2 players and 55 with 4 players. Players pay or receive according to their difference from the tie score, and in addition a player who
captures a red 5 with the other red 5 is paid 10 points by each of the other players.
Red Frog Black Frog (Gob Dum Gob Dang - กบดำกบแดง)
This Thai game, described to me by Kantaphon Tongmee, is played the same way as Chinese Ten, but the scoring is somewhat different.
• Red cards are always worthless.
• Black cards 2-9 are worth face value.
• Black 10-J-Q-K are worth 10 points each.
• The Ace of Clubs is worthless.
• The Ace of Spades is worth 50 points. | {"url":"http://www.pagat.com/fishing/chinten.html","timestamp":"2014-04-21T00:31:07Z","content_type":null,"content_length":"16033","record_id":"<urn:uuid:cb71f444-f510-4c3a-95f3-ab77c34696cd>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00542-ip-10-147-4-33.ec2.internal.warc.gz"} |
Langhorne Prealgebra Tutor
...I have written several short stories(unpublished) and I have been working on a full length novel for approximately three years now. I have experience tutoring students in English for SATs, and
for high school and college essay writing. I am an accounting major at Fox School of Business at Temple University.
18 Subjects: including prealgebra, English, writing, calculus
...I'm not too shabby at reading and writing, either. Unlike the one-size-fits-all test-prep courses, and the overly-structured national tutoring companies, I always customize my methods and
presentation for the student at hand. There is a big difference between a student striving for 700s on the SAT and one hoping to reach the 500s.
23 Subjects: including prealgebra, English, calculus, geometry
...I am flexible and will work around the student's schedule. If students have an Individualized Education Plan (IEP), I am adept at accommodating students who struggle with understanding math
concepts. In addition to my business management (computer) and math background, I played basketball and baseball in high school.
8 Subjects: including prealgebra, geometry, algebra 1, algebra 2
I taught high school mathematics in New Jersey for over 20 years. I have experience tutoring students from my former high school and I have worked as a tutor for Sylvan Learning Center and other
organizations. My favorite math subjects are pre-calculus and statistics.
14 Subjects: including prealgebra, calculus, algebra 1, algebra 2
...Geometry being a prerequisite for algebra 2, I feel very confident that I can successfully help students in geometry. I have a strong math background from college (I was a chemistry major). I
have always enjoyed math and I have been successfully tutoring several students from 5th to 8th grade ...
7 Subjects: including prealgebra, chemistry, French, geometry | {"url":"http://www.purplemath.com/Langhorne_Prealgebra_tutors.php","timestamp":"2014-04-17T04:53:04Z","content_type":null,"content_length":"24188","record_id":"<urn:uuid:3e5b45c3-7406-45cb-890a-62eb0a3712a1>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00278-ip-10-147-4-33.ec2.internal.warc.gz"} |
Observational Issues in Radiometric and Interferometric Detection
and Analysis of the SZ Effects - M. Birkinshaw & K. Lancaster
1.5. Interferometric techniques
The technique of interferometry was originally developed with the intention of achieving high angular resolution, and instruments such as the VLA are examples of this. The SZ effect is primarily a
large angular-scale feature on the sky, so high resolution is not of interest and is, in fact, detrimental in this context. However, interferometers offer improvements in the control of systematic
effects compared with single-dish telescopes. First, interferometers experience a loss of coherence (and thus a loss of sensitivity) away from the pointing centre, meaning that ground spillover,
terrestrial interference and other spurious signals will be attenuated. Second, structures on the sky are modulated by a fringe pattern at a different rate than most contaminating sources. This
allows rejection of signals from astronomical sources such as the Sun and bright planets. Finally, interferometers with a wide range of baselines allow simultaneous observations of confusing, small
angular-scale, possibly-variable radio sources, so that their effects can be separated from the SZ signal.
Figure 4. A simple one-dimensional interferometer. Radiation from the source must travel an extra distance b sin
We can understand the basic principles of interferometry by discussion of a simple case. Consider a two-element interferometer, with antennas separated by a distance b, observing a source at an angle
Fig. 4. Each antenna receives a signal which produces a time-varying voltage, and the product of these voltages is measured. Due to the path difference for radiation travelling from a distant source
to the two antennas, there will be a phase difference between the received signals given by
The correlated output is then
In practice the output S, and the area of each antenna is a, the interferometer response is
Phases are usually not measured absolutely, but relative to some reference direction, [0]. For a source offset by a small angle [0], we have [0] +
since we are dealing with small angles. The correlated output differs at different antenna separations, so that the angular resolution of this simple interferometer is proportional to b. A more
complex multi-baseline instrument is sensitive to a range of scales determined by the set of baseline lengths defined by the antenna locations. The shortest baseline defines the maximum scale which
can be sampled. Sky structures on larger angular scales will not modulate [0] (and hence with time), and so will not produce a detected signal.
Figure 5. The same simple interferometer as Fig. 4, where the field centre is specified by r[0] and the source position is r. [0] and
The interferometer response can be expressed more generally -- we consider the main points here, but a full treatment is given in Thompson, Moran and Swenson (1986). If we now specify the source
position by a vector r (see Fig. 5) and the baseline by the vector b, the phase difference from (12) can be written b.r. The reference direction may be specified by a vector r[0], so that r = r[0] +
s, where s describes the shift between the two. After some manipulation, the response to all sources within the solid angle
It is conventional to specify the baseline vector b in terms of right-handed coordinates (u, v, w), where w is in the direction of the source, u and v point East and North respectively as seen from
the source position, and distances are measured in wavelengths. Additionally, the position of the source on the sky is usually described in terms of co-ordinates (l, m, n). We see that b.r[0] = wb.r
= (ul + vm + wn) b.s = (ul + vm + w(n - 1)) ddl dm) / n, we find
where a(l, m) is the effective total area of the antennas in the direction (l, m) and I(l, m) is the brightness distribution on the sky. n = (1 - l^2 - m^2)^1/2 I(l, m). A map made from
interferometer data contains structures which are modulated by the synthesized beam. This is given by the Fourier transform of the telescope aperture, which is eq (17) above with the sky brightness
replaced by a two-dimensional | {"url":"http://ned.ipac.caltech.edu/level5/Sept04/Birkinshaw/Birk1_5.html","timestamp":"2014-04-19T06:57:01Z","content_type":null,"content_length":"9688","record_id":"<urn:uuid:09b7e61c-61b7-4646-bbfa-1d7db601b1b3>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00652-ip-10-147-4-33.ec2.internal.warc.gz"} |
Bazaraa Nonlinear Programming Solutions Manual
Bazaraa Nonlinear Programming Solutions Manual PDF
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Nonlinear Programming 2nd Edition Solutions Manual Dimitri P. Bertsekas Massachusetts Institute of Technology Athena Scientific, Belmont, Massachusetts
Title: Download Solutions Manual to Accompany Linear Programming and Network Flows, Mokhtar S. Bazaraa Author: Solutions Manual to Accompany Linear Programming and Network Flows, Mokhtar S. Bazaraa
Introduction to nonlinear programming. 4. ... [BSS] M.S. Bazaraa, H.D. Sherali, and C.M. Shetty, Nonlinear Programming: Theory and Algo-rithms (2nd Ed.), Wiley, 1993. ... The manual of CPLEX is
available at the course web site.
... Solutions Manual to Accompany Nonlinear Programming. Theory and Algorithms ISBN: 978-1-118-76237-0 2013. Softcover. RRP € 26.90 Discounted price € 21.50 Bazaraa, M. S.: Nonlinear Programming.
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For the nonlinear programming part of the course we will use the lecture notes and ... M.S. Bazaraa, H.D. Sherali, and C.M. Shetty, Nonlinear Programming: Theory and Algo- ... must, however, write up
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Hanif D. Sherali, C. M. Shetty. The Scaled Boundary Finite Element Method.
... since we were able to obtain many local optimum solutions for instances ... this decision making problem as a nonlinear programming problem, highlight its ... 1 Bazaraa, M. S., Sherali, H. D.,
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*Integer Programming and *Nonlinear Programming ( chapter 11, 12 ) 12. *Metaheuristics ... Bazaraa, M. S., Linear Programming and Network Flows, 3rd ed., Hoboken, NJ : Wiley-Interscience, c2005. ...
User's Manual [PDF] 2. Download LINGO User's Manual ... | {"url":"http://ebookily.org/pdf/bazaraa-nonlinear-programming-solutions-manual","timestamp":"2014-04-24T16:30:17Z","content_type":null,"content_length":"19603","record_id":"<urn:uuid:a2c38c95-b200-4c32-a97f-a86132900f66>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00230-ip-10-147-4-33.ec2.internal.warc.gz"} |
Inflection Points
An Inflection Point is where a curve changes from Concave upward to Concave downward (or vice versa)
So what is concave upward / downward ?
Concave upward is when the slope increases:
Concave downward is when the slope decreases:
Here are some more examples:
Learn more at Concave upward and Concave upward.
Finding where ...
So our task is to find where a curve goes from concave upward to concave downward (or vice versa).
Derivatives help us!
The derivative of a function gives the slope.
The second derivative tells us if the slope increases or decreases.
• When the second derivative is positive, the function is concave upward.
• When the second derivative is negative, the function is concave downward.
And the inflection point is where it goes from concave upward to concave downward (or vice versa).
Example: y = 5x^3 + 2x^2 − 3x
Let's work out the second derivative:
• The derivative is y' = 15x^2 + 4x − 3
• The second derivative is y'' = 30x + 4
And 30x + 4 is negative up to x = −4/30 = −2/15, positive from there onwards. So:
f(x) is concave downward up to x = −2/15
f(x) is concave upward from x = −2/15 on
And the inflection point is at x = −2/15
A Quick Refresher on Derivatives
In the previous example we took this:
y = 5x^3 + 2x^2 − 3x
and came up with this derivative:
y' = 15x^2 + 4x − 3
There are rules you can follow to find derivatives, and we used the "Power Rule":
• x^3 has a slope of 3x^2, so 5x^3 has a slope of 5(3x^2) = 15x^2
• x^2 has a slope of 2x, so 2x^2 has a slope of 2(2x) = 4x
• The slope of the line 3x is 3
Another example for you:
Example: y = x^3 − 6x^2 + 12x − 5
The derivative is: y' = 3x^2 − 12x + 12
The second derivative is: y'' = 6x − 12
And 6x − 12 is negative up to x = 2, positive from there onwards. So:
f(x) is concave downward up to x = 2
f(x) is concave upward from x = 2 on
And the inflection point is at x = 2: | {"url":"http://www.mathsisfun.com/calculus/inflection-points.html","timestamp":"2014-04-18T16:46:00Z","content_type":null,"content_length":"8244","record_id":"<urn:uuid:691d0c91-dde0-41d7-a9f3-b695d51fbff7>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00153-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum Discussions - Finding asymptote of a field data
Date: Dec 8, 2012 11:26 PM
Author: Marc Sean
Subject: Finding asymptote of a field data
Given a set of data collected from field, how do you find the asymptotes?
I'm particularly interested in finding the horizontal asymptotes.
Is there a proper mathematical way to compute? I was thinking to just sample a group of data and once the variances dont differ by certain amount, that is the asymptote?
Any pointer please.
Many thanks. | {"url":"http://mathforum.org/kb/plaintext.jspa?messageID=7934344","timestamp":"2014-04-17T07:58:53Z","content_type":null,"content_length":"1341","record_id":"<urn:uuid:0dafefe3-d5da-418e-936f-f0d81381f35c>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00470-ip-10-147-4-33.ec2.internal.warc.gz"} |
Alfred Tarski
1. Hodges (1985/6) claims that no notion defined by Tarski in the monograph on truth possesses all the crucial features of the notion of truth in a structure, and thus that this notion is not really
defined by Tarski at this time. (See also Milne 1999. Gómez-Torrente 2001 contains a critique of Hodges’s claim.)
2. The claim that in the 1936 paper Tarski is primarily thinking of languages with the conventions of LAr is controversial. It is defended in Gómez-Torrente (1996), (2009) and disputed e.g. in
Mancosu (2006), (2010). According to Mancosu, Tarski is paradigmatically thinking of languages like LAr*: LAr* is like LAr except that quantifications need not be relativized to “N”, and the range of
the variables is the set of all individuals of some underlying type theory. See the next footnote.
3. A very similar definition can be given for languages like LAr* (see the preceding footnote): an interpretation <A, a, R> of LAr* satisfies the formula function X with respect to a sequence f (that
assigns arbitrary values from the set of individuals of an underlying type theory to the original variables of LAr*) if and only if:
(a) (i) X is Px[n] (for some n) and f(x[n])∈A; or X is Py and a∈A; or
(ii) X is Yx[n]x[m] (for some m and n) and <f(x[n]),f(x[m])>∈R; or X is Yyx[n] (for some n) and <a,f(x[n])>∈R; or X is Yx[n]y (for some n) and <f(x[n]),a>∈R; or X is Yyy and <a,a>∈R; or
(b) there is a formula function Y such that X is ¬Y and <A, a, R> does not satisfy Y with respect to sequence f; or
(c) there are formula functions Y and Z such that X is (Y→Z) and either <A, a, R> does not satisfy Y with respect to sequence f or <A, a, R> satisfies Z with respect to sequence f; or, finally,
(d) there is a formula function Z and a number n such that X is ∀x[n]Z and every sequence g that assigns arbitrary values to the (original) variables of LAr and that differs from f at most in what it
assigns to x[n] is such that <A, a, R> satisfies Z with respect to g. | {"url":"http://plato.stanford.edu/entries/tarski/notes.html","timestamp":"2014-04-16T13:41:19Z","content_type":null,"content_length":"15889","record_id":"<urn:uuid:06f2efd5-b520-4b8c-9571-b04ffa8de4d9>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00501-ip-10-147-4-33.ec2.internal.warc.gz"} |
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21 Subjects: including algebra 1, reading, geometry, Chinese | {"url":"http://www.purplemath.com/solana_beach_algebra_tutors.php","timestamp":"2014-04-17T15:34:54Z","content_type":null,"content_length":"24091","record_id":"<urn:uuid:e1f0e629-e468-4900-a133-3f8ba843d305>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00245-ip-10-147-4-33.ec2.internal.warc.gz"} |
The budget and affordability
Anonymous wrote:
The students will suffer if the budget is not passed. The contingency budget cuts too deep into their lives. Stop this nonsense already. I have read all of the sites you post, I attend BOE meetings,
and the onlybresponsible thing to do FOR THE KIDS is to pass this budget. Youbcan call them all the names you want, the facts are the facts, and numbers are numbers, but the students should not be
short changed because people have an ax to grind. Don't make today's students pay for the decisions made by yesterdays adults.
In 2009 the voters passed a $274 Million budget… the BOE increased it to $277 Million
In 2010 the votes passed a $280 Million budget…the BOE increased it to $282 Million
If the 20011-12 Budget passes the BOE will increased it to $291 Million
What has a passed or failed budget to do with Students?
Next Year’s Tax Increase is 4.5%...$5,422,460
Declared fund Balance (budget fat) 1s $7 million
$1.2 million of the budget is capital improvements
For $7 million fund balance See last page http://www.sachem.edu/board/budget11/pd ... on%203.pdf and http://www.sachem.edu/board/budget11/pdf/2011-2012%20Budget%20Status%20Report.pdf
The Board sees no need for honesty in dealing with voters or taxpayers. Just listen to this tape http://www.sachemunspun.com/10106timo.mp3
This page will show you ten years of budget increases, actual expenses, taxes and fund balance http://www.sachem.edu/board/budget11/pd ... istory.pdf
You need to understand every line on this one page http://www.sachemunspun.com/10-6Auditpg21.pdf
This year the Board is hiding $25 Million of fund balance and hording $40 million of reserves and available funds. They are building fund balance at the expense of taxpayers, employees and programs
to banking money because they fear the Tax Cap.
This year the district will spend $259 million and collect $287 million
Currently the District has;
$28 million in reserves (13 million ) and liability accounts ($15 million)
$11 million in un-appropriated fund balance
$25 million of budget surplus
$10 million in capital funds that can be used to restore programs
Next Year’s Tax Increase is 4.5% and they don’t need it;
The 2011-12 Tax increase is $5,422,460
Declared budget fat 1s $7 million
$1.2 million of the budget is capital improvements
And the are firing 90 people
For $7 million fund balance See last page http://www.sachem.edu/board/budget11/pd ... on%203.pdf and http://www.sachem.edu/board/budget11/pd ... Report.pdf
The Sachem Board can only deal in fear and deception they have no trust in the community or a moral compass. At the 4/13 meeting the Blackmailers went over the worst cuts possible so they could put
them in the budget news letter. Timo, Like Scavo the year before and Falco the year before that said, These cuts are horrible but ewe don't have to do them.
Go to www.Sachemunspun.com to hear the Board members confirm all the below
Current Budget, $282 Million … Passed Budget $287 Million … Contingent Budget $285 Million
Current Taxes, $145 Million … Passed Budget $151 Million … Contingent Budget $149 Million
Fund Balance, $17 Million … Passed Budget $23 Million … Contingent Budget $23 Million
Fund balance is the money you were overtaxed this year that is used to reduce taxes next year
CAN YOU TRUST THIS BOARD WITH A PASSED BUDGET? Twice in the last two years the Board increased the voter approved budget for $ 6 million of unforeseen circumstances. This year their meetings are full
of “emergency repairs”. If the Budget passes expect a $4million increase for unforeseen circumstances.
A PASSED BUDGET IS AN OPEN CHECKBOOK! It permits the Board to increase spending and taxes for unforeseen circumstances. It also increases every future budget’s base by $6 million.
A DEFEATED BUDGET LIMITS SPENDING TO $285 MILLION AND NOT A DIME MORE In the last two years the Board has found excuses to increase spending and taxes beyond what the voters approved to avoid
returning over taxation to the voters. Last year they collected $282,111,665 and only spent $265,509,366
The Board picks taxpayer pockets every year to level future taxes and bank money
2006-07 Budget 266,665,361 Expenses 262,000,417 Fund Balance 25,471,388
2007-08 Budget 274,707,921 Expenses 265,060,293 Fund Balance 32,148,964
2008-09 Budget 278,801,607 Expenses 267,926,888 Fund Balance 36,110,757
2009-10 Budget 277,694,940 Expenses 265,509,366 Fund Balance 41,687.472
2010-11 Budget 282,111,665 Expenses 259,700,000 Fund Balance 61,700.000 est. inc Excel BAN
2011-12 Budget 287,834,125 Expenses 270,000,000 Fund Balance 72,700.000 est.. inc Excel BAN
For 2011-12 http://www.sachem.edu/board/budget11/pd ... ndices.pdf
For 2010-11 http://www.sachem.edu/board/budget10/pdf/Appendices.pdf
For 2009-10 http://www.sachem.edu/board/budget09/pd ... 09to10.pdf
For 2008-09 http://www.sachem.edu/board/budget08/pd ... NDICES.pdf
For 2007-08 http://www.sachem.edu/board/budget07/20 ... ndices.pdf
For 2006-07 http://www.sachemunspun.com/documents/06-7Apendix.pdf
Now Listen to the 4 one minute clips and understand why I call the Board Pickpockets and deceivers.
ransomechief wrote:
For the past few years the Board budget letters have touted holding the line on taxes - NO TAX INCREASE. Yet Sachem's school taxes go up each year - last year mine went up $195. The previous year
they went down $44. The previous year they went down $38. The previous year they were up $290.Over the past nine years they are up $1069 to $4205 and STAR relief is diminishing year by year. Somehow
costs and expenditures must be contained, curtailed, or consolidated.
As the energy costs increase we constantly hear that the consumer should be able to absorb these SHORT TERM increases. Yet the oil companies make no such claim for themselves. Costs are passed right
along to the consumer. one lI gas station was posting gas for $4.079. Heating oil for a 100 gallon delivery is now costing on average $160-$170.
Yes, schools must make sdjustments for energy increases, but avenues for cutting must also be explored. Teachers' contracts must be up for review BEFORE the budget comes up for vote, and all details
must be released for public scrutiny. Only then can a truly informed person make an intelligent vote . Fudging the budget with overtaxation and hidden surpluses is unacceptable.
As for board members in the union's pocket - no one can fight a stacked deck. We need unbiased and impartial members to represent the taxpayers and put the interests of the taxpayers above their own
self-serving ones.
Tripi, Duncan, Scavo & Falco
As the energy costs increase we constantly hear that the consumer should be able to absorb these SHORT TERM increases. Yet the oil companies make no such claim for themselves. Costs are passed right
along to the consumer. one lI gas station was posting gas for $4.079. Heating oil for a 100 gallon delivery is now costing on average $160-$170.
Yes, schools must make sdjustments for energy increases, but avenues for cutting must also be explored. Teachers' contracts must be up for review BEFORE the budget comes up for vote, and all details
must be released for public scrutiny. Only then can a truly informed person make an intelligent vote . Fudging the budget with overtaxation and hidden surpluses is unacceptable.
As for board members in the union's pocket - no one can fight a stacked deck. We need unbiased and impartial members to represent the taxpayers and put the interests of the taxpayers above their own
self-serving ones.
ransomechief wrote:
For the past few years the Board budget letters have touted holding the line on taxes - NO TAX INCREASE. Yet Sachem's school taxes go up each year - last year mine went up $195. The previous year
they went down $44. The previous year they went down $38. The previous year they were up $290.Over the past nine years they are up $1069 to $4205 and STAR relief is diminishing year by year. Somehow
costs and expenditures must be contained, curtailed, or consolidated.
As the energy costs increase we constantly hear that the consumer should be able to absorb these SHORT TERM increases. Yet the oil companies make no such claim for themselves. Costs are passed right
along to the consumer. one lI gas station was posting gas for $4.079. Heating oil for a 100 gallon delivery is now costing on average $160-$170.
Yes, schools must make sdjustments for energy increases, but avenues for cutting must also be explored. Teachers' contracts must be up for review BEFORE the budget comes up for vote, and all details
must be released for public scrutiny. Only then can a truly informed person make an intelligent vote . Fudging the budget with overtaxation and hidden surpluses is unacceptable.
As for board members in the union's pocket - no one can fight a stacked deck. We need unbiased and impartial members to represent the taxpayers and put the interests of the taxpayers above their own
self-serving ones.
There is ONE Reason why your Taxes keep going up-YOUR TEACHER CONTRACTS are Killing you.
Plus I suspect the SACHEM BOE Is socking away money for when those STATE CAPS Become Law. Once a Budget has a Surplus Built into to it- do you think they will ever give it back?
b]TEACHERS’ SALARIES:[/b]
Average Full-Time Worker- DAY LENGTH= 8 Hr
Average Full-Time Teacher- DAY LENGTH=7 Hr
Average Full-Time Worker WORK DAY- Working at job hired for= 7.5 Hr/day
Average Full-Time Teacher WORK DAY (Working with students): 5 periods of 45 minutes= 225 minutes/day (3.75Hr /day) The rest of their day spent as follows: Lunch- 1 period (45 Minutes), 2 Preps(45
minutes each) + 1 Duty Period (Babysitting the cafeteria, hall, playground, busses etc. for 45 minutes)
Average Full-Time Worker WORK YEAR = 260 Days per Year
Average Full-Time Teacher WORK YEAR =185 Days per Year
Average Full-Time Worker YEARLY SALARY(after 6 Years)= [Plug in a career]
Average Full-Time Teacher YEARLY SALARY(after 6 Years)=$60,000 ADDITIONAL BENEFITS:
Average Full-Time Worker VACATION DAYS=10 to 20 Days dependent upon career & length of Service
Average Full-Time Teacher VACATION DAYS=75 DAYS (includes ALL HOLIDAYS)
Average Full-Time Worker PAID SNOW DAYS=NONE
Average Full-Time Teacher PAID SNOW DAYS= As many as the District gives Students off
Average Full-Time Worker SICK DAYS= 7 to 10 per year
Average Full-Time Teacher SICK DAYS= 10 to 15 per year
Average Full-Time Worker PERSONAL DAYS = 3 per Year
Average Full-Time Teacher PERSONAL DAYS= 5 per Year
Average Full-Time Worker BEREAVEMENT DAYS= 3 days for immediate family members
Average Full-Time Teacher BEREAVEMENT DAYS= 5 days for EACH immediate & extended family member
Average Full-Time Worker JOB SECURITY= Can be fired AT WILL By their Employer
Average Full-Time Teacher JOB SECURITY= Receive Tenure after 3 years and CANNOT be fired short of conviction of criminal activity PLUS: Average Full-Time Teacher= EXTRA PAY for after School clubs,
sports, etc- ALL add to TEACHER PENSION
Average Full-Time Teacher= if work in Summer for a municipality or School District- ALL add to TEACHER PENSION
Average Full-Time Worker In-Service/Review Courses= Paid for by employee & taken on their own time
Average Full-Time Teacher In-Service= Paid for by District & Taken on District Time
Teacher's ABILITY TO increase their SALARY(AND PENSION) by Taking All types of College Credit- even online IF Teacher Yearly SALARY= $35,000 THEN ACTUAL DAILY rate =$189
IF Teacher Yearly SALARY= $40,000 THEN ACTUAL DAILY rate =$216
IF Teacher Yearly SALARY= $45,a00 THEN ACTUAL DAILY rate =$243
IF Teacher Yearly SALARY= $50,000 THEN ACTUAL DAILY rate =$270
IF Teacher Yearly SALARY= $55,000 THEN TEACHER'S ACTUAL DAILY rate =$297
IF Teacher Yearly SALARY= $60,000 THEN TEACHER'S ACTUAL DAILY rate =$324
IF Teacher Yearly SALARY= $65,000 THEN TEACHER'S ACTUAL DAILY rate =$350
IF Teacher Yearly SALARY= $70,000 THEN TEACHER'S ACTUAL DAILY rate =$378
IF Teacher Yearly SALARY= $75,000 THEN TEACHER'S ACTUAL DAILY rate =$405
IF Teacher Yearly SALARY= $80,000 THEN TEACHER'S ACTUAL DAILY rate =$432
IF Teacher Yearly SALARY= $85,000 THEN TEACHER'S ACTUAL DAILY rate =$459
IF Teacher Yearly SALARY= $90,000 THEN TEACHER'S ACTUAL DAILY rate =$486
IF Teacher Yearly SALARY= $95,000 THEN TEACHER'S ACTUAL DAILY rate =$514
IF Teacher Yearly SALARY= $100,000 THEN TEACHER'S ACTUAL DAILY rate =$541
IF Teacher Yearly SALARY= $105,000 THEN TEACHER'S ACTUAL DAILY rate =$568
IF Teacher Yearly SALARY= $110,000 THEN TEACHER'S ACTUAL DAILY rate =$595
IF Teacher Yearly SALARY= $115,000 THEN TEACHER'S ACTUAL DAILY rate =$622
IF Teacher Yearly SALARY= $120,000 THEN TEACHER'S ACTUAL DAILY rate =$649
IF Teacher Yearly SALARY= $125,000 THEN TEACHER'S ACTUAL DAILY rate =$676
IF Teacher Yearly SALARY= $130,000 THEN TEACHER'S ACTUAL DAILY rate =$703
IF Teacher Yearly SALARY= $135,000 THEN TEACHER'S ACTUAL DAILY rate =$730
IF Teacher Yearly SALARY= $140,000 THEN TEACHER'S ACTUAL DAILY rate =$757
IF Teacher Yearly SALARY= $145,000 THEN TEACHER'S ACTUAL DAILY rate =$784
IF Teacher Yearly SALARY= $150,000 THEN TEACHER'S ACTUAL DAILY rate =$811
IF Teacher Yearly SALARY= $155,000 THEN TEACHER'S ACTUAL DAILY rate =$838
IF Teacher Yearly SALARY= $160,000 THEN TEACHER'S ACTUAL DAILY rate =$865
IF Teacher Yearly SALARY= $165,000 THEN TEACHER'S ACTUAL DAILY rate =$892
IF Teacher Yearly SALARY= $170,000 THEN TEACHER'S ACTUAL DAILY rate =$919
IF Teacher Yearly SALARY= $175,000 THEN TEACHER'S ACTUAL DAILY rate =$946
IF Teacher Yearly SALARY= $180,000 THEN TEACHER'S ACTUAL DAILY rate =$973
IF Teacher Yearly SALARY= $185,000 THEN TEACHER'S ACTUAL DAILY rate =$1000
IF Teacher Yearly SALARY= $1905,000 THEN TEACHER'S ACTUAL DAILY rate =$1027
IF Teacher Yearly SALARY= $195,000 THEN TEACHER'S ACTUAL DAILY rate =$1070
IF Teacher Yearly SALARY= $200,000 THEN TEACHER'S ACTUAL DAILY rate =$1081
That's for a 185 DAY WORK YEAR (The average Maximum amount of days a teacher works in almost all School Districts in NEW YORK STATE)
That is NOT even taking into account that a teacher WORKS LESS THAN 4 HOURS a DAY WITH STUDENTS! AND THE REAL Point is that teachers (effectively) CANNOT BE FIRED! LET"S ANALYZE THAT SALARY EVEN
A Teacher TEACHES 5 periods/TEACHING workday x 45 minutes/period x 185 workdays /year =694 TEACHING Hours/year
A Full-time Worker who has 4 WEEKS VACATION/Year WORKS (That is doing what they were hired for): 7.5 WORK Hours/day x 240 WORK days/year= 1800 WORK Hours/year THEREFORE, a NON-Teacher Worker WORKS
160% MORE in a year than a TEACHER!
THEREFORE, (PLUG in) TEACHER SALARY x 260% = Equivalent FULL-Time Salary PAY (Had the Teacher WORKED the FULL YEAR & FULL DAY!!!)
Example: If a Teacher worked a FULL DAY & FULL YEAR, the EQUIVALENT YEARLY SALARY of the
Teacher: Making $30,000 is the EQUIVALENT OF $78,000
Teacher: Making $40,000 is the EQUIVALENT OF $104,000
Teacher: Making $60,000 is the EQUIVALENT OF $156,000
Teacher: Making $80,000 is the EQUIVALENT OF $208,000
Teacher: Making $100,000 is the EQUIVALENT OF $260,000
Teacher: Making $130,000 is the EQUIVALENT OF $338,000
Teacher: Making $140,000 is the EQUIVALENT OF $338,000
Teacher: Making $150,000 is the EQUIVALENT OF $390,000
Teacher: Making $160,000 is the EQUIVALENT OF $416,000
Teacher: Making $170,000 is the EQUIVALENT OF $442,000
Teacher: Making $180,000 is the EQUIVALENT OF $468,000
Teacher: Making $190,000 is the EQUIVALENT OF $494,000
Teacher: Making $200,000 is the EQUIVALENT OF $520,000 AND if a Teacher worked a FULL DAY & FULL YEAR, the EQUIVALENT DAILY RATE of the Teacher: Making $30,000 is
the EQUIVALENT
DAILY RATE OF $422 Teacher: Making $40,000 is
the EQUIVALENT
DAILY RATE OF $562 Teacher: Making $60,000 is
the EQUIVALENT
DAILY RATE OF $843 Teacher: Making $80,000 is
the EQUIVALENT
DAILY RATE OF $1,124 Teacher: Making $100,000 is
the EQUIVALENT
DAILY RATE OF $1,405 Teacher: Making $130,000 is
the EQUIVALENT
DAILY RATE OF $1,827 Teacher: Making $140,000 is
the EQUIVALENT
DAILY RATE OF $1,968 Teacher: Making $150,000 is
the EQUIVALENT
DAILY RATE OF $2,108 Teacher: Making $160,000 is
the EQUIVALENT
DAILY RATE OF $2,249 Teacher: Making $170,000 is
the EQUIVALENT
DAILY RATE OF $2,389 Teacher: Making $180,000 is
the EQUIVALENT
DAILY RATE OF $2,530 Teacher: Making $190,000 is
the EQUIVALENT
DAILY RATE OF $2,670 Teacher: Making $200,000 is
the EQUIVALENT
DAILY RATE OF $2,811
AND THAT DOES not Take into consideration the other benefits that Private Sector jobs DO NOT GET- Not to mention TENURE(ie PERMANENT JOB SECURITY) DO YOU GET THE POINT!!!!
They need to start
WORKING a FULL-YEAR for their CURRENT PAY!!!
That alone with a staggered school year for STUDENTS would SAVE the Taxpayers AT LEAST 30% SCHOOL TAXES ( by being able to reduce the current Staffing needs!!!
The EQUIVALENT TEACHER SALARY is what is Killing LI!!!!!!!!!!! | {"url":"http://www.sachemunspun.com/bulletinboardupdated/viewtopic.php?f=1&t=4179&p=41157","timestamp":"2014-04-21T12:09:28Z","content_type":null,"content_length":"53576","record_id":"<urn:uuid:e96417dc-8c00-492c-a2ed-8530b4a36ec8>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00611-ip-10-147-4-33.ec2.internal.warc.gz"} |
Inboard Stall Delay Due to Rotation
1. Introduction
In the design process of improved rotor blades the need for accurate aerodynamic predictions is very important. During the last years a large effort has gone into developing CFD tools for prediction
of wind turbine flows (Duque et al., 2003; Fletcher et al., 2009; Sørensen et al.,2002). However, there are still some unclear aspects for engineers regarding the practical application of CFD, such
as computational domain size, reference system for different computational blocks, mesh quality and mesh number, turbulence, etc. Thus, in the design process and in the power curve prediction of wind
turbines, the aerodynamic forces are calculated with some form of the blade element method (BEM) and its extensions to the three-dimensional wing aerodynamics. The results obtained by the standard
methods are reasonably accurate in the proximity of the design point, but in stalled condition the BEM is known to underpredict the forces acting on the blades (Himmelskamp, 1947). The major
disadvantage of these methods is that the airflow is reduced to axial and circumferential flow components (Glauert, 1963). Disregarding radial flow components present in the bottom of separated
boundary layers of rotating wings leads to alteration of lift and drag characteristics of the individual blade sections with respect to the 2-D airfoils (Bjorck, 1995).
Airfoil characteristics of lift (C [ L ]) and drag (C [ D ]) coefficients are normally derived from two-dimensional (2-D) wind tunnel tests. However, after stall the flow over the inboard half of the
rotor is strongly influenced by poorly understood 3-D effects (Banks & Gadd, 1963; Tangler, 2002). The 3-D effects yield delayed stall with C [ L ] higher than 2.0 near the blade root location and
with correspondingly high C [ D ] . Now the design of constant speed, stall-regulated wind turbines lacks adequate theory for predicting their peak and post-peak power and loads.
During the development of stall-regulated wind turbines, there were several attempts to predict 3-D post-stall airfoil characteristics (Corrigan & Schlichting, 1994; Du & Schling, 1998; Snel et al.,
1993), but these methods predicted insufficient delayed stall in the root region and tended to extend the delayed stall region too far out on the blade.
The present work aims at giving a conceptualization of the complex 3-D flow field on a rotor blade, where stall begins and how it progresses, driven by the needs to formulate a reasonably simple
model that complements the 2-D airfoil characteristics used to predict rotor performance.
Understanding wind turbine aerodynamics (Hansen & Butterfield, 1993) in all working states is one of the key factors in making improved predictions of their performance. The flow field associated
with wind turbines is highly three-dimensional and the transition to two-dimensional outboard separated flow is yet not well understood. A continued effort is necessary to improve the delayed stall
modeling and bring it to a point where the prediction becomes acceptable. In the sequel, based on previous computed and measured results, a comprehensible model is devised to explain in physical
terms the different phenomena that play a role and to clarify what can be modeled quantitatively in a scientific way and what is possible in an engineering environment.
In 1945 Himmelskamp (Himmelskamp, 1947) first described through measurements the 3-D and rotational effects on the boundary layer of a rotating propeller, finding lift coefficients much higher moving
towards the rotation axis. Further experimental studies confirmed these early results, indicating in stall-delay and post-stalled higher lift coefficient values the main effects of rotation on wings.
Measurements on wind turbine blades were performed by Ronstend (Ronsten, 1992), showing the differences between rotating and non-rotating pressure coefficients and aerodynamic loads, and by Tangler
and Kocurek (Tangler & Kocurek, 1993), who combined results from measurements with the classical BEM method to properly compute lift and drag coefficients and the rotor power in stalled conditions.
The theoretical foundations for the analysis of the rotational effects on rotating blades come at the late 40’s with Sears (Sears, 1950), who derived a set of equations for the potential flow field
around a cylindrical blade of infinite span in pure rotation. He stated that the spanwise component of velocity is dependent only upon the potential flow and it is independent of the span (the
so-called independence principle). Then, Fogarty and Sears (Fogarty & Sears, 1950) extended the former study to the potential flow around a rotating and advancing blade. They confirmed that, for a
cylindrical blade advancing like a propeller, the tangential and axial velocity components are the same as in the 2-D motion at the local relative speed and incidence. A more comprehensive work was
made once more by Fogarty (Fogarty, 1951), consisting of numerical computations on the laminar boundary layer of a rotating plate and blade with thickness. Here he showed that the separation line is
unaffected by rotation and that the spanwise velocities in the boundary layer appeared small compared to the chordwise, and no large effects of rotation were observed in contrast to (Himmelskamp,
1947). A theoretical analysis done by Banks and Gadd (Banks &Gadd, 1963), focused on demonstrating how rotation delays laminar separation. They found that the separation point is postponed due to
rotation, and for extreme inboard stations the boundary layer is completely stabilized against separation.
In the NASA report done by McCroskey and Dwyer (McCroskey & Dwyer, 1969), the so-called secondary effects in the laminar incompressible boundary layer of propeller and helicopter rotor blades are
widely studied, by means of a combined, numerical and analytical approach. They showed that approaching the rotational axis, the Coriolis force in the cross flow direction becomes more important. On
the other hand the centrifugal pumping effect is much weaker than was generally thought before, but its contribution increases particularly in the region of the separated flow. The last two decades
have known the rising of computational fluid dynamics and the study of the boundary layer on rotating blade has often been carried on through a numerical approach. Sørensen (Sørensen, 1986)
numerically solved the 3-D equations of the boundary layer on a rotating surface, using a viscous-inviscid interaction model. In his results the position of the separation line still appears the same
as for 2-D predictions, but where separations are more pronounced a larger difference between the lift coefficient calculated for the 2-D and 3-D case is noticed. A quasi 3-D approach, based on the
viscous-inviscid interaction method, was introduced by Snel et al (Snel et al., 1993) and results were compared with measurements. They proposed a semi-empirical law for the correction of the 2-D
lift curve, identifying the local chord to radii (c/r) ratio of the blade section as the main parameter of influence. This result has been confirmed by Shen and Sørensen (Shen & Sørensen, 1999), and
by Chaviaropoulos and Hansen (Chaviaropoulos & Hansen, 2000), who performed airfoil computations applying a quasi 3-D Navier-Stokes model, based on the streamfunction-vorticity formulation and
respectively a primitive variables form. Du and Seling (Du & Selling, 2000), and Dumitrescu and Cardoş (Dumitrescu & Cardoş, 2010), investigated the effects of rotation on blade boundary layers by
solving the 3-D integral boundary-layer equations with the assumed velocity profiles and a closure model. Dumitrescu and Cardoş studying the boundary layer behavior very close to the rotation center
(r/c< 1) stated that the stall delay depends strongly on the leading edge separation bubbles formed on inboard blade segment due to a suction effect at the root area of the blade (Dumitrescu &
Cardoş, 2009).
Recently, with the advent of the supercomputer, computational fluid dynamics (CFD) tools have been employed to investigate the stall-delay for wind turbines (Duque et al., 2003; Fletcher et al., 2009
; Sørensen et al., 2002). These calculations in addition to Narramore and Vermeland’s results (Narramore & Vermeland, 1992) showed that the 3-D rotational effect was particularly pronounced for the
inboard sections, but the genesis of this phenomenon is a problem still open. The present concern aims at giving explanations in physics terms of the different features widely-observed experimentally
and computationally in wind turbine flow.
2. Flow at low wind speeds
At low wind speed conditions, i.e. at high speed ratios (TSR>3), the visualization of the computed flow indicates that the flow is well-behaved and attached over much of the rotor, Fig. 1 . Figure
1 shows the separated area and radial flow on the suction side of a commercial blade with 40 m length at the design tip speed ratio (TSR=5); the secondary flow is strongest at approximately 0.17R and
reaches up to 0.31R, where R is the rotor radius. The local air velocity relative to a rotor blade consists of free-wind velocity V [ w ] defined as the wind speed if there were no rotor present,
that due to the blade motion Ω[ b ] r and the wake induced velocities; at high TSR, a weak wake (Glauert, 1963) occurs and its rotational induction velocity can be neglected. Wind turbine blade
sections can operate in two main flow regimes depending on the size of the rotation parameter defined as the ratio of wind velocity to the local tangential velocity Vw/Ωbr . If the rotation parameter
is less than unity along the entire span, and for properly twisted blades, the flow is mostly two-dimensional and attached, while for rotation parameter greater than unity the flow is neither
two-dimensional nor steady, and is strongly affected by rotation. At high tip speed ratios the subunitary Vw/Ωbr condition is accomplished and the blade sections usually operate at prestall
incidences. Then, the boundary layer is attached all the way to the trailing edge on the outside blade and is separated at trailing edge only on the inside blade, Fig. 1. At the root area of the
blade the flow close to the hub behaves like a rotating disk in a fluid at rest (Fig. 2a), where the centrifugal forces induce a spinning motion in the separated flow; and a radial velocity field is
more or less uniform (Fig. 2b). Beginning at the hub, this secondary flow generates the so-called centrifugal pumping mechanism, acting in separated trailing edge flow. On the other hand, the
Coriolis force acts in the chordwise direction as a favorable pressure that mitigates separation at the trailing edge along the whole span.
Therefore, at low wind speed, the main rotational effect is due to the Coriolis force which delays the occurrence of separation to a point further downstream towards the trailing edge, and by this
the suction pressures move towards higher levels as r/c decreases. The pumping effect is much weaker than was generally thought before.
The pressure field created by the presence of the turbine is related to the incoming flow field around the blade, taken as being composed of the free wind velocity and the so-called induction
velocity due to the rotor and its wake. Thus, the incoming field results from a weak interaction between two different flows: one axial and the other rotational (Vw/Ωbr1) . In such a weak interaction
flow, the basic assumptions made are:
• the radial independence principle is applied to flow effects, i.e. induction velocities used at a certain radial station depend only on the local aerodynamic forces at that same station;
• the mathematical description of the air flow over the blades is based on the 2-D flow potential independent of the span, and on corrections for viscosity and 3-D rotational effects.
These assumptions suitable to BEM methods reduce the complexity of the problem by an order of magnitude yielding reliable results for the local forces and the overall torque in the proximity of the
design point, at high tip speed ratios. In order to estimate the 3-D rotational effects, usually neglected in the traditional BEM model, the flow around a hypothetic blade with prestall/stall
incidence and chord constant along the whole span is considered in the sequel.
2.1. Representation of flow elements
The set of equations including a simplified form of the inviscid flow and the full three-dimensional boundary-layer equations are used to identify the influence of the3-D rotational effects at low
wind speeds.
A. Inviscid flow. In order to find the velocity at the airfoil surface in absence of viscous effects, the reference velocity at a point on a rotating wind turbine blade is
where V [ w ] is the wind speed, λ=ΩbRVw is the tip-speed ratio (TSR), R is the radius of the turbine and a is the axial induced velocity interference, a function of the speed ratio λ (Burton et al.,
2001). Starting from the idea of Fogarty and Sears (Fogarty & Sears, 1950), an inviscid edge velocity can be calculated as
where U, V and W represent the velocity components in the cylindrical coordinate system (θ,r,z) , which rotates with the blade with a constant-rotational speed Ω[ b ] (Fig. 3). ϕ=ϕ(θ,z) denotes the
2-D potential solution, that is constant at all radial positions. The interesting point regarding this set of equations (2) is that the spanwise component V can be derived from the local 2-D velocity
potential. However, this spanwise component is very small and thus neglected in the present work. The potential edge velocity components can be approached as
where the non-dimensional velocity U2D could be obtained by a viscous-inviscid interaction procedure for flow past a 2-D airfoil (Drela, 1989). Since the primary concern of the present work is to
investigate the rotational 3-D effects on prestalled and stalled blades by means of the boundary layer method, the inviscid pressure distribution is simply considered as
where Cp=p−p∞ρ/2(Vw2+Ωbr2) is the local pressure coefficient, x/c is the airfoil abscissa by airfoil chord and subscripts U and L indicate properties on the upper surface and lower surface
respectively. The scale parameter |Cpm| is equal to maximum value of C [ p ] and is only dependent on the local aerodynamic field: the airfoil shape, incidence and Reynolds number. This one-parameter
pressure distribution family ranges a large gamut from the prestall distributions, values of |Cpm|=2−6 , until the stall and post-stall distributions with |Cpm|=7−12 . Examples of distributions and
experimental comparisons are shown in Fig. 4.
; b)
Because a wind turbine can operate a long time at low tip speed ratios, the inboard regions of the blade are stalled, and there leading-edge separation bubbles can occur (Dumitrescu & Cardoş, 2010).
Behind the mild separation it is assumed that the flow relaxes with the vanishing skin friction. Therefore, the following flow is applied to solve
where δ2x and H are, respectively, the momentum thickness and the boundary-layer shape parameter in the streamwise direction.
Figures 5 and 6 illustrate such chordwise inviscid velocity distributions and the corresponding variations of the peripheral skin-friction coefficient C [ fx ] and the boundary layer shape parameter
H, calculated for various values of |Cpm| (a velocity gradient like parameter) and r/c = ∞ (2-D). Laminar separation takes place at Cfx=0 , and a given value of the shape factor, H=3.4 , is used as
the criterion for turbulent separation/reattachment (Cebeci & Cousteix, 1999). The “transition point” x [ t ] is the point that corresponds to the minimum skin-friction, which sometimes can be the
point of laminar separation.
Fig 6. Variation of a) the peripheral skin-friction coefficient; b) the boundary-layer shape parameter.
B. Viscous flow
Momentum integral equations. The flow in the boundary layer on a rotating blade in attached as well as in stalled conditions is represented using the integral formulation developed in (Dumitrescu &
Cardoş, 2010) for analyzing separated and reattaching turbulent flows involving leading-edge separation bubbles. The 3-D incompressible steady boundary layer equations are written in the cylindrical
coordinate system (θ,r,z) which rotate with the blade at a constant rotational speed Ωz=Ωb (Fig. 3); θ denotes the peripheral, z the axial and r the radial (spanwise) direction. The equations used
early (Cebeci & Cousteix, 1999) are shown in their conservative laminar form:
Momentum, θ component:
Momentum, r component:
where u, v and w stand for velocity components in θ, r and z directions, respectively, r for the local radius measured from the centre of rotation, ρ for the fluid density, ν for the kinematic
viscosity, and P is a pressure like term including the centrifugal effect
with p denoting the static pressure.
The equations (7)-(9) are integrated with respect to z (normal to blade) from 0 to δ (boundary layer thickness) and the integral forms of equations are obtained as
where (U [ e ], V [ e ]=0) are the inviscid freestream velocity components and (Cfx,Cfr) are the skin-friction coefficient components. The various boundary layer thicknesses are defined as
An order of magnitude analysis shows that ∂δ2xr∂r=O(εδ2x) and ∂δ2r∂r=O(ε2δ2x) , and thereby these terms can be neglected as a first approximation. The widely used Pohlhausen (Schlichting, 1979) and
Mager (Mager, 1951) velocity profiles and the associated closure relations are introduced to solve the laminar integral boundary layer equations (Dumitrescu et al., 2007). For the turbulent boundary
layer a power law-type of velocity profile is assumed for the mean stream velocity profile
where H is the local boundary layer shape factor.
The Mager cross flow profile is assumed (Mager, 1951)
where εw is the limiting streamline parameter (tan βw ).
Equations (11) and (12) can now be written in terms of the parameters δ2x , εw , H and Cfx ,
The skin-friction relation for flows with pressure gradients and rotation effects is based on the experimental data for a turbulent boundary layer in a rotating channel (Lakshminarayana & Govindan,
This correlation is a modified version of the correlation developed by Ludwieg and Tillmann (Ludwieg & Tillman, 1949), which includes the effect of rotation. In this relation B1 is an empiric
constant (a value of 0.52 is used), Reδ2x is the Reynolds number, based on the streamwise velocity at the edge of the boundary layer and the streamwise momentum thickness δ2x , and xt is the distance
between the leading edge and the transition point along the streamwise direction (the laminar separation point is used).
Closure model-entrainment
The pressure gradients cause large changes in velocity profiles and, consequently, in the shape parameter H. The variation of H cannot be neglected and an additional equation is required. Out of the
available auxiliary equations, only the energy integral equation and the entrainment equation have been suitable for the turbulent boundary layers.
The entrainment equation is chosen to model the rotor boundary layer, which in the coordinate system used here can be written as
where the entrainment coefficient CE≡(∂δ∂x−WeUe) is a function of the factor H1=(δ−δ1x)/δ2x . CE represents the volume flow rate per unit area through the surface δ(x,r) and is the rate of
entrainment of inviscid external flow into the boundary layer.
The entrainment function CE for 3-D flow is not yet available and hence, the correlation for 2-D flow is used (Head, 1958):
Also, the similarity solutions have shown that H1 is a function of the streamwise boundary-layer shape parameter H. This relationship, which results from a best fit to experimental data (Lock &
Williams, 1987) is
, for
< 4.(21)
Then, it is assumed that the variation of the entrainment rate with H1 follows the same relationship for three-dimensional flows.
Equation (19) is written in a form similar to Eqs. (15) and (16)
Equations (15), (16) and (22) are to be solved for δ2x , εw , and H ( H1 is related to H, Eq. (21)) with the prescribed boundary conditions. At the leading edge, δ2x and εw are assumed to be zero,
and an initial value of 2.55 (laminar flow) is assumed for H.
2.2. Application
A hypothetic blade, with constant incidence and chord along the span is chosen to analyze the rotational effects on the boundary layer of a wind turbine blade. In this example, an external flow with
the distribution of velocity on the surface of blade given by Eqs. (5) and (6) is imposed.
The curves plotted in Fig. 7 are calculated on the prestall ( |Cpm|=6 ) and stall ( |Cpm|=10 ) conditions, with a Reynolds number of 10^6. Relationships between the separation points and the blade
radius are shown in Fig. 7, in which two apparent trends can be seen. Firstly, as compared to the 2-D prestall condition (r/c = ∞), the separation point is postponed because of the effect of Coriolis
force, which acts as a favorable pressure gradient tending to delay the separation to a point further downstream towards the trailing edge. It is shown that, as the radial location r/c decreases, the
separation point moves towards the trailing edge, and consequently the suction side distribution of the pressure coefficient moves towards lower levels (Chaviaropoulos & Hansen, 2000), Fig. 8. The
drop of the pressure coefficient along the suction side can be related to the separated area on the blades, which reduces in comparison with the 2-D (r/c = ∞).
Secondly, in the stall incidence condition, leading edge separation bubbles are formed on the upper surface of the blade, at the root area, which delays the occurrence of massive separation.
Beginning at the root of the blade, the bubble continues to stretch towards the trailing edge and at approximately r/c = 4 where the bubble stretches all the way to the trailing edge, the flow
separates over the whole airfoil (Dumitrescu & Cardoş, 2010) In the next section it is shown that there is a suction effect at the hub up to the mid-span which reduces the leading-edge bubble volume
and produces a significant pressure drop along the suction side of the airfoils increasing, thus, the loading of the blade.
3. Flow at high wind speeds
At high wind speeds conditions ( TSR≤3.0 ), though the stall incidences are exceeded on the whole blade, the flow is partially separated on the inboard half-span of the blade and massively separated
on the outboard half-span, Fig. 9 .
This behavior, which is characterized by significantly increased lift coefficients as compared to the corresponding 2-D case, and by a delay of the occurrence of flow separation to higher angles of
attack, is caused by the rotational augmentation at the root area of the blade. The knowledge extracted from experimental and computational visualizations of these rotational effects on the flow
field along the blade can be used to develop a model for their prediction or the so-called stall-delay phenomenon.
The wind turbine blade sections, often operate with stall at low tip speed ratios and can undergo a delayed stall phenomenon on the inner part of span. As the wind velocity increases, the inboard
regions of the blades are stalled, having Vw/Ωbr1 for much of their operational time. The flow is neither two-dimensional nor steady, and is affected by rotation. As the root is approached the flow
causes high values of tangential velocity and it behaves rather like the rotating flow over a stationary disk, Fig. 10 a), b). This is the result of a strong nonlinear interaction between the high
wind speed and the rotational flow induced by a constant-speed rotor, which can be written as
a form that suggest an outer rotational flow with angular velocity higher than that of the constant-speed rotor, ΩvΩbandΩv/Ωv=−Ωb/Ωb .
In contrast with the Glauert weak wake (Glauert, 1963), the strong potential vortex-like wake is responsible for all the 3-D rotational effects, affecting the boundary layer on the turbine blades
once the phenomenon of stall is initiated. The enhanced rotational outer flow and the bottom boundary layer are the cause of the high-lift effect of the airfoils at high angle of attack, i.e. the
stall delay.
3.1. Boundary layer beneath a Rankine-like vortex
The axisymmetric angular velocity (Ekman) boundary layer developing on the suction side of the rotor disk is used to illustrate the contribution of the suction effect of a strong wake on the
chordwise surface pressure distribution and volume flux in the radial direction. Thus, consider an axisymmetric rotating flow of a viscous incompressible fluid over a disk of radius R [0] ( Fig. 11
The angular velocity of the disk Ω[ b ] is constant and that of the fluid far away from the disk Ω[ v ] may be an arbitrary function of the radius r, provided it is stable, i.e. d(Ωvr2)dr≥0 and near
the axis the outer flow behaves locally like a rigid body rotation, i.e. r→0:dΩvdr→0,Ωv→Ωb . The Rankine-like vortex assumed for the outer rotational flow satisfies both conditions and can be
described by the vortex circulation Γv≡Ωvr2 given by the empirical formula (Vatistas et al., 1991)
where r [ h ] is the radius of forced vortex core.
This vortex model avoids the discontinuity of the velocity derivatives at the point of transition from free to forced modes and gives the possibility of the description of the flow near the hub.
For the inertial system of cylindrical coordinates (r,θ,z) considered in Fig. 10, the laminar boundary layer equations write as (Rott & Lewellen, 1966)
, where
are the velocity components, p is the static pressure, ρ is the fluid density and
is the kinematic viscosity.
The boundary conditions at the surface of the disk and the edge of boundary layer are
The outer flow radial momentum equation becomes then V2r=1ρdpdr and the pressure in the outer flow is
where C [ ps ] is the suction pressure coefficient normalized by the pressure at the vortex center, and the subscripts c and 0 indicate properties at the vortex center and the edge of disk,
The tangential and radial momentum-integral equations may be derived from equations (26) and (27) for radial and angular velocity profiles which satisfy the boundary conditions on the disk surface
and the smoothness requirements at the edge of the boundary layer
Outside the boundary layer, the tangential velocity V(r)=Γvr is assumed to be some given function, Eq. (24), while the radial velocity vanishes. The variables are the boundary-layer thickness Z [δ]
and the radial volume flux Q=∫0Zδrudz .The momentum-integral equations under these conditions are
where λ[1], λ[2], λ[3] and λ[4] are profile form parameters defined in Table 1 (Rott & Lewellen, 1966)
Flow f'(η) g(η) λ1 λ2 λ3 λ4 C1 C2
Laminar 6.75η(1−η)2 η(2−η) 2.5 1.372 0.467 0.667 2 12
Turbulent 1.69η1/7(1−η)2 η1/7 4.93 1.63 0.222 0.250 0.0225 0.0513
The Blasius shear laws for three-dimensional flow (Schlichting, 1979) cover both the laminar and turbulent case
where Um is equal to the maximum value of u and following Blasius, μ=1 (laminar flow), μ=1/4 and C = 0.0225 (turbulent flow).
This is the final result of the analytical formulation and the equations (32) and (33) must be integrated numerically for a particular circulation function Γ[ v ] (Eq. (24)) with an appropriate
choice of the boundary layer form parameters.
The onset of vortical flows like tornadoes can be visible on the constant-rotational speed rotor at low speed ratios (λ < 3.0). At the root area of the blade these flows generally behave like a
vacuum pump which is featured by volume flow rate (Q) and suction pressure (C [ ps ]).
Figures 12 and 13 show such characteristics induced by the wake modeled as a Rankine-like vortex with its axis normal to the rotor disk for different tip speed ratios. The suction characteristics
show that the effects of the disk defined by Vw/Ωbr≥1 decrease rapidly with the radius towards the edge of the disk ( Vw/Ωbr=1 ) and have maximum values near the hub (here rh=0.2R ).
On the other hand, large radial volume fluxes close to the hub along with the previous results (Fig. 7a) indicate a 3-D attached boundary layer with small leading-edge separation bubbles at high
angles of attack.
Therefore, at low tip speed ratios and close to the hub, there is essentially a 3-D attached flow field and a theoretical description in this region can be performed by viscous-inviscid interactive
flow models.
3.2. Blade-wake interaction
Flow at low tip speed ratios is dominated by a vortex-like wake which triggers the stall delay phenomenon on the inboard half-rotor disk. The dual working mode of the rotor blades as a stationary
disk in a fast rotating flow near the hub-sucking mode, and as the rotating disk in the outboard separated flow-pumping mode, are clearly visible from their pressure distributions when the wind
turbine operates at λ≤3.0 .
On the upper side of the inner blade sections there are two suction pressure fields: one is azimuthally uniform generated by circulation augmentation of wake and the other is the aerodynamic non
uniform pressure field with a suction spike at the leading-edge (Fig. 4). These two pressure fields interact and give rise to the radial effects which contribute to stall delay, as well as to a
higher lift coefficient at the root area of the blade. There are two main effects: superimposed pressure fields of wake ( Fig. 12 ) and of blade (Fig. 4), which decrease the suction spikes at the
leading edge until they cancel, and a radial flow effect induced by the volume flow merger in the boundary layers of wake and blade, resulting in velocity profile skewing and boundary-layer
energizing (no separation). Assuming the conservation of circulation once the phenomenon of stall is initiated (here at λ=3.0 ) the wake-blade pressure interaction for the increased wind speed ( λ3.0
) is a pressure redistribution process of pressures ( CpU,Cps ) along the chord at constant circulation (lift). The rule of pressure redistribution is sketched as is shown in Fig. 14.
As we will show in the next section, the phenomenon of stall-delay can be described as a three step process: generation of the strong wake by the clustering of shed vortices from the inboard blade
segments at λ=3.0 , chordwise pressure redistribution at the constant value of circulation at the start regime ( λ=3.0 ), for λ3.0 , followed by circulation decay for increasing radius. The pressure
redistribution process is companied by the move of the pressure center to the midchord.
3.3. Application
The sucking suction effect is affected by two key non-dimensional parameters (VWΩbr,rc) . Figures 15a and 15b illustrate the influence of these parameters on the chordwise pressure
Fig 14. The rule of the suction pressure with superimposed rotation effect in a normalized plan:
distribution for the previous hypothetic blade, with constant post-stall incidence and chord along the span (|Cpm|=10) . The figures also indicate the boundary layer state on the upper surface: S
-separation and R- reattachment.
4. Model for delayed stall regime
Stall occurs at an airfoil when flow separates at high angles of attack, typically>15^0, beyond this angle the 2-D lift coefficient drops significantly. It has long been known (Himmelskamp, 1947;
Ronsten, 1992; Snel et al., 1993), however, that the observed power curve under conditions where most parts of the blade are stalled is consistent with significantly higher lift coefficients. The
lack of a conceptual model for the complex 3-D flow field on the rotor blade, where stall begins, and how it progresses, has hindered the finding of an unanimously accepted solution. This section
aims at giving a better understanding of the delayed stall events.
To interest an engineer in stall-delay one might cite the fact that the inboard regions of the blades are stalled for much of their operational time. Thus predicting blade loads and the power output
during stall is very important in making good predictions of wind turbine performance. With a physicist one could reason that rotational stall admits interesting transient states from 3-D inboard
delayed stall in partially separated flow to 2-D outboard massively separated flow. Here, the detailed physical understanding of the viscous flow surrounding the hub area is dominated by the
interaction between the vorticity and pressure fields and their dynamics. However, all the existent models have been developed more or less intuitively without a basic fluid dynamics support. Six
different models mostly used to correct the airfoil characteristics for stall delay, including a wide range of different assumptions, were recently analyzed (Breton et al., 2008). The conclusion was
that none of the six models studied correctly represented the flow physics, and that this was ultimately responsible for their lack of generality.
4.1. New physics-based model
Flow Topology of Delayed Stall. There are two main flow regimes pertinent to wind turbine rotors: one is a flow close to rotating disk in axial flow at low velocity, practically at rest, (λ>3, Fig. 2
), and another is a rotating flow over a stationary or slowly rotating disk (Fig. 9). The last can produce delayed stall on the wind turbine blade sections operating at low tip speed ratios (λ≤3).
The key to the stall-delay phenomenon is the rotational flow surrounding the root area of the blade, i.e. the wake rotation. At low tip speed ratios (or high wind speeds), just as a vortex is shed
from each tip blade, a vortex is also shed beginning from the section {VwΩbr=1} of each blade. Then the blade vortices from {VwΩbr≥1} span will each be a line vortex running axially to the center of
the rotor. The direction of rotation of all {VwΩbr≥1} span vortices will be the same, forming a core (or root) vortex of total strength Γv≫Γb=Ωbr2 . The root vortex is primarily responsible for
inducing the rotational 3-D effects on blades. The acquisition of the tangential component of air velocity is compensated for by a fall in the static pressure (suction) in the wake.
There are two common characteristics which most strong potential vortices exhibit, like those produced at the blade root; the first is that the vortex velocity field above the surface boundary layer
is always dominated by the tangential velocity component, while the second shows that the balance of forces between the flow outside the boundary layer and vortex core is governed by a balance
between the centrifugal force and radial pressure gradient (1ρ∂p∂r=V2r) . In this working mode the centrifugal force has a strong stabilizing effect on the boundary layer through the favorable radial
pressure gradient which occurs. In contrast with the pumping - working mode at high tip speed ratios, now the centrifugal forces produce a centerwise suction effect, resulting mainly in boundary
layer stabilizing against the separation. On the other hand, in the presence of a solid surface (blade), this balance of forces is disrupted by the friction retardation of the tangential flow close
to the boundary. Under such a condition, the radial pressure gradient proceeds to drive the retarded boundary flow along the surface towards the center, resulting in large radial velocities residing
close to the surface. Above the boundary layer where the cyclostrophic balance is still intact, the radial velocity inevitably falls to zero. This gives rise to an inflexion point in the radial
velocity profile. At some location close to the axis, this inward flowing air moves away from the solid boundary and effuses into a vortex core. The radial pressure gradient has a very strong
stabilizing effect on the boundary layer and acts to revert it to its reattaching, and even more to its laminar state upstream of the effusing core.
Modeling of Stall-Delay. The acquisition of the enhanced rotational flow (tangential component of velocity) by the increase in the kinetic energy of wind, associated with a strong suction of the air
at the hub is governed by conservation of angular momentum, like the potential vortex, and conservation of energy:
Then, as the blade section moves through the air a circulation Γairfoil develops around it. In order to comply with Kelvin’s theorem,
a starting vortex Γwake must exist such that the total circulation around a line that surrounds both the airfoil and the wake remains unchanged. Since the circulation of the sectional airfoil, where
the phenomenon of stall was initiated at λ=3.0 , is practically the blade circulation due to rotation, Γairfoil=Γb=Vw2/Ωb=ΩbR2/λ2 (at r/R=λ−1 the sectional airfoil is stalled for λ=3.0 ), then
Γwake,0=−Γb(r=R/3.0) .
The wake’s circulation, which is constant up to the hub induces a suction pressure along the upper side of blade (Fig. 16) stabilizing the boundary layer against the separation at post-stall angles
of attack (angles of 30 degrees exist currently at the blade root), followed by forming of a leading-edge separation bubble. The bubble at the innermost sectional airfoil has almost no effect on
integrated loads, because it is never more than a few percent of the chord in length, and the airfoil acts as in an ideal fluid flow.
The phenomenon of stall-delay can be described as a three step process: rise of strong wake by clustering of vortices shed from the inboard blade span at λ=3.0 , inviscid (pressure) load
redistribution along the airfoil chord at constant circulation or CL,INV of the start regime ( λ=3.0 ) for increased suction pressure at λ3.0 , followed by spanwise circulation or lift decay
involving the stretching of separation bubble surface all the way to the trailing edge for increasing radius; beyond that blade section ( r/R=λ−1 ) the flow is separated over the whole airfoil and
the leading-edge stall occurs.
The specific mechanism for circulation (or lift) decay is not presently known. However, a measure of the degree of radial instability of the boundary layer and implicitly the lift decay can be
obtained from the sucked volume flow rate (Fig. 13) into the boundary layer beneath the Rankine vortex-like wake. Thus, the maximum sucked flow rate would correspond to the minimum volume of the
separation bubble and CL,INV . Consequently C [ L ] decay can be supposed to follow the flow rate decrease, dCLdr∼dQdr , where the zero flow rate indicates the full stall occurring. More accurately,
the curves of the volume flow rate show the state of
boundary layer, namely larger values of the flow rate signify smaller bubble volumes and thinner boundary layers for reducing tip speed ratios. Further, it can be easily found that a thicker boundary
layer with distributed vorticity ( λ=3.0 ) is more stable, while a thinner boundary layer with more concentrated vorticity ( λ→1.0 ) is less stable. Since the lift decay law depending on the
stability of the boundary layer, is different from that suggested by Fig. 13, tending to be slower for higher λ and steeper for lower λ, an average decay corresponding to λ=2.0 is assumed. Therefore,
the decay is assumed to be terminated for all regimes at midspan, i.e. the end of the flow regime for λ=2.0 . This conclusion is consistent with the result found in (Dumitrescu & Cardoş, 2010), which
indicates the full separation at r/c=4 .
Point Flow Structure Forces
1 The attached 3-D boundary layer with leading-edge separation Exceed 2-D CLmax, extrapolate linear regime
2 The separation bubble is sucked and its volume reduces Maximum lift, CL,INV atλ=3.0, followed by suction pressure redistribution and movement of pressure center to midchord; no
correction to drag
3 Thick skewed 3-D boundary layer with stretching bubble Gradual decay of lift forrr0; no correction to drag
4 The bubble breaks away from the trailing edge and forms a free Readjust to stall regime
shear layer
The various stages of the delayed stall process are summarized schematically in Fig. 17. and Table 2. Stage 1 represents the delay in the onset of separation in response to reduction in adverse
pressure gradients produced by the influence of the strong vortex wake at the root. The phenomenon is initiated at λ=3.0 , where the suction pressure equalizes to the kinetic energy of wake (Fig. 16)
resulting mainly in boundary layer reattaching and the rise of a leading edge separation bubble, a stable focus as topological entity.
Stage 2 of the delayed stall process involves the suction pressure redistribution of the start pressure distribution with the conservation condition of its integrated load and the simultaneous move
of pressure center to the midchord; as the suction/tip speed ratio increases/decreases the volume of the leading-edge separation bubble is reduced. The start pressure distribution is known from the
inviscid aerodynamic loading solution over the chord; the pressure redistribution continues up to no suction peak exists at the leading-edge and even more no suction surface pressure gradient exists.
On the other hand, the pressure redistribution is associated with a vorticity concentration process, which influences the stability of boundary layer.
Stage 3 is the sectional circulation (lift) decay which occurs at greater radii than the start location, where the separation bubble surface stretches all the way to the trailing edge. Beyond that
zero flow rate location the flow is separated over the whole airfoil so that full stall occurs. By reason of the boundary layer stability it is inferred that the stall-delay effects are present up to
a 50% of span for all the regimes.
Stage 4 is the relaxation to stall regime, which begins just as the integrated normal load of 2-D stall, C[NS], is reached (end of delayed stall) and continues as far as the recovery of the specific
pressure distribution (suction peak at the leading edge) is achieved.
4.2. Application
The combined experimental rotor (Schepers et al. 1997) of the Renewable Energy Laboratory (NREL) is chosen to illustrate the stall-delay model proposed for a wind turbine blade. The rotor uses the
NREL S 809 airfoil and a simpler BEM method. In this example, the one parameter pressure distribution on the upper surface of blade at the root with |Cpm|=11(α=22deg) is imposed and the hub is at
approximately 0.20R. The curves plotted in Fig. 19 are calculated with these conditions: the onset of stall-delay at λ=3.0 , r0/R=0.268 , CL,INV(λ=3)=2.35 and a linear approximate distribution of
normal load decay (Fig. 18).
The comparison of predicted and available measured C [ N ], C [ T ] and P are shown in Fig. 19 (a, b, c), for wind speeds 13 m/s, 15 m/s and 25 m/s. Excepting predicted C [ T ] at the blade root, the
agreement between predictions and measurements is reasonably good. Most likely, at the root area of the blade there are computational and experimental uncertainness, and better more exist
uncertainness for predictions. However, the present stall-delay model is the most comprehensive one able to capture much of the actual physics of flow.
5. Chapter review
The phenomenon of stall-delay has been shown to be an important consideration in wind turbine design because, in the presence of turbulence, it ultimately generates the loads with the highest
peak-to-peak fatigue cycles for both blade and rotor shaft bending (Tangler, 2004). It has been shown that stall-delay is characterized by a favorable delay in onset of flow separation to higher
angles of attack for the inboard regions of the blades. This is followed by the less favorable event of leading-edge separation bubble stretching. As long as the bubble surface stays over airfoil, it
acts to decay the lift gradually. Then, the leading-edge separation occurs. The stall-delay begins at the hub and crosses varied partially separated flow regions over the inboard half-rotor disk,
depending on the wind speed. Therefore, the consideration of stall-delay phenomenon represents a necessary refinement in the rotor design process which will more accurately define the performance and
power control at high wind speeds.
While the prediction of the conditions for stall-delay onset and their subsequent effects clearly forms an essential part of any rotor design process, it has been shown that this is a problem not yet
fully understood, nor easily predicted. For engineering analyses, the modeling of stall-delay still remains a particularly challenging problem. This is mainly because of the need to balance physical
accuracy with computational efficiency and/or the need to formulate a model of stall-delay in a particular mathematical form. To this end, a number of semi-empirical models have been developed for
use in wind turbine design work. A discussion of these semi-empirical methods has been recently presented, along with the conclusion that none of the six most known models correctly represented the
physics of flow (Breton et al., 2008). Generally, predictions are good when measurements are available for validation or empirical refinement of the model, but their capabilities for general blade
parameters (twist, taper and airfoil shape) are less certain.
A conceptualization of the complex 3-D flow field on a rotor blade, where stall begins, and how it progress, has been proposed, along with a reasonably simple model that complements the 2-D airfoil
characteristics used to predict rotor performance. This concept considers that the 3-D flow field near the hub behaves like the rotational flow over a stationary disk, inducing a strong suction
pressure towards the center. Here, the centrifugal force has a strong stabilizing effect on the boundary layer, in which the rotor is acting as a vacuum pump on the separated volume of bubble. The
rough model has shown encouraging results, but some uncertainties remain in the prediction of the sectional circulation decay and also in the proper validation of predictions with more measured and
CFD computed airloads on the rotor. The more correct physical description provided by this present model should give more coherence and consistence of future experimental and computational studies on
the stall-delay phenomenon. | {"url":"http://www.intechopen.com/books/fundamental-and-advanced-topics-in-wind-power/inboard-stall-delay-due-to-rotation","timestamp":"2014-04-16T17:17:40Z","content_type":null,"content_length":"252092","record_id":"<urn:uuid:edec5915-7d72-431f-a3d8-73271deef51d>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00485-ip-10-147-4-33.ec2.internal.warc.gz"} |
simplify expression
May 16th 2008, 02:00 AM
Mr Rayon
simplify expression
According to my math teacher, when we factorise -15w+20, we always have to get -5(3w-4) and not the other possible alternative, e.g. 5(-3w+4).
Just out of curiosity, why is this so?
EDIT: Putting this into consideration, if I were to factorise -2t^2-8t, would t(-2-8) be correct?
May 16th 2008, 02:43 AM
According to my math teacher, when we factorise -15w+20, we always have to get -5(3w-4) and not the other possible alternative, e.g. 5(-3w+4).
Just out of curiosity, why is this so?
EDIT: Putting this into consideration, if I were to factorise -2t^2-8t, would t(-2-8) be correct?
I think this is only true because he considers that "simplest form" for the sake of your grade I'd go along with it, but in reality, you can factor it however is most useful to you. Both of the
examples you showed are correct, but in the first one that your instructor has endorsed, you factored out a negative one and a 5. In the second you only factored a 5. I assume the reason your
instructor wanted you to do it this way is because he wanted the term with the variable to be positive, but I don't know for sure.
For example, you could factor it in any of these ways (and infinitely more):
$-15w+20 = -\left(15w - 20\right)$
$-15w+20 = 15\left(-w + \frac 43\right)$
$-15w+20 = -20\left(\frac{3w}{4} -1\right)$
$-15w+20 = w\left(-15 + \frac{20}w\right)$
$-15w+20 = -w^2 \left(\frac{15}w -\frac{20}{w^2}\right)$
and so on.
May 16th 2008, 03:28 AM
It is not so, they are equally acceptable though the second could be rewitten 5(4-3w). Your maths teacher may be using some particular convention that HS maths teachers have invented but it is
not universal.
EDIT: Putting this into consideration, if I were to factorise -2t^2-8t, would t(-2-8) be correct?
(but again this may not meet with your teachers approval). | {"url":"http://mathhelpforum.com/algebra/38488-simplify-expression-print.html","timestamp":"2014-04-19T17:26:34Z","content_type":null,"content_length":"7624","record_id":"<urn:uuid:be731b0e-2eef-404a-9037-a8072fb0f6ee>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00007-ip-10-147-4-33.ec2.internal.warc.gz"} |
IntroductionRelated WorkSystem EnvironmentBeacon Point SelectionLocation EstimationLoMoB Localization SchemeSystem EnvironmentBeacon Point SelectionLocation EstimationObtaining potential locationsEstimating the Location of a Sensor Node Using the Weighted MeanPerformance EvaluationsSimulation SetupMetricsSimulation ResultsConclusionsReferencesFigures
For a long time, there has been significant interest in monitoring underwater environments to collect oceanographic data and to explore underwater resources. These harsh underwater environments have
limited human access and most of them remain poorly understood. Recent advances in hardware and network technology had enabled sensor networks capable of sensing, data processing, and communication.
A collection of sensor nodes, which have a limited sensing region, processing power, and energy, can be randomly deployed and connected to form a network in order to monitor a wide area. Sensor
networks are particularly promising for use in underwater environments where access is difficult [1–3]. In underwater acoustic sensor networks (UASN), localization of sensor nodes is essential
because it is difficult to accurately deploy sensor nodes to predetermined locations. In addition, all the information collected by sensor nodes may be useless unless the location of each sensor node
is known [4].
To localize unknown nodes, static beacons [5–11] or a mobile beacon [12–16] can be used; typically, buoys are used for static beacons and an autonomous underwater vehicle (AUV) for a mobile beacon.
The use of a mobile beacon has a similar effect to the use of many static beacons. For this reason, localization using a mobile beacon is inherently more accurate and cost-effective than localization
using static beacons. Further, the vehicle can be conveniently maneuvered on the sea surface or underwater, based on a planned or random path, as shown in Figure 1.
Localization schemes can also be classified as range-based or range-free schemes. Range-based schemes use the distance and/or angle information measured by the Time of Arrival (ToA), Time Difference
of Arrival (TDoA), Angle of Arrival (AoA), or Received Signal Strength Indicator (RSSI) techniques. In underwater environments, range-based schemes that use the ToA and TDoA techniques have been
proposed [5–10,13–15]. Time synchronization between sensor nodes is typically required for the ToA and TDoA techniques [6]. However, precise time synchronization is challenging in underwater
environments [17]. In addition, even though the ToA and TDoA techniques are used without time synchronization, the techniques are based on the speed of sound and this varies with temperature,
pressure, salinity, and depth. In contrast to range-based schemes, range-free schemes localize a sensor node by deducing distance information instead of measuring distances [11,12,16]. Because
range-free schemes do not require additional devices in order to measure distances and do not suffer from distance measurement errors, range-free schemes are a promising approach for underwater
sensor networks [16].
Localization is one of the challenges associated with underwater sensor networks and many studies have focused on localization in recent years. Among these studies, LDB is a range-free localization
scheme with a mobile beacon [16]. In LDB, sensor nodes can localize themselves through passively listening to beacons sent by an AUV that has an acoustic directional transceiver. Because sensor nodes
just receive the beacons from the AUV without inter-node communications, LDB is energy-efficient because it reduces the energy consumption due to transmitting; the power consumed by transmitting is
often 100 times higher than the power consumed by receiving [18]. In addition, because a sensor node can localize itself using its own received beacons from the AUV independently of the other sensor
nodes, LDB has fine-grained accuracy even in sparse networks. The depth of a sensor node is directly determined by using a cheap pressure sensor. After the first and last received beacons, which are
called beacon points, from the AUV are projected onto the horizontal plane on which a sensor node resides, the 2D location of a sensor node can be determined based on the distance between the two
projected beacon points. At this point, the location of the sensor node remains ambiguous, because the two projected beacon points provide two possible node locations. If the sensor node obtains two
more beacon points, the ambiguity can be resolved. The first two beacon points are used for location computation and the other two beacon points are used for resolving the ambiguity. However, if the
two beacon points for location computation have a large error, the error in the estimated sensor location also increases. Even though the location of a sensor node can be simply computed from the
first two beacon points, the use of just two points out of all the beacon points can make localization very error-prone. To improve the localization accuracy in challenging underwater environments,
we propose a new localization scheme LoMoB that obtains a set of potential locations as candidates for a node's location using the bilateration method and then localizes the sensor node through the
weighted mean of all the potential locations with the weights computed based on residuals.
In this section, we briefly explain LDB in the viewpoint of system environment, beacon point selection, and location estimation [16].
In LDB, an AUV with a directional transceiver moves at a fixed depth in water and broadcasts its location, which called a beacon, and the angle of its transceiver's beam (a) at regular intervals,
called beacon distance (d), as shown in Figure 2(a). When the AUV sends a beacon, the sensor nodes that fall in the conical beam receive the beacon; e.g., in Figure 2(a), the sensor node in red can
receive the beacon B[2] and the sensor node in blue can receive the beacons B[1], B[2], and B[3]. The conical beam forms circles with different radii according to the depth of the sensor nodes, as
shown in Figure 2(a). When the z coordinates of the AUV and a sensor node are z[A] and z[s] and the angle of the transceiver's beam is a, the radius of the circle formed by the beam for the sensor
node is given as follows: r b = tan ( a / 2 ) × | z A − z s |
From the viewpoint of a sensor node located at (x, y, z[s]), when a beacon is within the circle centered at (x, y, z[A]) with a radius r[b], the sensor node can receive the beacon; e.g., in Figure 2
(b), the sensor node receives the beacons B[1] to B[6]. Among the series of received beacons, the first beacon is defined as the first-heard beacon point and the last beacon is defined as the
last-heard beacon point. The location of a sensor node is estimated using only the beacon points, and not all the received beacons.
Because the sensor depth is known from the pressure sensor, the beacon points can be projected onto the horizontal plane in which the sensor node resides, as shown in Figure 2(b). After projection,
3D localization is transformed into a 2D localization problem.
We describe location estimation based on the projected beacon points. As shown in Figure 3(a), the projected first-heard beacon point is denoted by F and the projected last-heard beacon point is
denoted by L. The projected point of the beacon received just before the first-heard beacon point is defined as the projected prior-heard beacon point F′ (e.g., in Figure 2(b), the projected point of
B[0]), and the projected point of the beacon after the last-heard beacon point is defined as the projected post-heard beacon point L′ (e.g., in Figure 2(b), the projected point of B[7]).
Four circles are then drawn with radius r[b] centered at the four points F′, F, L, and L′. Because the sensor node should be located outside the circles centered at F′ and L′ and inside the circles
centered at F and L, the intersection areas ABCD and A′B′C′D′ represent possible locations of the sensor node. One of the two points, S̃ and S̃′, within the two areas will be estimated as the location
of the sensor node. Here, S̃ is the midpoint of B and D and S̃′ is the midpoint of B′ and D′. When just one projected first-heard beacon point and one projected last-heard beacon point are used, two
possible node locations are found. The choice between the two points S̃ and S̃′ can be made after an additional two beacon points are obtained, as shown in Figure 3(b). In LDB, the first two beacon
points are used to compute two possible points for the location of a sensor node based on geometric constraints and the next two beacon points are used for the choice between the two possible points.
In this section, we explain the system environment, beacon point selection, and location estimation method for LoMoB. Because LoMoB can be applied to systems that use either a directional transceiver
or an omnidirectional transceiver, we explain LoMoB for both systems. In addition, because LoMoB improves LDB, we explain LoMoB by comparing and contrasting it with LDB.
LoMoB considers a system that uses an omnidirectional transceiver in addition to a system that uses a directional transceiver [19]. Because the system environment using a directional transceiver was
explained in Section 2, we explain here the system environment using an omnidirectional transceiver.
A mobile beacon moves on the sea surface or underwater. When a mobile beacon has an omnidirectional transceiver, 3D movement of the mobile beacon is possible whereas a mobile beacon that has a
directional transceiver is restricted to 2D movement at a fixed depth. The mobile beacon is assumed to know its own location and to broadcast a beacon containing its location information at regular
distance intervals, which are called the beacon distance d. When the mobile beacon transmits a beacon, the sensor nodes that are located within the communication range of the mobile beacon receive
the beacon; e.g., in Figure 4(a), the sensor node in red can receive the beacon B[2] and the sensor node in blue can receive the beacons B[1] and B[2]. Here, the communication range r is assumed to
be constant in 3D [11,12,20]. In addition, because movement of a mobile beacon in a straight line is more controllable than curved movement, the mobile beacon is assumed to follow the random waypoint
(RWP) model [21]; the mobile beacon moves in a series of straight paths to random destinations.
From the viewpoint of a sensor node, the sensor node can receive a beacon when the beacon is within the communication range of the sensor node; e.g., in Figure 4(b), the sensor node receives the
beacons B[1] to B[6]. The selection of beacon points from the received beacons is performed as in LDB. When a sensor node receives the first beacon from a mobile beacon, the first beacon is selected
as a beacon point; i.e., B[1] in Figure 4(b). If the sensor node receives no further beacons during a predefined time after receiving its last beacon, the last beacon is selected as a beacon point;
i.e., B[6] in Figure 4(b). The above process is repeated each time the mobile beacon passes through the communication sphere of the sensor node.
As shown in Figure 4(b), the beacon points do not lie exactly on the communication sphere of the sensor node, because the sensor node receives a beacon at every beacon distance d. When the
communication range of the sensor node is r, the beacon points are located between the distances r − d and r from the sensor node. Based on the distance range between a beacon point and a sensor
node, the middle value of the range, r − d/2, is estimated as the distance between the beacon point and sensor node in order to minimize the error in the estimated distance.
Sensor nodes are assumed to have a pressure sensor, and therefore they are assumed to know their depth [22]. With this depth information, the beacon points can be projected onto the horizontal plane
in which the sensor node resides, as shown in Figure 4(b). When the z coordinates of a sensor node and the i^th beacon point are z[s] and z[i], the distance r i ′ between the sensor node and the i^th
projected beacon point is ( r − d / 2 ) 2 − ( z i − z s ) 2.
For a system using a directional transceiver, all the projected beacon points are located between the distances r[b] − d and r[b] from a sensor node; here, r[b] is the radius of the circle formed by
the beam for the sensor node. Subsequently, all the estimated distances between the sensor node and projected beacon points are identically r[b] − d/2.
In this subsection, we explain how to estimate the location of a sensor node based on the distances between the sensor node and the projected beacon points. First, potential locations are obtained as
candidates for the location of the sensor node. Second, the location of the sensor node is estimated using the weighted mean of the potential locations.
After estimating the distances between a sensor node and projected beacon points, the following equations need to be solved for localization: { ( x − x 1 ′ ) 2 + ( y − y 1 ′ ) 2 = r 1 ′ ( x − x 2 ′ )
2 + ( y − y 2 ′ ) 2 = r 2 ′ ⋮ ( x − x N ′ ) 2 + ( y − y N ′ ) 2 = r N ′ ⇔ { ( x − x 1 ′ ) 2 + ( y − y 1 ′ ) 2 = r 1 ′ 2 ( x − x 2 ′ ) 2 + ( y − y 2 ′ ) 2 = r 2 ′ 2 ⋮ ( x − x N ′ ) 2 + ( y − y N ′ ) 2
= r N ′ 2where (x, y) is the location of a sensor node, ( x i ′ , y i ′ ) is the location of the i^th projected beacon point, N is the number of the projected beacon points, and r i ′ is the
estimated distance between a sensor node and the i^th projected beacon point.
Let us explain how to estimate the location of a sensor node based on the bilateration method with Figures 5 and 6. When N projected beacon points are obtained, N circles can be drawn satisfying
Equation (2); i.e., four circles with centers B P 1 ′, B P 2 ′, B P 3 ′, and B P 4 ′ and radii r 1 ′, r 2 ′, r 3 ′, and r 4 ′ can be drawn, as shown in Figure 5. The points of intersection of the N
circles that are near the sensor node are obtained as potential locations for the sensor node's estimated location, as shown in Figure 5; some or all of the potential locations may overlap. However,
because a potential location can be either of the two intersection points of two circles, a decision process is needed; here, because the intersection points are obtained using just two of many
projected beacon points, this method is called bilateration [23]. One of the two intersection points of the two circles with centers at two projected beacon points is closer to the circles with
centers at the other projected beacon points than the other intersection point; i.e., C[12](1) is closer to the two circles with centers B P 3 ′ and B P 4 ′ than C[12](2) in Figure 6. When C[jk](1)
and C[jk](2) are the intersection points of the circles with centers B P j ′ and B P k ′ and the following inequality is satisfied, C[jk](1) is selected as the potential location P̂[jk]; otherwise, C
[jk](2) is selected.
∑ i = 1 , i ≠ j , k N | ‖ B P i ′ − C j k ( 1 ) ‖ − r i ′ | < ∑ i = 1 , i ≠ j , k N | ‖ B P i ′ − C j k ( 2 ) ‖ − r i ′ |
In rare cases, the number of intersection points between two circles can be one or zero. If there is one intersection point, this single point becomes the potential location. If there is no
intersection point, which occurs when the distance between the two beacon points B P j ′ and B P k ′ is larger than r j ′ + r k ′, the point that divides the line segment B P j ′ B P k ′ joining the
points B P j ′ and B P k ′ into a r j ′ : r k ′ ratio becomes the potential location. In these two cases, a decision process is not needed.
The location of a sensor node can be estimated using all the potential locations. The location of a sensor node can simply be estimated as the mean of all the potential locations. However, to improve
the estimation accuracy, a weighted mean of the potential locations can be used instead. Here, the weight is determined based on a residual, which we define here. Given x̂ as a solution of x at f(x) =
b, the residual is b − f(x̂), which indicates how far f(x̂) is from the correct value of b, and the error is x̂ − x. As we do not know x, we cannot compute the error, but we can compute the residual.
From Equation (2) N residuals associated with the potential location P[jk](x̂[jk], ŷ[jk]) can be computed as follows: { ɛ ^ j k ( 1 ) = | ( x ^ j k − x 1 ′ ) 2 + ( y ^ j k − y 1 ′ ) 2 − r 1 ′ | ɛ ^ j
k ( 2 ) = | ( x ^ j k − x 2 ′ ) 2 + ( y ^ j k − y 2 ′ ) 2 − r 2 ′ | ⋮ ɛ ^ j k ( N ) = | ( x ^ j k − x N ′ ) 2 + ( y ^ j k − y N ′ ) 2 − r N ′ |Because the accuracy of a potential location is expected
to be inversely proportional to the sum of the residuals, the reciprocal of the sum of the residuals represents a weighting factor. The weight for the potential location P[jk](x̂[jk], ŷ[jk]) is
defined as follows: w j k = 1 ∑ i = 1 N ɛ ^ j k ( i )Finally, the location of a sensor node is estimated as the weighted mean of all the potential locations.
( x ^ , y ^ ) = ∑ j = 1 N − 1 ∑ k = j + 1 N w j k ( x ^ j k , y ^ j k ) ∑ j = 1 N − 1 ∑ k = j + 1 N w j k
Less than three beacon points may be acquired for some sensor nodes in certain circumstances. In this case, because the location of a sensor node cannot be estimated through Equation (6), another
approach is needed. When just two beacon points B P 1 ′ and B P 2 ′ are acquired for a sensor node, the intersection points C[12](1) and C[12](2) of two circles with centers B P 1 ′ and B P 2 ′ and
radii r 1 ′ and r 2 ′ are obtained. Because there are no other beacon points available to the decision-making process, the neighboring sensor nodes can play the role of an additional beacon point in
order to estimate the sensor location. The sensor node can receive the location information of neighboring sensor nodes by sending a request to the neighboring sensor nodes that know their location
after localization. The sensor node just knows that the neighboring sensor nodes are within the communication range without the distance information between the sensor node and the neighboring sensor
nodes. If one of the intersection points is close to the sensor node, more sensor nodes among neighboring sensor nodes of the sensor node are expected to be within the communication range of the
intersection point, compared to the other intersection point. By comparing the number of neighboring sensor nodes of the sensor node within the communication range for each intersection point, one of
the two intersection points that has more neighboring sensor nodes of the sensor node is estimated as the location of the sensor node. When a sensor node has less than two beacon points, the
estimated locations of the neighboring sensor nodes can be used. In this case, the location of the sensor node is estimated as the midpoint of all the neighboring sensor nodes.
It is noteworthy that the main feature of LoMoB that can improve LDB is the use of the weighted mean of all the potential locations. In LDB, when a sensor node has four projected beacon points F[1],
L[1], F[2], and L[2], as shown in Figure 3(b), two possible points are obtained based on F[1] and L[1] and one point between the two is estimated as the location of the sensor node based on F[2] and
L[2]. Here, the error in the estimated location depends largely on the distance between F[1] and L[1] [16]. If F[1] and L[1] are located very close to each other, the location error is expected to be
large. In this case, a better choice may be that F[2] and L[2] (or another two projected beacon points) are used for the two possible points, rather than F[1] and L[1]. However, the selection of two
projected beacon points for the possible points has not yet been studied. If the errors of the beacon points F[1] and L[1] increase because of some factors such as irregularities in the radius of the
circle formed by the transceiver's beam and the location error of a mobile beacon, the errors of the two possible points also increase, which results in an increase of the location error. Even though
more beacon points, in addition to F[1], L[1], F[2], and L[2], are obtained, the localization accuracy is not improved because the additional beacon points are not used for localization. As in LDB,
LoMoB obtains two intersection points based on two beacon points and determines one point between the two as a potential location for the location of the sensor node based on the other beacon points.
However, compared to LDB, LoMoB estimates the location of a sensor node based on all the potential locations, not just one potential location. In addition, potential locations with a high weight
contribute more to the estimation of the location of the sensor node than other potential locations. For these reasons, LoMoB is expected to improve LDB in challenging underwater environments even
though LoMoB uses just four beacon points, as does LDB. If more than four beacon points are used, the localization accuracy is expected to improve further.
In this section, we first describe the simulation parameters, then introduce the metric to evaluate the localization accuracy and, finally, we compare the localization accuracy of LoMoB with that of
LDB through simulations.
The sensing space for UASN is a rectangular parallelepiped of 1 km × 1 km × 100 m, in which 100 sensor nodes are randomly deployed. A mobile beacon is assumed to move linearly at a velocity of 1 m/s
and broadcasts a beacon at every beacon interval, i.e., each 1 s. To fairly compare LoMoB and LDB, we assume that a mobile beacon has a directional transceiver and just four beacon points are used
for localization because LDB works for a system using a directional transceiver and uses four beacon points. The angle of the directional transceiver's beam is 60°. Because it is more convenient to
move a mobile beacon on the sea surface than underwater in real underwater environments, the mobile beacon is assumed to move on the sea surface.
In LoMoB and LDB, localization begins by selecting beacon points. Any projected beacon points that are not exactly on the circle formed by the transceiver's beam for a sensor node cause an estimation
error of the distance between the projected beacon point and the sensor node, which results in an error in the estimated location. The beacon distance causes projected beacon points to be located not
exactly on the circle formed by the beam. In addition, the phenomena such as reflection, diffraction, and refraction in underwater environments cause irregularities in the radius of the circle formed
by the beam. To improve the reality of our simulations, irregularities in the radius of the circle are modeled as r̂ = r + e[r], where e[r] is a random value with the Gaussian distribution N (0, σ r
2). Because the GPS error in a mobile beacon, water fluctuations, and tidal currents can cause errors in the location of the mobile beacon, we assume the location information broadcasted from a
mobile beacon is error prone [24]. The location error of a mobile beacon is modeled as x̂[m] = x[m] + e[m], where x̂[m] = [x̂, ŷ]^T is the measured location of the mobile beacon located at x[m] = [x, y]
^T and e[m] = [e[x], e[y]]^T is a random vector whose e[x] and e[y] are independent and identical random values with the Gaussian distribution N(0, σ m 2). We use Matlab to perform these simulations.
To compare the localization accuracy of LoMoB and LDB, we define the average location error as follows: e average = 1 N ∑ k = 1 N ‖ x ^ k − x k ‖where x̂[k] = (x̂[k], ŷ[k]) is the estimated location of
a sensor node that is located at x[k] = (x[k], y[k]) and N is the number of sensor nodes.
Because the average location error may be seriously affected by a small number of large location errors, the ratio of localized sensor nodes with location errors below a given threshold can be used
as an additional metric for evaluating the localization accuracy. The ratio of localized sensor nodes below a given threshold allows the localization accuracy to be assessed in greater depth than the
average location error alone.
Because the beacon distance, irregularities in the radius of the circle formed by the transceiver's beam, and the location error of a mobile beacon all influence the accuracy of distance estimation
between a sensor node and a projected beacon point, the localization accuracy is analyzed with respect to these factors.
Figure 7(a) compares the average location error as a function of the beacon distance and Figure 7(b) compares the ratio of localized sensor nodes with location errors below 10 m. The average location
error of LDB is shown to fluctuate more and to be larger than that of LoMoB. As the beacon distance increases, the performance gap between LoMoB and LDB tends to widen. Figure 7 demonstrates that
LoMoB is less vulnerable to the beacon distance than LDB.
In real underwater environments, the radius of the circle formed by the transceiver's beam is expected to fluctuate. Irregularities in the radius of the circle cause sensor nodes to provide erroneous
estimates of distances between the projected beacon points and a sensor node, in addition to the error associated with the beacon distance. Figure 8(a) compares the average location error as a
function of the standard deviation σ[r] in the radius of the circle and Figure 8(b) compares the ratio of localized sensor nodes with location errors below 10 m. As shown in the figures, the average
location error increases as irregularities in the radius of the circle increase. LoMoB has 32.4%–45.8% smaller location errors than LDB. The ratios of localized sensor nodes show a maximum difference
of 7.4%. LoMoB is verified to be more tolerant to irregularities in the radius of the circle formed by the transceiver's beam than LDB.
A mobile beacon is generally assumed to know its own location. In reality, the location information of a mobile beacon may be erroneous. As well as irregularities in the radius of the circle formed
by the transceiver's beam, the location error of a mobile beacon needs to be considered because this also causes erroneous estimation of the distances between the projected beacon points and a sensor
node. Figure 9(a) compares the average location error according to the standard deviation σ[m] in the location error of a mobile beacon and Figure 9(b) compares the ratio of localized sensor nodes
with location errors below 10 m. LoMoB improves the average location error by 21.5%–32.4% compared to LDB. The ratios of localized sensor nodes show a maximum difference of 4.3%. In addition, Figure
10 shows the average location error of LDB and LoMoB applying a simple mean and the weighted mean; if all the weights are equal, then the weighted mean is the same as the simple mean. LoMoB applying
a simple mean and the weighted mean improves the average location error by 14.3%–26.9% and 21.5%–32.4%, respectively. This indicates that the use of all the potential locations in LoMoB is the main
factor to improve LDB and the use of the weights results in additional increase in the localization accuracy.
The localization accuracy performance of LoMoB and LDB was compared through the average location error and the ratio of localized sensor nodes with location errors below 10 m according to the beacon
distance, irregularities in the radius of the circle formed by the transceiver's beam, and the location error of a mobile beacon. The simulation results verify that LoMoB is more tolerant to
estimation errors of the distances between the projected beacon points and a sensor node. Subsequently, LoMoB is more promising than LDB in harsh underwater environments. | {"url":"http://www.mdpi.com/1424-8220/12/5/5486/xml","timestamp":"2014-04-21T16:12:24Z","content_type":null,"content_length":"84440","record_id":"<urn:uuid:5110fec1-feab-4298-8325-6ce35e1a250a>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00506-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Construction of Bézier points of quadrilateral forms those of triangles, Computer-Aided Design 12,1(1980
, 1993
"... In view of the fundamental role that functional composition plays in mathematics, it is not surprising that a variety of problems in geometric modeling can be viewed as instances of the
following composition problem: given representations for two functions F and G, compute a representation of the fu ..."
Cited by 20 (1 self)
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In view of the fundamental role that functional composition plays in mathematics, it is not surprising that a variety of problems in geometric modeling can be viewed as instances of the following
composition problem: given representations for two functions F and G, compute a representation of the function H = F ffi G: We examine this problem in detail for the case when F and G are given in
either B'ezier or B-spline form. Blossoming techniques are used to gain theoretical insight into the structure of the solution which is then used to develop efficient, tightly codable algorithms.
From a practical point of view, if the composition algorithms are implemented as library routines, a number of geometric modeling problems can be solved with a small amount of additional software.
This paper was published in TOG, April 1993, pg 113-135 Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling - curve, surface, and object
representations; J.6 [...
- CURVES AND SURFACES , 1997
"... CAD systems are usually based on a tensor product representation of free form surfaces. In this case, trimmed patches are used for modeling non rectangular zones. Trimmed patches provide a
reasonable solution for the representation of general topologies, provided that the gap between equivalent trim ..."
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CAD systems are usually based on a tensor product representation of free form surfaces. In this case, trimmed patches are used for modeling non rectangular zones. Trimmed patches provide a reasonable
solution for the representation of general topologies, provided that the gap between equivalent trimming curves in the euclidean space is small enough. Several commercial CAD systems, however,
represent certain non rectangular surface regions through degenerate rectangular patches. Degenerate patches produce rendering artifacts and can lead to malfunctions in the subsequent geometric
operations. In the present paper, two algorithms for converting degenerate tensor-product patches into triangular and trimmed rectangular patches are presented. The algorithms are based on specific
degree reduction algorithms for B'ezier curves. In both algorithms, the final surface approximates the initial one in a quadratic sense while inheriting its boundary curves. In the second one, " \
Gamma G 1 cont...
, 2000
"... This paper presents an explicit formula that converts a triangular Bézier patch of degree n to a degenerate rectangular Bézier patch of degree n × n by reparametrization. Based on this formula,
we develop a method for approximating a degenerate rectangular Bézier patch by three nondegenerate Bézier ..."
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This paper presents an explicit formula that converts a triangular Bézier patch of degree n to a degenerate rectangular Bézier patch of degree n × n by reparametrization. Based on this formula, we
develop a method for approximating a degenerate rectangular Bézier patch by three nondegenerate Bézier patches; more patches can be introduced by subdivision to meet a user-specified error
, 2011
"... in co-operation with ..." | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=2846270","timestamp":"2014-04-24T06:45:34Z","content_type":null,"content_length":"19848","record_id":"<urn:uuid:6554efd1-6828-4b77-a1d8-351cd4294d10>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00596-ip-10-147-4-33.ec2.internal.warc.gz"} |
question on notation for partial derivatives
October 27th 2009, 04:28 PM #1
Aug 2008
question on notation for partial derivatives
what does it mean when a partial derivative has a prime on it? usually i just see partial derivatives as f and a subscript x but i recently came across f with a subscript x and a prime on the
top. just curious, what does that mean?
One possibility: f is a function of position and time (so it is a physics problem) and they are using the ' to distinguish differentiation with respect to time, t, while using the partial
derivative to indicate differentiation with respect to a space variable.
Another possibility: there are (at least) two different "f" functions, one called f and the other called f' and the prime is used only to distinguish functions.
Which, if either, of those is correct would depend on the context.
October 27th 2009, 04:45 PM #2
Super Member
Jun 2009
United States
October 27th 2009, 09:37 PM #3
Aug 2008
October 28th 2009, 08:22 AM #4
MHF Contributor
Apr 2005 | {"url":"http://mathhelpforum.com/calculus/110897-question-notation-partial-derivatives.html","timestamp":"2014-04-16T05:32:32Z","content_type":null,"content_length":"39118","record_id":"<urn:uuid:967b4b74-79a1-46e2-9635-ae6bd7e80044>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00580-ip-10-147-4-33.ec2.internal.warc.gz"} |
Elements of a Hypothesis Test - Boundless Open Textbook
A statistical hypothesis test is a method of making decisions using data from a scientific study. In statistics, a result is called statistically significant if it has been predicted as unlikely to
have occurred by chance alone, according to a pre-determined threshold probability—the significance level. Statistical hypothesis testing is sometimes called confirmatory data analysis, in contrast
to exploratory data analysis, which may not have pre-specified hypotheses. Statistical hypothesis testing is a key technique of frequentist inference.
Statistical hypothesis tests define a procedure that controls (fixes) the probability of incorrectly deciding that a default position (null hypothesis) is incorrect based on how likely it would be
for a set of observations to occur if the null hypothesis were true. Note that this probability of making an incorrect decision is not the probability that the null hypothesis is true, nor whether
any specific alternative hypothesis is true. This contrasts with other possible techniques of decision theory in which the null and alternative hypothesis are treated on a more equal basis.
The Testing Process
The typical line of reasoning in a hypothesis test is as follows:
1. There is an initial research hypothesis of which the truth is unknown.
2. The first step is to state the relevant null and alternative hypotheses. This is important as mis-stating the hypotheses will muddy the rest of the process.
3. The second step is to consider the statistical assumptions being made about the sample in doing the test—for example, assumptions about the statistical independence or about the form of the
distributions of the observations. This is important because invalid assumptions will mean that the results of the test are invalid.
4. Decide which test is appropriate, and state the relevant test statisticT.
5. Derive the distribution of the test statistic under the null hypothesis from the assumptions.
6. Select a significance level (α), a probability threshold below which the null hypothesis will be rejected. Common values are 5% and 1%.
7. The distribution of the test statistic under the null hypothesis partitions the possible values of T into those for which the null hypothesis is rejected, the so called critical region, and those
for which it is not. The probability of the critical region is α.
8. Compute from the observations the observed value t[obs] of the test statistic T.
9. Decide to either reject the null hypothesis in favor of the alternative or not reject it. The decision rule is to reject the null hypothesis H[0] if the observed value t[obs] is in the critical
region, and to accept or "fail to reject" the hypothesis otherwise.
An alternative process is commonly used:
7. Compute from the observations the observed value t[obs] of the test statistic T.
8. From the statistic calculate a probability of the observation under the null hypothesis (the p-value).
9. Reject the null hypothesis in favor of the alternative or not reject it. The decision rule is to reject the null hypothesis if and only if the p-value is less than the significance level (the
selected probability) threshold.
The two processes are equivalent. The former process was advantageous in the past when only tables of test statistics at common probability thresholds were available. It allowed a decision to be made
without the calculation of a probability. It was adequate for classwork and for operational use, but it was deficient for reporting results. The latter process relied on extensive tables or on
computational support not always available. The calculations are now trivially performed with appropriate software. | {"url":"https://www.boundless.com/statistics/estimation-and-hypothesis-testing/hypothesis-testing-one-sample/elements-of-a-hypothesis-test/","timestamp":"2014-04-20T00:38:49Z","content_type":null,"content_length":"74654","record_id":"<urn:uuid:af71fb09-6452-414a-89d4-40610c73550c>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00528-ip-10-147-4-33.ec2.internal.warc.gz"} |
On multiply connected wandering domains of meromorphic functions
Rippon, P. J. and Stallard, G. M. (2008). On multiply connected wandering domains of meromorphic functions. Journal of the London Mathematical Society, 77(2) pp. 405–423.
We describe conditions under which a multiply connected wandering domain of a transcendental meromorphic function with a finite number of poles must be a Baker wandering domain, and we discuss the
possible eventual connectivity of Fatou components of transcendental meromorphic functions. We also show that if f is meromorphic, U is a bounded component of F(f) and V is the component of F(f) such
that F(U) is contained in V, then f maps each component of the boundary of U onto a component of the boundary of V in C^. We give examples which show that our results are sharp; for example, we prove
that a multiply connected wandering domain can map to a simply connected wandering domain, and vice versa.
Item Type: Journal Article
Copyright Holders: 2008 The London Mathematical Society
ISSN: 1469-7750
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 22450
Depositing User: Philip Rippon
Date Deposited: 28 Jul 2010 15:22
Last Modified: 02 Dec 2010 21:01
URI: http://oro.open.ac.uk/id/eprint/22450
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Display Class Syllabus
Calculus I
CLASS CODE: MATH 112 CREDITS: 4
DIVISION: PHYSICAL SCIENCE & ENGINEERING
DEPARTMENT: MATHEMATICS
GENERAL This course fulfills a General Education - Math requirement.
CATALOG Limits, continuity, derivatives, integrals, transcendental functions and applications. Properties and applications of the above.
DESCRIPTION: The first of a three-course sequence, Calculus I is intended for students in mathematics, computer science, engineering, the physical sciences, economics and related fields. It is
essentially a traditional course, though it incorporates the use of electronic technology as a body of tools for learning and calculation. Additionally, many faculty members
incorporate some of the tools and methods that have grown out of nationwide efforts to reform Calculus education.
TOPICS: One- and two-sided limits, both finite and infinite, and limits at infinity. Continuity and the Intermediate Value Theorem. First and higher derivatives, including their
interpretations and rules for calculating them. Rolle’s Theorem and the Mean Value Theorem. Implicit differentiation and related rates. Analysis of functions through their graphs,
using such tools as the first and second derivative tests. Optimization, including applications to Economics. The differential and local linearization. Area under curves, the Riemann
sum, antiderivatives, definite integrals and the Fundamental Theorem of Calculus. Integration by substitution. Calculus of transcendental functions, including hyperbolic functions. As
may of the following applications of integration as time allows: the area between curves, volume by slices, disks and shells, arc length, the area of a surface of revolution, center
of mass, work and static fluid force.
OBJECTIVES: Students in Calculus I will develop enough of the basic tools of Calculus to solve a wide variety of problems involving functions of one variable. They will also learn how to
construct their own solutions to such problems, and verify their solutions. As they learn to view the world in terms of Calculus, they will prepare themselves for a broad range of
further educational experiences in many disciplines.
REQUIREMENTS: All students must have their own text (currently Larson, Hostetler & Edwards, Calculus, 7th. edition) and a graphing calculator. Homework assignments and exams are required by all
faculty. Students must attend classes and may be required to participate in computer labs, projects or other forms of learning and assessment, as determined by their instructor.
Math 110 and Math 111 or high school or college preparation in Algebra and Trigonometry (or in Precalculus) equivalent to Math 110 and Math 111. For more information the student
PREREQUISITES: should consult with the instructor or an adviser in the Mathematics Department. Students entering Calculus I are expected to know how to use those features of their graphing
calculators that are typically used in precalculus courses.
OTHER: (1) Those who teach Calculus I use graphing calculators to varying degrees. Some only discuss them occasionally while others require their students to be able to do everything in the
course using the calculator, as well as by hand. Some encourage their students to use Scientific Notebook, which includes basic features of the Maple computer algebra system. The
Department of Mathematics and Computer Science operates a computer lab in which Calculus I students may use such software.
(2) Some faculty teach delta-epsilon proofs; others don’t. Some faculty teach Newton’s Method and numerical integration; others don’t.
(3) Calculus I meets five hours per week, for four semester credits. The fifth hour, traditionally a recitation hour, is used nowadays in the same manner as the other four.
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Dynamic Binding
Dynamic binding, binding evaluation contexts, and (delimited) control effects
Following Moreau (HOSC 1998), we call a dynamic variable a variable whose association with a value exists only within the dynamic extent of an expression, that is, while control remains within that
expression. If several associations exist for the same variable at the same time, the latest one takes effect. Such association is called a dynamic binding. The expression dynamic scope is convenient
to refer to the indefinite scope of a variable with a dynamic extent. Examples of dynamic variables include exception handlers, stdin, and hostname (for mobile code). Given this abstract
specification, dynamic binding can be implemented in many ways: by adding an environment argument to each function (so-called environment-passing style), using global mutable cells (shallow binding),
and so on.
Dynamic variables are also called `fluid variables' or `parameters'. For history, terminology, and thorough motivation of dynamic binding, please see
Luc Moreau: A Syntactic Theory of Dynamic Binding. Higher-Order and Symbolic Computation, 11, 233-279 (1998).
This is joint work with Chung-chieh Shan and Amr Sabry.
Delimited Dynamic Binding
[The Abstract of the paper]
Dynamic binding and delimited control are useful together in many settings, including Web applications, database cursors, and mobile code. We examine this pair of language features to show that
the semantics of their interaction is ill-defined yet not expressive enough for these uses.
We solve this open and subtle problem. We formalize a typed language DB+DC that combines a calculus DB of dynamic binding and a calculus DC of delimited control. We argue from theoretical and
practical points of view that its semantics should be based on delimited dynamic binding:
□ capturing a delimited continuation closes over part of the dynamic environment, rather than all or none of it;
□ reinstating the captured continuation supplements the dynamic environment, rather than replacing or inheriting it.
We introduce a type- and reduction-preserving translation from DB+DC to DC, which proves that delimited control macro-expresses dynamic binding. We use this translation to implement DB+DC in
Scheme, OCaml, and Haskell.
We extend DB+DC with mutable dynamic variables and a facility to obtain not only the latest binding of a dynamic variable but also older bindings. This facility provides for stack inspection and
(more generally) folding over the execution context as an inductive data structure.
This paper answers Moreau's call for ``a single framework integrating continuations, side-effects, and dynamic binding''. Our delimited dynamic binding is a new and useful form of abstraction --
over parts of the dynamic environment, just as lambda lets us abstract over parts of the lexical environment.
The current version is 1.1, Jul 2006.
DDBinding.pdf [269K]
``Delimited Dynamic Binding.'' Proceedings of the ICFP 2006, pp. 26-37.
Chung-chieh Shan. Slides of the ICFP 2006 Presentation, Sep 18, 2006.
< http://www.cs.rutgers.edu/~ccshan/dynscope/talk.pdf >
DBplusDC.tar.gz [25K]
DBplusDC-README.txt [5K]
The code accompanying the paper
The archive includes numerous examples (in various Scheme systems and SML) illustrating the ill-defined interaction between dynamic variables and shift/reset. The code includes implementations --
in OCaml, Scheme and Haskell -- of dynamic binding in terms of delimited continuations, which is consistent with the combined context-reduction semantics of DB+DC . A separate, Scheme48-specific
code implements delimited continuations (shift/reset) that are aware of dynamic-wind and dynamic-environment.
Please see the README file for the full description.
Mutable bindings in evaluation contexts: formalization of dynamic binding
We demonstrate a mechanized formalization of mutable dynamic binding in a direct-style semantics, without environment- or state-passing but relying instead on the evaluation context to play the
prominent double role of the dynamic environment.
Our language is DB (as defined in Figure 1 and Section 2 of the `Delimited Dynamic Binding' paper), which is a simply-typed lambda-calculus with mutable dynamically-scoped variables. The language
is based on Moreau's syntactic theory of dynamic binding, to which we added a sound type system. Our formalization of type safety uses the reduction semantics exactly as published by Moreau,
Figure 6 of his paper. Our type system is conventional, except that judgments are parameterized by a signature Sigma, which associates every dynamic variable with its type. This type system does
not prevent access to an unbound dynamic variable, which matches the existing practice.
Our Twelf code closely follows the small-step approach of `Syntactic Approach to Type Soundness' with contexts, focusing and redexing. Evaluation contexts are modeled as meta-level functions. The
notable complication is that contexts and pre-values are mutually dependent, and so the pre-val and focus functions are mutually recursive. Another complication is that our type system
specifically lets a well-typed expression try to access an unbound dynamic variable, resulting in an error. This reflects the behavior of most real systems that use dynamic variables. We have to
model this error behavior and account for it in our progress proofs.
Our formalization points out a deep similarity between lexical variables in the typing context and dynamic (mutable) variables in the evaluation context.
The current version is 1.2, Sep 20, 2006.
db-lang.elf [23K]
The commented Twelf code
Mechanising dynamic binding
A message posted on the POPLmark mailing list on Wed Sep 20 14:43:52 EDT 2006
< http://lists.seas.upenn.edu/pipermail/poplmark/2006-September/000368.html >
Haskell's `implicit parameters' are not dynamically scoped
Implicit parameters of Haskell are often regarded as similar to dynamically scoped variables, such as `special' variables of Common Lisp, current-input-port of Scheme, stdin, and exception
handlers in many languages. It is important to realize however that implicit parameters lack truly dynamic scope, as the following code illustrates.
(let ((p (make-parameter #f)))
(let ((x
(parameterize ((p 1))
(lambda () (+ (p) 1)))))
(parameterize ((p 2)) (x))))
This scheme code uses parameter objects, SRFI-39, which are natively implemented in some Scheme systems. The code evaluates to 3. The equivalent Haskell code
let x = (let ?p = 1 in (\ () -> ?p + 1)) in let ?p = 2 in x()
yields the value 2
The current version is Spring 2005.
Simply typed lambda-calculus with dynamic binding is not strongly normalizing
Simply typed lambda-calculus has strong normalization property: the sequence of reductions of any term terminates. If we add (typed) dynamic variables, however, as in the language DB above, the
strong normalization property no longer holds. Indeed, the following term (written with OCaml dynamic binding library) is essentially fun () -> Omega: its inferred type is unit -> 'a, and
consequently, the evaluation of loop () loops forever.
let loop () =
let p = dnew () in
let delta () = dref p () in
dlet p delta delta
This term constitutes probably the simplest proof of the result, shown in a different setting by Moreau, that dynamic binding brings new expressiveness to the language.
The current version is September 29, 2006.
Delimited Dynamic Binding
The paper that introduced the DB calculus, simply typed lambda-calculus with dynamic binding.
Printing the outline of a pruned tree
This practical problem has been posed by the LtU user `rici' as follows: ``You have a labeled tree
type 'a tree =
Leaf of string * 'a | Branch of string * 'a tree list
that might correspond to a normal file system... where the leaves are files and the branches are directories. You are to produce an outline view of this tree, showing only those leaves which
satisfy some predicate function and the branches necessary to reach those leaves.... I find this problem interesting, in part because it is practical (my original implementation was to produce a
view of an Apache configuration file showing only directives which pertained to a given module)...''
We show the solution to this problem in Theta(tree-depth) space using dynamic binding with the extension to access the previous binding of a dynamic variable. This extension dupp is the part of
our dynamic binding library for OCaml, Haskell, and Scheme. It lets us fold over the execution context as an inductive data structure.
The current version is December 6, 2006.
pruned-tree-outline.ml [5K]
The commented OCaml code, with examples and the test run output at the end.
Delimited Dynamic Binding
The description of our extended dynamic binding library, and its source code. The dupp extension is described in Section 6.2 of the paper and illustrated in the `nub' example.
Last updated January 1, 2007
This site's top page is http://okmij.org/ftp/
oleg-at-pobox.com or oleg-at-okmij.org
Your comments, problem reports, questions are very welcome!
Converted from HSXML by HSXML->HTML | {"url":"http://www.okmij.org/ftp/Computation/dynamic-binding.html","timestamp":"2014-04-17T21:24:02Z","content_type":null,"content_length":"13191","record_id":"<urn:uuid:a3182d50-100e-44fe-9026-eb7af22077a8>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00662-ip-10-147-4-33.ec2.internal.warc.gz"} |
I have a few questions regarding the following piece from the MathML 3.0 spec (section 4.3.3.1 Uses of <domainofapplication>, <interval>, <condition>, <lowlimit> and <uplimit>)
The general technique of using a condition element together with domainofapplication is quite powerful. For example, to extend the previous example to a multivariate domain, one may use an extra
bound variable and a domain of application corresponding to a cartesian product:
Note that the order of the inner and outer bound variables is significant.
My questions:
1. Does this exaple try to denote a double integral ? If yes, is this the recommended way to do it ?
2. Why is the <set> wrapper inside <domainofapplication> necessary at all? Only because the later happens to just take a single argument ?
In a set, however, I could put arbitrary elements. From the example I conclude that this use of <set> is very specific to the <domainofapplication> parent and hence requires a very specific
element list. I could not find any specification about the required structure of this <set>.
3. What purpose serves the last <set> element <list>... ? We already have declared <bvar> t and u (in that order) as set element 1 and 2. Why list them again ? Does the final note of this section
refer in any way to the use of the <list> element ?
Any comments or expalantions would be greatly appreciated.
Thank you in advance, | {"url":"http://lists.w3.org/Archives/Public/www-math/2011Jul/att-0034/00-part","timestamp":"2014-04-25T05:02:02Z","content_type":null,"content_length":"4036","record_id":"<urn:uuid:9c41be7c-211f-410e-ba4a-bcc25c500f98>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00472-ip-10-147-4-33.ec2.internal.warc.gz"} |
definite integrals, upper limit variable
February 13th 2009, 10:13 PM #1
definite integrals, upper limit variable
it is correct to write:
$\int^x_0 f(x)\,dx$
or i need write ?
$\int^x_0 f(t)\,dt$
why ?
The question is :
if I need a substitution of f(x) in a formula :
$y=f(x) + \int^x_0 f(x)\,dx$
I can directly replace f(x)
otherwise :
$y=f(x) + \int^x_0 f(t)\,dt$
don't seem me elegant imho..
Thanks for the possible answers !
it is correct to write:
$\int^x_0 f(x)\,dx$
or i need write ?
$\int^x_0 f(t)\,dt$Mr F says: This is the one you want.
why ? Mr F says: Let me re-write your two choices and then you tell me which one doesn't make sense:
${\color{red}\int^{x = x}_{x = 0} f(x)\,dx}$
${\color{red}\int^{t = x}_{t = 0} f(t)\,dt}$
The question is :
if I need a substitution of f(x) in a formula :
$y=f(x) + \int^x_0 f(x)\,dx$
I can directly replace f(x)
otherwise :
$y=f(x) + \int^x_0 f(t)\,dt$
don't seem me elegant imho..
Thanks for the possible answers !
As Mr. Fantastic indicatses, the second is far less confusing. In the first, you would have to recognize that you are using "x" to mean two different things- and that's never a good idea.
why ?
The question is :
if I need a substitution of f(x) in a formula :
$y=f(x) + \int^x_0 f(x)\,dx$
I can directly replace f(x)
otherwise :
$y=f(x) + \int^x_0 f(t)\,dt$
don't seem me elegant imho..
Thanks for the possible answers !
Would you say that,, when x= 1, $y(1)= f(1)+ \int_0^1 f(1)d1$? That doesn't even make sense: there is no such thing as "d1"! If you write instead $y(1)= f(1)+\int_0^1 f(x)dx$ then you are
recognizing that the "x" outside the integral and in the upper limit is NOT the same as the "x" inside the integral- and so you shouldn't use the same symbol!
Far better to write $y(x)= f(x)+ \int_0^x f(t)dt$ so that when x= 1, $y(1)= f(1)+ \int_0^1 f(t)dt$
Last edited by mr fantastic; February 14th 2009 at 03:58 AM. Reason: Fixed quote tag
February 14th 2009, 02:36 AM #2
February 14th 2009, 03:47 AM #3
MHF Contributor
Apr 2005 | {"url":"http://mathhelpforum.com/math-topics/73556-definite-integrals-upper-limit-variable.html","timestamp":"2014-04-19T03:02:19Z","content_type":null,"content_length":"45583","record_id":"<urn:uuid:5bff1807-75e0-4a88-99a7-a284ea9d34ac>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00435-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Memoirs of the American Mathematical Society
2002; 198 pp; softcover
Volume: 158
ISBN-10: 0-8218-2935-1
ISBN-13: 978-0-8218-2935-6
List Price: US$67
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Order Code: MEMO/158/750
We deal with a map \(\alpha\) from a finite group \(G\) into the automorphism group \(Aut({\mathcal L})\) of a factor \({\mathcal L}\) satisfying (i) \(G=N \rtimes H\) is a semi-direct product, (ii)
the induced map \(g \in G \to [\alpha_g] \in Out({\mathcal L})=Aut({\mathcal L})/Int({\mathcal L})\) is an injective homomorphism, and (iii) the restrictions \(\alpha \! \! \mid_N, \alpha \! \! \
mid_H\) are genuine actions of the subgroups on the factor \({\mathcal L}\). The pair \({\mathcal M}={\mathcal L} \rtimes_{\alpha} H \supseteq {\mathcal N}={\mathcal L}^{\alpha\mid_N}\) (of the
crossed product \({\mathcal L} \rtimes_{\alpha} H\) and the fixed-point algebra \({\mathcal L}^{\alpha\mid_N}\)) gives us an irreducible inclusion of factors with Jones index \(\# G\). The inclusion
\({\mathcal M} \supseteq {\mathcal N}\) is of depth \(2\) and hence known to correspond to a Kac algebra of dimension \(\# G\).
A Kac algebra arising in this way is investigated in detail, and in fact the relevant multiplicative unitary (satisfying the pentagon equation) is described. We introduce and analyze a certain
cohomology group (denoted by \(H^2((N,H),{\mathbf T})\)) providing complete information on the Kac algebra structure, and we construct an abundance of non-trivial examples by making use of various
cocycles. The operator algebraic meaning of this cohomology group is clarified, and some related topics are also discussed.
Sector technique enables us to establish structure results for Kac algebras with certain prescribed underlying algebra structure. They guarantee that "most" Kac algebras of low dimension (say less
than \(60\)) actually arise from inclusions of the form \({\mathcal L} \rtimes_{\alpha} H \supseteq {\mathcal L}^{\alpha\mid_N}\), and consequently their classification can be carried out by
determining \(H^2((N,H),{\mathbf T})\). Among other things we indeed classify Kac algebras of dimension \(16\) and \(24\), which (together with previously known results) gives rise to the complete
classification of Kac algebras of dimension up to \(31\). Partly to simplify classification procedure and hopefully for its own sake, we also study "group extensions" of general (finite-dimensional)
Kac algebras with some discussions on related topics.
Graduate students and research mathematicians interested in functional analysis.
• Introduction
• Actions of matched pairs
• Cocycles attached to the pentagon equation
• Multiplicative unitary
• Kac algebra structure
• Group-like elements
• Examples of finite-dimensional Kac algebras
• Inclusions with the Coxeter-Dynkin graph \(D^{(1)}_6\) and the Kac-Paljutkin algebra
• Structure theorems
• Classification of certain Kac algebras
• Classification of Kac algebras of dimension 16
• Group extensions of general Kac algebras
• 2-cocycles of Kac algebras
• Classification of Kac algebras of dimension 24
• Bibliography
• Index | {"url":"http://www.ams.org/bookstore?fn=20&arg1=memoseries&ikey=MEMO-158-750","timestamp":"2014-04-20T16:05:49Z","content_type":null,"content_length":"17113","record_id":"<urn:uuid:7549b66b-e02a-4c7e-a5c4-77b4a4e7107b>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00094-ip-10-147-4-33.ec2.internal.warc.gz"} |
Shocking statistic..
Shocking statistic..
"50% of universities are below average according to a recent survey."
This wasn't meant to be a joke, it was actually reported on the news!
Re: Shocking statistic..
Hi studymaths;
It is not shocking, just expected. Over here we even have a phrase for it. They call it the "dumbing down of America." They have been very successful so it is just natural that they would want to
expand their results to other countries.
Glad I was immune to it. I mean how could they dumb me down? I was never in school. I sure showed them!
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Shocking statistic..
I am impressed my your intelligence from reading posts on this forum. Did you really never attend school?!
I told my class about the headline to see if they understood the error the news team had made. They got it eventually.
I then explained to them that the average person on the planet has less than two legs
Re: Shocking statistic..
Did you really never attend school?!
Based on my results I wonder if I ever was there at all. Because I am a person, I can easily prove that the average person on the planet has less than one brain.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Legendary Member
Re: Shocking statistic..
Are you serious?
And to think I went to an Open Day today. And it seemed like such a nice place. They gave me free food.
People don't notice whether it's winter or summer when they're happy.
~ Anton Chekhov
Cheer up, emo kid.
Re: Shocking statistic..
Oh, I see what you are saying, yes, I am very serious.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Power Member
Re: Shocking statistic..
. .
Re: Shocking statistic..
bobbym, I hope you do get the humor
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Shocking statistic..
What humor did I miss?
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Shocking statistic..
The average of some data cannot be less than half of it
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Shocking statistic..
That is not the point. I said I could prove that humanity had on the average less than 1 brain. That is humor.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Shocking statistic..
Okay, prove it. I'll post a couple of questions in Help Me the next morning
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Shocking statistic..
Post #4 proves it most elegantly.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Shocking statistic..
Maybe it's me (or some language thing ..), but i don't see an error in the initial post
Re: Shocking statistic..
How is it possible that half of the data is less than its mean?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Shocking statistic..
I like this one:
Did you hear about the statistician who took the Dale Carnegie course? He improved his confidence level from .95 to .99.
97.3% of all statistics are made up.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Shocking statistic..
Agnishom wrote:
How is it possible that half of the data is less than its mean?
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Shocking statistic..
Look at {1,2,3,10}. 3 / 4's of the data is below the mean.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Shocking statistic..
Then why is this a joke?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Shocking statistic..
The joke works only if by average you mean median.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Shocking statistic..
Or mode
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Shocking statistic..
Or if you understand what those terms mean.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Shocking statistic..
Agnishom wrote:
Or mode
Not quite.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Shocking statistic..
anonimnystefy wrote:
The joke works only if by average you mean median.
Ah i see. guess someone might find that funny.
Median never came to my mind as a translation .. | {"url":"http://www.mathisfunforum.com/viewtopic.php?pid=174770","timestamp":"2014-04-18T08:06:51Z","content_type":null,"content_length":"36480","record_id":"<urn:uuid:d977cb83-9315-485b-8fd8-003866a225d8>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00655-ip-10-147-4-33.ec2.internal.warc.gz"} |
FOM: Natural examples
Joe Shipman shipman at savera.com
Tue Oct 12 16:04:53 EDT 1999
Stephen Fenner wrote:
> I don't have a reference, but I strongly suspect that the
> degrees of the Friedberg-Muchnik sets are very sensitive to the choice of
> the G"odel numbering (equivalently, universal TM) of computable partial
> functions. There are so many different, "natural" models of computation
> (besides TMs) that one really cannot favor one over all others. The
> choice of computational model is a source of much additional Kolmogorov
> complexity in the F-M construction.
This is one of the things it would be really useful to know. Can anyone
clarify how sensitive the degrees of sets constructed in various priority
arguments are to the particular choice of a basis for computation? Since the
standard bases for computation (TMs, general recursive functions, combinators,
etc.) are recursively isomorphic in a strong sense, I think it plausible that
the simpler priority arguments might not lead to different degrees for any of
these. Can any recursion theorists can help us here?
> BTW, the solution of Post's problem presented first in Soare's book is the
> construction of a noncomputable low c.e. set, and is simpler than the F-M
> construction.
I don't have this book -- can you sketch this proof for us?
-- Joe Shipman
More information about the FOM mailing list | {"url":"http://www.cs.nyu.edu/pipermail/fom/1999-October/003414.html","timestamp":"2014-04-20T03:12:38Z","content_type":null,"content_length":"3504","record_id":"<urn:uuid:42569b85-30f3-491d-8ab8-95c07d3e77e2>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00171-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Multiplication of matrices in GF(2) and R
up vote 0 down vote favorite
$H$ is an $n \times n$ matrix with elements in $ \{ -1,1 \}$
$G$ is an $n \times k$ matrix with elements in $GF(2)$ and also upper triangular, invertable
$m$ is an $k \times 1$ vector with elements in $GF(2)$
How can we perceive the output of $HGm$ where $Gm$ multiplication is in $GF(2)$ and $H$ multiplication is a normal real multiplication. Actually I want to combine $HG$ transformation into one $P$
transformation. How can I multiply two matrices while elements in one are in $GF(2)$ and other is in $R$ ? (We can also restrict the entries in $H$ to be one of $-1$ and $1$ but the output can be in
Motivation: It is a digital communication problem. $Gm$ is output codeword with 1 being mapped to -1 and 0 bit being mapped to +1. This codeword is multiplied to a channel convolution matrix $H$ e.g.
write in MATLAB
H = [1 -1 0 0 0;0 1 -1 0 0; 0 0 1 -1 0; 0 0 0 1 -1; 0 0 0 0 1]'
Now I want to impose a restriction on the complete transformation $P=HG$ and want to know if it is upper triangular or not ?
galois-theory finite-fields
The result will generally not be a linear transformation. If you provide some motivation, I may have a good idea for you. Right now, it looks to me like a lookup table of column sums. Gerhard "Ask
Me About System Design" Paseman, 2011.10.28 – Gerhard Paseman Oct 28 '11 at 20:45
As long as your entries are integers (or rationals with odd denominators) you can reduce mod 2. You should tell us what you want to do with this information, so we can help you better. – S.
Carnahan♦ Oct 28 '11 at 21:04
I added the description Carnahan. Let me know if you need any other information. – Aitezaz Abdullah Oct 28 '11 at 21:06
The "transformation" P is a general set mapping, not a linear mapping. It would make sense to call it upper triangular only if it or some representation seemed upper triangular. You can use
Carnahan's suggestion and ask if P' = (P then mod 2) is upper triangular; P will not look like matrix multiplication when the dust settles, except in special circumstances. Gerhard "Ask Me About
0-1 Matrices" Paseman, 2011.10.28 – Gerhard Paseman Oct 28 '11 at 21:16
Also, P' won't always be upper triangular. If G is upper triangular then it is routine to show that P' will be, because P' looks like H'G, where H' is the GF(2) version of H, if that makes sense.
Gerhard "Ask Me About System Design" Paseman, 2011.10.28 – Gerhard Paseman Oct 28 '11 at 21:19
show 2 more comments
1 Answer
active oldest votes
It seems to me that there may be some confusion here. As pointed out by all the commenters, the overall mapping cannot be linear. For me the most compelling reason for that would be that the
only additive homomorphism from $GF(2)^k$ to $\mathbf{R}^n$ is the trivial map sending everything to all zeros. Anyway, here is my attempt to make sense out of the question. This is just
educated guessing based on the scant given data.
It seems to me that the channel state matrix $H$ is upper triangular. This may have the following natural interpretation. The output of $Gm$ is a transmitted signal (usually mapped from $GF
(2)$ to reals as follows: $0\mapsto +1$, $1\mapsto -1$, but other conversion mappings from binary to real are possible). This is transmitted sequentially: the first component in the first
time slice, then the next,... The shape of the example matrix $H$ (as well as mentioning the term "channel convolution") suggests that what may be happening is that at instant of time $n$,
the receiver sees a linear combination $r(n)=\sum_{i=0}^d w(i) t(n-i)$, where $t(n)$ is the signal transmitted at time $n$ and $w(i)$ is a weight assigned to a signal "delayed" by $i$ time
units. I guess other interpretations are possible. Please comment!
up vote If the above holds, then you in some sense equate 'upper triangular' with 'causal', in the sense that only the things transmitted in the past can affect what is being received now. If this
2 down (or another equivalent interpretation) is what you wanted to ask, then I can finally give an answer: "Yes". Your encoding matrix $G$ was specified to be upper triangular, meaning that the
vote payload sequence of bits $m_1,m_2,\ldots,m_k$ is turned into an encoded sequence $c_1,c_2,\ldots,c_n$ of bits in such a way that for all $i$, the bit $c_i$ is a (binary linear) function of
the bits $m_j, j\le i$, and does not depend on the "future" bits $m_j, j>i$. Several encoder have this property (e.g. convolutional), and all linear codes have an encoder like this
(systematic part in the beginning).
But the relation between the $m_i$-sequence and the $r(i)$-sequence is not linear. With the $0\mapsto +1, 1\mapsto -1$ it could be expressed as something like $$ r(n)=\sum_{i=0}^n w(i)(-1)^
{\sum_{j\ge0}c(i,j,n) m(n-i-j)}, $$ and the causality is kept, but I don't think this is saying very much?
I also think that you might be better off asking this at, e.g. dsp.stackechange. I don't think this is really a research level math question.
add comment
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Rcpp and inline example
December 20, 2009
By dirk.eddelbuettel
Following up on yesterday's
Rcpp 0.7.0 release
, here is a quick example of why the new feastures can be so useful.
The following R code defines a character variable gslrng. This variable contains a short C++ code segment, which is then transformed by the function cfunction into a function of two arguments
assigned to funx:
## now use Rcpp to pass down a parameter for the seed, and a vector size gslrng <- ' int seed = RcppSexp(s).asInt(); int len = RcppSexp(n).asInt(); gsl_rng *r; gsl_rng_env_setup(); std::vector<double> v(len); r = gsl_rng_alloc (gsl_rng_default); gsl_rng_set (r, (unsigned long) seed); for (int i=0; i<len; i++) { v[i] = gsl_rng_get (r); } gsl_rng_free(r); return RcppSexp(v).asSexp();
' ## turn into a function that R can call ## compileargs redundant on Debian/Ubuntu as gsl headers are found anyway funx <- cfunction(signature(s="numeric", n="numeric"), gslrng, includes="#include <gsl/gsl_rng.h>", Rcpp=TRUE, cppargs="-I/usr/include", libargs="-lgsl -lgslcblas") print(funx(0, 5))
The signature argument to cfunction defines two variables s and n -- which the C++ function then reads in from R and converts to two integers seed and len. seed is used to initialize the
random-number generator, and len draws are then taken and stored in the STL vector v which returned at the end.
As the R level, we now have a function of two arguments returning a vector of RNG draws of the given lenth and using the given seed.
Also note how we tell cfunction to add the GSL include line, specify that we want to compile and link against Rcpp and provide -I and -L arguments to compile and link with the GSL. (The include
statement is not needed as the compiler would have found them in /usr/include anyway, but it shows how to set this if needed.)
Finally, we simply call our freshly compiled, linked and loaded C++ function with arguments zero for the seed and five for the length, and print the results vector returned to R from C++.
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Proof Please: Context Free Grammar
April 28th 2011, 07:10 PM #1
Junior Member
May 2010
Proof Please: Context Free Grammar
Find context-free grammar for the following languages (with $n\geqslant0$, $m\geqslant0$, $k\geqslant0$).
$(a) L = \{a^n b^m c^k : n + 2m = k\}$
What i got so far is:
$S \to aSc | A$
$A \to bAcc | \lambda$
$(b) L = \{a^n b^m c^k : n = m \or\ m ot= k\}$
No idea
Last edited by extatic; April 28th 2011 at 07:24 PM.
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osggiveaways at gmail dot com | {"url":"http://blog.thanksmama.com/2010/04/thanks-mama-giveaway-we-will-be-giving.html?showComment=1272002363470","timestamp":"2014-04-20T19:02:43Z","content_type":null,"content_length":"366495","record_id":"<urn:uuid:d838ed43-d3a2-41d9-9288-bacb849bae68>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00026-ip-10-147-4-33.ec2.internal.warc.gz"} |
Heat- Work interconversion
The way you wrote those equations and presented those equations doesn't prove that you have showed a way that heat can be turned into work.
All you really wrote was that if the change in potential energy is 0, then the change in heat is equal to the change in work -- according to the equation.
However, an exception was created to that equation which is now called the second law of thermodynamics. This law, in essence, states that heat cannot be changed completely into work, based on the
principles of entropy. | {"url":"http://www.physicsforums.com/showthread.php?p=3904355","timestamp":"2014-04-20T08:42:30Z","content_type":null,"content_length":"28809","record_id":"<urn:uuid:b2a855a5-25fe-49f4-a70a-f594f273c8a5>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00609-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Child Swings Back And Forth On A Swing Suspended ... | Chegg.com
A child swings back and forth on a swing suspended by 3.0 m-long ropes.
Find the turning-point angles if the child has a speed of 0.80 m/s when the ropes are vertical.
Express your answer using two significant figures. | {"url":"http://www.chegg.com/homework-help/questions-and-answers/child-swings-back-forth-swing-suspended-30-m-long-ropes-find-turning-point-angles-child-sp-q973025","timestamp":"2014-04-17T07:35:27Z","content_type":null,"content_length":"20431","record_id":"<urn:uuid:41cd3572-ad9a-41e2-9358-e35578fb6c15>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00454-ip-10-147-4-33.ec2.internal.warc.gz"} |
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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. | {"url":"http://openstudy.com/users/quietstorm93/asked","timestamp":"2014-04-17T18:27:12Z","content_type":null,"content_length":"120495","record_id":"<urn:uuid:179a1edc-ae67-4dbd-bf11-3b46abc5bbf1>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00493-ip-10-147-4-33.ec2.internal.warc.gz"} |
DOCUMENTA MATHEMATICA, Vol. 16 (2011), 781-868
DOCUMENTA MATHEMATICA
, Vol. 16 (2011), 781-868
K.R. Davidson and E.G. Katsoulis
Dilation Theory, Commutant Lifting and Semicrossed Products
We take a new look at dilation theory for nonself-adjoint operator algebras. Among the extremal (co)extensions of a representation, there is a special property of being fully extremal. This allows a
refinement of some of the classical notions which are important when one moves away from standard examples. We show that many algebras including graph algebras and tensor algebras of
C*-correspondences have the semi-Dirichlet property which collapses these notions and explains why they have a better dilation theory. This leads to variations of the notions of commutant lifting and
Ando's theorem. This is applied to the study of semicrossed products by automorphisms, and endomorphisms which lift to the C*-envelope. In particular, we obtain several general theorems which allow
one to conclude that semicrossed products of an operator algebra naturally imbed completely isometrically into the semicrossed product of its C*-envelope, and the C*-envelopes of these two algebras
are the same.
2010 Mathematics Subject Classification: 47L55, 47L40, 46L05, 37B20
Keywords and Phrases: dilation, extremal coextension, semicrossed product, commutant lifting, Fuglede property, C*-envelopes, Ando's theorem
Full text: dvi.gz 135 k, dvi 352 k, ps.gz 455 k, pdf 605 k.
Home Page of DOCUMENTA MATHEMATICA | {"url":"http://www.maths.soton.ac.uk/EMIS/journals/DMJDMV/vol-16/28.html","timestamp":"2014-04-16T16:07:39Z","content_type":null,"content_length":"2322","record_id":"<urn:uuid:4a041703-58c7-48f4-95a6-a2968f168c98>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00025-ip-10-147-4-33.ec2.internal.warc.gz"} |
Is There A Difference... [Archive] - MacRumors Forums
View Full Version : Is There A Difference...
Apr 10, 2009, 09:06 AM
Between a Set and an Array? any reason to use one over the other? or is it simply a matter of taste?
Apr 10, 2009, 09:22 AM
A set is unordered, and its elements are unique. An array is ordered, and elements can be duplicated. You'll have to decide which fits your needs per problem. The array is the hammer in the
expression "when all you have is a hammer, everything looks like a nail.". A lot of times other data structures are ignored when they are a better fit than an array.
Apr 10, 2009, 09:33 AM
I would say the difference is that sets are associative containers where the element is both the key and value. Arrays on the other hand use an index for accessing elements.
b e n
Apr 10, 2009, 10:00 AM
sets are associative containers where the element is both the key and value.
In Cocoa, what you're describing is called a dictionary. In other languages/environments they are often called hashes or associative arrays. Sets are unordered collections like lee said.
Apr 10, 2009, 12:46 PM
In Cocoa, what you're describing is called a dictionary. In other languages/environments they are often called hashes or associative arrays. Sets are unordered collections like lee said.
I'll stick by what I said: a set is, (or if you prefer) is like, an associative container. The difference between a set and a map (or dictionary or associative array) is that in a set, the value of
an element ***is also*** the key.
Also depending on what language you're using, a set can have duplicate entries.
b e n
Apr 10, 2009, 02:49 PM
That doesn't really make sense to me. In Cocoa, there's no notion of "association" in a set. The same goes for the mathematical definition of a set.
I'll accept that fact that perhaps in some environments a set is implemented as an associative collection where both the key and value are the same object. (My guess is PHP, because isn't every
collection implemented as an associative array?) This would work because the mathematical definition of a "set" is implemented with the keys, and since you have to fill the value, why not make it the
same as the value (which is redundant)?
But defining a set thus to a new-comer is confusing. The mathematical definition is the clearest: An unordered collection in which each element is unique.
Apr 10, 2009, 03:24 PM
A set is most often used when you need to see if one object is part of a group.
For example, let's say you had an array with 100,000 random numbers. If you wanted to figure out if the array contained a certain number, you would have to walk through the array, checking each
element--this could potentially take 100,000 operations!
In a set, the elements are kept in a special data structure that allows fast lookup. Determining if a certain number is present in a group 100,000 will be much, much faster if you use a set than if
you used an array.
Since each element must be unique, a set is basically a dictionary of keys with no associated values. Likewise, a dictionary is a set where each element can have a value associated with it.
An NSCountedSet (also known in the computer science world as a "bag" or multiset) is a mix between a set and a dictionary. Each element has a count associated with it that keeps track of how many
times that element occurs in the collection. For example, the values "a", "b", "c", "b", "c", "c" would produce a counted set where "a" => 1, "b" => 2, and "c" => 3. Lookups in a counted set are just
as fast as with a set or dictionary.
tl;dr: use an NSSet/NSCountedSet/NSDictionary when you need to perform lots of lookups/membership tests and they need to be fast. Use an NSArray for practically everything else. And google
"computational complexity theory."
Apr 10, 2009, 03:53 PM
But defining a set thus to a new-comer is confusing. The mathematical definition is the clearest: An unordered collection in which each element is unique.
Umm, how does a mathematical description of a set make clear what the difference is between a set and an array in a programming context? All I wanted to get over was that with arrays items are
referenced by their index, and in a set items are referenced by their value - not something immediately obvious from the statement that a set is an unordered collection! :)
b e n
Apr 10, 2009, 05:37 PM
Between a Set and an Array? any reason to use one over the other? or is it simply a matter of taste?
They are different things.
NSSet and CFSet are like mathematical sets: They have elements, but the same value can be contained in a set only once. If you try to add the same object or an object with the same value again,
nothing happens (but it takes a bit of time, because whenever you add an object, the set checks whether it has already an element with that value). Note that the objects are compared for equality: If
you have two NSNumber objects, say when created with a double 5.0 and another created with an int 5, then they have the same value, and only one of them can be in the set. For that reason, it is also
illegal to change an object that is stored in any set (because changing it could make it equal to another object in the set).
NSArray and CFArray just store pointers to objects (at least that is how they are usually used), possibly using retain and release to hold on to objects). You can store the same object multiple
times, and the objects are indexed, so you can easily iterate through objects. | {"url":"http://forums.macrumors.com/archive/index.php/t-683916.html","timestamp":"2014-04-25T06:27:46Z","content_type":null,"content_length":"13367","record_id":"<urn:uuid:0f18395b-18c8-46b0-a063-2962dc6540e5>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00545-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Given that the density of the supplied vinegar solution in this experiment is 1.025 g/mL, calculate the % acetic acid found in the vinegar sample.
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candies. What is the maximum number of canies that this variety of R&Ns will yi...
R&N is a new candy with magical (yet mathematical) properties. Not all candies are the same size. When removing the largest R&N from a new bag, the bag becomes 1/2 full; removing the second largest
makes the bg 1/3 full; removing the third largest makes it 1/4 full, and so on....
In most geometry courses, we learn that there's no such thing as "SSA Congruence". That is, if we have triangles ABC and DEF such that AB = DE, BC = EF, and angle A = angle D, then we cannot deduce
that ABC and DEF are congruent. However, there are a few special ...
h(t)=tan theta (5pie/3,3pie) graph the function on the given interval
A gas at 54◦C occupies 4.54 L. At what tem- perature will the volume be 3.04 L, assuming the same pressure?
A sample of helium gas has a volume of 328 mL at 0.69 atm. What pressure is needed to reduce the volume at constant temperature to 32 mL?
You and some friends go bungee jumping, but you are afraid of oscillations which go too fast. Which one of these options would provide the slower oscillations? a) Tying two bungee cords to your feet
such that each bungee cord is attached to your feet and the object which you a...
One side of the barn counts as an enclosure.
A farmer is making a pig pen next to the barn. He has enough materials to build 50 feet of fencing. a)what is the largest area he can fence in? b)what are the dimensions of the pen with the largest
Use the Rational Zeros Theorem to write a list of all possible rational zeros of the function. -2x^4+4x^3+3x^2+18
1-tan^2 theta/1+tan^2 theta + 2sin^2 theta need to simply to sin and cos
So (650)(25)/248=353(x)/298 65.5241=.8441/(x) 55.3=x got it right thanks Dr.Bob
If a sample of gas occupies 25.0 mL at -25°C and 650 mmHg, what is the volume at 25°C and 353 mmHg? (Could you please show me step by step?)
Calculate the gravitational force between the Sun and Saturn. The mass of the Sun is 1.99×1030 kg. The mass of Saturn is 5680×1023 kg. The distance between them is 143×1010 m. In Newtons
I have $35,000 in holding since 1997 earning an interest rate of 3%. How much is this worth today in $
strayer university
The Lo Company earned $ 2.60 per share and paid a dividend of $ 1.30 per share in the year just ended. Earnings and dividends per share are expected to grow at a rate of 5 percent per year in the
future. Determine the value of the stock: A. if the required rate of return is 12...
2 nacl (aq) + h2so4 ==> na2so4 (aq) + 2 hci (g)
if the capacitance meter applies a voltage of 3.0 V across the plates and reads a capacitance of 30 pF, what is the charge on the plates?
Pages: 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | Next>> | {"url":"http://www.jiskha.com/members/profile/posts.cgi?name=JOHN","timestamp":"2014-04-20T19:07:07Z","content_type":null,"content_length":"31126","record_id":"<urn:uuid:e88979f3-3245-4a6d-8203-c48bf58c354b>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00056-ip-10-147-4-33.ec2.internal.warc.gz"} |
Group items tagged
These practice items provide examples of the new content and increased rigor represented by the revised Standards of Learning (SOL) and illustrate the new Technology-Enhanced Item types for the
mathematics, reading, science, and writing SOL tests. Technology-Enhanced Items (TEI) require students to indicate their responses in ways other than a multiple-choice format.
Practice makes perfect, and IXL makes math practice fun. With unlimited math questions in more than 1,000 topics, students improve their skills and confidence and always have new challenges to
meet. Click a grade below to get started!
This was shared by Martha Miranda. She's one of my Spec. Ed. , Smart Board, iPod using teachers.
1 - 5 of 5
items per page
Selected Tags
Related tags | {"url":"https://groups.diigo.com/group/apsitcs/content/tag/practice","timestamp":"2014-04-16T11:40:31Z","content_type":null,"content_length":"51861","record_id":"<urn:uuid:fff4de37-03e0-44ca-a6ce-9415e5c93a32>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00192-ip-10-147-4-33.ec2.internal.warc.gz"} |
Here is the midterm from 2012. Make yourself do this test under test conditions: no book, 50 minutes. After you have taken the test look at the solutions and check your work.
The following practice problems are good preparation material for the midterm. Some of these problem sets have material which will not be covered on the midterm (these problem sets are from previous
years), so use your common sense!
Practice problem set 1.
Practice problem set 2.
Practice problem set 3.
Practice problem set 4.
Practice problem set 5.
Practice problem set 6.
Practice problem set 7.
Practice problem set 8. | {"url":"http://www.math.ubc.ca/~jbryan/2013T1math253/MT1review/study.html","timestamp":"2014-04-19T14:31:25Z","content_type":null,"content_length":"1624","record_id":"<urn:uuid:881d52ef-9777-4ef8-b8b9-a9627058ed4e>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00065-ip-10-147-4-33.ec2.internal.warc.gz"} |
Finding the eq of a normal to the parabola
January 4th 2010, 06:38 PM #1
Jan 2010
Finding the eq of a normal to the parabola
Find the eq of the normal at a pt on the parabola $y^2 = 2x$ whose ordinate is 4
Eq of tangent to parabola at $(x_1,y_1)$ is $\Rightarrow yy_1 = 2a(x+x_1)$
Eq of parabola is $y^2 = 2x \therefore a = \frac{1}{2}$
\therefore the eq of a tangent to the parabola at $(x_1,4)$ is $\Rightarrow 4y = (x+x_1) \Rightarrow y = \frac{1}{4}(x+x_1)$
since the normal i perpendicular to the tangent then its slope would be -4
$\therefore$ eq of the normal to the parabola at a pt $(x_1, 4)$ is
$y = -4 (x+x_1)$
Am i correct !!!!!
Find the eq of the normal at a pt on the parabola $y^2 = 2x$ whose ordinate is 4
Eq of tangent to parabola at $(x_1,y_1)$ is $\Rightarrow yy_1 = 2a(x+x_1)$
Eq of parabola is $y^2 = 2x \therefore a = \frac{1}{2}$
\therefore the eq of a tangent to the parabola at $(x_1,4)$ is $\Rightarrow 4y = (x+x_1) \Rightarrow y = \frac{1}{4}(x+x_1)$
since the normal i perpendicular to the tangent then its slope would be -4
$\therefore$ eq of the normal to the parabola at a pt $(x_1, 4)$ is
$y = -4 (x+x_1)$
Am i correct !!!!!
No. and the fact that you have $x_1$ in your answer makes this obvious. the only variables should be x and y.
$y^2 = 2x$. When $y = 4$, $x = 8$, so the point the normal line passes through is (8,4). Now we need to find the slope.
Differentiating implicitly we see that $y' = \frac 1y$. Which means the slope when y = 4 is 1/4. which implies the slope of the normal line is -4.
I think you can take it from here.
(did you make an error? should this be in precalc or calc?)
thanks for the reply
January 4th 2010, 06:46 PM #2
January 4th 2010, 06:51 PM #3
Jan 2010 | {"url":"http://mathhelpforum.com/pre-calculus/122450-finding-eq-normal-parabola.html","timestamp":"2014-04-17T11:46:54Z","content_type":null,"content_length":"40663","record_id":"<urn:uuid:e1241eda-2d7a-408e-b005-cc883a137933>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00357-ip-10-147-4-33.ec2.internal.warc.gz"} |
Blog series: User-Centered Data Visualization. Part 2 - The Pizza Effect
In one of the recent editions, Time magazine featured this graphic:
The underlying data set is pretty simple. It consists of just 8 numbers which are used for four target/actual comparisons: how many swine flu vaccinations 4 organizations have requested and how many
they actually received. And still, the way the data was visualized is problematic.
Estimating the area of a circle is hard. The area of a circle grows quadratically with the radius. You double a circle’s radius and the area gets four times as large. You may know this from ordering
pizza: the difference between a 14-in pizza (medium) and a 16-in pizza (large) doesn’t really sound significant if you just look at these two numbers. After all, it’s just a 2 inch difference. But
eating a large pizza is quite a different challenge than a medium pizza – the effect of the area growing quadratically with the radius.
Columbia University requested twice as many vaccines than Goldman Sachs. This is very hard to see in the circle sizes: with both red circles being the same size (showing that they both received the
same number of vaccines), Columbia’s large dotted circle does not look twice as large as Goldman’s – but it is. That’s the pizza effect. Were the actual numbers not shown below the circles, the
visualization would be not very expressive. Because estimating the area of circles is so difficult, it's also tough to compare the size of the red circles against the dotted circles for all of the
Here’s one suggestion to visualize the data in an easier to interpret manner:
Because it’s much easier to estimate the length of a bar than the area of a circle, this graph makes it easy to see that Columbia ordered twice as many vaccines than Goldman – even without data
labels next to the bars. It's still a challenge to estimate the number of received vaccines for Goldman and Columbia. A logarithmic scale could resolve this problem, but people have hard times
understanding log scales (see the first blog of this series).
People often say that bar charts or column charts are too standard and therefore boring ("not sexy"). These chart types are standard because they are superior to other ways of visualizing data. There
are still ways (within limits) to make them attractive, but from a user-centered design perspective I believe that usability has the top priority.
Here’s another interesting fact that neither the original visualization nor the bar chart reveal. It doesn’t seem like it was the intention of Time to focus on this, but since the ratios between
requested and received vaccines are shown, it’s straight forward to calculate the actual value which then makes it very easy to see which of the four organizations got better off than the others.
Here’s what the result looks like:
Again I didn’t put data labels next to the bars. Not necessary. It’s obvious that the cancer center and Goldman Sachs have approximately the same ratio between requested and received vaccines –
roughly 5% and that’s a third of what Citigroup got and twice as much as Columbia. It’s interesting that, unlike pie charts, bar charts require you to think about how to scale the x-axis. I put it to
100% max to show the distance between the optimum relative return (100%) and the best achieved relative return (17%). So the message is that none of the four organizations got a good portion of what
they had asked for. If I had put the max on the x-axis to 20% it would’ve shown more clearly the differences between the organizations. So the message had been: Citigroup got a much higher relative
return than the others. Which scaling is better? There is no right or wrong answer. It depends on the underlying question at hand and what you want to express. Doesn’t this demonstrate how
manipulative data visualization can be? You betcha.
More blogs in this series to come!
Round Edges
on Mon, Jun 18 2012 5:08 PM
Recently, TIME magazine published this info graphic about top-earning movies. It’s kinda interesting | {"url":"http://www.infragistics.com/community/blogs/ux/archive/2010/01/05/blog-series-user-centered-data-visualization-part-2-the-pizza-effect.aspx","timestamp":"2014-04-21T04:34:01Z","content_type":null,"content_length":"46581","record_id":"<urn:uuid:fded8173-7c95-49e7-9a56-e3806d7f395c>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00156-ip-10-147-4-33.ec2.internal.warc.gz"} |
Powerball Example (Generating Random Numbers)
Join Date
Feb 2010
Rep Power
I have to do an assignment where I have to write a program that simulates the selection of Powerball lottery numbers. I have to use the Math.random() function to simulate the random selection
of a numbered ball. There has to be five white balls and one red ball. The five white numbers must be unique and between 1 and 49. (After a ball is removed from the container, it can’t be
selected again.) The one red number must be between 1 and 42. The red number can possibly be the same number as one of the selected
white balls.
I'm so close to figuring this one out; the only issue is that when I enter a number, I get an endless loop (it compiles fine). I can't seem to figure out what's wrong with my code. Any
advice? Thanks.
Java Code:
import java.util.Random;
public class PowerBall
public static void main(String [] args)
int w = Integer.parseInt(args[0]);
int r = Integer.parseInt(args[1]);
int g = Integer.parseInt(args[2]);
for (int i = 1; i <= w; i++)
if (w != g)
System.out.println("You already picked this number");
for (int random = 1; random <= r; random++)
int rand = (int) Math.random();
random = (rand + 1);
System.out.println("You picked numbers " + w + " and " + r);
Join Date
Jun 2008
Blog Entries
Rep Power
You're biggest mistake is here:
Java Code:
for (int random = 1; random <= r; random++) {
int rand = (int) Math.random();
random = (rand + 1);
Your loop index is an int called random, and you allow this index to be changed to a random number within the loop itself which will cause unpredictable behavior. Solution: in general don't
do this, don't change the loop index (usually called "i") from within the loop itself.
Join Date
Feb 2010
Rep Power
I made a mistake with my problem description; my program doesn't execute endlessly. It asks for three numbers when it should only ask for two, and whatever numbers I post, it just executes
the first number by its amount (for example, if I input the numbers 4, 15, and 9, my output is:)
I know the number issue involves the three variable args at the top, but I need all three variables up there. I don't know why the output is doing what it's doing. :confused:
Join Date
Jun 2008
Blog Entries
Rep Power
What changes have you made to your program based on my recommendations above?
I remember to have read this problem from mister rich raposa book "Java in 60 Minutes a Day" and here is his solution:
Java Code:
public class Powerball
public static void main(String [] args)
int white1, white2, white3, white4, white5, red;
red = (int) (Math.random() * 42 + 1);
white1 = (int) (Math.random() * 49 + 1);
white2 = (int) (Math.random() * 49 + 1);
}while(white1 == white2);
white3 = (int) (Math.random() * 49 + 1);
}while(white1 == white3 || white2 == white3);
white4 = (int) (Math.random() * 49 + 1);
}while(white1 == white4 || white2 == white4 || white3 == white4);
white5 = (int) (Math.random() * 49 + 1);
}while(white1 == white5 || white2 == white5 || white3 == white5 || white4 == white5);
System.out.println("White balls: " + white1 + " " + white2 + " " + white3 + " " + white4 + " " + white5);
System.out.println("Red ball: " + red);
I've solved the problem in a different way but I think his solution is quite self-explanatory. Hope this helps.
Join Date
Feb 2010
Rep Power
I was able to figure it out, but thank you so much. My class is actually using Raposa's book (every assignment is their exercise). I'm going to have to ask you for all the answers haha jk.
My solution looked very similar after I got it. It makes a lot more sense after seeing it this way. My previous code made it way too complex than it had to be.
I don't want you to become a cheater, but if allowed from your teacher you can download all solutions from here Wiley::Java in 60 Minutes A Day
Last edited by j2me64; 02-12-2010 at 11:41 AM.
Join Date
Feb 2010
Rep Power
Wow, every solution, huh? I'm really grateful for the link. I don't plan on cheating my way through the class, but it's nice to have a reference guide so if I screw up a program I can figure
out what I did wrong. | {"url":"http://www.java-forums.org/new-java/25388-powerball-example-generating-random-numbers.html","timestamp":"2014-04-16T20:18:35Z","content_type":null,"content_length":"96706","record_id":"<urn:uuid:c8cf9b8a-b855-46bc-9ef5-b26e4b608952>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00243-ip-10-147-4-33.ec2.internal.warc.gz"} |
UMBC High Performance Computing Facility
Please note that this page is under construction. We are documenting the 240-node cluster maya that will be available in Spring 2014. Currently, the cluster tara is still available. Please see the
2013 Resources Pages under the Resources tab.
How to use PETSc on tara
The PETSc (Portable, Extensible Toolkit for Scientific Computation) library "is a suite of data structures and routines for the scalable (parallel) solution of scientific applications modeled by
partial differential equations." It provides a useful framework of parallel data structures, solvers, preconditioners, etc, so that you can focus on solving your PDE problem. In this tutorial, we'll
show how to build and run one of the examples that comes with PETSc, which solves a "variable Poisson" problem. We will not discuss the details of problem here, we will just focus on getting the
example to run on tara. Before you begin, make sure to read the tutorial for compiling C programs.
Where to find documentation
PETSC is thoroughly documented by its authors at the PETSc home page. On tara, PETSc is installed in the following locations (corresponding to the currently available MPI/compiler combinations):
/usr/cluster/contrib/gcc-mvapich2-1.4rc2/petsc-3.0.0-p9/cxx-opt/ /usr/cluster/contrib/gcc-openmpi-1.3.3/petsc-3.0.0-p9/cxx-opt/ /usr/cluster/contrib/pgi-9.0-mvapich2-1.4rc2/petsc-3.0.0-p9/cxx-opt
/ /usr/cluster/contrib/pgi-9.0-openmpi-1.3.3/petsc-3.0.0-p9/cxx-opt/
Each location has an "include" directory with PETSc header files which you can browse.
The example we'll use is ex13.c . An easy way to get the example to your tara account is with wget. Notice that we are retrieving the plain C file, and not the HTML-ized version.
[araim1@tara-fe1 petsc-poisson]$ wget http://www.mcs.anl.gov/petsc/petsc-as/snapshots/petsc-current/
(The command should be one continuous line). The next thing we'll need is a Makefile
EXECUTABLE := petsc_ex13
OBJS := ex13.o
CFLAGS := -O3 -c99 -Minform=warn -fastsse
INCLUDES := -I$(PETSC_DIR)/include
LIBLOCS := -L$(PETSC_DIR)/lib
LDFLAGS := -lpetsc -lpetscvec -lpetscmat -lpetscksp -lpetscdm
CC := mpicc $(INCLUDES)
%.o: %.c %.h
$(CC) $(CFLAGS) $(DEFS) $(INCLUDES) -c $< -o $@
$(EXECUTABLE): $(OBJS)
$(CC) $(CFLAGS) $(DEFS) $(INCLUDES) $(OBJS) -o $@ $(LIBLOCS) $(LDFLAGS)
-rm -f *.o $(EXECUTABLE)
The most important lines, which will allow us to use the PETSc library are:
INCLUDES := -I$(PETSC_DIR)/include
LIBLOCS := -L$(PETSC_DIR)/lib
LDFLAGS := -lpetsc -lpetscvec -lpetscmat -lpetscksp -lpetscdm
Notice that the environment variable PETSC_DIR is used, but not defined in this file. We will define it in our session before proceeding (if you forget to do this, compiling and execution will
fail!). We will assume you're using the default PGI+MVAPICH2 switcher setting, so execute the following commands.
[araim1@tara-fe1 petsc-poisson]$ export PETSC_ARCH=cxx-opt
[araim1@tara-fe1 petsc-poisson]$ export PETSC_DIR=/usr/cluster/contrib/pgi-9.0-mvapich2-1.4rc2/petsc-3.0.0-p9/cxx-opt
[araim1@tara-fe1 petsc-poisson]$ export LD_LIBRARY_PATH=:/lib
[araim1@tara-fe1 petsc-poisson]$
Note that you need to set the environment variables PETSC_ARCH, PETSC_DIR, and LD_LIBRARY_PATH in each tara session, any time you want to use PETSc. To automate this, add the three above lines into /
Now we should be able to compile the example
[araim1@tara-fe1 petsc-poisson]$ make
mpicc -I/usr/cluster/contrib/pgi-9.0-mvapich2-1.4rc2/petsc-3.0.0-p9/cxx-opt/include
-O3 -c99 -Minform=warn -fastsse -c -o ex13.o ex13.c
mpicc -I/usr/cluster/contrib/pgi-9.0-mvapich2-1.4rc2/petsc-3.0.0-p9/cxx-opt/include
-O3 -c99 -Minform=warn -fastsse
ex13.o -o petsc_ex13 -L/usr/cluster/contrib/pgi-9.0-mvapich2-1.4rc2/petsc-3.0.0-p9/cxx-opt/lib
-lpetsc -lpetscvec -lpetscmat -lpetscksp -lpetscdm
[araim1@tara-fe1 petsc-poisson]$
Since this is a small example, we can try to run it directly on the command line
[araim1@tara-fe1 petsc-poisson]$ ./petsc_ex13
m 6 n 7 error norm 0.000925256
m 6 n 7 error norm 0.000925256
[araim1@tara-fe1 petsc-poisson]$
The example ran, but all we got at the end was the error norm. Let's see what other kinds of things we can do with the example by specifying the "-help' flag.
[araim1@tara-fe1 petsc-poisson]$ ./petsc_ex13 -help
Solves a variable Poisson problem with KSP.
Petsc Release Version 3.0.0, Patch 9, Fri Oct 23 13:37:28 CDT 2009
The PETSc Team
See docs/copyright.html for copyright information
See docs/changes/index.html for recent updates.
See docs/troubleshooting.html for problems.
See docs/manualpages/index.html for help.
Libraries linked from /usr/cluster/contrib/pgi-9.0-mvapich2-1.4rc2/petsc-3.0.0-p9/cxx-opt/lib
Options for all PETSc programs:
... (tons of options) ...
[araim1@tara-fe1 petsc-poisson]$
To illustrate one option, let's try "-ksp_view" to view the parameters of the linear solver
[araim1@tara-fe1 petsc-poisson]$ ./petsc_ex13 -ksp_view
KSP Object:
type: gmres
GMRES: restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
GMRES: happy breakdown tolerance 1e-30
maximum iterations=10000, initial guess is zero
tolerances: relative=1e-05, absolute=1e-50, divergence=10000
left preconditioning
PC Object:
type: lu
LU: out-of-place factorization
matrix ordering: nd
LU: tolerance for zero pivot 1e-12
LU: factor fill ratio needed 2.38043
Factored matrix follows
Matrix Object:
type=seqaij, rows=42, cols=42
package used to perform factorization: petsc
total: nonzeros=438, allocated nonzeros=438
not using I-node routines
linear system matrix = precond matrix:
Matrix Object:
type=seqaij, rows=42, cols=42
total: nonzeros=184, allocated nonzeros=210
not using I-node routines
m 6 n 7 error norm 0.000925256
[araim1@tara-fe1 petsc-poisson]$
Running the example through the batch system requires no special effort. As long as your environment variables are set correctly, your executable should be able to find the PETSc runtime libraries.
As a quick example, the following batch script should be sufficient.
#SBATCH --job-name=petsc_ex13
#SBATCH --output=slurm.out
#SBATCH --error=slurm.err
#SBATCH --partition=develop
#SBATCH --nodes=1
#SBATCH --ntasks-per-node=1
srun ./petsc_ex13 | {"url":"http://www.umbc.edu/hpcf/resources-tara-2010/how-to-PETSC.html","timestamp":"2014-04-20T11:34:20Z","content_type":null,"content_length":"14108","record_id":"<urn:uuid:27fa4c5c-c1bd-4edf-89c1-f72699f77922>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00472-ip-10-147-4-33.ec2.internal.warc.gz"} |
Nita Kurniasari, Maria and S., R. Heri PELABELAN TOTAL TITIK AJAIB GRAF HASIL KALI KARTESIUS DARI GRAF SIKEL. Jurnal Matematika dan Komputer, 13 (1). pp. 52-60. ISSN 1410-8518
A vertex-magic total labeling of graph , with the vertices and the edges is the bijection from to the set of integers , and for each vertex in satisfying , is the vertex that adjacent with , then
named a magic constant in . The sum of the label of and the labels of all edges incident to the is the same for all vertices of and named vertex-magic total graph. Vertex-magic total labeling of
cartesian products of cycles, with the type , with and is odd are the labeling to the and the concept used to label is -vertex antimagic total labeling and to label it is used vertex magic total
labeling of cycles, with the cycle is odd.
Repository Staff Only: item control page | {"url":"http://eprints.undip.ac.id/26878/","timestamp":"2014-04-20T08:46:16Z","content_type":null,"content_length":"16201","record_id":"<urn:uuid:19ad55ab-351e-4aed-a01a-2a78d6df52bc>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00490-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum Discussions - Matheology § 158
Date: Nov 22, 2012 8:35 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Matheology § 158
Matheology § 158
One user of transfinite set theory, i.e., a man who arrives where he
cannot arrive, is Mohamed El Naschie.
Then came the next quantum jump, around 1990, when M.S. El Naschie who
was originally working on elastic and fluid turbulence began to work
on his Cantorian version of fractal space-time. He showed that the n-
dimensional triadic Cantor set has the same Hausdorff dimension as the
dimension of a random inverse golden mean Sierpinski space to the
power n-1. [...] Sometime later El Naschie using the work of Prigogine
on irreversibility showed that the arrow of time may be explained in a
fractal space-time. A few years later two of El Naschie?s papers on
the subject were noted by Thompson essential science indicators as the
most cited New frontier paper in physics and as Hot paper in
engineering. {{That seems to come a bit too early. At the univeryity
of applied sciences Augsburg the theories of El Naschie have not yet
been taught.}} [...]
In E-infinity theory El Naschie admit formally infinite dimensional
??real? space-time. This infinity is hierarchical in a strict
mathematical way and he was able to show that E-infinity has finite
number of dimensions when observed from a distance. At low resolution
or equivalently at low energy the E-infinity Cantorian space-time
appear as a four dimensional space-time manifold. [...] The
eigenvalues like equation have a very simple interpretation: Dim E8 E8
= 496 represent all fundamental interactions. Thus it is equal to
particle physics 339 symmetries plus the R(4) = 20 of gravity plus
aleph_0. From that we deduce aleph:_0 = 496 - 339 - 20 = 137. {{The
correct value 137.036 has exorcized some number mysticist and
numerologist.}} [...] The author is indebted to the many members of
the fractal-Cantorian space-time community {{Cantor's idea of
countably many body-atoms and uncountably many ether-atoms gains new
impetus. The fatal Space-Time-Community grows, it seems, above all
[L. Marek-Crnjac: "A short history of fractal-Cantorian space-time",
Chaos, Solitons and Fractals 41 (2009) 2697?2705]
Regards. WM | {"url":"http://mathforum.org/kb/plaintext.jspa?messageID=7926728","timestamp":"2014-04-19T20:43:27Z","content_type":null,"content_length":"3461","record_id":"<urn:uuid:da785cd3-adb9-474b-aeec-a071c1f0cb24>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00197-ip-10-147-4-33.ec2.internal.warc.gz"} |
From Gerris
Functions can be used in most objects which require a numerical parameter. A function can be a constant or a piece of C code taking coordinates (x,y,z), time t or any of the domain variables as
arguments and returning a floating-point value.
The syntax in parameter files is as follows:
or a C function
double a = sin (x + y);
double b = cos (x - z);
double c = sin (M_PI*t);
return a + b + c;
or a C expression
40.*(P - 1.)
or a GTS file
The GTS file must be a planar (preferably Delaunay) triangulation of a set of points. The value of the function at a given (x,y) coordinate is then calculated by computing the z-coordinate of the
intersection of a vertical line passing through the point at (x,y,0) with the triangulation defined by the GTS file.
Gradients of variables can be computed using the dx(), dy() and dz() functions. For example, the z-component of the vorticity would be computed as:
(dx("V") - dy("U"))
More details on C functions
Comments should use the C syntax; i.e. opening /* and closing */ not necessarily on the same line rather than the usual parameter file syntax of a line beginning with a #. This is to allow the use of
C preprocessor directives in C functions in GfsFunctions. | {"url":"http://gfs.sourceforge.net/wiki/index.php?title=GfsFunction&oldid=1808","timestamp":"2014-04-17T10:13:48Z","content_type":null,"content_length":"12052","record_id":"<urn:uuid:30872766-aa28-4144-a0ea-37f08b77dff3>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00260-ip-10-147-4-33.ec2.internal.warc.gz"} |
Patent US20040001554 - Antenna signal processing systems
[0056] Referring now to FIG. 8, this shows a schematic diagram of an antenna processing system 800 for calculating weights for an adaptive antenna of an OFDM receiver.
[0057] An array antenna 802 comprises n directional antenna elements of which four 802 a, b, c, d are shown. To each of the respective antenna elements is coupled a downconverter 804 a-d and these
downconverters in turn provide received signal inputs for a set of fast fourier transform (FFT) blocks 806 a-d. Thus up to this point in the signal processing architecture there is, in effect, a
conventional OFDM receiver front end for each antenna element 802. Each FFT block 806 provides a plurality of frequency-domain outputs, one for each subcarrier. For simplicity, in FIG. 8 outputs for
only four such subcarriers are shown but, in practice, there are typically more than ten such subcarriers often more than one hundred subcarriers and sometimes more than one thousand subcarriers.
[0058] In FIG. 8 lines 808 a-d carry frequency domain signals for versions of a first subcarrier derived from received signals from antenna elements 802 a-d respectively. Similarly lines 810 a-b
carry versions of a second subcarrier from respective antenna elements 802 a-d, lines 812 a-d carry third subcarrier signals from these antenna elements and lines 814 a-d carry versions of a fourth
subcarrier, one from the signal from each of the antenna elements 802 a-d. Each set of each subcarrier has an associated set of multipliers for weighting the subcarrier versions prior to their
combination. In the case of the first subcarrier, multipliers 816 a-d are provided to multiply the version of the first subcarrier from each antenna element by an appropriate weight prior to
summation of these weighted versions of the first subcarrier in summer 818. Likewise multipliers 820 a-d are provided for applying weights to the fourth subcarrier on lines 814 a-d, the weighted
versions of the fourth subcarrier being combined by summer 822. One set of multipliers and one corresponding summer is provided for each set of versions of each subcarrier of the OFDM signal but, for
simplicity, only the summers for the first and fourth subcarriers are shown in FIG. 8.
[0059] The weights multiplying each version of a subcarrier are generally complex numbers, comprising an amplitude and a phase. The values of the weights are selected to provide an antenna response
with one or more beams or lobes pointing towards the direction (or directions) having the greatest received signal power or, preferably, the greatest signal-to-interference ratio.
[0060] For an ideal narrow band system with only one signal path and one interference path it is straightforward for an adaptive array antenna processing system to maximise the desired signal by
nulling the interference in the spatial domain, employing a single set of weights. However where the relevant frequency band is relatively wide the application of only a single set of coefficients
across the whole band results in a spatial response for the antenna which varies across the frequency band. A conventional way to deal with this problem is to treat each subcarrier in an OFDM system
individually, as a narrow-band signal, and calculate the weights for multipliers 816 and 820 accordingly. However this approach is complex and processor-intensive.
[0061] The applicants have recognised that, in theory, the subcarrier (or subband) weights should be chosen to reproduce the spatial characteristics of the antenna at the frequency of each subcarrier
to mitigate the interference. When looked at in this way it can be recognised that calculation of a separate set of weights for each subcarrier or group of subcarriers (i.e. subband) ab initio is
unnecessary as the use of spatial information is repeated, that is, the spatial characterisation at each subcarrier frequency should, ideally, be the same. Thus the signal processing system can be
simplified if the weights are calculated in stages, a first stage in which the required spatial response of the antenna is determined, and a second stage in which sets of weights for the versions of
each subcarrier (or group of subcarriers) are determined making use of the desired spatial response.
[0062] In the arrangement of FIG. 8, as will be described further below, a time-domain process is used to determine an approximate spatial response required for the adaptive antenna, and the set of
weights determined by this process is then modified to provide a set of weights appropriate to each subcarrier. This significantly reduces the redundancy present in prior art weight calculation
[0063] Referring again to FIG. 8, the signals input to FFT blocks 806 a-d are tapped at points 824 a-d to provide inputs to a further set of multipliers 826 a-d. The outputs from the multipliers are
combined by summer 828 and provide an input 830 to a conventional beamsteering algorithm 832. The beamsteering algorithm provides a set of outputs 834 comprising weights for input to multipliers 826
a-d, one for each antenna element 802 a-d. In this way a conventional beamsteering algorithm is applied to the wideband, time-domain OFDM signals from each antenna elements, prior to the FFT
[0064] The time-domain adaptive antenna process aims to spatially separate the desired signal from any interference by controlling the antenna beam pattern. The signal direction of arrival (DoA)
information can be determined by employing the same, conventional algorithms and procedures as are used for DoA determination in narrow-band systems. Such methods include the Fourier Method (FM),
Least Means Squares (LMS) method, Multiple Signal Classification (MUSIC) method, and the ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) method.
[0065] The purpose of algorithm 832 is to detect the direction of arrival of one signal or of a group of signals, in the presence of multipath, and to spatially separate the signal from the
interference. The assumption is made that the spatial characterisation of the signal and interference is approximately constant across the applicable frequency band. Making this assumption a matrix
of weights or coefficients for each subcarrier (or for a group of subcarriers) can be calculated in a frequency domain to substantially maintain the spatial characteristics of the antenna.
[0066] In a preferred embodiment the antenna beamsteering is based upon a determined signal-to-interference ratio rather than an absolute level of received power. Such a signal-to-interference ratio
may be calculated by making use of a known transmitted data sequence such as a preamble portion of an OFDM signal, correlating the known and received signals as explained in more detail below. For
this reason the signal processing system preferably includes a signal generator 836 to provide a version of the known sequence, as transmitted. The beamsteering algorithm may be chosen to maximise
the signal-to-interference ratio (or, alternatively, to minimise the interference-to-signal ratio), as described, for example, in M Fujimoto, N Kikuma and N Inagaki, “Performance of CMA adaptive
array optimised by Marquardt method for suppressing multipath waves”, IEICE Trans., vol. J74-B-II, no. 11, 1991, page 599 which is hereby incorporated by reference. Since a measurement of
signal-to-interference ratio based upon a preamble at the start of an OFDM data frame is not a continuous measurement, the assumption is made that the channels for the desired signal and interference
(that is amplitude and phase) are quasi-static.
[0067] Referring now to FIG. 9a, this illustrates an exemplary signal and interference detector 900 for the signal processing system of FIG. 8. The input signal on line 903 is provided to first and
second cross-correlators 901, 902, the outputs of which are provided to a signal strength detector 904 and to an interference strength detector 906 respectively to provide signal 908 and interference
910 strength outputs for calculating a signal-to-interference strength ratio. Cross-correlator 901 cross-correlates the input signal with a known reference signal comprising a time-domain OFDM signal
generated, for example, from a packet data frame preamble sequence. Cross-correlator 902 cross-correlates the input signal with a signal which is at least partially orthogonal to the known reference
signal, and thus extracts an interference component from the input signal, as will be explained in more detail below. Where the reference signal is derived from a preamble data sequence the output of
the signal and interference strength detector 900 will only be valid when the preamble sequence is present. In this case the cross-correlators 901, 902 may be arranged to operate over a time window
during which the preamble sequence is present. The time window may be determined, for example, by a synchronisation process such as an FFT window control process (not shown in FIG. 8). Alternatively
one or more pilot signals transmitting a known data sequence substantially continuously may be extracted from the received signal and used in the cross-correlation process.
[0068]FIG. 9b shows an exemplary signal generator 920 for providing the reference and orthogonal reference signals 912, 914. The signal generator 920 comprises, in the illustrated example, a preamble
sequence generator 922, to generate a preamble sequence 924 and a signal 926 orthogonal to the preamble sequence. These two signals are then inverse fast fourier transformed by IFFT 928 to provide
reference signals 912 and 914. In a HIPERLAN 2 system the preamble sequence p may be generated using the polynomial s(x)=x^7+x^4+1 with an initial all ones state, replacing all “1”s with −1 and all
“0”s with the number 1.
[0069] The operation of the signal and interference detector may be understood by considering a signal (R+I) where R is a known reference symbol and I is an interference contribution. The correlation
of (R+I) with R is R^2, that is the result is a measure of the desired signal strength, assuming that the cross-correlation between the reference signal and the interference is low. The
cross-correlation between (R+I) and R*, where R* is a signal orthogonal to R, is equal to the cross-correlation of R* with I as the cross-correlation of R* with R is by definition zero. The
cross-correlation of R* with I is a measure of the interference strength. Thus the outputs from the signal and interference strength detector 900 can be used to calculate a form of
signal-to-interference ratio, which can then be used by the beamsteering algorithm 832 to direct one or more antenna beams towards received signals with the least interference to the desired signal.
[0070] The correlation of two orthogonal sequences may be referred to as a zero-value-correlation, in mathematical terms,
[0071] where, a[i ]is the first correlation sequence and b[i ]is the second correlation sequence (both of length N). When the second correlation sequence is correlated with noise or co-channel or
multipath interference the result is non-zero. The sequence a[i ]is inserted into the data frame, for example, in the preamble. One way to format a zero-value-correlated sequence is from a pair of
highly correlated sequences. A highly correlated sequence has the property:
[0072] By combining a pair of this kind, zero-value-correlation sequences can be defined. Total received signal energy can be found by calculating the received power, and one measure of the level of
unwanted interference energy is indicated by the peak level of the zero-value-correlated signal.
[0073] The signal and interference detection operation may be carried out at the beginning of data reception and/or every successive packet, or frame-by-frame. The beamforming approach allows the
formation of flexible beam patterns and, where signals from three or more elements are combined, multiple directional lobes. Suitable beamforming methods are described in more detail in J. E. Hudson,
“Adaptive Array Principles”, Peter Peregrinus Limited, 1981, which is hereby incorporated by reference.
[0074] Referring again to FIG. 8, the weights on line 834 from beamsteering algorithm 832 are provided to a frequency domain (i.e. post-FFT) weight calculation procedure 838. Also input to procedure
838 is channel information from a channel information calculation block 840. Channel information block 840 has inputs from FFT procedures 806 a-d and provides an amplitude and phase value for each
subcarrier. As the skilled person will be aware, a number of conventional channel estimation techniques exist for OFDM based upon, for example, the use of a preamble comprising one or more known OFDM
training symbols and/or pilot subcarriers.
[0075] The weight for co-efficient calculation procedure 838 determines weights to be applied to each version of each subcarrier, in FIG. 8 by means of exemplary multipliers 816 a-d and 820 a-d.
However there is no need to apply an iterative process such as algorithm 832 again as since the frequency domain weights are chosen to reproduce the same physical direction (or directions) as
determined in the time domain a simple matrix multiplication will suffice. Thus with, for example, an array antenna with eight elements only eight multiplies are required.
[0076] The pre-FFT direction of arrival (DoA) calculation determines a set of weights on the assumption that the distance between each array antenna element is approximately half a wavelength. This
assumption is correct at one frequency, typically the centre frequency, but to either side of this frequency the element spacing is no longer half a wavelength. Thus the weights may be adjusted by
recalculation based upon an antenna array element spacing (in wavelengths) at the wavelength for which the set of weights is required. In one embodiment this calculation is performed by procedure 838
. The skilled person will understand that the determination of a set of weights for each subcarrier does not need to make use of any one specific matrix equation.
[0077] Sets of weights for each subcarrier are extracted by demultiplexer 840 and provide (complex) inputs to, in FIG. 8, multipliers 816 a-d and 820 a-d. It will be appreciated that the functional
elements of FIG. 8 may be implemented either in hardware or in software. Generally the downconverters 804 will be implemented in hardware, following which the received signals are digitised by
analogue-to-digital converters (not shown), the remainder of the signal processing preferably taking place in the digital domain. Thus the functions of elements 806-840 will generally be implemented
by software running on a digital signal processing integrated circuit, often a DSP tailored to radio applications.
[0078] The above-described method for extrapolating from pre-FFT weights to post-FFT weights assumes an approximately linear receiver response across the wideband OFDM signal. However in practice
this assumption is not necessarily completely true as the frequency response of amplifiers and other signal processing elements in the receive chain is typically not completely flat. For example an
amplifier may provide 10 dB of gain at 5 GHz but only 7 dB gain at 4.5 GHz and 5.5 GHz. It is therefore preferable to provide means for compensating for such gain and other variations in response
across the wideband channel. This can be done by dividing the wideband OFDM channel into a plurality of sub-bands and determining, in the pre-FFT domain, a set of weights for each sub-band, based
upon direction of arrival techniques as previously described. Each of these sets of weights can then be used to derive weights for use in the post-FFT domain, by allocating each subcarrier to one of
the sub-bands. It will be appreciated that each of these sub-bands will generally comprise a plurality of subcarriers, for example 100 subcarriers, so that, in effect, the wideband OFDM signal is
being sub-divided and the above-described technique applied within each sub-band. Thus, for example, the wideband OFDM signal may be sub-divided into between 4 and 10 sub-bands so that different sets
of pre-FFT weights can be calculated for each of these sub-bands. In this way the burden of post-FFT weight calculation is still significantly reduced since each sub-band will typically contain many
tens of subcarriers, without the rigid assumption of the applicability of a single set of weights across the wideband bandwidth. The sub-bands may be statically or dynamically allocated and may, for
example, be based upon pilot frequencies within the OFDM signal.
[0079] Equation 3, below, shows the format of a weight vector V[weights ]for one beam direction, direction i. A first beam direction is labelled by i=1; i=2 for a second beam direction and so on. The
application of the above-described techniques where more than one direction of arrival is defined is discussed in more detail below with reference to FIGS. 10 and 11.
[0080] In equation 3 the superscript f denotes the frequency of one of the sub-bands, so that f[1 ]denotes the frequency of the first sub-band and f[m ]denotes the frequency of the mth sub-band. The
subscripts 1 to n label the antenna array elements. Thus, for example, XXX denotes the pre-FFT weight for the first antenna element at the centre frequency of the first sub-band, for the ith formed
[0081] Referring to FIG. 10, this shows the architecture of an OFDM receiver 1000 with pre-FFT digital beam forming (DBF) and post-FFT diversity, helpful for understanding the operation of the
receiver of FIG. 11. The hybrid architecture of FIG. 10 includes time-domain DoA 1002 and DBF 1004 processing for forming two beams, each in a direction with a high signal-to-interference ratio. As
in the arrangement of FIG. 8, an array antenna with n elements is employed so that up to n−1 directions may be defined although, in the arrangement of FIG. 10, only two directions are defined. The
DoA process 1002 is based upon a Fourier method and operates in a corresponding manner to that for a narrow band system.
[0082] In the arrangement of FIG. 10 the DoA 1002 and DBF 1004 processing operates in the pre-FFT domain to generate two sets of weights 1006, 1008, each set multiplying 1010, 1012 signals from each
of the antenna elements so that after summation 1014, 1016 of the signals two separate beam directions are defined. The signals 1018, 1020 from each of these beam directions are then FFT processed
1022, 1024 in a conventional manner to provide two pluralities of sub-carrier signals 1026, 1028, one set of subcarriers from each beam direction. The subcarriers are weighted 1030, 1032 according to
an angular diversity process 1034 and then diversity combined by summing 1036 a version of each sub-carrier from each beam direction. The subcarriers are then processed 1038 in a conventional manner.
It will be understood that the sub-carrier weightings 1030, 1032 will, in general, be complex, that is comprise both amplitude and phase values.
[0083] Referring now to FIG. 11, this shows a receiver architecture 1100 broadly similar to that of FIG. 8 but incorporating angular diversity along the lines illustrated in the receiver of FIG. 10.
The portion of the receiver 1100 above dashed line 1102 broadly corresponds to the receiver 800 of FIG. 8. Thus like elements have been given like reference numerals and, in particular, the DoA and
DFB processing blocks correspond to the beam steering process 832 of FIG. 8. However by contrast with FIG. 8, and similarly to FIG. 10, these processing blocks calculate two sets of weights vectors
1104, 1106, defining first and second beams. The weights vector 1104 is calculated in a corresponding manner to the FIG. 8 arrangement, to define a first beam but, in addition, a second weights
vector 1106 is also determined to define a second beam direction of arrival. This is schematically illustrated by weights 1108, although a single input into DoA processing 832 is sufficient to
determine more than one beam direction so that the conceptually illustrated signals from array antenna elements 802, as weighted by second weights vector 1106, are not needed by the DoA and DBF
processing. As before, the beam directions are determined to maximise the signal-to-interference ratio and thus the two weights vectors 1104, 1106 preferably each define a beam direction with a high
signal-to-interference ratio, for example the directions with the highest and next highest signal-to-interference ratio. It will be appreciated that although the arrangement of FIG. 11 only
determines two different beam directions diversity reception may employ a plurality of such beam directions, up to n−1 directions, where n is the number of antenna elements.
[0084] As previously described the two weights vectors 1104 and 1106 are each used to generate a set of post-FFT weights, for each subcarrier within each beam direction processed. Thus below dashed
line 1102 the receiver is provided with corresponding elements 816′, 818′, 840′ to the receiver of FIG. 8 which, for conciseness, will not be described again. A set of subcarrier signals 1109, 1110
is thus obtained for each beam direction (for simplicity only three subcarrier signals are shown in FIG. 11). These are weighted using respective multipliers 1112, 1114 by applying (complex) weights
determined by diversity processing unit 1116, which operates in a broadly similar manner to diversity processing unit 1034 of FIG. 10. Each weighted subcarrier from one beam direction is then
combined with its counterpart from one or more other defined beam directions, in summers 1118, before further processing in a conventional manner, beginning with parallel to serial conversion 1120 to
provide a data output 1122 for decoding. It will be appreciated that although, for simplicity, FIG. 11 shows the processing required for two beam directions, subcarriers from more than two beam
directions may be combined for greater diversity gain.
[0085] The invention has been described with reference to a receiver but the skilled person will appreciate that once a set (or sets) of weights has been determined in the frequency domain, this same
set (or sets) of weights may be applied when transmitting signals. In other words the weights may be employed to provide a spatial pattern for a transmit antenna which mitigates co-channel
interference since, so far as concerns the radio channel, transmission and reception are symmetrical. This is particularly the case where transmission and reception are at the same frequency, for
example in a time division duplex (TDD) system.
[0086] No doubt many effective alternatives will occur to the skilled person and the invention is not limited to the described embodiments but encompasses modifications within the spirit and scope of
the attached claims.
[0044] These and other aspects of the invention will now be further described, by way of example only, with reference to the accompanying figures in which:
[0045]FIGS. 1a and 1 b show, respectively, an OFDM symbol and an exemplary OFDM transmitter;
[0046]FIGS. 2a and 2 b show, respectively, a receiver front end, and signal processing blocks of a HIPERLAN 2 OFDM receiver;
[0047]FIG. 3 shows an exemplary Media Access Control frame of a packet data communications system;
[0048]FIGS. 4a to 4 e show, respectively, a broadcast burst, a downlink burst, an uplink burst with a short preamble, an uplink burst with a long preamble, and a direct link burst of a HIPERLAN 2
physical layer signal;
[0049]FIG. 5 shows, schematically, uses of the preamble portion of a HIPERLAN 2 broadcast burst in a mobile terminal OFDM receiver;
[0050]FIG. 6 shows a frequency-time plot of a HIPERLAN 2 OFDM signal including preamble and pilot signals;
[0051]FIG. 7 shows an OFDM receiver in which a separate set of weights is applied to each sub-carrier in the frequency domain;
[0052]FIG. 8 shows a schematic diagram of an antenna signal processing system according to an embodiment of the present invention;
[0053]FIGS. 9a and 9 b show, respectively, a schematic diagram of a signal and interference detector, and a schematic diagram of a signal generator for the signal and interference detector of FIG. 9
[0054]FIG. 10 shows an architecture of an OFDM receiver with pre-FFT digital beam-forming and post-FFT diversity; and
[0055]FIG. 11 shows an OFDM receiver for beam direction diversity reception according to an embodiment of the present invention.
[0001] This invention relates to improved antenna techniques, particularly for Orthogonal Frequency Division Multiplexed (OFDM) communication systems.
[0002] Orthogonal frequency division multiplexing is a well-known technique for transmitting high bit rate digital data signals. Rather than modulate a single carrier with the high speed data, the
data is divided into a number of lower data rate channels each of which is transmitted on a separate subcarrier. In this way the effect of multipath fading is mitigated. In an OFDM signal the
separate subcarriers are spaced so that they overlap, as shown for subcarriers 12 in spectrum 10 of FIG. 1a. The subcarrier frequencies are chosen that so that the subcarriers are mutually
orthogonal, so that the separate signals modulated onto the subcarriers can be recovered at the receiver. One OFDM symbol is defined by a set of symbols, one modulated onto each subcarrier (and
therefore corresponds to a plurality of data bits). The subcarriers are orthogonal if they are spaced apart in frequency by an interval of 1/T, where T is the OFDM symbol period.
[0003] An OFDM symbol can be obtained by performing an inverse fourier transform, preferably an Inverse Fast Fourier Transform (IFFT), on a set of input symbols. The input symbols can be recovered by
performing a fourier transform, preferably a fast fourier transform (FFT), on the OFDM symbol. The FFT effectively multiplies the OFDM symbol by each subcarrier and integrates over the symbol period
T. It can be seen that for a given subcarrier only one subcarrier from the OFDM symbol is extracted by this procedure, as the overlap with the other subcarriers of the OFDM symbol will average to
zero over the integration period T.
[0004] Often the subcarriers are modulated by QAM (Quadrature Amplitude Modulation) symbols, but other forms of modulation such as Phase Shift Keying (PSK) or Pulse Amplitude Modulation (PAM) can
also be used. To reduce the effects of multipath OFDM symbols are normally extended by a guard period at the start of each symbol. Provided that the relatively delay of two multipath components is
smaller than this guard time interval there is no inter-symbol interference (ISI), at least to a first approximation.
[0005]FIG. 1b shows an exemplary OFDM transmitter 100 (here in a mobile terminal, MT) and an exemplary OFDM receiver 150 (here in an access point, AP). In the transmitter 100 a source 102 provides
data to a baseband mapping unit 104, which optionally provides forward error correction coding and interleaving, and which outputs modulated symbols such as QAM symbols. The modulated symbols are
provided to a multiplexer 108 which combines them with pilot symbols from a pilot symbol generator 106, which provides reference amplitudes and phases for frequency synchronisation and coherent
detection in the receiver (in other arrangements differential detection may be employed). The combination of blocks 110 converts the serial data stream from multiplexer 108 to a plurality of
parallel, reduced data rate streams, performs an IFFT on these data streams to provide an OFDM symbol, and then converts the multiple subcarriers of this OFDM symbol to a single serial data stream.
This serial (digital) data stream is then converted to an analogue time-domain signal by digital-to-analogue converter 112, up-converted by up-converter 114, and after filtering and amplification
(not shown) output from an antenna 116. Antenna 116 may comprise an omni-directional antenna, a sectorised antenna or an array antenna with beamforming.
[0006] The signal from antenna 116 of transmitter 100 is received by an antenna 152 of receiver 150 via a “channel” 118. Typically the signal arrives at antenna 152 as a plurality of multipath
components, with a plurality of different amplitudes and phases, which have propagated via a plurality of different channels or paths. These multipath components combine at the receiver and interfere
with one another to provide an overall channel characteristic typically having a number of deep nulls, rather like a comb, which generally change with time (particularly where the transmitter or
receiver is moving). Often there will be a number of transmitters in the same general location, for example an office, and this gives rise to co-channel interference, which can be more problematic
than multipath.
[0007] The antenna 152 of receiver 150 is coupled to a down-converter 154 and to an analogue-to-digital converter 156. Blocks 158 then perform a serial-to-parallel conversion, FFT, and
parallel-to-serial re-conversion, providing an output to demultiplexer 160, which separates the pilot symbol signal 162 from the data symbols. The data symbols then demodulated and de-mapped by
base-band de-mapping unit 164 to provide a detected data output 166. Broadly speaking the receiver 150 is a mirror image of the transmitter 100. The transmitter and receiver may be combined to form
an OFDM transceiver.
[0008] OFDM techniques may be employed in a variety of applications and are used, for example, for military communication systems and high definition tv. Here, applications of the invention will be
discussed with particular reference to the HIPERLAN (High Performance Radio Local Area Network) Type 2 standard (www.etsi.org/technicalactiv/hiperlan2.htm, and DTS/BRAN-0023003 v 0.k). Although
applications of the invention are not limited to this environment HIPERLAN 2 wireless local area network communications are managed by a common node, the access point.
[0009] The receiver of FIG. 1b is somewhat simplified as, in practice, there is a need to synchronise the FFT window to each OFDM symbol in turn, to avoid introducing non-orthogonality and hence
Inter-Carrier Interference (ICI). This may be done by auto-correlating an OFDM symbol with the cyclic extension of the symbol in the guard period but it is generally preferable, particularly for
packet data transmission, to use known OFDM (training) symbols which the receiver can accurately identify and locate, for example using a matched filter. It will be appreciated that this matched
filter operates in the time domain, that is before the FFT is carried out (as opposed to the post-FFT frequency domain). In a packet data system data packets may be provided with a preamble including
one or more of these training symbols.
[0010]FIGS. 2a and 2 b show, respectively, a receiver front end 200 and receiver signal processing blocks 250 of a HIPERLAN 2 mobile terminal (MT) OFDM receiver. The receiver 250 shows some details
of the analogue-to-digital conversion circuitry 252, the synchronisation, channel estimation and control circuitry 254 and the de-packetising, de-interleaving and error correcting circuitry 256.
[0011] The front end 200 comprises a receive antenna 202 coupled to an input amplifier 204 and a mixer 206, which has a second input from an IF oscillator 208 to mix the RF signal to IF. The IF
signal is then provided to an automatic Automatic Gain Control (AGC) amplifier 212 via a band pass filter 210, the AGC stage being controlled by a line 226 from control circuitry 254, to optimise
later signal quantisation. The output of AGC 212 provides an input to two mixers 214, 216, which are also provided with quadrature signals from an oscillator 220 and splitter 218 to generate
quadrature I and Q signals 222, 224. These I and Q signals are then over-sampled, filtered and decimated by analogue-to-digital circuitry 254. The over-sampling of the signal aids the digital
filtering, after which the signal is rate reduced to the desired sample rate.
[0012] It is desirable (but not absolutely essential) to compensate for the effects of the transmission channel. This can be done using a known symbol, for example in preamble data or one or more
pilot signals. In the receiver 250 of FIG. 2 a known preamble symbol, referred to as the “C symbol”, is used to determine a channel estimate. The receiver synchronises to the received signal and
switch 258 is operated to pass the received C symbol to channel estimator 260. This estimates the effect of the channel (rotation of the symbols in the sub-carriers) on the known C symbol so that the
effects of the channel can be compensated for, by multiplying by the complex conjugate of the channel response. Alternatively the one or more pilot signals (which also contain known symbols) can be
used to determine a channel estimate. Again the phase rotation and amplitude change required to transform the received pilot into the expected symbol can be determined and applied to other received
symbols. Where more than one pilot is available at more than one frequency improved channel compensation estimates can be obtained by interpolation/extrapolation to other frequencies using the
different frequency pilot signals.
[0013] In FIG. 2 the receiver front end 200 will generally be implemented in hardware whilst the receiver processing section 250 will often be implemented in “software”, as illustrated schematically
by Flash RAM 262 using, for example, ASICs, FPGAs or one or more DSP (digital signal processor) chips. A similar division between hardware and software will generally be present in the transmitter.
However the skilled person will recognise that all the functions of the receiver of FIG. 2 (or of an equivalent transmitter) could be performed in hardware. Similarly the exact point at which the
signal is digitised in a software radio will generally depend upon a cost/complexity/power consumption trade-off, as well as upon the availability of suitable high speed analogue/digital converters
and processors, and that the RF signal could be digitised at IF or a higher frequency.
[0014]FIG. 3 shows an example of a Media Access Control (MAC) frame 300 of a packet data communications system including preamble sequences. The MAC frame includes a broadcast channel (BCH) burst 302
, a frame channel (FCH) burst 304, an access feedback channel (ACH) burst 306, a down-link (DL) burst 308, an up-link (UL) burst 310, a direct link (DiL) burst 312, and a random access (RCH) burst
314, all of which contain a preamble sequence.
[0015]FIGS. 4a to e show, respectively, a broadcast burst, downlink burst, an uplink burst with a short preamble, uplink burst with a long preamble, and a direct link burst of a HIPERLAN 2 physical
layer signal. Each of these bursts comprises a preamble portion 400 and a data payload portion 402. The preamble portions 400 comprise one or more of three basic OFDM symbols, denoted A, B and C. The
values of these symbols are known and A and B (and, if desired, C) can be recovered in the time domain (pre-FFT). These symbols are generally used to establish the frame and frequency synchronisation
and to set the FFT window for the data following the symbols; they may also be employed to control AGC stage 212. In the receiver of FIGS. 2 A and B are recovered in the time domain and C is
recovered in the frequency domain, that is post-FFT.
[0016]FIG. 5 illustrates, schematically, the use of these (known) preamble symbols for frame detection 502 based on RSSI (Received Signal Strength Indication), automatic gain control 504, frame
synchronisation 506, and frequency synchronisation 508; a schematic illustration of the preamble portion of an MAC frame 500 is also illustrated for comparison.
[0017]FIG. 6 shows a plot 600 in the frequency and time domain illustrating the relative positions of preamble sequences 602, pilot signals 604, and data signals 606 for HIPERLAN 2, which has 48 data
sub-carriers and 4 pilots (and one unused, central carrier channel 608). As can be seen from FIG. 6 the first four OFDM symbols comprise preamble data, and the pilot signals 604 continue to carry
their preamble symbols. However on the remaining (data-bearing) sub-carriers OFDM symbols 5 onwards carry data. In other OFDM schemes similar plots can be drawn, although the preamble and pilot
positions may vary (for example, the pilots need not necessarily comprise continuous signals).
[0018] It has previously been mentioned that OFDM is a useful technique for alleviating the effects of frequency selective fading caused by multipaths. However with particularly high data rates or in
particularly severe multipath environments OFDM communications systems can still suffer from the effects of multipath fading. Moreover in indoor wireless environments, such as small office wireless
LANs, there will often be a number of similar systems operating simultaneously in the same frequency band, because of limited spectrum availability. This can result in severe co-channel interference.
[0019] One technique which has been proposed for combatting such multipath and co-channel interference is the use of a sectorised transmit and/or receive antenna. The region to be covered is divided
into a number of sectors, typically 3, 4 or 6, and one antenna (or more where diversity is employed) is provided for each sector, the patterns of the antennas being arranged to each cover mainly just
one sector. In effect the main beam of each of the sector antennas points in a different direction and by selecting the transmit and/or receive direction the effects of multipath components and/or
co-channel interference arriving from unwanted directions can be reduced. HIPERLAN 2, for example, supports the use of up to seven sectors at the Access Point. Some of the benefits of employing a
sectorised switching array antenna in a HIPERLAN 2 environment are described in “Performance of HIPERLAN 2 using Sectorised Antennas” A. Dufexi, S. Armour, A. Nix, P. Karlsson and D. Bull, IEE
Electronics Letters Feb. 15, 2001, volume 37 no. 4, page 245.
[0020] Another approach employed to mitigate the effects of multipath and co-channel interference uses a beamforming antenna array, such as a linear array of antenna elements in which the
inter-antenna spacing is approximately one half a (carrier) wavelength. Signals from the antennas are combined, with appropriate phase and amplitude weightings, to provide a combined response with
one or more lobes or beams. An array comprising n elements can be arranged to provide up to n−1 beams.
[0021] There are a number of different beamforming algorithms which may be applied to such an adaptive antenna array and details of these will be well known to the skilled person. One commonly used
algorithm is the Constant Modulus Algorithm (CMA), described in J. R. Treichler and B. G. Agee, “A New Approach to Multipath Correction of Constant Modulus Signals”, IEEE Trans. Acoust. Speech and
Signal Process., vol. ASSP-31, no. 2, page 459, 1983, which is hereby incorporated by reference. Broadly speaking this algorithm iteratively determines the weights for combining the signals from the
antenna elements based upon a cost function chosen to make the spectrum of the combined signals approximately flat. The phase angles of the weights are chosen so that the beams point in the direction
of maximum signal power, or, alternatively, so that nulls are formed in the directions of the unwanted multipath components or co-channel interference.
[0022] Determining appropriate weights for the antenna array elements is relatively straightforward in a narrow band system but in an OFDM receiver, where the bandwidth occupied by the group of
sub-carriers is normally >1 MHz and in many cases >10 MHz, a single set of weights is unlikely to be optimal across the entire bandwidth and may only be valid, for example, at the centre of the
frequency band. This can be understood, for example, from the consideration that the antenna element spacing, in terms of fractions of a sub-carrier wavelength, varies across the OFDM frequency band.
In the receiver of FIG. 1 adaptive array weights may be applied at points 168, 170, or 172 but applying the array weights at positions 168 or 170 (pre-FFT) will not normally result in a good set of
estimated weights across the frequency band.
[0023] One solution to this problem is therefore to apply weightings after the FFT, at point 172, where different sets of weights can be applied to each sub-carrier. FIG. 7 shows an OFDM receiver 700
in which a separate set of weights is applied to each sub-carrier in this way. However it will be appreciated that with K sub-carriers and L antenna elements a total of K×L weights must be
determined, which is a lengthy and processor-intensive task adding considerably to the receiver complexity. EP 0 852 407 describes an arrangement in which an operational band is partitioned into four
equal sub-bands, one set of weights being calculated for each sub-band rather than for each sub-carrier, to reduce the number of weights to be calculated. However this is still a relatively
complicated procedure and, moreover, produces a sub-optimal result. An alternative approach is described in Fujimoto et al, “A Novel Adaptive Array Utilising Frequency Characteristics”, IEICE Trans.
Commun., vol. E 83-B, no. Feb. 2, 2000, page 371, which is hereby incorporated by reference, in which the post-FFT separated sub-carriers are used to determine a single set of pre-FFT time domain
weights using CMA. This approach provides a considerable simplification of the weight determining procedure but, again, the weights are sub-optimal.
[0024] U.S. Pat. No. 6,249,250 describes OFDM adaptive antenna weight determination techniques for use in either the time or frequency domain (but not both simultaneously). Other weight-determination
techniques involving sub-carrier clustering are described in Japanese patent application number 2000-391221 filed on Dec. 22, 2000, inventor Hidehiro Matsuoka, and British patent application number
0108026.6 filed on Mar. 30, 2001 entitled “Adaptive Antenna”.
[0025] The above-described weight calculation techniques for array antennas each have pros and cons, some providing relatively accurate weight determination at the expense of complex and
time-consuming processing, others having simpler, faster weight determination algorithms but in general providing poorer weight estimates. There is therefore a need for improved array antennas and
weight determination techniques which can provide good weight estimates without imposing an excessive signal processing burden.
[0026] According to the present invention there is therefore provided a signal processing system for determining weights for an adaptive antenna, the antenna having a plurality of antenna elements
each for receiving a signal comprising a plurality of subcarriers, the system comprising, a plurality of inputs for receiving signals from said plurality of antenna elements, a time-to-frequency
domain transformer for each input for transforming the signal from each input to a plurality of subcarrier signals, a first weight determiner coupled to said inputs, before the time-to-frequency
domain transformer in the signal path, and configured to determine a first set of weights, one for each input, and a second weight determiner coupled to said first weight determiner and configured to
determine, from the determination of said first set of weights, at least one second set of weights comprising a frequency-domain weight for a subcarrier signal derived from each of said inputs.
[0027] The signal processing system greatly simplifies the calculation of weights for each subcarrier (or for a subcarrier group) in the frequency domain as compared with prior art techniques. This
in turn allows the power consumption and/or cost of providing an adaptive antenna with digital beamforming in the frequency domain to be significantly reduced. The system is also flexible enough to
allow additional adaptive signal processing methods to be applied in the frequency domain to further improve system performance where necessary.
[0028] In one embodiment the time domain or pre-FFT processing to determine the first set of weights has the effect of defining a direction, for example a direction with the highest
signal-to-interference ratio for the band. One or more directions may alternatively be determined by identifying those directions which are above a threshold of received power level. This direction
or directions, or more generally a spatial property or beam pattern defined by the first set of weights, is preferably substantially maintained when calculating the second set of weights. The
algorithm used to determine the second set or sets of weights aims to maintain the beam direction or pattern and, in a simple embodiment, comprises a small number of multiply operations, for example
one for each array antenna element. Thus the second sets of weights are chosen so that the antenna beam for each subcarrier has the same direction as that determined when calculating the first set of
weights. The values of the sets of weights will be different because the subcarriers are at different frequencies.
[0029] The first weight determiner effectively functions as a direction of arrival (DoA) detector, the output of which is used by the second weight determiner. Thus in the invention the function of
the first weight determiner may be performed by any form of direction of arrival detector.
[0030] Various measures may be used to determine the initial beam direction (or directions), such as a simple measure of received power, but preferably the system uses the first weight determiner to
determine a direction (or directions) based upon signal-to-interference ratio. This helps discriminate against co-channel interference, which can have a relatively large received power level. The
first weight determiner may determine a set of weights which aims to null out multipath and/or co-channel interference, that is an antenna response pattern best characterised as having one or more
nulls rather than one or more main lobes. However this embodiment is less preferable as, particularly in a wide-band system, there may be a large number of multipath components to attenuate whereas
generally speaking, only one or a few lobes will serve to pick out the signals with the best signal-to-interference ratios. The first set of weights may be determined conventionally by minimising a
cost function which increases as the signal-to-interference ratio goes down.
[0031] The signal processing system may be applied at an Access Point or base station or at the mobile end, for example in a Mobile Terminal. The system may be implemented in hardware, for example in
silicon, or in software, or in a combination of the two. Preferably the subcarriers are substantially mutually orthogonal, although the system may also be employed where the subcarriers do not
overlap and thus do not need to be orthogonal. However, the system is particularly suited to processing OFDM (Orthogonal Frequency Division Multiplexed) signals.
[0032] The second weight determiner preferably calculates the second sets of weights from the first set of weights, but in embodiments it may instead use the results from an intermediate calculation,
and in this case it may not be necessary to complete the calculation to determine a usable first set of weights. One second set of weights is preferably provided for each subcarrier signal (a set of
weights comprising a weight for each version of a subcarrier, one version derived from the signal from each of the antenna elements) but, where a still further simplified calculation is required,
groups of subcarriers may share sets of weights. Thus the subcarriers may be arranged in a number of groups each of which has one of the second sets of weights. Each of the weights in a second set of
weights will generally, however, have been derived from the received signals from all the array antenna elements. Preferably the second sets of weights are modified by a channel estimate determined
from the signal from each of the antenna elements.
[0033] In embodiments all the signals for each subcarrier are combined after weighting by one of the weights from the second sets of weights. Thus, for example, weighted versions of the first carrier
derived from each of the antenna elements are combined to provide a combined first subcarrier output, and the remaining subcarriers are likewise combined. Where one or a group of subcarriers has a
quality which is significantly inferior to that of the other subcarriers, for example a significantly worse signal-to-interference ratio, a post-FFT narrowband-type beamforming algorithm may be used
separately on the one (or the group of) subcarriers. Techniques for this are further described in the related case, “Adaptive Antenna”, British patent application number 0108026.6, filed by the
present applicant on Mar. 30, 2001, the contents of which are hereby incorporated by reference.
[0034] Preferably the first weight determiner implements a direction of arrival detection algorithm which controls a directional response of the antenna, such as a digitial beamforming or
beamsteering algorithm. In embodiments where the antenna elements are directional direction of arrival detection may simply comprise selection of one of the elements based upon a power or
signal-to-interference ratio threshold.
[0035] A preferred embodiment of the invention incorporates a signal and interference detector, which may provide separate signal and interference outputs or a combined output comprising, for
example, a signal-to-interference ratio. Preferably the signal and interference detector operates in the time domain, that is before the received signal has been transformed from the time domain to a
frequency domain, typically by a fourier transform for an OFDM signal. One signal and interference detector may be provided for each antenna element or a single signal and interference detector may
be shared between the elements, for example on a time-multiplexed basis.
[0036] Preferably the signal and interference detector determines the strength of the wanted signal and unwanted interference using a correlation technique. A known portion of the received signal may
be correlated with a reference signal to determine the wanted signal strength, and with a signal orthogonal to the reference signal to determine the interference strength. The known portion of the
received signal may comprise, for example, a preamble sequence such as a pseudo-noise sequence. The correlation is performed in the time-domain - that is the correlation is between two OFDM symbols
and hence an inverse fourier transform is performed on the known (preamble or pseudo-noise) data prior to the correlation. Preferably, therefore, the antenna element selector includes a signal
generator to generate the reference signal and signal orthogonal to the reference signal, although this is not essential as an appropriate reference may be extracted from the received data.
[0037] The signal processing system may incorporate means for angular diversity reception, combining signals with a high signal-to-interference ratio received from different directions. Thus a
plurality of post-FFT second sets of weights may be determined for each sub-carrier and for each beam direction and then the corresponding sub-carriers received from each direction may be combined
using diversity reception techniques. As previously described the post-FFT subcarrier weights may be derived from the pre-FFT weight determination used for defining the beam directions.
[0038] In some cases it may be preferable to divide the incoming OFDM signal into a plurality of sub-bands, each sub-band preferably including a number of subcarriers. Separate pre-FFT weight
calculations may then be made for each sub-band, which allows an improved accuracy in post-FFT weight determination, particularly where the receiver response is not flat across the bandwidth of
interest. Thus preferably a number of subcarriers are allocated to each of the sub-bands and the post-FFT weights for the sub-carriers within a sub-band use the pre-FFT weights determined for that
sub-band. The signal processing system may include means for sub-dividing the OFDM signal bandwidth into sub-bands, for example based upon the OFDM pilot signals.
[0039] In a related aspect the invention provides a method of determining weights for an adaptive antenna, the antenna having a plurality of antenna elements each for receiving a signal comprising a
plurality of subcarriers, the method comprising, receiving a plurality of signals from said plurality of antenna elements, transforming each of said plurality of received signals into a plurality of
subcarrier signals, determining a first set of weights, one for each received signal, and determining, using the determination of said first set of weights, at least one second set of weights
comprising a frequency-domain weight for a subcarrier signal derived from each of said received signal inputs signals.
[0040] The invention also provides a method of adaptive beamforming for an OFDM antenna comprising a plurality of antenna elements, the method comprising, applying a first beamforming algorithm in
the time domain to determine at least one antenna beam direction, and calculating a set of weights for application in the frequency domain to versions of at least one subcarrier, each version of the
subcarrier being received by a corresponding one of the antenna elements, said set of weights being calculated using said at least one antenna beam direction.
[0041] The invention further provides a method of calculating weights for an array antenna for receiving an orthogonal frequency division multiplexed (OFDM) signal comprising a plurality of
subcarriers, the method comprising, determining, in a time domain, a direction of arrival of the OFDM signal at the array antenna, calculating at least one set of weights for application to the
subcarriers of the OFDM signal in a frequency domain using said direction of arrival.
[0042] Here “time domain” is used to refer to a part of the signal path preceding a time-to-frequency or fourier transform and “frequency domain” is used to refer to a part of the signal path
following a time-to-frequency or fourier transform of the received OFDM signal.
[0043] In a related aspect the invention provides a carrier carrying processor control code to implement the above-described signal processing systems and methods. This processor control code may
comprise computer programme code, for example to control a digital signal processor, or other code such as a plurality of register values to set up a general purpose integrated circuit to implement
the selector or method. The carrier may comprise a data carrier or storage medium such as a hard or floppy disk, CD- or DVD-ROM, or a programmed memory such as a read-only memory, or an optical or
electrical signal carrier. As the skilled person will appreciate the control code may be also be distributed between a plurality of coupled components, for example on a network. The skilled person
will further recognise that the invention may be implemented by a combination of dedicated hardware and functions implemented in software. | {"url":"http://www.google.com.au/patents/US20040001554","timestamp":"2014-04-16T06:01:38Z","content_type":null,"content_length":"143525","record_id":"<urn:uuid:86d05811-160d-4657-b223-ff43321116c7>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00014-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mandelbrot Set ... in the 13th Century?
Re: Mandelbrot Set ... in the 13th Century?
That is what I missed, that it came from Yale . If I had noticed that I would have suspected something.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof. | {"url":"http://www.mathisfunforum.com/viewtopic.php?pid=131907","timestamp":"2014-04-19T17:28:57Z","content_type":null,"content_length":"14335","record_id":"<urn:uuid:3558540a-b8db-4b46-a3f2-ad05ddc32b72>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00390-ip-10-147-4-33.ec2.internal.warc.gz"} |
Solutions To Mathematics Textbooks
This wikibook aims to provide many solutions and explanations to questions and assignments posed in mathematics textbooks. Few mathematics textbooks (especially towards university / college level
mathematics) have the complete set of solutions available, often favouring answers to selected problems, or perhaps odd or even numbered questions exclusively. Even where answers are provided, they
may be succinct in order to save printed space.
New pages devoted to an entire book should take the following naming convention:
Book Name (Edition) (ISBN)
For instance; Calculus (3rd) (0521867444). If there is not an edition number, omit it. The ISBN and name must appear, however. Please refer to the books below for some more examples. Below the book
level, this wikibook follows the structure of:
/Book Name (Edition) (ISBN)/Chapter/Section
For instance; /Basic Mathematics (0387967877)/Chapter 5/Section 2. Naturally, this is open to interpretation. Some books may have a handful of chapter exercises with no subsections, in which case the
solutions will fit nicely in the Chapter page. Other books may have chapters divided into a series of subchapters or sections with individual exercises, and may benefit from a deeper structure.
Other adviceEdit
Existing solutionsEdit
If one produces solutions to a book problem that contains the answer already, there is nothing wrong with providing the solutions again in these pages. In fact, it is encouraged that you expand on
the original answer provided in a book, particularily if you feel the solution or proof is not sufficiently clear. Many books, in the interest in saving space, may provide the end result of a
calculation with no indication on how to arrive at the answer.
It is likely that subsequent editions of books contain almost the same solutions. Should you wish to provide solutions to a particular edition, check to see if the answer has been provided in a
subsequent edition first.
Please use LaTeX to format these guides.
You may find it easier to use the following for lines of math: \begin{array} .... \end{array}
$\begin{array}{l} line\,1 = something, \\ line\,2 = something\ else \\ line\,3 = whatever \end{array}$
• Algebra, by Gelfand, I.M. (ISBN 9780817636777)
• Basic Mathematics, by Lang, S. (ISBN 0387967877)
• Calculus (3rd Edition), by Spivak, M. (ISBN 0521867444)
• Calculus: Early Transcendentals (6th Edition), by Stewart, J. (ISBN 0495011665)
• Classic Set Theory, by Goldrei, D. (ISBN 0412606100)
• Discrete Mathematics and its Applications (5th Edition), by Rosen, K. H. (ISBN 0072424346)
• Discrete Mathematics and its Applications (6th Edition), by Rosen, K. H. (ISBN 0072880082)
• Introduction to Mathematical Statistics (5th Edition), by Hogg, R. and Craig, A. (ISBN 8178086301)
• Mathematical Methods in the Physical Sciences (3rd Edition), by Mary L. Boas (ISBN-13: 978-0471198260)
• Principles of Mathematical Analysis (3rd edition), by Walter Rudin (ISBN 0070856133)
• Probability and Statistics for Engineering and the Sciences (7th ed), by Jay L. Devore (ISBN-10: 0-495-38217-5)
• Proofs and Fundamentals, by Ethan Bloch (ISBN 0817641114)
Last modified on 11 June 2012, at 18:53 | {"url":"http://en.m.wikibooks.org/wiki/Solutions_To_Mathematics_Textbooks","timestamp":"2014-04-21T07:25:53Z","content_type":null,"content_length":"21469","record_id":"<urn:uuid:b581a8de-964f-42cc-acb8-6098263f645a>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00112-ip-10-147-4-33.ec2.internal.warc.gz"} |
"Moving flaps" for different flow configurations
Short description and remarks
The following simulation addresses the problem of two-dimensional channel flow around two moving flaps. From the left side, a parabolic flow of different velocity magnitude (v=1 or 4) is injected.
The movement of the flaps is explicitly prescribed to be periodic and continuous, its angular velocity w being in the range of w=0.5 up to w=8. The flaps are moving with the same speed and can
overlap with the channel walls. The minimum distance between the flaps is 0.6 as compared to the total channel height of 3. The movement of the flaps is prescribed by the "Fictitious Boundary
Conditions" only: All calculations are performed on ONE SINGLE mesh! Due to the nature of the calculation, it is difficult to visualize streamfunction and pressure plots because of possible
singularities at the boundaries. Therefore, only velocity plots are presented here. The simulation is a pre-study for a planned simulation of heart flap movement and was performed as part of an
advanced laboratory study ("Fortgeschrittenen-Praktikum") at Heidelberg 1998 by the author of this document and by S. Turek. Click here for some detailed comparisons of the simulated configurations.
All simulations were done on level 6 unless noted otherwise.
The angular velocity of the flaps (in multiples of Pi) is denoted by "w" , while the inflow velocity is denoted by "v". | {"url":"http://www.featflow.de/album/catalog/movingflaps_2d/data.html","timestamp":"2014-04-18T23:36:56Z","content_type":null,"content_length":"13795","record_id":"<urn:uuid:c71929f5-b7f6-4c71-b441-3eca0f37fd3c>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00055-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality
The Annals of Probability
The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality
Let $\hat{F}_n$ denote the empirical distribution function for a sample of $n$ i.i.d. random variables with distribution function $F$. In 1956 Dvoretzky, Kiefer and Wolfowitz proved that $P\big(\sqrt
{n} \sup_x(\hat{F}_n(x) - F(x)) > \lambda\big) \leq C \exp(-2\lambda^2),$ where $C$ is some unspecified constant. We show that $C$ can be taken as 1 (as conjectured by Birnbaum and McCarty in 1958),
provided that $\exp(-2\lambda^2) \leq \frac{1}{2}$. In particular, the two-sided inequality $P\big(\sqrt{n} \sup_x|\hat{F}_n(x) - F(x)| > \lambda\big) \leq 2 \exp(-2\lambda^2)$ holds without any
restriction on $\lambda$. In the one-sided as well as in the two-sided case, the constants cannot be further improved.
Article information
Ann. Probab. Volume 18, Number 3 (1990), 1269-1283.
First available: 19 April 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62E15: Exact distribution theory
Secondary: 62G15: Tolerance and confidence regions
Brownian bridge empirical process Kolmogorov-Smirnov statistics
Massart, P. The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality. The Annals of Probability 18 (1990), no. 3, 1269--1283. doi:10.1214/aop/1176990746. http://projecteuclid.org/euclid.aop/ | {"url":"http://projecteuclid.org/euclid.aop/1176990746","timestamp":"2014-04-16T04:17:59Z","content_type":null,"content_length":"29091","record_id":"<urn:uuid:28799d62-5633-4e16-88bc-3b7443e24e94>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00626-ip-10-147-4-33.ec2.internal.warc.gz"} |
Pulleys and Lifting - Important Formulas
PULLEYS AND LIFTING - IMPORTANT FORMULAS
V. Ryan © 2004 2010
When using pulleys for lifting the formulas for mechanical advantage and velocity ratio are very important. The formulas are shown below.
Mechanical advantage is defined as the ratio of load to effort. Pulley systems rely on this important relationship between load and effort. The formula seen below is best understood by writing it
within a triangle. This helps when it is necessary to change the formula to find either; mechanical advantage or the load or the effort. In this way three formulas can be generated from the single
formula inside the triangle.
Velocity Ratio (sometimes called movement ratio)- is defined as the ratio of the distance moved by the effort to the distance moved by the load. The formula seen below is best understood by writing
it within a triangle. This helps when it is necessary to change the formula to find either; velocity ratio or the distance moved by the load or the distance moved by effort. In this way three
formulas can be generated from the single formula inside the triangle.
CLICK HERE FOR GEARS AND PULLEYS INDEX PAGE | {"url":"http://www.technologystudent.com/gears1/pulle12.htm","timestamp":"2014-04-18T11:24:30Z","content_type":null,"content_length":"9106","record_id":"<urn:uuid:88b75beb-24ab-4bbf-9655-f06cab6ff0b8>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00063-ip-10-147-4-33.ec2.internal.warc.gz"} |
how to find the sum terms in of binomial expansion
2 Answers
Edited by
Ahmed A. Selman
on 11 Apr 2013
Accepted answer
No products are associated with this question.
how to find the sum terms in of binomial expansion
I need to find the sum of few terms in binomial expansion...more precisely i need to find the sum of this expression:
∑ (nCr) * p^r * q^(n - r)
and limits for summation are from r = 2 to 15. and n=15
any help would be most welcome.
0 Comments
The general steps to find such a summation are:
- Start a loop over r,
- Calculate each term as a function of (r),
- In the loop, add the terms one by one to a unique matrix,
- After the loop is finished, sum over the added terms.
The code should be something as :
% ∑ (nCr) * p^r * q^(n - r)
clc; clear all
p =...;
q =...;
n = 15;
i = 0;
for r = 2:15
i = i+1;
nCr = ...;% calculate the coefficients here
Terms(i) = nCr * p.^r .* q.^(n-r);
SumOfTerms = sum(Terms)
The output is in (SumOfTerms), which should be a single value.
I assumed that (nCr) is not a constant, as I expect, it must be a function of (n and r). If it was a constant of (n and r), then define it outside the loop.
I also didn't understand the meaning of ( *| ) at the beginning and end of the formula in your question. If they mean some operation s ( complex conjugate or absolute) then add any of the commands at
the very end of the code:
finalSum = conj(SumOfTerms);% for complex conjugate value.
finalSum = real(SumOfTerms);% for real value.
and in this case the output will be in (finalSum).
If you tried it and it didn't work, please let me know in which line it made an error, and the error message.
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@ Image Analyst : Thank You Sir.
I am new to Matlab, so i did not inspect the Workspace.My answer is listed there in Workspace.I was only bothering the behavior of Command Window.
@ Ahmed A Selman:
thank You Mr. Ahmed A Selman for helping me out.
@ Ahmed A. Selman
Sir, the code you provided works well...Can you please explain me how this loop is executing?
With that range of r I would think it would be more efficient to compute
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Axiom of choice
From Uncyclopedia, the content-free encyclopedia
The Axiom of Choice (a.k.a. AC) is the classiest, most desired axiom of set theory (hence its name). At some point, the other axioms rebelled against it in jealous rage, leaving mathemagicians with
no longer any constructive way to use its high-horse results. It now lives independently from those other axioms (if you know what I mean), and prolonged consideration of its truth will probably
result in Christians knocking down your door.
edit Formal Statement
For any set of disjoint pregnancies, the mother is able to choose one child from each pregnancy to abort or not to abort. Naturally, this was found highly offensive by Pro-Lifers and Pro-Deathers,
but was finally upheld in Roe vs. Wade (the issue of twin or triplet abortions being addressed somewhat later in Roe vs. Wade vs. Paddle). In finalizing that decision, the Supreme Court had to
consider, among other sets of possible pregnancies, sets that contained themselves or that contained an infinite nest of abortable babies-within-babies, making it a historic landmark decision.
The Axiom of Choice was accidentally discovered by the Irish mathematician James Choice in 1939. Its first formulation was deeply embedded in the undergraduate novel Finnegans Wedge, but since no one
was able to decipher the symbolism of the book, the discovery went unnoticed until 1977.
In the distant future of nonstandard mathematics, AC will be replaced by MULTIVAC (MULTIple Vacuous Axioms of Choice), which will allow us to choose, in turn, from different Axioms of Choices
themselves. MULTIVAC, which first appeared as a hypothetical supercomputer in an Asimov et al paper on LAAThSFSLPhJ, will have the astounding property of separating milk from coffee by reversing
entropy. (George M. Cohen successfully wrote a song in 1963 about how espresso may be chosen independently of milk and sugar.)
edit Equivalent Formulations
The Axiom of Choice has an uncountable number of equivalent formulations (of which a bullshit attribute can ironically be selected from each one). For the sake of ennumerability, we will list the
following ones for no apparent reason:
• Tychonoff's Theorem: the product of compact discs is always a compact disc
• The Existence of Marxist Ideals: every ideal actually belongs to a greater Communist Ideal.
• The Existence of Bases: every baseball field has at least one base.
• Hausdorff Maximal Principle: Every partially-ordered meal contains a totally-ordered side dish.
• Well-ordering Principle: Any Chinese meal is well-ordered with rice (to which the Chinese Remainder Theorem is a corollary).
• Porn's Lemma: every non-empty porn site in which every chained girl has an upper bound contains at least one maximally annoying popup window
• The Resistance of Vector Bases: all your vector base are belong to us
• Lobotomy's Law: either I do not know this law, or you are too stupid to understand it, or both of them.
• Tarski's Theorem: For every infinite A, we have A=A^2, which in turn implies A=0 or A=.999999999
• Zorro's Lemma: no matter how bad the current situation is, Zorro will get out of prison and kick Sgt. Garcia's ass.
• The Axiom of Choice for Dummies™: among an infinite number of pairs of nonchiral socks, one can select one sock out of every pair, even though nobody knows how to go about doing this. (Note that
the above is obviously false for a finite number of nonchiral socks, as there is always a missing one.)
edit Weak-Ass Formulations
These are not as cool as the Axiom of Choice, but rely on it for dear life and generally hang out in the same clique.
• Banach-Tarski Paradox: an implementation of the Fundamental Principle of Wiki - "duplicate material excessively".
• From Category Theory: Every small category has a skeleton in its closet.
• Ultrafilter Lemma: Every toilet filter able to block out your bullshit is, in fact, contained within a maximal filter: a plug. | {"url":"http://uncyclopedia.wikia.com/wiki/Axiom_of_choice","timestamp":"2014-04-19T10:56:18Z","content_type":null,"content_length":"37122","record_id":"<urn:uuid:39fb8589-0a1f-4862-a48d-34a227d35450>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00499-ip-10-147-4-33.ec2.internal.warc.gz"} |
Q. A person while swimming crosses a 200 m wide river in a minimum time 40 seconds and when he crosses the river... - Homework Help - eNotes.com
Q. A person while swimming crosses a 200 m wide river in a minimum time 40 seconds and when he crosses the river straight to opposite bank he took 50 seconds.
A)The speed of the man is 5m/s.
B)The speed of the river water is 3 m/s.
C)The horizontal drift of the man while crossing the river in 40 s is 160 m.
D)The speed of the man is 4m/s.
Let the river run from West to East with parallel sides 200m apart. Let the swimmer begin at A and end at B. Note that B is some distance downstream from A due to the current. Then let the swimmer
cross to C perpendicular from B.
The swimmers speed is 5m/s as all effort from A to B is expended perpendicular to the bank. (It is the least time.) Since he crosses 200m in 40 sec he travels at 5m/s.
When crossing from B to C he must aim at an angle `theta` upstream. Thus his forward progress is `5costheta` (cosine of `/_ABC` ). Since we know his time from B to C we know that `5costheta=4 ==>
costheta=4/5` or `theta~~36.9^@` .
Since `costheta=4/5` we find that ``AB=250m and AC=150m.
The current's speed can be found since the swimmer went downstream 150m in 40 sec so the current is 3.75m/s.
Check: the trip from A to B has 2 components: 5m/s directly across the river and 3.75m/s downstream with a resultant 6.25m/s. The distance from A to B is 250m, covered in 40 sec which is a resultant
speed of 6.25m/s.
So A is correct.
B is incorrect -- the current is 3.75m/s
C is incorrect as the horizontal drift is 150m
D is incorrect as A is correct.
The correct answer is A
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