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sketching a function
September 14th 2009, 03:40 PM
sketching a function
Hi, been a little while since ive done some math, and need help with a probably basic problem.
It asks to sketch p=(1+x)^2 & p=1+2x for -1<x<1
i would assume the first one is just like the parabola x^2 shifted to the left 1 unit. but with the restricted domain when i pop it in my graphic calc to check it looks bizarre. So, what does it
look like?
September 14th 2009, 05:57 PM
Hi, been a little while since ive done some math, and need help with a probably basic problem.
It asks to sketch p=(1+x)^2 & p=1+2x for -1<x<1
i would assume the first one is just like the parabola x^2 shifted to the left 1 unit. but with the restricted domain when i pop it in my graphic calc to check it looks bizarre. So, what does it
look like?
graph of both attached. domain restricted -1 < x < 1 ...
September 14th 2009, 06:13 PM
Hi, thanks thats great! could you remind me, what steps in order to get this?
Also, there is a follow up question asking what range of values of v is the second function good within 5%? and you have to be accurate with a calc.. hmm | {"url":"http://mathhelpforum.com/pre-calculus/102272-sketching-function-print.html","timestamp":"2014-04-16T04:17:56Z","content_type":null,"content_length":"5004","record_id":"<urn:uuid:6576abb3-6d71-4756-8584-a8ceb2c484bb>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00061-ip-10-147-4-33.ec2.internal.warc.gz"} |
Partially Additive Categories and Fully Complete Models of Linear Logic
, 2000
"... this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics
in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by S ..."
Cited by 44 (10 self)
Add to MetaCart
this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the
symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by Stefanescu (Stefanescu 2000).) However, the first author realized, following a stimulating
discussion with Gordon Plotkin, that traced monoidal categories provided a common denominator for the axiomatics of both the Girard-style and Abramsky-Jagadeesan-style versions of the Geometry of
Interaction, at the basic level of the multiplicatives. This insight was presented in (Abramsky 1996), in which Girard-style GoI was dubbed "particle-style", since it concerns information particles
or tokens flowing around a network, while the Abramsky-Jagadeesan style GoI was dubbed "wave-style", since it concerns the evolution of a global information state or "wave". Formally, this
distinction is based on whether the tensor product (i.e. the symmetric monoidal structure) in the underlying category is interpreted as a coproduct (particle style) or as a product (wave style). This
computational distinction between coproduct and product interpretations of the same underlying network geometry turned out to have been partially anticipated, in a rather di#erent context, in a
pioneering paper by E. S. Bainbridge (Bainbridge 1976), as observed by Dusko Pavlovic. These two forms of interpretation, and ways of combining them, have also been studied recently in (Stefanescu
2000). He uses the terminology "additive" for coproduct-based (i.e. our "particle-style") and "multiplicative" for product-based (i.e. our "wave-style"); this is not suitable for our purposes,
because of the clash with Linear Logic term...
- Mathematical Structures in Computer Science , 2001
"... We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of K othe sequence spaces. In this
setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The ..."
Cited by 31 (9 self)
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We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of K othe sequence spaces. In this setting,
the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings.
This work provides a simple setting where typed -calculus and dierential calculus can be combined; we give a few examples of computations. 1
, 2005
"... We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units
which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a v ..."
Cited by 4 (1 self)
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We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which
applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a version of partial traces and trace ideals (related to previous work of Abramsky, Blute, and
Panangaden); (ii) we do not require the existence of a reflexive object for our interpretation (the original GoI 1 and 2 were untyped and hence involved a bureaucracy of domain equation
isomorphisms); (iii) we introduce an abstract notion of orthogonality (related to work of Hyland and Schalk) and use this to develop a version of Girard’s theory of types, datum and algorithms in our
setting, (iv) we prove appropriate Soundness and Completeness Theorems for our interpretations in partially traced categories with orthogonality; (v) we end with an application to completeness of
(the original) untyped GoI in a unique decomposition category. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1982572","timestamp":"2014-04-17T06:26:08Z","content_type":null,"content_length":"19155","record_id":"<urn:uuid:330bb5d8-6969-4190-ba3c-ad9bdce92241>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00279-ip-10-147-4-33.ec2.internal.warc.gz"} |
Permutations of 1234567890
Date: 09/23/2001 at 06:09:19
From: Leeanna blyton
Subject: Combinations
I'm trying to find a pattern in combinations and how many combinations
there are in 1234567890.
I have found that
7 = 1 46 = 2 123 = 6 4567 = 24
I am now trying to find out how many combinations there are in
5-, 6-, 7-, 8-, 9-, and 10-digit numbers.
Date: 09/23/2001 at 07:19:34
From: Doctor Mitteldorf
Subject: Re: Combinations
Dear Leeanna,
You're doing it just right. Doing examples with small numbers of
digits will help you to see a pattern, and then you'll know how to
calculate the combinations for any number of digits. (Actually, these
are "permutations," not "combinations." For a review of Permutations
and Combinations, see the Dr. Math FAQ:
http://mathforum.org/dr.math/faq/faq.comb.perm.html )
The hard part about it is that the numbers get big so fast. You can
list the permutations for 4 without too much trouble, but there are
enough of them for 5 that you might have trouble listing them all
without making mistakes. So the key is to come up with some kind of
system so you're sure you have them all.
Here's a suggestion for a system. Say you want to find the
permutations of the digits 12345. Let's keep the 5 anchored at the end
and list all the permutations that end in 5. Well, what you're left
with is the digits 1234, and you can put them in any order. So you
know how many there will be - you just did this problem, and found 24.
Now let's count how many there are that end in 4. The other digits are
1235, and they can be in any order. But this is really the same as the
problem you just did: there are 24 ways to order the numbers 1235, so
there are 24 permutations of the digits 12345 that all end in 4.
Now you're starting to see a pattern, I think. Follow the reasoning
and you can count how many permutations there are going to be
altogether without having to list them all.
Once you've done this, you know the answer for 5 digits. Let's move on
to 6. Can you apply the same kind of reasoning? How many permutations
are there that end in 6? Well, you just calculated that by your astute
and careful reasoning in the problem above. Then you list each of the
other digits that the permutations could end in, and you have an idea
how to get the total for 6 digits.
Getting really abstract, you might formulate a rule: If I know how
many permutations there are for N digits, then I can figure how many
there are for N+1 digits. Describe the rule for how you do that. Make
a chart, like the one you started, for the number of permutations of
1,2,3,4,5 6 and 7 digits.
(Now I don't want you to peek. Finish the whole process that I've
described above, and understand the patterns that you see. Only after
you understand just what you've done and why it comes out the way it
does, go to your dictionary and look up the word "factorial.")
- Doctor Mitteldorf, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/56235.html","timestamp":"2014-04-20T04:14:45Z","content_type":null,"content_length":"7945","record_id":"<urn:uuid:cf8c7993-6390-438a-b98a-29bac48eba76>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00252-ip-10-147-4-33.ec2.internal.warc.gz"} |
Gary Garber's Blog
If you were to ask a second grader, they might tell you that gravity is down and buoyancy is up. For a long time, my son thought gravity pulled his bicycle down the hill and buoyancy was responsible
for it to go up the hill. Not quite.
But what is buoyancy anyways? A textbook definition might be that the force of buoyancy is equal to the weight of the displaced fluid. A way to think of this is that classically, we cannot have two
objects occupy the same space. Do if an objects sinks into a fluid (liquid OR gas) then it is taking up space and that fluid is pushed up. This is known as Archimedes Principle.
That displaced fluid wants to be as close to the ground as it can get. The fluid wants to be in a state of lower gravitational potential energy. The only way for the fluid to be lower is to push
that invasive object out of the way, or UP! Thus we have the force of buoyancy.
Your text might present the equation for buoyancy as
F = ρgV
Now there is some lack of clarity as to what these variable represent. To be correct, ρ is the density of the displaced fluid, g is the acceleration due to gravity, and V is the volume of the
displaced fluid. But to clarify, we need to know if the object is floating on the top of the surface of the fluid or completely submerged.
If the object is 100% submerged, then we can assume the volume of the displaced fluid is equal to the volume of the object itself. This works nice whether or not the net force on the 100% submerged
object is zero or not.
However, if the object is floating on the surface, the volume of the objects is NOT equal to the volume of the displaced fluid. In this case, a neat trick we can use is to assume the net force
equals zero (otherwise it would be sinking, not floating). Thus the force of buoyancy is equal to the weight of the object.
F = mg
Where m is the mass of the object. Now using our equation for density where
ρ = m/V,
we can equate the mass of the object equal to the density of the object times the volume
Substituting this in, we actually get the force of gravity (and thus the buoyancy) depending on the density and volume of the object, even if the equation is deceptively similar.
F = ρgV
Some things to consider is how could you experimentally measure the force of buoyancy on an object in water?
Where do you experience the buoyant force if you want to describe this to your little brother or sister?
What happens to the buoyant force on a SCUBA diver as the descend and ascend?
You can play around with this PHET simulation to gain some experience with buoyancy. | {"url":"http://blogs.bu.edu/ggarber/archive/bua-py-25/states-of-matter-and-thermodynamics/bouyancy/","timestamp":"2014-04-17T09:59:22Z","content_type":null,"content_length":"28645","record_id":"<urn:uuid:b5eeeb16-086d-424e-ba57-9811368369bb>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00255-ip-10-147-4-33.ec2.internal.warc.gz"} |
Weeknumbers using VBA in Microsoft Excel
by admin About
The function WEEKNUM() in the Analysis Toolpack addin calculates the correct week number for a given date,
if you are in the U.S. The user defined function below will calculate the correct week number depending
on the national language settings on your computer.
Function UDFWeekNum(InputDate As Date)
UDFWeekNum = DatePart("ww", InputDate, vbUseSystemDayOfWeek, vbUseSystem)
End Function
The function above can also be modified to calculate the weeknumber the European way:
Function UDFWeekNumISO(InputDate As Date)
UDFWeekNumISO = DatePart("ww", InputDate, vbMonday, vbFirstFourDays)
End Function
The two functions above can, due to a bug, return a wrong week number. This occurs for dates around
New Year for some years, e.g. the years 1907, 1919, 1991, 2003, 2007, 2019 and 2091. You can use this
worksheet formula to calculate the correct week number (Thanks to George Simms, [email protected],
for pointing this out):
=INT((A1-(DATE(YEAR(A1+(MOD(8-WEEKDAY(A1),7)-3)),1,1))-3+ MOD(WEEKDAY(DATE(YEAR(A1+(MOD(8-WEEKDAY(A1),7)-3)),1,1))+1,7))/7)+1
The formula above assumes that cell A1 contains a valid date for which you want to return the week number.
To calculate the correct week number with a user-defined VBA function, you can use the function below:
Function WEEKNR(InputDate As Long) As Integer
Dim A As Integer, B As Integer, C As Long, D As Integer
WEEKNR = 0
If InputDate < 1 Then Exit Function
A = Weekday(InputDate, vbSunday)
B = Year(InputDate + ((8 - A) Mod 7) - 3)
C = DateSerial(B, 1, 1)
D = (Weekday(C, vbSunday) + 1) Mod 7
WEEKNR = Int((InputDate - C - 3 + D) / 7) + 1
End Function | {"url":"http://www.exceltip.com/custom-functions/weeknumbers-using-vba-in-microsoft-excel.html","timestamp":"2014-04-18T20:55:04Z","content_type":null,"content_length":"19172","record_id":"<urn:uuid:1185e2b1-4379-4719-8859-b39249e4d4ef>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00305-ip-10-147-4-33.ec2.internal.warc.gz"} |
A few problems encountered in exam revision
January 19th 2010, 11:44 AM
A few problems encountered in exam revision
Hi. Any help with these would be much appreciated.
1. By considering the infinite series (SUMMATION SIGN with infinity on top and n=0 on bottom) of x^n for x< 1 show that:
(SUMMATION SIGN with infinity on top and n=0 on bottom) (n^2)(x^n) = x(1-x^2)/(1-x)^4.
I know this is to do with differentiating the previous derivative (x/(1-x)^2), but i got in a mess with it.
2.Find the indefinate integral of x^2*e^(x^3)
This is integration by parts but the x^3 is confusing me as i don't know how to deal with it. Basically my question is; how do you integrate e^(x^3)?
3. Use your calculator to find the value of the following expression when x=0.01 to 3dp:
y(x) = (e^(2x) - 2(1 +2x)^(1/2) +1) / (cos (x/2) -1)
The previous part of the question asked me to obtain Maclaurin series and re-write the above expression which i did. However when i substituted x=0.01 the 2 answers from the series and the exact
did not match. I maybe approaching this question incorrectly. What is the meothod for this
Again any help would be much appreciated.
January 19th 2010, 12:02 PM
Hi. Any help with these would be much appreciated.
1. By considering the infinite series (SUMMATION SIGN with infinity on top and n=0 on bottom) of x^n for x< 1 show that:
(SUMMATION SIGN with infinity on top and n=0 on bottom) (n^2)(x^n) = x(1-x^2)/(1-x)^4.
I know this is to do with differentiating the previous derivative (x/(1-x)^2), but i got in a mess with it.
This is what I would do
$\sum_{n=0}^{+\infty} x^n = \frac{1}{1-x}$
Differentiating with respect to x
$\sum_{n=0}^{+\infty} (n+1) x^n = \frac{1}{(1-x)^2}$
Differentiating again with respect to x
$\sum_{n=0}^{+\infty} (n+2)(n+1) x^n = \frac{2}{(1-x)^3}$
Using $(n+2)(n+1) = n^2 + 3n + 2 = n^2 + 3(n+1) - 1$ or $n^2 = (n+2)(n+1) - 3(n+1) + 1$ you are able to find
$\sum_{n=0}^{+\infty} n^2 x^n$
January 19th 2010, 12:12 PM
No need for integration by parts.
the substitution $u=x^3$ will solve it easily :) .
By the way, It is not easy to evaluate $\int e^{(x^3)} dx$
since it has unelementary functions.
see this:
integrate e^(x^3) - Wolfram|Alpha | {"url":"http://mathhelpforum.com/calculus/124447-few-problems-encountered-exam-revision-print.html","timestamp":"2014-04-17T21:40:10Z","content_type":null,"content_length":"8642","record_id":"<urn:uuid:40da0f41-d518-45e1-a1cc-7d33d70fcdfe>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00446-ip-10-147-4-33.ec2.internal.warc.gz"} |
[FOM] Infinity and the "Noble Lie"
praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Sat Jan 7 02:21:37 EST 2006
> > "Are you prepared to say that the question of the "truth" of an
> > arithmetical statement proved using the axiom of infinity is also
> > ridiculous?"
I wonder whether everyone here is using the notion of "an axiom of
infinity" in the same sense.
Often, in logic, it means any sentence which forces the domain to be
infinite (also the standard axioms of successor are together an axiom of
infinity in this sense). In set theory, on the other hand, the axiom of
infinity is the axiom which says that there is an infinite set. The axioms
of ZFC without this axiom* already make domain infinite, but it is this
axiom which gives ZFC its extreme power. It is much stronger assumption
than an axiom of infinity in the first sense.
(There may be also other senses...)
[*This is an infinite set of axioms, but one can give a finite
conservative extension of them. The latter (or, a conjunction of them) is
an axiom of infinity in the first sense, but obviously, not in the second
Best, Panu
Panu Raatikainen
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy
P.O. Box 9
FIN-00014 University of Helsinki
E-mail: panu.raatikainen at helsinki.fi
More information about the FOM mailing list | {"url":"http://www.cs.nyu.edu/pipermail/fom/2006-January/009528.html","timestamp":"2014-04-20T23:30:43Z","content_type":null,"content_length":"3854","record_id":"<urn:uuid:380b9784-b122-4f68-ae7f-250bf5c12440>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00579-ip-10-147-4-33.ec2.internal.warc.gz"} |
Abelian group
October 20th 2009, 02:28 PM #1
Sep 2009
Abelian group
a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then $(ab)^{mn}= e$ .
b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).
c) Give an example of an Abelian group G and elements a and b in G such that o(ab) $eq\$o(a)o(b). Compare part (b)
I suspect I could figure out, or at least know where to start, part b if I knew part a (If t is an integer, then $a^t=e$ iff n is a divisor of t)
But, I just don't know what to do with part a...
From what I understand, $a^m=e$ and $b^n=e$
I have done all sorts of random inserts of $a^m$, $a^m * a^-m$ , etc. Nothing seems to give me what I need though... erm, I was wondering, since $a^m=e$ does $a^{-m}=e$?
a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then $(ab)^{mn}= e$ .
b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).
c) Give an example of an Abelian group G and elements a and b in G such that o(ab) $eq\$o(a)o(b). Compare part (b)
I suspect I could figure out, or at least know where to start, part b if I knew part a (If t is an integer, then $a^t=e$ iff n is a divisor of t)
But, I just don't know what to do with part a...
From what I understand, $a^m=e$ and $b^n=e$
I have done all sorts of random inserts of $a^m$, $a^m * a^-m$ , etc. Nothing seems to give me what I need though... erm, I was wondering, since $a^m=e$ does $a^{-m}=e$?
The trick is that in an abelian group, $(ab)^m=a^mb^m$...as simple as that!
a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then $(ab)^{mn}= e$ .
b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).
c) Give an example of an Abelian group G and elements a and b in G such that o(ab) $eq\$o(a)o(b). Compare part (b)
I suspect I could figure out, or at least know where to start, part b if I knew part a (If t is an integer, then $a^t=e$ iff n is a divisor of t)
But, I just don't know what to do with part a...
From what I understand, $a^m=e$ and $b^n=e$
I have done all sorts of random inserts of $a^m$, $a^m * a^-m$ , etc. Nothing seems to give me what I need though... erm, I was wondering, since $a^m=e$ does $a^{-m}=e$?
1) $a^m=e, b^n=e$, which follows $a^{mn}=e, b^{mn}=e$. Since G is abelian, $(ab)^{mn}=a^{mn}b^{mn}=e$.
2) Let o(a)=m, o(b)=n. By using the basic number theory property, we see that mn=gcd(m,n)lcm(m,n). We know that $a^{lcm(m,n)}=e$ and $b^{lcm(m,n)}=e$. Since G is abelian, it follows that $(ab)^
{lcm(m,n)}=e$. Thus o(ab) | lcm(m,n) and o(ab)|mn.
3) Check 2 and 3 in $\mathbb{Z}_4$.
October 20th 2009, 10:15 PM #2
Oct 2009
October 20th 2009, 10:26 PM #3
Senior Member
Nov 2008 | {"url":"http://mathhelpforum.com/advanced-algebra/109281-abelian-group.html","timestamp":"2014-04-19T02:26:09Z","content_type":null,"content_length":"43336","record_id":"<urn:uuid:519a2fd3-cb58-4051-85c1-92553a82f90d>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00328-ip-10-147-4-33.ec2.internal.warc.gz"} |
Game Theory
Welcome to E-Books Directory
This page lists freely downloadable books.
Game Theory
E-Books for free online viewing and/or download
e-books in this category
Game Theory Relaunched
by Hardy Hanappi (ed.) - InTech , 2013
New simulation tools and network analysis have made game theory omnipresent these days. This book collects recent research papers in game theory, which come from diverse scientific communities all
across the world, and combine many different fields.
(1312 views)
Introduction to Game Theory
by Christian Julmi - BookBoon , 2012
This textbook provides an overview of the field of game theory which analyses decision situations that have the character of games. The book is suitable as an introductory reading and is meant to
sharpen the reader's strategic thinking abilities.
(2838 views)
Games in Verification
by Moshe Y. Vardi - ESSLLI , 2001
Games have shown to provide a useful paradigm for reasoning about reactive systems. This text demonstrates the power of the game-theoretic approach, by showing how it gives rise to a unifying
algorithmic framework through the use of tree automata.
(3516 views)
Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations
by Yoav Shoham, Kevin Leyton-Brown - Cambridge University Press , 2008
Multiagent systems consist of multiple autonomous entities having different information and diverging interests. This comprehensive introduction to the field offers a computer science perspective,
but also draws on ideas from game theory.
(5762 views)
Algorithmic Game Theory
by Noam Nisan, at al. - Cambridge University Press , 2007
The subject matter of Algorithmic Game Theory covers many of the hottest area of useful new game theory research, introducing deep new problems, techniques, and perspectives that demand the attention
of economists as well as computer scientists.
(6705 views)
Game Theory: A Nontechnical Introduction to the Analysis of Strategy
by Roger McCain - Drexel University , 2010
Striking an appropriate balance of mathematical and analytical rigor, this book teaches by example. Learners typically relate better to examples from their own fields, and McCain provides
illustrations everyone can relate to.
(8539 views)
An Introduction to Quantum Game Theory
by J. Orlin Grabbe - arXiv , 2005
This essay gives a self-contained introduction to quantum game theory, and is primarily oriented to economists with little or no acquaintance with quantum mechanics. It assumes little more than a
basic knowledge of vector algebra.
(3807 views)
More Games of No Chance
by Richard J. Nowakowski - Cambridge University Press , 2002
This book is a state-of-the-art look at combinatorial games, that is, games not involving chance or hidden information. The book contains articles by some of the foremost researchers and pioneers of
combinatorial game theory.
(4751 views)
Games of No Chance 3
by Michael H. Albert, Richard J. Nowakowski - Cambridge University Press , 2009
This fascinating look at combinatorial games, that is, games not involving chance or hidden information, offers updates on standard games such as Go and Hex, on impartial games, and on aspects of
games with infinitesimal values.
(8609 views)
The Compleat Strategyst: Being a Primer on the Theory of Games of Strategy
by John D. Williams - RAND Corporation , 2007
When this book was originally published in 1954, game theory was an esoteric and mysterious subject. Its popularity today can be traced at least in part to this book, which popularized the subject
for amateurs and professionals throughout the world.
(7685 views)
Games of Strategy: Theory and Applications
by Melvin Dresher - RAND Corporation , 1961
This book introduces readers to the basic concepts of game theory and its applications for military, economic, and political problems, as well as its usefulness in decisionmaking in business,
operations research, and behavioral science.
(6932 views)
Strategy and Conflict: An Introductory Sketch of Game Theory
by Roger A. McCain , 2007
This document may be useful as a means of making the basic ideas a bit more accessible. The author tried to limit these pages to fairly elementary topics and to avoid mathematics other than numerical
tables and a very little algebra.
(6137 views)
Game Theory
by Thomas S. Ferguson - UCLA , 2008
In this text, the author presents various mathematical models of games and study the phenomena that arise. The book covers impartial combinatorial games, two-person zero-sum games, two-person
general-sum games, and games in coalitional form.
(13641 views)
Graduate-Level Course in Game Theory
by Jim Ratliff , 1997
Lecture notes from a game-theory course the author taught to students in their second year of the economics PhD program. The material is also helpful to first-year PhD students learning game theory
as part of their microeconomic-theory sequence.
(8245 views)
Strategic Foundations of General Equilibrium
by Douglas Gale - Cambridge University Press , 2000
This is a book on strategic foundations of the theory of competition. Using insights from game theory, the author develops a model to explain what actually goes on in markets and how a competitive
general equilibrium is achieved.
(5758 views)
Games, Fixed Points and Mathematical Economics
by Christian-Oliver Ewald , 2007
These are lecture notes for a course in game theory. The text covers general concepts of two person games, Brouwer’s fixed point theorem and Nash’s equilibrium theorem, more general equilibrium
theorems, cooperative games and differential games.
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Posts about science on The Wandering Glitch
Last year, a company called DWave Systems announced their quantum computer (the ‘Orion’) – another milestone on the road to practical quantum computing. Their controversial claims seem worthy in
their own right but they are particularly important to the semantic web (SW) community. The significance to the SW community was that their quantum computer solved problems akin to Grover’s Algorithm
speeding up queries of disorderly databases.
Semantic web databases are not (completely) disorderly and there are many ways to optimize the search for matching triples to a graph pattern. What strikes me is that the larger the triple store, the
more compelling the case for using some kind of quantum search algorithm to find matches. DWave are currently trialing 128qbit processors, and they claim their systems can scale, so I (as a layman)
can see no reason why such computers couldn’t be used to help improve the performance of queries in massive triple stores.
What I wonder is:
1. what kind of indexing schemes can be used to impose structure on the triples in a store?
2. how can one adapt a B-tree to index each element of a triple rather than just a single primary key – three indexes seems extravagant.
3. are there quantum algorithms that can beat the best of these schemes?
4. is there is a place for quantum superposition in a graph matching algorithm (to simultaneously find matching triples then cancel out any that don’t match all the basic graph patterns?)
5. if DWave’s machines could solve NP-Complete problems, does that mean that we would then just use OWL-Full?
6. would the speed-ups then be great enough to consider linking everyday app data to large scale-upper ontologies?
7. is a contradiction in a ‘quantum reasoner’ (i.e. a reasoner that uses a quantum search engine) something that can never occur because it just cancels out and never appears in the returned
triples? Would any returned conclusion be necessarily true (relative to the axioms of the ontology?)
Any thoughts?
DWave are now working with Google to help them improve some of their machine learning algorithms. I wonder whether there will be other research into the practicality of using DWave quantum
computing systems in conjunction with inference engines? This could, of course, open up whole new vistas of services that could be provided by Google (or their competitors). Either way, it gives me a
warm feeling to know that every time I do a search, I’m getting the results from a quantum computer (no matter how indirectly). Nice.
Weird Wired Clouds
Wired has a very interesting article on strange or rare weather formations.
Here’s an example – morning glory clouds from Cape York, Australia.
LinqToRdf v0.7.1 and RdfMetal
I’ve just uploaded version 0.7.1 of LinqToRdf. This bug fix release corrects an issue I introduced in version 0.7. The issue only seemed to affect some machines and stems from the use of the GAC by
the WIX installer (to the best of my knowledge). I’ve abandoned GAC installation and gone back to the original approach.
Early indications (Thanks, Hinnerk) indicate that the issue has been successfully resolved. Please let me know if you are still experiencing problems. Thanks to 13sides, Steve Dunlap, Hinnerk
Bruegmann, Kevin Richards and Paul Stovell for bringing it to my attention and helping me to overcome the allure of the GAC.
Kevin also reported that he’s hoping to use LinqToRdf on a project involving the Biodiversity Information Standards (TDWG). It’s always great to hear how people are using the framework. Please drop
me a line to let me know how you are using LinqToRdf.
Kevin might find feature #13 useful. It will be called RdfMetal in honour of SqlMetal. It will automate the process of working with remotely managed ontologies. RdfMetal will completely lower any
barriers to entry in semantic web development. You will (in principle) no longer need to know the formats, protocols and standards of the semantic web in order to consume data in it.
Here’s an example of the output it generated from DBpedia.org for the FOAF ontology:
./RdfMetal.exe -e:http://DBpedia.org/sparql -i -n http://xmlns.com/foaf/0.1/ -o foaf.cs
Which produced the following source:
namespace Some.Namespace
[assembly: Ontology(
BaseUri = "http://xmlns.com/foaf/0.1/",
Name = "MyOntology",
Prefix = "MyOntology",
UrlOfOntology = "http://xmlns.com/foaf/0.1/")]
public partial class MyOntologyDataContext : RdfDataContext
public MyOntologyDataContext(TripleStore store) : base(store)
public MyOntologyDataContext(string store) : base(new TripleStore(store))
public IQueryable<Person> Persons
return ForType<Person>();
public IQueryable<Document> Documents
return ForType<Document>();
// ...
[OwlResource(OntologyName="MyOntology", RelativeUriReference="Person")]
public partial class Person
[OwlResource(OntologyName = "MyOntology", RelativeUriReference = "surname")]
public string surname {get;set;}
[OwlResource(OntologyName = "MyOntology", RelativeUriReference = "family_name")]
public string family_name {get;set;}
[OwlResource(OntologyName = "MyOntology", RelativeUriReference = "geekcode")]
public string geekcode {get;set;}
[OwlResource(OntologyName = "MyOntology", RelativeUriReference = "firstName")]
public string firstName {get;set;}
[OwlResource(OntologyName = "MyOntology", RelativeUriReference = "plan")]
public string plan {get;set;}
[OwlResource(OntologyName = "MyOntology", RelativeUriReference = "knows")]
public Person knows {get;set;}
[OwlResource(OntologyName = "MyOntology", RelativeUriReference = "img")]
public Image img {get;set;}
[OwlResource(OntologyName = "MyOntology", RelativeUriReference = "myersBriggs")]
// ...
[OwlResource(OntologyName="MyOntology", RelativeUriReference="Document")]
public partial class Document
[OwlResource(OntologyName = "MyOntology", RelativeUriReference = "primaryTopic")]
public LinqToRdf.OwlInstanceSupertype primaryTopic {get;set;}
[OwlResource(OntologyName = "MyOntology", RelativeUriReference = "topic")]
public LinqToRdf.OwlInstanceSupertype topic {get;set;}
// ...
As you can see, it’s still pretty rough, but it allows me to write queries like this:
public void TestGetPetesFromDbPedia()
var ctx = new MyOntologyDataContext("http://DBpedia.org/sparql");
var q = from p in ctx.Persons
where p.firstName.StartsWith("Pete")
select p;
foreach (Person person in q)
Debug.WriteLine(person.firstName + " " + person.family_name);
RdfMetal will be added to the v0.8 release of LinqToRdf in the not too distant future. If you have any feature requests, or want to help out, please reply to this or better-still join the LinqToRdf
discussion group and post there. | {"url":"http://aabs.wordpress.com/category/science/","timestamp":"2014-04-21T14:45:01Z","content_type":null,"content_length":"31471","record_id":"<urn:uuid:068708ac-776e-442e-a6aa-f45bb74ca1e5>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00221-ip-10-147-4-33.ec2.internal.warc.gz"} |
Matches for: Author/Editor=(Cheer_A_Y)
Contemporary Mathematics
1993; 586 pp; softcover
Volume: 141
ISBN-10: 0-8218-5148-9
ISBN-13: 978-0-8218-5148-7
List Price: US$89
Member Price: US$71.20
Order Code: CONM/141
This book contains nearly all the papers presented at the AMS-IMS-SIAM Joint Summer Research Conference on Biofluiddynamics, held in July 1991, at the University of Washington, Seattle. The lead
paper, by Sir James Lighthill, presents a comprehensive review of external flows in biology. The other papers on external and internal flows illuminate developments in the protean field of
biofluiddynamics from diverse viewpoints, reflecting the field's multidisciplinary nature. For this reason, the book appeals to mathematicians, biologists, engineers, physiologists, cardiologists,
and oceanographers.
The papers highlight a number of problems that have remained largely unexplored due to the difficulty of addressing biological flow motions, which are often governed by large systems of nonlinear
differential equations and involve complex geometries. However, recent advances in computational fluid dynamics have expanded opportunities to solve such problems. These developments have increased
interest in areas such as the mechanisms of blood and air flow in humans, the dynamic ecology of the oceans, animal swimming and flight, to name a few. This volume addresses many of these flow
Biologists with interest in fluid dynamics, applied mathematicians, engineers, physiologists, and oceanographers.
• J. Lighthill -- Biofluiddynamics: A survey
• M. R. Koehl -- Hairy little legs: Feeding, smelling, and swimming at low Reynolds numbers
• M. W. Denny -- Disturbance, natural selection, and the prediction of maximal wave-induced forces
• L. J. Fauci -- Computational modeling of the swimming of biflagellated algal cells
• M. Murase -- Mechanical approach toward flagellar motility
• C. Loudon and D. N. Alstad -- Mechanical analysis of particle capture by rectangular-mesh nets
• S. L. Sanderson and A. Y. Cheer -- Fish as filters: An empirical and mathematical analysis
• C. S. Peskin and D. M. McQueen -- Computational biofluid dynamics
• J. Lighthill -- Acoustic streaming in the ear itself
• C. Kiris, S. Rogers, D. Kwak, and I.-D. Chang -- Computation of incompressible viscous flows through artificial heart devices with moving boundaries
• A. A. Mayo and C. S. Peskin -- An implicit numerical method for fluid dynamics problems with immersed elastic boundaries
• A. L. Fogelson -- Continuum models of platelet aggregation: Mechanical properties and chemically-induced phase transitions
• D. Halpern and J. B. Grotberg -- Surface-tension instabilities of liquid-lined elastic tubes
• E. B. Pitman, H. E. Layton, and L. C. Moore -- Dynamic flow in the nephron: Filtered delay in the TGF pathway
• H. Huang, V. J. Modi, and B. R. Seymour -- A new finite-difference scheme and its application to flows in stenosed arteries
• J. M. V. Rayner -- On aerodynamics and the energetics of vertebrate flapping flight
• G. R. Spedding -- On the significance of unsteady effects in the aerodynamic performance of flying animals
• H. de la Cueva and R. W. Blake -- Mechanics and energetics of ground effect in flapping flight
• D. Weihs -- Stability of aquatic animal locomotion
• C. P. van Dam, K. Nikfetrat, and P. M. H. W. Vijgen -- Lift and drag calculations for wings and tails: Techniques and applications
• S. A. Berger -- Flow in large blood vessels
• T. W. Secomb -- The mechanics of blood flow in capillaries
• C. G. Yam and H. A. Dwyer -- Unsteady flow in a curved pipe
• M. LaBarbera -- Optimality in biological fluid transport systems | {"url":"http://www.ams.org/cgi-bin/bookstore/booksearch?fn=100&pg1=CN&s1=Cheer_A_Y&arg9=A._Y._Cheer","timestamp":"2014-04-21T03:01:15Z","content_type":null,"content_length":"17512","record_id":"<urn:uuid:faaaac3b-cfe3-4408-ac7b-667ec527889a>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00316-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Catechism of the Steam Engine Summary
A.—With the velocity which a body would acquire by falling from the height of a homogeneous atmosphere, which is an atmosphere of the same density throughout as at the earth’s surface; and although
such an atmosphere does not exist in nature, its existence is supposed, in order to facilitate the computation. It is well known that the velocity with which water issues from a cistern is the same
that would be acquired by a body falling from the level of the head to the level of the issuing point; which indeed is an obvious law, since every particle of water descends and issues by virtue of
its gravity, and is in its descent subject to the ordinary laws of falling bodies. Air rushing into a vacuum is only another example of the same general principle: the velocity of each particle will
be that due to the height of the column of air which would produce the pressure sustained; and the weight of air being known, as well as the pressure it exerts on the earth’s surface, it becomes easy
to tell what height a column of air, an inch square, and of the atmospheric density, would require to be, to weigh 15 lbs. The height would be 27,818 feet, and the velocity which the fall of a body
from such a height produces would be 1,338 feet per second.
15. Q.—How do you determine the velocity of falling bodies of different kinds?
A.—All bodies fall with the same velocity, when there is no resistance from the atmosphere, as is shown by the experiment of letting fall, from the top of a tall exhausted receiver, a feather and a
guinea, which reach the bottom at the same time. The velocity of falling bodies is one that is accelerated uniformly, according to a known law. When the height from which a body falls is given, the
velocity acquired at the end of the descent can be easily computed. It has been found by experiment that the square root of the height in feet multiplied by 8.021 will give the velocity.
16. Q.—But the velocity in what terms?
A.—In feet per second. The distance through which a body falls by gravity in one second is 16-1/12 feet; in two seconds, 64-4/12 feet; in three seconds, 144-9/12 feet; in four seconds, 257-4/12
feet, and so on. If the number of feet fallen through in one second be taken as unity, then the relation of the times to the spaces will be as follows:—
Number of seconds | 1| 2| 3| 4| 5| 6|
Units of space passed through | 1| 4| 9|16|25|36| &c.
so that it appears that the spaces passed through by a falling body are as the squares of the times of falling.
17. Q.—Is not the urging force which causes bodies to fall the force of gravity?
A.—Yes; the force of gravity or the attraction of the earth.
18. Q.—And is not that a uniform force, or a force acting with a uniform pressure? | {"url":"http://www.bookrags.com/ebooks/10998/5.html","timestamp":"2014-04-18T19:14:08Z","content_type":null,"content_length":"34197","record_id":"<urn:uuid:7f2e2ace-19a8-47b2-84e0-362d9d1456f0>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00260-ip-10-147-4-33.ec2.internal.warc.gz"} |
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please explain thi i dont get it:( Which of the following are the x and y intercepts of the equation 4x - 2y = 8?
• one year ago
• one year ago
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change it to slope-intercept form first
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divide everything by -2, then you'll be in slope-intercept form
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y = 2x - 4*
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so now you can tell what the y intercept is. what is it?
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what is a y intercept?
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Basically, what you do is you change it to y=mx+c first. If we rearrange it like that, it'd look like this: \[y=2x-4\] Then you know that the y-intercept is -4. To find the x-intercept, you make
y=0 and solve for x. \[0=2x-4\] \[4=2x\] \[x=2\] So the y-intercept is -4 and the x-intercept is 2.
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ok what makes it a y intercept?
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We know that when y=mx+c. C is always the y-intercept and m is the slope :)
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ok so the answer is the last one?(3, 0) and (0, 4) (2, 0) and (0, 4) (-2, 0) and (0, -4) (2, 0) and (0, -4)
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Yes I would assume so!
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i have a computer block thing i cant watch videos sorry=(
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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/updates/50a8214fe4b082f0b8531742","timestamp":"2014-04-20T06:28:36Z","content_type":null,"content_length":"61043","record_id":"<urn:uuid:0cae68cd-e5ae-49c3-b4f8-b90368af2b9a>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00439-ip-10-147-4-33.ec2.internal.warc.gz"} |
Experiment 1
Hooke’s Law and Simple Harmonic Motion
1. To verify Hooke’s law for a linear spring, and
2. To verify the formula for the period, T, of an oscillating mass-spring system
A linear spring, slotted weights, a stop watch, a spring hanger, a meter stick or a 30.0-cm ruler, a mass scale, a C-clamp and rod attachment, a skew clamp, a regular weight hanger and a few sheets
of Cartesian graph paper
Hooke’s law simply states that for a linear spring the spring force, F[s ], is proportional to the change in length, x, that the spring undergoes. (Here, the term “spring force” means the force
exerted by the spring on the object attached to it. The object is often called the “mass.”) Mathematically, F[s] = - kx, where k is the spring constant. The reason for the (-) sign is that F[s] and x
always have opposite signs. If a spring is pulled to the right, Fig. (a), the externally applied force, F[appl.], is to the right, but the spring force, F[s] , acts to the left. On the other hand,
when the spring is pushed to the left, Fig. (c), F[s] acts to the right.
If mass M is hung from a spring as shown below, it stretches the spring of initial length y[1] , and the spring attains an equilibrium length of y[o] + y[1]. If the mass is pushed up a distance A and
then released, it oscillates above and below that equilibrium level. Distance A, that is the maximum deviation from equilibrium, is called the “amplitude” of oscillations.
This formula is a result of the solution to a 2^nd order linear differential equation with constant coefficients. The differential equation is set up very easily as follows. At any instant of
oscillation, it is the spring force (F[s] = - ky) that accelerates mass M at a rate a = d^2y /dt^2 . According to Newton’s 2^nd law, F[s] = Ma. This may be written as:
- ky = Ma, or - ky = Md^2y /dt^2, or d^2y /dt^2 + (k / M) y = 0.
This may be written as:
d^2y /dt^2 + ω^2 y = 0 where ω^2 = k / M from which ω = (k / M)^(1/2).
The value of k, the spring constant, can be measured in two ways. One method is to use Hooke’s law. The other method is to measure the period (T) of oscillations of a mass-spring system. The values
of k determined by the two methods may then be compared and used as a verification of the validity of the theories involved.
I. The Hooke’s Law Method:
The mass-spring system acts similar to a spring scale. It has a vertical ruler that measures the spring’s elongation.
1. Measure the mass of the hanger without the spring.
2. Attach the spring and hanger to the support. Zero the system by sliding the ruler against the needle. The ruler slides easily once its collar or slider (at the back of the ruler) is squeezed with
two fingers. By zeroing the system with its small weight hanger attached, you do not have to take its mass into account for this part of the experiment.
3. Place a 100.-g mass (M[1]) on the hanger and measure the change in length of the spring (Δy). It is better to use two 50.-g slotted masses instead of a single 100.-g mass. Make sure that the slots
are exactly parallel and opposite to each other such that the weights hang perfectly vertical. If the slots are not opposite to each other, the weights hang tilted, making the needle tilted and
causing an incorrect reading of the scale. Record the measured values of M and Δy, and the calculated value of F, in a table similar to the one shown below.
4. Repeat the above step for two or three additional values of mass up to about 250. g. Again use smaller slotted weights with the slots configured to avoid tilting.
2. Plot F versus Δy and find the slope of the graph. The spring constant k is equal to the slope.
II. The Oscillation Period Method:
1. Place the first recommended mass on the weight hanger.
2. Add the mass of the weight hanger to this mass and record it in the appropriate space in a table similar to the one shown below.
3. Pull the mass with its weight hanger down to about 2 to 3 cm below its equilibrium level and release. Start counting oscillations when the mass reaches either the highest or the lowest point.
Start counting at zero while starting a stopwatch. The greater the number of oscillations, the more accurate is the measurement of the period. Count about 25 to 50 oscillations, and stop the
stopwatch. Record the total time in the table, and calculate the period T and T^2. Repeat this procedure for all recommended masses.
4. Plot T^2 versus M, and find the slope of the graph. The spring constant k is given by k = {(2π)^2 / slope}, an equation which can be obtained from ω = 2 π / T. Calculate the spring constant.
5. Calculate the percent difference for the k values obtained by the two methods.
Note that the oscillating mass is not just the mass of the slotted weights in each case. In each calculation, the mass of the weight hanger must be taken into account.
│ M │Total Time │ T │ T^2 │
│ │ │ │ │
│(kg)│ (s) │(s)│(s^2) │
M[1] = 100. g M[2] = 150. g M[3] = 200. g M[4] = 250. g
Mass of Weight Hanger =
Perform the calculations in the tables, and calculate k in method II using
k = (2π)^2 / slope.
Comparison of the Results:
Calculate a percent difference using
To be explained by students
To be explained by students
1. Does the solution to a second order linear differential equation {d^2y/dt^2 + ω^2 y = 0} have the following form: y = A cos (ω t) + B sin (ω t) ? If yes, what is the role of ω in the equation?
What unit should it have if t is in seconds?
2. If a second order linear differential equation has the form: {d^2y/dt^2 + (k / M) y = 0}, what should the value of ω be?
3. If ω is the angular frequency in (rad/s), how are f in (cycles/s) and T in (s) related to it? | {"url":"http://www.pstcc.edu/departments/natural_behavioral_sciences/Web%20Physics/Expm%2001.htm","timestamp":"2014-04-19T04:58:13Z","content_type":null,"content_length":"21829","record_id":"<urn:uuid:da719e51-a369-482c-9e90-f841c1b1564c>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00600-ip-10-147-4-33.ec2.internal.warc.gz"} |
The speed of light is 186,000 miles per second. How far can light travel in 600 years. Give my answer in miles.
60x60x24x365x186,282 x 600, do the math. My calculator is too small. It's an irrational number, ending in e + , I'm sure.the answer will be in light year miles.
Report as You have reported this | {"url":"http://www.ask.com/answers/505100741/the-speed-of-light-is-186-000-miles-per-second-how-far-can-light-travel-in-600-years-give-my-answer-in-miles","timestamp":"2014-04-16T22:42:29Z","content_type":null,"content_length":"127469","record_id":"<urn:uuid:07c3173a-5c9c-4462-b876-cfb086ec901d>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00358-ip-10-147-4-33.ec2.internal.warc.gz"} |
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An electron in a P.P.capacitor is shot from the neg.plate to the pos. plate. Why the book says that the negative plate is the lower potential? I thought that like charges repel thus, the electron
would have a high potential when close to the neg.plate.
• one year ago
• one year ago
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No. Higher potential can be thought of as a hill and lower can be thought of as a valley FOR A POSITIVE CHARGE. Electrons have negative charge, so everything flips for them.
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So the book is wrong?
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No. As I said, for an electron, the negative plate will have lower potential, since that means it will be a hill. The electron rolls off the hill to the valley, the positive plate.
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** for an electron, the negative plate will be a hill since it has lower potential, not the other way around.
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I get you. So to escape the valley the electron would need kinetic energy exceeding the potential. An electron next to a negative plate needs less kinetic energy to escape than when its near a
positive one.
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Again, if a I have my negative plate at say x = 1 and my positive plate at x = 3, then voltage looks something like this: |dw:1349756069232:dw| Since \(\vec E = - \nabla V\), and, given the
charge density \(\sigma\) on the left plate (and an equivalent opposite charge on the right) we can simplify it to \(E = \frac{\sigma}{\epsilon_0} = -\frac{dV}{dx}\). So \(V=c (x-1)\) where c is
some positive constant. Put a proton anywhere on that line, and it will roll to the left. Put an electron on it, and you have to switch where it goes, since \(U=qV\), and with an electron q Is
negative. Technically you have to plot \(U\), not \(V\) (which is independent of the charge in quesiton) if you want to look at kinetic energy and all that stuff about being able to roll from a
valley, etc.
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Re: st: Imputing values for categorical data
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Re: st: Imputing values for categorical data
From "Renzo Comolli" <renzo.comolli@yale.edu>
To <statalist@hsphsun2.harvard.edu>
Subject Re: st: Imputing values for categorical data
Date Thu, 8 Apr 2004 23:29:44 -0400
Hi Jennifer,
I have one piece of advice: be very careful when using -impute-
It is not suitable to impute categorical variables, and I am surprise the
manual does not mention that.
When I actually "ripped the ado file open" an saw what it does I gave up on
imputing categorical variables, but I had never done imputations before so I
have very little knowledge of the field
At its core, -impute- does a simple OLS projection.
Let me explain with a simplified case first and then with a more complicated
Simplifying assumption: only one variable (denoted by y) necessitates to be
imputed, all the other variables (denoted by matrix X) have no missings.
Without loss of generality assume that you have ordered the variable y so
that all the cases for which you have observations appear at the top (denote
this part of the vector y'), and all the missings at the bottom, denote this
part of the vector y by y". Also denote by X' and X" the corresponding
values of X (remember that X has no missings, X" just contains the X values
corresponding to the observation y")
Then -impute- trivially does OLS of y'=X'beta+epsilon where beta is the OLS
vector of coefficients. It saves it and imputes y" by doing X"beta
So of course this is completely unsuitable for cases categorical variables.
Even with continuous variables you have to be careful not to predict "out of
range". Let's assume that you are predicting "number of weeks of work", it
might well happen that -impute- predicts that the interviewee worked -1
weeks last year
The case is not that simple when the matrix X contains missing variables
itself. If so, -impute- looks for the best subset of regressors. In practice
-impute- repeats the procedure explained here above several times trying to
keep as many regressors as possible (exactly how I did not understand either
from the ado file or from the manual, but I did not spend much time on it,
because I did not care that much.
Said that, I did not know of these other methods you mentioned (hotdeck,
Amelia) and I would be glad to read what others have to say about it.
Renzo Comolli
*From Jennifer Wolfe Borum <jjfrog@bellsouth.net>
To <statalist@hsphsun2.harvard.edu>
Subject st: Imputing values for categorical data
Date Thu, 8 Apr 2004 18:50:21 -0400
I am working with a data set composed of responses to survey questions which
contains some categorical variables such as gender and ethnicity. The data
has missing values and I have decided that it would be best to keep all
observations due to a pattern in the missing values. I have decided to use
the impute command in Stata to handle this as I've had some difficulty and
am not familiar enough with the hotdeck and Amelia imputations. I've found
that impute works fine for the continuous variables, however for the
categorical variables I am obtaining values for which I am unsure how to
interpret. For example, I will get an imputed value of .35621 for gender
which is coded 1 or 0. Would anyone be able to help with the interpretation
of the values I am obtaining for the categorical data?
Also, I would be interested in knowing which approach other Stata users
prefer for imputing values as this is the first time I have encountered
missing values and I am just beginning to research the various methods of
Thanks in advance,
Graduate Student
Florida International University
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
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Math Tools Discussion: All Topics in Algebra on Computer, Feedback requested for online Algebra book
Discussion: All Topics in Algebra on Computer
Topic: Feedback requested for online Algebra book
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Subject: RE: Feedback requested for online Algebra book
Author: rfant
Date: Apr 8 2008
My friend, I must agree with LFS. As much work and effort as I put into my
classes and my teaching, this book's title is offensive.
Mathematics is effortless only for the elite few. However, it should not be a
subject for the elite or the few, it should be a topic for the masses.
What is the average student going to do if he/she doesn't "get it" while
expecting to put in little to no effort? Most of them are going to quit. Not
only am I trying to teach mathematics but I'm also trying to teach my students
the value of a STRONG WORK ETHIC! Mathematics is a beautiful endeavor but it
sometimes requires MUCH EFFORT. (My marriage is a great endeavor and requires
much effort. Bottom line, almost everything WORTH doing or achieving requires a
great deal of effort.)
On the other side of the coin; After changing the title, let some folks on this
forum download a copy of your book and I for one will give you an honest
critique of the material. I did thumb through some of it but the "Issusu
Viewer" is just too distracting.
Please don't read us (me and LFS) too harshly but we work too hard to overlook
such a thing. I've got too many students at stake.
Reply to this message Quote this message when replying?
yes no
Post a new topic to the All Content in Algebra on Flash discussion
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Problems from Another Time
Individual problems from throughout mathematics history, as well as articles that include problem sets for students.
Given a semicircle, Prove that if O is the circle's center, DO=OE.
Discussion of 15th century French manuscript, with translation of its problems, including one with negative solutions
In a circle whose circumference is 60 units, a chord is drawn forming a segment whose sagitta is 2 units. What is the length of the chord?
Of a collection of mango fruits, the king took 1/6; the queen 1/5 the remainder, and the three princes took 1/4, 1/3 and 1/2 (of the same remainder); and the youngest child took the remaining 3
Three vertical posts along a straight canal, each rising to the same height above the surface of the water. By looking at the line of vision, determine, to the nearest mile, the radius of the earth.
Find two numbers, x and y, such that their sum is 10 and x/y + y/x = 25
One person possesses 7 asavas horses, another 9 hayas horses, and another 9 camels. Each gives two animals away, one to each of the others.
The answer to the following question is obtained using as optimum strategy-the farmer is getting the "best deal" possible. Can you figure out the solution strategy?...
I am a brazen lion; my spouts are my 2 eyes, my mouth, and the flat of my foot. My right eye fills a jar in 2 days, my left eye in 3, and my foot in 4. | {"url":"http://www.maa.org/publications/periodicals/convergence/problems-another-time?page=33&device=mobile","timestamp":"2014-04-19T01:03:02Z","content_type":null,"content_length":"25119","record_id":"<urn:uuid:8f73bfd8-b583-4e2f-b0ad-350fa568d5ff>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00462-ip-10-147-4-33.ec2.internal.warc.gz"} |
Formal Theories of Truth
In eight lectures I will provide an introduction to formal theories of truth.
The slides (at least some) for the lectures can be found here on WebLearn.
I shall start with an account of the diagonal lemma and the liar paradox. The main emphasis of the lectures will be on developments after 1975, that is after Kripke's Outline of the Theory of Truth.
The relevance of the formal results with respect to truth-theoretic deflationism will be discussed.
I intend to cover the following topics:
1. The theory of sytax and diagonalisation
2. Inconsistencies
3. Disquotation
4. Kripke's theory
5. Axiomatisations of Kripke's theory
6. Revision semantics
7. Other axiomatic theories
8. Truth and necessity
Cantini, Andrea (1996), Logical Frameworks for Truth and Abstraction. An Axiomatic Study, vol. 135 of Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam.
Feferman, Solomon (1991), ‘Reflecting on incompleteness’, Journal of Symbolic Logic 56, 1–49.
Gupta, Anil and Nuel Belnap (1993), The Revision Theory of Truth, MIT Press, Cambridge (Mass.) and London.
Halbach, Volker (2001), ‘How innocent is deflationism?’, Synthese 126, 167–194.
Halbach, Volker (2006b), ‘How not to state the T-sentences’, Analysis 66, 276–280. Correction of printing error in vol. 67, 268.
Halbach, Volker (Spring 2006a), Axiomatic theories of truth, in E. N.Zalta, ed., ‘Stanford Encyclopedia of Philosophy’. URL = http://plato.stanford.edu/archives/spr2006/entries/truthaxiomatic/.
Heck, Richard G. (2004), ‘Truth and disquotation’, Synthese 142, 317–352.
Herzberger, Hans G. (1982), ‘Notes on naive semantics’, Journal of Philosophical Logic 11, 61–102.
Ketland, Jeffrey (1999), ‘Deflationism and Tarski’s paradise’, Mind 108, 69–94.
Kripke, Saul (1975), ‘Outline of a theory of truth’, Journal of Philosophy 72, 690–712. reprinted in Martin (1984).
Leitgeb, Hannes (2001), ‘Theories of truth which have no standard models’, Studia Logica 21, 69–87.
Leitgeb, Hannes (2005), ‘What truth depends on’, Journal of Philosophical Logic 34, 155–192.
Leitgeb, Hannes (2007), What theories of truth should be like (but cannot be), in ‘Blackwell Philosophy Compass 2/2’, Blackwell, pp. 276–290.
Martin, Robert L., ed. (1984), Recent Essays on Truth and the Liar Paradox, Clarendon Press and Oxford University Press, Oxford and New York.
McGee, Vann (1985), ‘How truthlike can a predicate be? A negative result’, Journal of Philosophical Logic 14, 399–410.
McGee, Vann (1991), Truth, Vagueness, and Paradox: An Essay on the Logic of Truth, Hackett Publishing, Indianapolis and Cambridge.
McGee, Vann (1992), ‘Maximal consistent sets of instances of Tarski’s schema (T)’, Journal of Philosophical Logic 21, 235–241.
Shapiro, Stewart (1998), ‘Proof and truth: Through thick and thin’, Journal of Philosophy 95, 493–521.
Sheard, Michael (1994), ‘A guide to truth predicates in the modern era’, Journal of Symbolic Logic 59, 1032–1054.
Tarski, Alfred (1956), The concept of truth in formalized languages, in ‘Logic, Semantics, Metamathematics’, Clarendon Press, Oxford, pp. 152–278.
Yablo, Stephen (1993), ‘Paradox without selfreference’, Analysis 53, 251–252. | {"url":"http://users.ox.ac.uk/~sfop0114/lehre/truth.htm","timestamp":"2014-04-18T21:14:47Z","content_type":null,"content_length":"8007","record_id":"<urn:uuid:e5657c5f-ad79-42dd-902a-1ad1f47927ad>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00637-ip-10-147-4-33.ec2.internal.warc.gz"} |
San Quentin Statistics Tutor
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Probing the CP of the Higgs at a $\gamma \gamma$ collider using $\gamma \gamma \to t \bar t \to lX$
Godbole, RM and Rindani, SD and Singh, RK (2002) Probing the CP of the Higgs at a $\gamma \gamma$ collider using $\gamma \gamma \to t \bar t \to lX$. [Preprint]
Download (215Kb)
We present results of an investigation to study CP violation in the Higgs sector in $t\bar t$ production at a $\gamma\gamma$-collider. This is done in a model independent way in terms of six
form-factors $\{\Re(S_{\gamma}), \Im(S_{\gamma}), \Re(P_{\gamma}), \Im(P_{\gamma}), S_t, P_t\}$ which parameterize the CP mixing in Higgs sector. The angular distribution of the decay lepton from $t/
\bar t$ is shown to be independent of any CP violation in the $tbW$ vertex. Hence it can be used as a diagnostic of the CP mixing. We study how well one can probe different combinations of the form
factors by measurements of the combined asymmetries that we construct, in the initial state lepton (photon) polarization and the final state lepton charge, using only circularly polarized photons. We
show that the method can be sensitive to loop-induced CP violation in the Higgs sector in the MSSM.
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What are easy steps to find:
log(base 6)36x^2 ? - Homework Help - eNotes.com
What are easy steps to find: log(base 6)36x^2 ?
`log_6 36 + log_6 x^2`
`2+2log_6 x`
`2(1+log_6 x)`
This comes easily from knowing the log rules. So to make problems like this "easy" you need to have a good understanding of log rules. I'd start with this intro video and just work through the
Thanks, I absolutely get it now. I guess i was confused to find it or expand it, but i see the difference now.
Join to answer this question
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AES-0.2.1: Fast AES encryption/decryption for bytestrings Source code Contents Index
An occasionally pure, monadic interface to AES
type AES s a = AEST (ST s) a Source
Modes ECB and CBC can only handle full 16-byte frames. This means the length of every strict bytestring passed in must be a multiple of 16; when using lazy bytestrings, its component strict
bytestrings must all satisfy this.
Other modes can handle bytestrings of any length, by storing overflow for later. However, the total length of bytestrings passed in must still be a multiple of 16, or the overflow will be lost.
For OFB and CTR, Encrypt and Decrypt are the same operation. For CTR, the IV is the initial value of the counter.
class Cryptable a where Source
A class of things that can be crypted
The crypt function returns incremental results as well as appending them to the result bytestring.
crypt :: a -> AES s a Source
:: MonadUnsafeIO m
=> Mode The AES key - 16, 24 or 32 bytes
-> ByteString The IV, 16 bytes
-> ByteString
-> Direction
-> AEST m a
-> m (a, ByteString)
Run an AES computation
=> Mode The AES key - 16, 24 or 32 bytes
-> ByteString The IV, 16 bytes
-> ByteString
-> Direction
-> forall s. AES s a
-> (a, ByteString)
Run an AES computation
Produced by Haddock version 2.6.0 | {"url":"http://hackage.haskell.org/package/AES-0.2.1/docs/Codec-Crypto-AES-Monad.html","timestamp":"2014-04-21T12:48:53Z","content_type":null,"content_length":"13297","record_id":"<urn:uuid:9c48fa18-423f-47df-bade-c1e2b83bfee6>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00118-ip-10-147-4-33.ec2.internal.warc.gz"} |
Rope simulation with Position Based Dynamics
up vote 0 down vote favorite
First off I apologize if this is the wrong stack exchange for this question, it seems to be like halfway between programming and math. But it leans more on the math side so hopefully I'm not out of
place. I also apologize if this is long winded.
I'm referring to this paper which in turn references this other paper to create thread / rope simulations that run in real time.
In Müller's paper he talks about constraint functions being $C_j:\mathbb{R}^{3nj} \rightarrow \mathbb{R}$ Which makes sense because using the constraint solver you solve attempt to get each
constraint function either equal to 0 or greater than or equal to 0.
However if you look at fratarcangeli's paper he gives the contact constraint function as $C(p) = [p - (p_{n0} + p_v)]$ Where $p$ must be some vertex of the rope, $p_v$ is the penetration vector and
$p_{n0}$ is "the current position of point $p$". This is where things stop making sense for me. Because it appears that fratarcangeli's constraint equation is in $\mathbb{R}^3$ and not $\mathbb{R}$.
Perhaps I'm miss-understanding the equation?
My second issue with his constraint function is $p_v = (\|p_{n0} - p_{n1}\| - r)\cdot\frac{p_{n0} - p_{n1}}{\|p_{n0} - p_{n1}\|}$ according to the diagram in the paper it looks like $p_{n0}$ and $p_
{n1}$ are the two "closest points" of the two linesegments in the two capsule shapes. But he also says $p_{n0}$ is "the current position of point $p$". Though he never tells us what p should be with
respect to the colliding capsules.
Perhaps someone can explain what his constraint function is supposed to be?
I attempted to figure out what the constraint function should be, forgive me I'm not incredibly mathematically strong.
I assumed if two segments were colliding I would have to apply a contact constraint to all 4 points. Since I'm applying the constraint to both sides I only need to move each mass point halfway out of
the collision.
When a collision occurs between two line segments $\overline{p_1q_1}$ and $\overline{p_2q_2}$ And $p = p1$. I get the two closest points $c_1$ and $c_2$ on those segments respectively.Most of the
time $c_1 \neq p$. So I have to define my constraint function with that in mind.
I call the collision normal $n = \frac{c_1 - c_2}{\|c_1 - c_2\|}$ and an offset $o = (p - c_1)\cdot n$ which is the offset along the collision normal of $p$ down to the contact point. This handles
when $c_1 \neq p$ Note: $o$ is calculated once at the beginning of a collision, it's expected to stay constant.
The goal is to have the constraint function equal to 0 when $p$ has moved halfway to resolve the collision (the other segment will move the other half) and $>$ 0 when the point has moved further than
So I define $C(p) = -\frac{2r - ((p - c_2)\cdot n - o)}{2}$
Here's a poorly drawn diagram to illustrate my thoughts. http://i.imgur.com/rL4rU43.png (MO wouldn't let me put it in an image tag.)
[(Image added by J.O'Rourke)]
Which when I punch that through the method described in Müller's paper. I get $\Delta p = -2C(p)n$ However when I plug this into my simulation it's stable up until I tie a knot which given the nature
of the papers means I've done something wrong. Can anyone elaborate on where I'm going wrong?
P.S. I'm not incredibly familiar with all the math so my tags could be way off, again apologies.
ag.algebraic-geometry ds.dynamical-systems simulation
Thanks for the image assist. – Tocs Jun 13 '13 at 5:06
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Finding points on a plane
May 17th 2011, 08:27 PM
Finding points on a plane
I'm brand new here, so feel free to move this post if need be.
I have a plane defined by its unit normal (A, B, C) and a distance (D). I need to define this plane by three points instead. I know the equation for a plane is Ax+By+Cz=D, and it's easy enough if
two of the coefficients are zero (the third coefficient is one and the point is dist along that axis). Is there a reliable formula for finding an x, y and z that satisfies this equation,
particularly when A, B and C are all nonzero?
May 17th 2011, 08:39 PM
Subtract 2 pairs, maybe A - B and A - C. A Normal to the plane would ahve to be orthogonal to both these vectors. I think there is a cross product in your future.
If you are not convinced, try a different pair, maybe B - A and C - A, and see if you an manage the same equation.
May 17th 2011, 09:56 PM
Subtract 2 pairs, maybe A - B and A - C. A Normal to the plane would ahve to be orthogonal to both these vectors. I think there is a cross product in your future.
If you are not convinced, try a different pair, maybe B - A and C - A, and see if you an manage the same equation.
I don't quite understand. A, B and C are all scalars for i, j and k along their respective axes, correct? In which case, I don't quite understand how subtracting two scalars results in a vector
to perform the cross product on.
May 18th 2011, 02:44 AM
Am I understanding this correctly? You have found that the equation of the plane is Ax+ By+ Cz= D and you want to "define this plane by three points instead." In other words, you just want to
find three points that lie on that plane. It you take x= y= 0 then z= D/C so (0, 0, D/C) is a point on the plane. If x= z= 0, then By= D so y= D/B. (0, D/B, 0) is a point on that plane. I'll
leave the third point to you.
I think TKHunny was misunderstanding the problem, thinking that A, B, and C were three points that you wished to use to find the equation of the plane.
May 18th 2011, 07:46 AM
Am I understanding this correctly? You have found that the equation of the plane is Ax+ By+ Cz= D and you want to "define this plane by three points instead." In other words, you just want to
find three points that lie on that plane. It you take x= y= 0 then z= D/C so (0, 0, D/C) is a point on the plane. If x= z= 0, then By= D so y= D/B. (0, D/B, 0) is a point on that plane. I'll
leave the third point to you.
I think TKHunny was misunderstanding the problem, thinking that A, B, and C were three points that you wished to use to find the equation of the plane.
Thanks for that, I think that leap of logic was exactly what I was missing. Yes, I have the A, B, C and D and I need to find x, y and z to satisfy the equation. Didn't occur to me that if I set
all other axes of the point to 0 that the third one would be D divided by the remaining coefficient.
However, this method falls apart if A B or C are zero. So now I know how to find a point if two of them are zero, or if none of them are zero.
May 18th 2011, 08:14 AM
If, say, B is 0, y does not enter into the equation so y could be anything. Take x and y to be whatever you want, solve the eqation for z. A different value of y with the same x and z will be
another point on the plane. You could not do that with a third value of y and the same x and z because that would give three points on a line which do not define a plane, but you could pick a new
value for x and solve for z to get a point not on the line determined by the first two points.
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Year 8 maths assignment - Algebra
Number of results: 99,596
A school has eleven Year 9 Maths classes of 27 students. Each class has 4 extra students added to it. Find the total number of students studying Year 9 Maths.
Thursday, February 26, 2009 at 1:41am by Jessica
the cost of living last year went up 3% fortunately alice swanson got a 3% rais in her salery from last year. This year she is earning $22,660 how much did she make last year
Monday, November 22, 2010 at 4:39pm by nora
Every year a man is paid Rs 500more than the previous year. If he receives Rs 17800 over four years, what was he paid on the first year
Thursday, December 27, 2012 at 3:08am by Anonymous
Im in grd11 this year nd when i ws in grd10 i ws doing maths lit bt in grd11 they say tht this year all grds frm 10 to 12 wl change maths lit to pere bcoz of CAPS.So iz tht true?
Wednesday, January 15, 2014 at 12:29pm by Matebane Jack
can any one tell me wat are the topics for class 10th maths project for the year 2010-2011.
Saturday, May 15, 2010 at 12:54pm by prateek nayak
maths letaracy,history,geography and life science
im going to grade 10 next year,and i want to be a psychologist and im going for maths letaracy and history feild im realy not good with maths and i just found out a few weeks ago that i need pure
maths to do psychology.i dont know what to do now ,i also dont know the jobs ...
Saturday, October 26, 2013 at 8:02am by khanyisa maringa
In science, my teacher gave out an assignment where we had to group the months of the year, kind of like the seasons. I have absolutely no ideas, and was wondering if anyone else had some input on
the assignment.
Wednesday, February 6, 2008 at 9:56pm by Maddie
The linear equation y=0.15x+0.79 represents an estimate of the average cost of gas for year x starting in 1997. the year 1997 would be represented by x=1, for example, as it is the first year in the
study. similarly, 2005 would be year 9, or x=9 A) what year would be ...
Saturday, July 18, 2009 at 1:54pm by Patty G
College Algebra
The following graph shows Bob’s salary from the year 2002 to the year 2005. He was hired in the year 2002; therefore t = 0 represents the year 2002. These are the points on the graph! O year = $41000
1 year = $43000 2 year = $46000 3 year = $48500 List the coordinates of any ...
Saturday, November 17, 2007 at 10:41pm by Cupcake
A man borrows Rs 18000 at 5% per annum compound interest .If he repays Rs 6000 at the end of the first year and Rs 8000 at the end of the second year; how much should he pay at the end of the 3rd
year in order to clear the account?
Monday, February 4, 2013 at 6:03am by Anonymous
Thats all the info that was provided, It's an assignment and I am finding some of these questions ambiguous. Here's two more that are similar: (1)Cindy has income of $12000 in year 0. Calculate her
income in year 1 if she wants to consume $26,000 in year 0 and $14,000 in year ...
Saturday, February 2, 2008 at 2:36pm by Will Jr.
Year 12 Biology
What is a good and easy experiment I could do for my Year 12 biology assignment? I was thinking something with alcohol and the human body but I wouldn't know what my hypothesis would be. HELP!!
Friday, July 30, 2010 at 7:55am by Samantha
Algebra Maths
A local company is hiring trainees with less than 1 year of experience and managers with 5 or more years of experience. Solve the inequality and graph the solutions?
Tuesday, October 1, 2013 at 9:39pm by Raman
invest $2,000, part earns 6% year and reh rest earned 11% year. Interest at end of year = $155. How much is invested at each rate?
Saturday, October 15, 2011 at 1:55pm by BJ
The linear equation y=0.15x+0.79 represents an estimate of the average cost of gas for year x starting in 1997. the year 1997 would be represented by x=1, for example, as it is the first year in the
study. similarly, 2005 would be year 9, or x=9 A) what year would be ...
Friday, January 6, 2012 at 1:59pm by Marcus
This assignment requires you to use Excel. In question 1, you will use the charting features. In questions 2 and 3, use the regression tool from the analysis toolpack. There is no template for this
assignment. Make sure you explain your answers and provide the regression ...
Monday, August 20, 2012 at 11:47am by David
Mr Srinivasan invests $55 000 in a fixed deposit account. The interest rate is 3.3 % per year. How much money will he have in the account after 1 year?
Thursday, October 4, 2012 at 8:24am by Zayn
Maths Investments
A woman has a total of $9,000 to invest. She invests part of the money in an account that pays 8% per year and the rest in an account that pays 11% per year. If the interest earned in the first year
is $840, how much did she invest in each account?
Tuesday, June 26, 2012 at 7:17am by Juan
Academic 4-year plan: Zoology
I want to work in the field of zoology when I get older, and I'm filling out an "academic 4-year plan" our counselors gave to my class to complete and turn in on Monday. The science and math I'm
taking this year are physics and geometry. Can you take Calculus (a high school ...
Saturday, December 1, 2007 at 6:54pm by Emily
I need help in understanding how to go about creating a budget for this assignment, excluding the Director and training supervisor's salary. I understand as far as th writing part of the
assignment... I am lost when it comes to creating the budget where to put the numbers etc...
Monday, December 6, 2010 at 1:18pm by Angela
Doing a balance sheet. Trying to understand what Bal c/d, and Bal d/d means. It gives me the profit for the year then the payout also gives me the earnings for the year and then the two quotes above
which I cannot find the definition of anywhere in my reading assignment.
Sunday, January 10, 2010 at 3:19pm by mike
Doing a balance sheet. Trying to understand what Bal c/d, and Bal d/d means. It gives me the profit for the year then the payout also gives me the earnings for the year and then the two quotes above
which I cannot find the definition of anywhere in my reading assignment.
Sunday, January 10, 2010 at 4:16pm by mike
A) If the year 1997 is year x=1, then 1996 is year x = 0. Add 4 to 1996 to get year x = 4 1996 + 4 = 2000 B) I assume you mean 2018. Subtract 1996, which is year x = 0 2018 - 1996 = 22 So 2018 is
year x = 22 C,D) The equation is in the form y = mx + b. y = 0.l5 x + 0.79 y = mx...
Saturday, July 18, 2009 at 1:54pm by Marth
Maths (Trigonometry)
what is your question about this assignment?
Thursday, May 31, 2012 at 3:50am by bobpursley
I would do it the same up to and including r = 1/20 = 5% After that, I made a mistake which you pointed out. The derivation should finish with S*r = S*0.05 = 20 S = 400 Rs The first year's interest
earned is $20. If that is compounded in the second year, the interest earned in...
Tuesday, December 18, 2012 at 6:17am by drwls
3. The linear equation y=0.15x+.079 represents an estimate of the average cost of gas for year x starting in 1997. The year 1997 would be represented by x = 1, for example, as it is the first year in
the study. Similarly, 2005 would be year 9, or x = 9. c) What is the slope (...
Saturday, May 9, 2009 at 6:34pm by Anonymous
Maths algebra
The efficiency of an industrial pump decreases by a fixed percentage each year. The percentage decrease each year is with respect to the efficiency of the pump at the start of each year. Over each
four year period, the efficiency decreases by 80%. Let us write e = f(t) so that...
Wednesday, April 2, 2014 at 10:30pm by Jasmine
Maths A Year 12
Four years ago, an art collector bought a painting for $32 500. He knows that this painting appreciates at 15% a year. What is this painting worth now, to the nearest $100?
Sunday, June 9, 2013 at 5:26am by Anne
x + 8000(1.05) + 6000(1.05)^2 = 1800(1.05)^3 ... .... x =5822.25 ( For these I make a "time graph" on a line, mark 0 (now), year 1, year 2 , and year 3 Place 18000 above the "now" place 6000 at 1,
8000 at 2, and x at 3 "move" all monies to position 3 )
Monday, February 4, 2013 at 6:03am by Reiny
English 7 - Journal Entry Assignment
Yup! :) Like I'm working really hard this year and next year so at the 8th grade grad. I'm going to be one of the students getting a presidental golden award and may get a good award for being the
top student in school or something... Thank You for answering my question tho
Monday, January 30, 2012 at 6:42pm by Laruen
Grandpa places money in an account on your first birthday and will place that same amount in the account every year, ending with your 18th birthday. So there is a total of 18 deposits starting on
first birthday. He expects to earn 8% interest per year. How much does he have to...
Wednesday, February 12, 2014 at 9:19am by Sejul
college algebra
The following graph shows how a 4-color web printing press depreciates from the year 2006 to the year 2010. It was purchased new in the year 2006; therefore x = 0 represents the year 2006. X – axis
(horizontal) = years starting from 0 = 2006 and increasing by 0.5 years Y – ...
Sunday, June 20, 2010 at 7:43pm by Lucy
Exams help
Hello I am a Grade 11 student , I am willing to resit my maths exam this November , but I have to take my account that my IGCSE's are this year , my mocks will be in January and my real resit exams
will be at the beginning of November , which is just a month ahead Should I ...
Saturday, October 4, 2008 at 6:07am by Amy
Check these sites. http://www.google.com/search?source=ig&hl=en&rlz=1G1GGLQ_ENUS374&q=class+10th+maths+project+for+the+year+2010-2011
Saturday, May 15, 2010 at 12:54pm by Ms. Sue
The population of a country is 10 million in 1997 and increasing at a rate of 0.6 million per year. The average annual income of a person during 1997 was 16000 dollars per year and increasing at a
rate of 800 dollars per year. How quickly was the total income of the entire ...
Monday, October 29, 2007 at 1:23am by anshu
Real Estate Project (Maths)
I don't know if you can help me here or not, but I'll ask anyways. I've been given a Real Estate assignment where I had to compare the sale prices of properties of different suburbs and then compare
them, with the Mean, Standard Deviations, Histograms, Boxplots, Outliersd, ...
Saturday, May 31, 2008 at 3:02am by TP
do your Open university end of year assignment yourself
Thursday, September 15, 2011 at 1:11pm by Anonymous
Algebra 1A
The N of states and fedarl inmates in million during year x, where X ¡Ý 2002 can be approximated by the following formula. N=0.04x - 78.46 Determine the year in which there were 1.7 million inmates
in the year?
Wednesday, May 16, 2012 at 8:42pm by Nini
we do have a data set, it's for an assignment, could you figure out k any other way?
Sunday, June 5, 2011 at 8:02pm by Anonymous
YEAR 4
Tuesday, September 29, 2009 at 12:53am by BRENDAN
Find the total amount if you deposit $500 at a rate of 5% for two years using simple interest. For year one For year two I started with $500. for year one and $525. for year two and added the
interest and ended up with 551.25 - and I was wrong...can someone please help
Monday, February 25, 2013 at 10:19pm by Cassie
interest rate is i per year so it is i/2 for half a year every half hear you multiply by (1+i/2) so after 1 year, two half years, you multiply by (1+/2)(1+i/2) = (1+i/2)^2 say i = 20% = .20 that is
10% per half year or .10 added every half year so after a half year we have P...
Thursday, April 16, 2009 at 7:52pm by Damon
A light-year is the distance that light travels in one year. Find the number of miles in a light-year if light travels 1.86 × 105 miles/second. (Round to one decimal place, and assume that there are
365 days in a year.)
Friday, March 12, 2010 at 3:10pm by larry
Suppose the estimated quadratic model Yt = 500 + 20 t - t2 is the best-fitting trend of sales of XYZ Inc. using data for the past twenty years (t = 1, 2,.., 20). Which statement is incorrect? A. The
trend was higher in year 10 than in year 20. B. Latest year sales are no ...
Monday, August 30, 2010 at 9:38pm by BC
Actually is does make sense. drwls misinterpreted the question. It's asking for a decrease of 400% over 10 year and NOT a percent decrease. This means, that for year 1, the cost of a cell phone drop
by 40%. At year 2 it dropped by another 40%, and on and on until year 10. So ...
Sunday, May 15, 2011 at 2:18pm by Marcus
1. I am your student from last year. Somewhat OK 2. I was your student from last year. No; remove "from" and it'll be fine. 3. I am your student last year. No. The verb is in present tense, so "last
year" doesn't make sense. 4. I was your student last year. This one's the best...
Wednesday, March 25, 2009 at 8:27am by Writeacher
If the finance company is paying interest out, how can there be a return? You say nothing about the income. Return per half year to whom? The lender? The lender gets 7% of 1700 back in the first half
Wednesday, November 2, 2011 at 1:06am by drwls
Logan Academy students take part in four field trips during the school year. They visit a library, a museum, a park, and a zoo. How many different ways can the four trips be ordered for the school
Monday, May 13, 2013 at 11:21pm by BRYAN
hahaha! just casually finding answers on the internet for the math B assignment and coming across this.
Thursday, May 3, 2012 at 11:39pm by Nardine
College Algebra
There are two ways of doing this, and they give different answers. The change between 1975 and 1980 is $890 in 5 years, or $178 per year. That is 18.4% of the INITIAL amount per year. But it is not
the average annual rate of change during the five year peiod The best way to ...
Thursday, January 29, 2009 at 12:15am by drwls
So your equation is really y = 1777 (year -2000) + 27153 so for 2007 for example y = 1777 (2007-2000) + 27153 y = 12439 + 27153 = 39,592 in 2007 do other years the same way for the second part unless
you have a typo we are talking about before 2000 11,160 = 1777(year -2000...
Monday, July 5, 2010 at 6:54pm by Damon
Year 11 Maths B :)
Thursday, May 24, 2012 at 5:48am by MathMate
Natalie earns $2.50 for each CD she sells and $3.50 for each DVD she sells. Natalie sold 45 DVDs last year. She earned a total of $780 last year selling CDs and DVDs. Write an equation that can be
used to determine the number of CDs (c) Natalie sold last year. How many CDs did...
Saturday, February 16, 2013 at 10:46pm by Sylvie
Natalie earns $2.50 for each CD she sells and $3.50 for each DVD she sells. Natalie sold 45 DVDs last year. She earned a total of $780 last year selling CDs and DVDs. Write an equation that can be
used to determine the number of CDs (c) Natalie sold last year. How many CDs did...
Sunday, February 17, 2013 at 12:57pm by Amy
Max works 7 hours per day Monday to friday and 3 hours per day in the weekend. He does this for 48 weeks of the year. How many hours does he work in total for the year? Write the expression and
Thursday, February 26, 2009 at 1:37am by Jessica
Max works 7 hours per day monday to friday and 3 hours per day in the weekend. He does this for 48 weeks of the year. How many hours does he work in total for the year?
Thursday, February 26, 2009 at 2:09am by Jessica
Your opening sentence says: I had trouble finishing this assignment because ____ Obviously, you're having trouble doing this assignment because you "don't know what to write." You may want to write
about another assignment that had you stuck. Since you've posted several ...
Thursday, December 18, 2008 at 4:50pm by Ms. Sue
the time index t runs from a to A. if an investment produced a continuous stream of income over 10 years at a rate of $20,000 per year and the interest rate is 6% per year continuously compounded.
what is the present value? what is the integral function?
Tuesday, October 19, 2010 at 10:46am by May
the time index t runs from a to A. if an investment produced a continuous stream of income over 10 years at a rate of $20,000 per year and the interest rate is 6% per year continuously compounded.
what is the present value? what is the integral function?
Tuesday, October 19, 2010 at 11:50pm by May
Maths literacy,life science,history,geography
Hi am letty ntuli in grade11 this year and am studing geography,life science,tourism,maths literacy so idont now which career imust do so pleas help
Saturday, January 25, 2014 at 9:53am by Anonymous
Which specific part of this assignment do you need help with? Please tell us your thinking and figures on this and we'll try to help you.
Monday, March 10, 2008 at 12:27pm by Ms. Sue
Year 11 Maths
sec A is cos A
Monday, February 15, 2010 at 11:37pm by Jim
maths year 7
can u give me some help on recipicol fractions?
Friday, June 3, 2011 at 11:48pm by who wants 2 no?
Maths A Year 12
what is 32500(1.15)^4 ?
Sunday, June 9, 2013 at 5:26am by Reiny
Do you want good grades? The best way is to learn the subject. When you cheat, you only cheat yourself as you won't know the informstion that the next assignment or the next year's teacher will
expect you to know. This assignment is trying to get you to think about how to deal...
Friday, September 7, 2007 at 12:59pm by LehrerinSagt
A town has a population of 32,000 in the year 2002; 35,200 in the year 2003; 38,720 in the year 2004; and 42,592 in the year 2005. If this pattern continues, what will the population be in the year
Saturday, March 10, 2012 at 1:10am by Brittany
I have the same assignment and I am struggling on it to because I have only lived in this community for 4 months and anywhere else for less then a year and a half and the assignment wants us to
interview three people sigh, also can someone help me with this part: Student ...
Thursday, August 19, 2010 at 10:37pm by Aryan
Economics(Please respond, thank you)
The table below shows the market basket quantities and prices for the base year (year 1) Base year 1 Price in price Quantity base year yr 2 Product Pizza 15 $3 $3.75 t-shirts 4 $10 $9 rent 1 $500
$550 In year 1 the CPI was? In year 2 the CPI was? I know that cpi= (expenditures...
Sunday, March 18, 2012 at 12:59pm by Hannah
Annual sales (in million of units) of PC are expected to grow in accordance with the rate of f(t)=0.18t^2+0.16+2.64 ; (0<t<6) where t is measured in years t=0 correspomding to year 1997. how many pc
will be sold over the 6 year period between the begining of 1997 and end...
Sunday, September 4, 2011 at 9:10am by thila29
Suppose the first Friday of a new year is the fourth day of that year. Will the year have 53 Fridays regardless of whether or not it is a leap year? What is a rule that represents the sequence of the
days in the year that are Fridays? How many full weeks are in a 365 day year...
Sunday, November 17, 2013 at 5:34pm by Anonymous
The variation in size from year to year of a particular population can be modelled by an exponential model with annual proportionate growth rate 0.1246. The size of the population at the start of the
initial year is 360. Choose the TWO options that give, as predicted by the ...
Wednesday, May 11, 2011 at 7:00pm by albert
The variation in size from year to year of a particular population can be modelled by an exponential model with annual proportionate growth rate 0.1246. The size of the population at the start of the
initial year is 360. Choose the TWO options that give, as predicted by the ...
Thursday, May 12, 2011 at 6:29pm by albert
The variation in size from year to year of a particular population can be modelled by an exponential model with annual proportionate growth rate 0.1246. The size of the population at the start of the
initial year is 360. Choose the TWO options that give, as predicted by the ...
Thursday, May 12, 2011 at 6:41pm by albert
lol ur doing gauss! =]
Friday, May 1, 2009 at 9:19pm by Some year 9 student
Year 11 Maths
oops 1 over cos A
Monday, February 15, 2010 at 11:37pm by Jim
Ecomonics(Urgent,please respond)
The table below shows the market basket quantities and prices for the base year (year 1) Base year 1 Price in price Quantity base year yr 2 Product Pizza 15 $3 $3.75 t-shirts 4 $10 $9 rent 1 $500
$550 In year 1 the CPI was? In year 2 the CPI was? I know that cpi= (expenditures...
Sunday, March 18, 2012 at 1:55pm by Hannah
Maths - More than one question please help me
Reiny thanks a lot for helping me with the other quesiton too but i got different points because this is how i did it: when x = o y = ? and when y = 0 x = ? and i did that to every equation and got
three points and then i plotted the points which gave me a triangle and thats ...
Thursday, July 3, 2008 at 7:05am by sweetG
Algebra ll
Oh my pet mouse has three babies every year each of those has three every year each of those has three every year how many are born in year t ?
Saturday, March 8, 2014 at 2:01pm by Damon
Thompson stores is considering a project that has the following cash flow data. What is the project's IRR. Note that a project's projected IRR can be less than the WACC and even negative, in which
case it wll be rejected. Year 0 ($1,000), year 1 $300, year 2 $295, year 3 $290...
Saturday, February 6, 2010 at 11:15pm by vince
i have this same assignment..and i am stuck as well...as far as i understand..u need to get each individual parts in m^3/year.
Wednesday, October 28, 2009 at 1:14am by Easha
hi, i think u should send ur assignment on ehomeworksolution @ g mai l,co m i have worked with this site once and they are good in their work
Monday, June 17, 2013 at 12:51am by jiya
math VERY URGENT
first year 1 second year 2+1 third year 6 +2+1 fourth year=18+6+2+1 total= sum above
Tuesday, December 4, 2012 at 7:49pm by bobpursley
Fourteenth day of May,year two-thousand, and eleven.
Sunday, May 15, 2011 at 8:23am by Henry
College Algebra
These are the points on the graph! O year = $41000 1 year = $43000 2 year = $46000 3 year = $48500 The following graph shows Bob’s salary from the year 2002 to the year 2005. He was hired in the year
2002; therefore t = 0 represents the year 2002. List the coordinates of any ...
Saturday, November 17, 2007 at 7:29pm by Cupcake
College Algebra
These are the points on the graph! O year = $41000 1 year = $43000 2 year = $46000 3 year = $48500 The following graph shows Bob’s salary from the year 2002 to the year 2005. He was hired in the year
2002; therefore t = 0 represents the year 2002. List the coordinates of any ...
Saturday, November 17, 2007 at 9:41pm by Cupcake
Maths C
lol! i have the same assignment and im having trouble aswell. I cant find anything on weddles rule on the internet and it is not in my textbook.
Thursday, June 23, 2011 at 12:48am by marcus
prepare an assignment on pratical applications &co-relatives of implementation of discounts. sales tax and vat in steps of c.i &s.i in bank &post office
Thursday, December 15, 2011 at 2:34am by vishva
prepare an assignment on pratical applications &co-relatives of implementation of discounts. sales tax and vat in steps of c.i &s.i in bank &post office
Thursday, December 15, 2011 at 2:34am by vishva
I'm not a math teacher by any means, but have you actually read what you posted? This assignment involves drawing a graph ... which cannot be done on this website. Sorry.
Saturday, January 19, 2013 at 1:07am by Writeacher
English 12
That's your whole assignment, right? Keep in mind that no one here will do your assignment for you, but if you have questions about the assignment or about your topic, please ask.
Thursday, July 21, 2011 at 2:05pm by Writeacher
This is an exact copy of Appendix D from week 4 of MAT/117 at University of Phoenix. (Not an exam. But an entire homework assignment.) Every single assignment can be found posted to many different
sites by many different students. I sometimes look them up to check a few of my ...
Thursday, January 31, 2008 at 3:01pm by Anonymous
I'm trying to write a script that will allow a user to enter a year and then determine if the year is a leap year. I need to include a form with a single text box where the user can enter a year. I
also need to display and alert dialog box to the user starting wherther the ...
Tuesday, July 30, 2013 at 8:42pm by pep
You'll find out after you complete your algebra assignment.
Friday, February 8, 2013 at 11:57am by Ms. Sue
advance algebra
A construction company purchases a bulldozer for $160000. Each year the value of the bulldozer depreciates 20% of its value in the preceding year. a.Find a formula for the value of bulldozer. b.In
what year will the value of the bulldozer be less than $100000?
Monday, December 15, 2008 at 6:54pm by marie
I could post some of my views as a 77-year-old retired teacher. But I'm sure that would not fulfill your assignment.
Sunday, July 3, 2011 at 8:04pm by Ms. Sue
In Newtopia, inflation runs at 15%. A home entertainment unit currently sells for $2000. A)How much would you expect it to cost in a year`s time? B)How much would you expect it to cost in two year`s
Thursday, February 24, 2011 at 7:49pm by Anonymous
I am working on an assignment in which I need to put together a pro forma balance sheet. At the end of the assignment, I need to find the predicted values of accounts for the upcoming year. In 2011,
Notes Payable was 11.28 million; Long Term Bonds were 7 million; Common Stock ...
Monday, January 30, 2012 at 1:25pm by Jessica
1. I don't have any pets. In my free tim,e I go out with my friends. I also go to the cinema, and I play my guitar. Finally, I like to go swimming once a week. 2. I usually go horseback riding on the
weekend. I like music very much. 3. My sister is in her fourth year of ...
Friday, September 23, 2011 at 5:21pm by Writeacher
How do I even write an equation for this problem all i need someone to show me how to get an equation so i can solve it. It is estimated that the Earth is losing 4000 species of plants and animals
every year. If S represents the number of species living last year, how many ...
Monday, December 11, 2006 at 7:30pm by Jasmine20
Hi,I have a linear model with equation p = 5.1t - 28, where p is the number of breeding pairs of birds and t is the year. I need to use algebra to find the year in which the linear model predicts the
breeding pairs will exceed 500. Any help appreciated.
Thursday, July 15, 2010 at 3:14pm by Si
Pages: 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | Next>> | {"url":"http://www.jiskha.com/search/index.cgi?query=Year+8+maths+assignment+-+Algebra","timestamp":"2014-04-16T05:05:49Z","content_type":null,"content_length":"40937","record_id":"<urn:uuid:e5ba9b73-3707-4d0d-821f-8a73727a14fe>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00639-ip-10-147-4-33.ec2.internal.warc.gz"} |
Induction with inequality
April 30th 2008, 04:39 PM #1
Mar 2008
Induction with inequality
How do i use induction(effectively) to prove that n! < nn for n ≥ 2 ?
let p(n): n! < nn for n ≥ 2.
we see that p(2) holds true.
now i prove it holds for p(n+1) right?
(n+1)! = (n+1)n!<(n+1)nn by the induction hypothesis.
i think this is right.. can someone show me how to finish this correctly?
oops, n^n came out as nn...anywhere there is nn, it should read (n)^n.
How do i use induction(effectively) to prove that n! < nn for n ≥ 2 ?
let p(n): n! < nn for n ≥ 2.
we see that p(2) holds true.
now i prove it holds for p(n+1) right?
(n+1)! = (n+1)n!<(n+1)nn by the induction hypothesis.
i think this is right.. can someone show me how to finish this correctly?
assume $k! < k^k$
$(k+1)^{k+1}=(k+1)^k(k+1) > k^k(k+1)>(k!)(k+1)=(k+1)!$
thanks empty set.
April 30th 2008, 04:41 PM #2
Mar 2008
April 30th 2008, 05:03 PM #3
April 30th 2008, 08:42 PM #4
Mar 2008 | {"url":"http://mathhelpforum.com/discrete-math/36727-induction-inequality.html","timestamp":"2014-04-19T12:37:17Z","content_type":null,"content_length":"38114","record_id":"<urn:uuid:e7cf237b-48dc-408a-b50b-1797782f4cd8>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00098-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wall-Crossing and Quiver Invariants
We start with a one-slide review of the Kontsevich-Soibelman
(KS) solution to the wall-crossing problem and then proceed to direct and comprehensive physics counting of BPS states that eventually connects to KS. We also asks what input data is needed for
either approaches to produce complete BPS spectra, and this naturally leads to the BPS quiver representation of BPS states and the new notion of quiver invariants.
We propose a simple geometrical conjecture that can segregate BPS states in Higgs phases of the BPS quiver dynamics to those that experience wall-crossing and those that do not, and give
proofs for all cyclice Abelian quivers. We close with explanation of how physics distinguishes two such classes of BPS states. | {"url":"http://perimeterinstitute.ca/seminar/wall-crossing-and-quiver-invariants","timestamp":"2014-04-18T06:08:32Z","content_type":null,"content_length":"25676","record_id":"<urn:uuid:eb4865e8-60b9-4de4-acd4-f595b543a2b1>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00506-ip-10-147-4-33.ec2.internal.warc.gz"} |
CGTalk - Quaternion Rotation and MEL
09-07-2006, 03:34 AM
In an ascii model file I am currently writing a set of import/export scripts for there is the following line.
orientation 0.0 0.0 -1.0 -1.5708
This corresponds to a rotation of the object in question as follows
X: 0.0 Y: 0.0 Z: 90.0
Easy enough to figure out -1.5708 radians is -90 degress. The -1.0 I assume to mean rotate on the Z-axis the inverse of the angle of rotation.
Well that all seemed simple enough until I ran across this later in the same model file
orientation 0.51909 0.440471 -0.732482 -4.41171
Which corresponds to a rotation of the object it is attached to as follows
X: -169.0 Y: 66.0 Z: -82.5
I've read several articles on quaternion rotation and quaternions in general, looked at a few scripts and programs that actually perform quaternion rotation and I'm no closer to figuring out how the
four values in my orientation line come together to spit out the x,y,z rotation values.
I know MQuaternion is part of the API. How would I use MEL to pass the orientation values to the MQuaternion class so that I can end up with the rotation values to plug into the rotateAxis attribute?
Thanks for any help, | {"url":"http://forums.cgsociety.org/archive/index.php/t-402071.html","timestamp":"2014-04-20T01:33:05Z","content_type":null,"content_length":"11697","record_id":"<urn:uuid:2cb7a630-1f9b-4ab7-8e8f-f14ed6430927>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00516-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Parallel Hashed Oct-Tree N-Body Algorithm
Results 1 - 10 of 116
- International Journal of Parallel Programming , 2001
"... The performance of irregular applications on modern computer systems is hurt by the wide gap between CPU and memory speeds because these applications typically underutilize multi-level memory
hierarchies, which help hide this gap. This paper investigates using data and computation reorderings to i ..."
Cited by 89 (2 self)
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The performance of irregular applications on modern computer systems is hurt by the wide gap between CPU and memory speeds because these applications typically underutilize multi-level memory
hierarchies, which help hide this gap. This paper investigates using data and computation reorderings to improve memory hierarchy utilization for irregular applications. We evaluate the impact of
reordering on data reuse at different levels in the memory hierarchy. We focus on coordinated data and computation reordering based on space-filling curves and we introduce a new
architecture-independent multi-level blocking strategy for irregular applications. For two particle codes we studied, the most effective reorderings reduced overall execution time by a factor of two
and four, respectively. Preliminary experience with a scatter benchmark derived from a large unstructured mesh application showed that careful data and computation ordering reduced primary cache
misses by a factor of two compared to a random ordering.
, 1999
"... Programming languages that provide multidimensional arrays and a flat linear model of memory must implement a mapping between these two domains to order array elements in memory. This layout
function is fixed at language definition time and constitutes an invisible, non-programmable array attribute. ..."
Cited by 72 (5 self)
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Programming languages that provide multidimensional arrays and a flat linear model of memory must implement a mapping between these two domains to order array elements in memory. This layout function
is fixed at language definition time and constitutes an invisible, non-programmable array attribute. In reality, modern memory systems are architecturally hierarchical rather than flat, with
substantial differences in performance among different levels of the hierarchy. This mismatch between the model and the true architecture of memory systems can result in low locality of reference and
poor performance. Some of this loss in performance can be recovered by re-ordering computations using transformations such as loop tiling. We explore nonlinear array layout functions as an additional
means of improving locality of reference. For a benchmark suite composed of dense matrix kernels, we show by timing and simulation that two specific layouts (4D and Morton) have low implementation
costs (2--5% of total running time) and high performance benefits (reducing execution time by factors of 1.1-2.5); that they have smooth performance curves, both across a wide range of problem sizes
and over representative cache architectures; and that recursion-based control structures may be needed to fully exploit their potential.
- IEEE Transactions on Parallel and Distributed Systems , 1995
"... We discuss Inverse Spacefilling Partitioning (ISP), a partitioning strategy for nonuniform scientific computations running on distributed memory MIMD parallel computers. We consider the case of
a dynamic workload distributed on a uniform mesh, and compare ISP against Orthogonal Recursive Bisectio ..."
Cited by 56 (2 self)
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We discuss Inverse Spacefilling Partitioning (ISP), a partitioning strategy for nonuniform scientific computations running on distributed memory MIMD parallel computers. We consider the case of a
dynamic workload distributed on a uniform mesh, and compare ISP against Orthogonal Recursive Bisection (ORB) and a Median of Medians variant of ORB, ORB-MM. We present two results. First, ISP and
ORB-MM are superior to ORB in rendering balanced workloads---because they are more finegrained ---and incur communication overheads that are comparable to ORB. Second, ISP is more attractive than
ORB-MM from a software engineering standpoint because it avoids elaborate bookkeeping. Whereas ISP partitionings can be described succinctly as logically contiguous segments of the line, ORB-MM's
partitionings are inherently unstructured. We describe the general d-dimensional ISP algorithm and report empirical results with two- and three-dimensional, non-hierarchical particle methods. Scott
B. Bad...
- Computer Physics Communications , 1995
"... We describe our implementation of the parallel hashed oct-tree (HOT) code, and in particular its application to neighbor finding in a smoothed particle hydrodynamics (SPH) code. We also review
the error bounds on the multipole approximations involved in treecodes, and extend them to include general ..."
Cited by 53 (7 self)
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We describe our implementation of the parallel hashed oct-tree (HOT) code, and in particular its application to neighbor finding in a smoothed particle hydrodynamics (SPH) code. We also review the
error bounds on the multipole approximations involved in treecodes, and extend them to include general cell-cell interactions. Performance of the program on a variety of problems (including gravity,
SPH, vortex method and panel method) is measured on several parallel and sequential machines. 1 Introduction There are two strategies that can be applied in the quest for more knowledge from bigger
and better particle simulations. One can use the brute force approach; simple algorithms on bigger and faster machines (and bigger and faster now means massively parallel). To compute the
gravitational force and potential for a single interaction takes 28 floating point operations (here we count a division as 4 floating point operations and a square root as 4 floating point
operations). A typical grav...
- In Proceedings of Eleventh Annual ACM Symposium on Parallel Algorithms and Architectures , 1999
"... Matrix multiplication is an important kernel in linear algebra algorithms, and the performance of both serial and parallel implementations is highly dependent on the memory system behavior.
Unfortunately, due to false sharing and cache conflicts, traditional column-major or row-major array layouts i ..."
Cited by 48 (4 self)
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Matrix multiplication is an important kernel in linear algebra algorithms, and the performance of both serial and parallel implementations is highly dependent on the memory system behavior.
Unfortunately, due to false sharing and cache conflicts, traditional column-major or row-major array layouts incur high variability in memory system performance as matrix size varies. This paper
investigates the use of recursive array layouts for improving the performance of parallel recursive matrix multiplication algorithms. We extend previous work by Frens and Wise on recursive matrix
multiplication to examine several recursive array layouts and three recursive algorithms: standard matrix multiplication, and the more complex algorithms of Strassen and Winograd. We show that while
recursive array layouts significantly outperform traditional layouts (reducing execution times by a factor of 1.2--2.5) for the standard algorithm, they offer little improvement for Strassen's and
Winograd's algorithms;...
- IN FIFTH INTERNATIONAL SYMPOSIUM ON HIGH-PERFORMANCE COMPUTER ARCHITECTURE , 1999
"... This paper studies application performance on systems with strongly non-uniform remote memory access. In current generation NUMAs the speed difference between the slowest and fastest link in an
interconnect---the "NUMA gap"---is typically less than an order of magnitude, and many conventional para ..."
Cited by 35 (11 self)
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This paper studies application performance on systems with strongly non-uniform remote memory access. In current generation NUMAs the speed difference between the slowest and fastest link in an
interconnect---the "NUMA gap"---is typically less than an order of magnitude, and many conventional parallel programs achieve good performance. We study how different NUMA gaps influence application
performance, up to and including typical wide-area latencies and bandwidths. We find that for gaps larger than those of current generation NUMAs, performance suffers considerably (for applications
that were designed for a uniform access interconnect). For many applications, however, performance can be greatly improved with comparatively simple changes: traffic over slow links can be reduced by
making communication patterns hierarchical---like the interconnect. We find that in four out of our six applications the size of the gap can be increased by an order of magnitude or more without
- Computer Methods in Applied Mechanics and Engineering
"... . In many important computational mechanics applications, the computation adapts dynamically during the simulation. Examples include adaptive mesh refinement, particle simulations and transient
dynamics calculations. When running these kinds of simulations on a parallel computer, the work must be a ..."
Cited by 34 (2 self)
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. In many important computational mechanics applications, the computation adapts dynamically during the simulation. Examples include adaptive mesh refinement, particle simulations and transient
dynamics calculations. When running these kinds of simulations on a parallel computer, the work must be assigned to processors in a dynamic fashion to keep the computational load balanced. A number
of approaches have been proposed for this dynamic load balancing problem. This paper reviews the major classes of algorithms, and discusses their relative merits on problems from computational
mechanics. Shortcomings in the state-of-the-art are identified and suggestions are made for future research directions. Key words. dynamic load balancing, parallel computer, adaptive mesh refinement
1. Introduction. The efficient use of a parallel computer requires two, often competing, objectives to be achieved. First, the processors must be kept busy doing useful work. And second, the amount
of interprocess...
, 1999
"... The performance of both serial and parallel implementations of matrix multiplication is highly sensitive to memory system behavior. False sharing and cache conflicts cause traditional
column-major or row-major array layouts to incur high variability in memory system performance as matrix size var ..."
Cited by 31 (0 self)
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The performance of both serial and parallel implementations of matrix multiplication is highly sensitive to memory system behavior. False sharing and cache conflicts cause traditional column-major or
row-major array layouts to incur high variability in memory system performance as matrix size varies. This paper investigates the use of recursive array layouts to improve performance and reduce
variability. Previous work on recursive matrix multiplication is extended to examine several recursive array layouts and three recursive algorithms: standard matrix multiplication, and the more
complex algorithms of Strassen and Winograd. While recursive layouts significantly outperform traditional layouts (reducing execution times by a factor of 1.2--2.5) for the standard algorithm, they
offer little improvement for Strassen's and Winograd's algorithms. For a purely sequential implementation, it is possible to reorder computation to conserve memory space and improve performance
between ... | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=240119","timestamp":"2014-04-19T01:20:32Z","content_type":null,"content_length":"39577","record_id":"<urn:uuid:3507a20b-f60f-4bd8-800b-9fc0a9198961>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00546-ip-10-147-4-33.ec2.internal.warc.gz"} |
Explicit function definition
Thank you for your answer. I don’t have a worksheet yet, but here is what I want to do (I simplified the problem a bit). I have the following (x y) data set where y represents the value of the
derivate function at point x:
X Y
0.2 0.1987
0.4 0.3894
0.6 0.5646
0.8 0.7174
1 0.8415
1.2 0.932
1.4 0.9854
1.5708 1
What I want is to get the value of the function, by numerical integration, at these x points. I don’t need any curve fitting, interpolation function or any other kind of approximation. Could you give
me some hints on how to solve this problem with Mathcad? | {"url":"http://communities.ptc.com/message/176003","timestamp":"2014-04-17T18:26:01Z","content_type":null,"content_length":"153205","record_id":"<urn:uuid:324d198a-14d6-4fa2-a8fe-7347c34fb6aa>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00040-ip-10-147-4-33.ec2.internal.warc.gz"} |
Perform Hamming window and FFT with Java
I have spent sometimes to study and do the coding for FFT on audio samples.
My project is to study the FFT working by performing the FFT on audio sampled data.
After this is done, then only proceed to extract the pitch value in the audio data.
Below is the coding that i have done by referring to the provided steps:
In my main, i call this method and past the retrieved sampled sound data to this method.
public static void Analyze(int[] soundSample,float sample_rate ) {
int N = (int)sample_rate/5;
int Number_Sample = soundSample.length;
Complex[] fftBuffer = new Complex [2*N];
Complex[] fftResult = new Complex [2*N];
Complex [] lastN = new Complex [N]; // The array to save the last N sample
int delay = 0;
double delta = 2*Math.PI/(2*N);
// I have no idea how can i convert my sample array to double so that it will be in the range of [-1,+1]
while(delay <=soundSample.length){
//Extract the 2N sample for FFT analysis and convert the data to complex number.
for (int z=0; z<2*N; z++){
fftBuffer[z] = new Complex(soundSample[z+delay],0) ;
for (int i=N-1;i>=N/2; i-- ){
lastN[N-1-i] = fftBuffer;
for (int z=0; z<2*N; z++){
fftBuffer[z] = fftBuffer[z].times(0.54-0.46*Math.cos(z*delta));
fftResult = FFT1.fft(fftBuffer);
delay = 2*N + delay;
1) I was trying to perform FFT with 2N samples then keep on looping the FFT method until 2N reaches the ends of sampled data.
But the FFT that i am working with is radix 2... It doesn't work with my 2N samples... Please teach me how should i work out FFT regardless the number of sample?
2) The hamming window coefficient i m using is based on http://www.mathworks.com/help/toolbox/signal/hamming.html . I am working on index [0:2N] ..
Is it appropriate?
3) According to your No1 steps, the acceptable frequency resolution is 5Hz. May i know what is this representing? And is it application for most of the FFT application? How can i determine the frequency resolution that i should used in my project?
Sorry for late reply as i was trying to work out the thing..
Hereby attach to your the my coding.. and hopes to have your guidance and tutorial how to extract the pitch for recording audio file with Java.
I have done the pitch extraction with MATLAB.. but Matlab as built-in FFT function...
So i m now get stucked how to perform FFT on audio sound sample regardless the N value of sound sample for FFT buffer.
Many thanks for your former advise... and
Looking forward for your replies again.
Happy New Year 2011 :)
Edited by: 诸葛 on Dec 31, 2010 9:29 AM | {"url":"https://community.oracle.com/thread/2152136?tstart=75&start=0","timestamp":"2014-04-19T11:43:32Z","content_type":null,"content_length":"237566","record_id":"<urn:uuid:83967c26-b868-416d-914e-fab0b40698d7>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00017-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum - Problems Library - All Trig/Calc Problems of the Week
This page:
all trig/calc
About Levels
of Difficulty
law of sines
law of cosines
Browse all
About the
PoW Library
Browse all Trig/Calculus Problems of the Week
Participation in the Trig/Calculus Problems of the Week allows teachers and students to address the NCTM Problem Solving Standard for Grades 9-12, enabling students to build new mathematical
knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; and monitor and reflect
on the process of mathematical problem solving.
For background information elsewhere on our site, explore the High School Trigonometry and Calculus areas of the Ask Dr. Math archives. To find relevant sites on the Web, browse and search
Trigonometry and Calculus (Single Variable) in our Internet Mathematics Library.
Access to these problems requires a Membership. | {"url":"http://mathforum.org/library/problems/sets/calcpow_all.html","timestamp":"2014-04-16T19:16:50Z","content_type":null,"content_length":"26340","record_id":"<urn:uuid:2d4632c4-0ceb-4d95-84cb-71cf8ecf7863>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00604-ip-10-147-4-33.ec2.internal.warc.gz"} |
American Canyon Calculus Tutors
Looking for help in Calculus, Precalculus, Trigonometry, Algebra, or Geometry? I'm a mathematics teacher who is extremely passionate about the subject. I have been teaching in the North Bay Area
for several years and am excited about the states adoption of the common core curriculum for mathematics.
13 Subjects: including calculus, geometry, ASVAB, algebra 1
...I think this is a wonderful combination: I can relate to students, understand their frustrations and fears, and at the same time I deeply understand math and take great joy in communicating
this to reluctant and struggling students, as well as to able students who want to maximize their achieveme...
20 Subjects: including calculus, statistics, geometry, biology
...I also participated in several undergraduate and graduate student mentoring programs at UC Berkeley, which included group seminar presentations as well as one-on-one tutoring sessions of
students from science and engineering majors, including minority students. I actually co-founded the Bioengin...
24 Subjects: including calculus, chemistry, physics, geometry
...My tutoring methods vary student-by-student, but I specialize in breaking down problems and asking questions to guide the student toward discovering and truly understanding concepts which helps
with retention and effective test-taking. I take pride in the success of each and every one of my stud...
17 Subjects: including calculus, chemistry, physics, statistics
I just recently graduated from the Massachusetts Institute of Technology this June (2010) with a Bachelors of Science in Physics. While I was there, I also took various Calculus courses and
courses in other areas of math that built on what I learned in high school. I'm a definite believer in the value of knowing the ways the world works, and the value of a good education.
6 Subjects: including calculus, physics, algebra 1, algebra 2 | {"url":"http://www.algebrahelp.com/American_Canyon_calculus_tutors.jsp","timestamp":"2014-04-18T09:12:25Z","content_type":null,"content_length":"25240","record_id":"<urn:uuid:91d34102-a0b2-4385-befb-5cb34c8deb84>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00280-ip-10-147-4-33.ec2.internal.warc.gz"} |
Benjamin Mack
I try to use this programm but I am totally new to matlab (but have experience in R). When I call the function
mrmr_mid_d (variables, response, 25)
the following error occurs:
??? Undefined function or method 'mrmr_mid_d' for input arguments of type 'double'.
The same occures when I convert my data in int8 or single.
something similar happens when I try to use the demo_mi.m :
??? Undefined function or method 'estpab' for input arguments of type 'double'.
Error in ==> mutualinfo at 21
[p12, p1, p2] = estpab(vec1,vec2);
I think my problem is very a very basic one because I do not know anything about Matlab.
Could anybody please help me. It is very important for me to use this programm and the online version does not work with my data set because it is too large.
Thank you very much in advance, | {"url":"http://www.mathworks.com/matlabcentral/fileexchange/authors/82929","timestamp":"2014-04-18T00:24:59Z","content_type":null,"content_length":"32688","record_id":"<urn:uuid:c1c77ab9-bb6e-4d37-b518-aba70a3fadfb>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00320-ip-10-147-4-33.ec2.internal.warc.gz"} |
More useless statistics
August 22, 2011
By richierocks
Over at the ExploringDataBlog, Ron Pearson just wrote a post about the cases when means are useless. In fact, it’s possible to calculate a whole load of stats on your data and still not really
understand it. The canonical dataset for demonstrating this (spoiler alert: if you are doing an intro to stats course, you will see this example soon) is the Anscombe quartet.
The data set is available in R as anscombe, but it requires a little reshaping to be useful.
anscombe2 <- with(anscombe, data.frame(
x = c(x1, x2, x3, x4),
y = c(y1, y2, y3, y4),
group = gl(4, nrow(anscombe))
Note the use of gl to autogenerate factor levels.
So we have four sets of x-y data, which we can easily calculate summary statistics from using ddply from the plyr package. In this case we calculate the mean and standard deviation of y, the
correlation between x and y, and run a linear regression.
(stats <- ddply(anscombe2, .(group), summarize,
mean = mean(y),
std_dev = sd(y),
correlation = cor(x, y),
lm_intercept = lm(y ~ x)$coefficients[1],
lm_x_effect = lm(y ~ x)$coefficients[2]
group mean std_dev correlation lm_intercept lm_x_effect
1 1 7.500909 2.031568 0.8164205 3.000091 0.5000909
2 2 7.500909 2.031657 0.8162365 3.000909 0.5000000
3 3 7.500000 2.030424 0.8162867 3.002455 0.4997273
4 4 7.500909 2.030579 0.8165214 3.001727 0.4999091
Each of the statistics is almost identical between the groups, so the data must be almost identical in each case, right? Wrong. Take a look at the visualisation. (I won’t reproduce the plot here and
spoil the surprise; but please run the code yourself.)
(p <- ggplot(anscombe2, aes(x, y)) +
geom_point() +
facet_wrap(~ group)
Each dataset is really different – the statistics we routinely calculate don’t fully describe the data. Which brings me to the second statistics joke.
A physicist, an engineer and a statistician go hunting. 50m away from them they spot a deer. The physicist calculates the trajectory of the bullet in a vacuum, raises his rifle and shoots. The
bullet lands 5m short. The engineer adds a term to account for air resistance, lifts his rifle a little higher and shoots. The bullet lands 5m long. The statistician yells “we got him!”.
for the author, please follow the link and comment on his blog:
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Verification of Arithmetic Functions with Binary Moment Diagrams
Results 1 - 10 of 77
- IEEE TRANSACTIONS ON SOFTWARE ENGINEERING , 1998
"... In this paper, we present our experiences in using symbolic model checking to analyze a specification of a software system for aircraft collision avoidance. Symbolic model checking has been
highly successful when applied to hardware systems. We are interested in whether model checking can be effect ..."
Cited by 117 (6 self)
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In this paper, we present our experiences in using symbolic model checking to analyze a specification of a software system for aircraft collision avoidance. Symbolic model checking has been highly
successful when applied to hardware systems. We are interested in whether model checking can be effectively applied to large software specifications. To investigate this, we translated a portion of
the state-based system requirements specification of Traffic Alert and Collision Avoidance System II (TCAS II) into input to a symbolic model checker (SMV). We successfully used the symbolic model
checker to analyze a number of properties of the system. We report on our experiences, describing our approach to translating the specification to the SMV language, explaining our methods for
achieving acceptable performance, and giving a summary of the properties analyzed. Based on our experiences, we discuss the possibility of using model checking to aid specification development by
iteratively applying the technique early in the development cycle. We consider the paper to be a data point for optimism about the potential for more widespread application of model checking to
software systems.
- IN PROCEEDINGS OF THE 32ND ACM/IEEE DESIGN AUTOMATION CONFERENCE , 1995
"... Binary Moment Diagrams (BMDs) provide a canonical representations for linear functions similar to the way Binary Decision Diagrams (BDDs) represent Boolean functions. Within the class of linear
functions, we can embed arbitrary functions from Boolean variables to integer values. BMDs can thus model ..."
Cited by 93 (10 self)
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Binary Moment Diagrams (BMDs) provide a canonical representations for linear functions similar to the way Binary Decision Diagrams (BDDs) represent Boolean functions. Within the class of linear
functions, we can embed arbitrary functions from Boolean variables to integer values. BMDs can thus model the functionality of data path circuits operating over word-level data. Many important
functions, including integer multiplication, that cannot be represented efficiently at the bit level with BDDs have simple representations at the word level with BMDs. Furthermore, BMDs can represent
Boolean functions with around the same complexity as BDDs. We propose a hierarchical approach to verifying arithmetic circuits, wherecomponentmodulesare first shownto implement their word-level
specifications. The overall circuit functionality is then verified by composing the component functions and comparing the result to the word-level circuit specification. Multipliers with word sizes
of up to 256 bits hav...
, 1996
"... ACL2 is a mechanized mathematical logic intended for use in specifying and proving properties of computing machines. In two independent projects, industrial engineers have collaborated with
researchers at Computational Logic, Inc. (CLI), to use ACL2 to model and prove properties of state-of-the-art ..."
Cited by 68 (14 self)
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ACL2 is a mechanized mathematical logic intended for use in specifying and proving properties of computing machines. In two independent projects, industrial engineers have collaborated with
researchers at Computational Logic, Inc. (CLI), to use ACL2 to model and prove properties of state-of-the-art commercial microprocessors prior to fabrication. In the first project, Motorola, Inc.,
and CLI collaborated to specify Motorola's complex arithmetic processor (CAP), a single-chip, digital signal processor (DSP) optimized for communications signal processing. Using the specification,
we proved the correctness of several CAP microcode programs. The second industrial collaboration involving ACL2 was between Advanced Micro Devices, Inc. (AMD) and CLI. In this work we proved the
correctness of the kernel of the floating-point division operation on AMD's first Pentium-class microprocessor, the AMD5K 86. In this paper, we discuss ACL2 and these industrial applications, with
particular attention ...
- IEEE Transactions on CAD , 1996
"... Regarding finite state machines as Markov chains facilitates the application of probabilistic methods to very large logic synthesis and formal verification problems. In this paper we present
symbolic algorithms to compute the steady-state probabilities for very large finite state machines (up to 10 ..."
Cited by 66 (7 self)
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Regarding finite state machines as Markov chains facilitates the application of probabilistic methods to very large logic synthesis and formal verification problems. In this paper we present symbolic
algorithms to compute the steady-state probabilities for very large finite state machines (up to 10 27 states). These algorithms, based on Algebraic Decision Diagrams (ADDs) --- an extension of BDDs
that allows arbitrary values to be associated with the terminal nodes of the diagrams --- determine the steady-state probabilities by regarding finite state machines as homogeneous,
discrete-parameter Markov chains with finite state spaces, and by solving the corresponding Chapman-Kolmogorov equations. We first consider finite state machines with state graphs composed of a
single terminal strongly connected component; for this type of systems we have implemented two solution techniques: One is based on the Gauss-Jacobi iteration, the other one is based on simple matrix
multiplication. Then we...
- In Int'l Conf. on CAD , 1995
"... e�mail � emc�cs.cmu.edu e�mail � masahiro�eecs.berkeley.edu e�mail � xzhao�cs.cmu.edu Abstract � Functions that map boolean vectors into the in� tegers are important for the design and
veri�cation of arith� metic circuits. MTBDDs and BMDs have been proposed for representing this class of functions. ..."
Cited by 55 (3 self)
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e�mail � emc�cs.cmu.edu e�mail � masahiro�eecs.berkeley.edu e�mail � xzhao�cs.cmu.edu Abstract � Functions that map boolean vectors into the in� tegers are important for the design and veri�cation of
arith� metic circuits. MTBDDs and BMDs have been proposed for representing this class of functions. We discuss the relation� ship between these methods and describe a generalization called hybrid
decision diagrams which is often much more concise. We show how to implement arithemetic operations e�ciently for hybrid decision diagrams. In practice � this is one of the main limitations of BMDs
since performing arith� metic operations on functions expressed in this notation can be very expensive. In order to extend symbolic model check� ing algorithms to handle arithmetic properties � it is
essential to be able to compute the BDD for the set of variable as� signments that satisfy an arithmetic relation. In our paper� we give an e�cient algorithm for this purpose. Moreover� we prove that
for the class of linear expressions � the time complexity of our algorithm is linear in the number of vari� ables. 1
, 1997
"... This paper presents a new data structure called Boolean Expression Diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams
(BDDs) which can represent any Boolean circuit in linear space and still maintain many of the desirable proper ..."
Cited by 46 (5 self)
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This paper presents a new data structure called Boolean Expression Diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams (BDDs)
which can represent any Boolean circuit in linear space and still maintain many of the desirable properties of BDDs. Two algorithms are described for transforming a BED into a reduced ordered BDD.
One is a generalized version of the BDD apply-operator while the other can exploit the structural information of the Boolean expression. This ability is demonstrated by verifying that two di erent
circuit implementations of a 16-bit multiplier implement the same Boolean function. Using BEDs, this veri cation problem is solved in less than a second, while using standard BDD techniques this
problem is infeasible. Generally, BEDs are useful in applications, for example tautology checking, where the end-result as a reduced ordered BDD is small.
- Proceedings of the 27-th STOC , 1998
"... . We prove that read-once branching programs computing integer multiplication require size 2 ## # n) . This is the first nontrivial lower bound for multiplication on branching programs that are
not oblivious. By the appropriate problem reductions, we obtain the same lower bound for other arithmeti ..."
Cited by 34 (0 self)
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. We prove that read-once branching programs computing integer multiplication require size 2 ## # n) . This is the first nontrivial lower bound for multiplication on branching programs that are not
oblivious. By the appropriate problem reductions, we obtain the same lower bound for other arithmetic functions. Key words. multiplication, read-once, branching programs, BDD, verification AMS
subject classifications. 68Q05, 68Q25, 68M15 PII. S0097539795290349 1. Introduction and background. It is well known that many functions, some of them very simple, cannot be computed by read-once
branching programs of polynomial size [We88, Za84, Du85, We87, BHST87, Ju88, Kr88]. Interest in whether integer multiplication can be so computed has been created by recent developments in the field
of digital design and hardware verification. 1.1. Hardware verification and branching programs. The central problem of verification is to check whether a combinational hardware circuit has been
correctly designe...
- In ARTS, LNCS 1601 , 1999
"... . Stochastic process algebras have been introduced in order to enable compositional performance analysis. The size of the state space is a limiting factor, especially if the system consists of
many cooperating components. To fight state space explosion, various proposals for compositional aggregatio ..."
Cited by 27 (13 self)
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. Stochastic process algebras have been introduced in order to enable compositional performance analysis. The size of the state space is a limiting factor, especially if the system consists of many
cooperating components. To fight state space explosion, various proposals for compositional aggregation have been made. They rely on minimisation with respect to a congruence relation. This paper
addresses the computational complexity of minimisation algorithms and explains how efficient, BDD-based data structures can be employed for this purpose. 1 Introduction Compositional application of
stochastic process algebras (SPA) is particularly successful if the system structure can be exploited during Markov chain generation. For this purpose, congruence relations have been developed which
justify minimisation of components without touching behavioural properties. Examples of such relations are strong equivalence [22], (strong and weak) Markovian bisimilarity [16] and extended
Markovian bisimi...
- Formal Hardware Verification , 1996
"... ion The main problem with model checking is the state explosion problem -- the state space grows exponentially with system size. Two methods have some popularity in attacking this problem:
compositional methods and abstraction. While they cannot solve the problem in general, they do offer significa ..."
Cited by 26 (6 self)
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ion The main problem with model checking is the state explosion problem -- the state space grows exponentially with system size. Two methods have some popularity in attacking this problem:
compositional methods and abstraction. While they cannot solve the problem in general, they do offer significant improvements in performance. The direct method of verifying that a circuit has a
property f is to show the model M satisfies f . The idea behind abstraction is that instead of verifying property f of model M , we verify property f A of model MA and the answer we get helps us
answer the original problem. The system MA is an abstraction of the system M . One possibility is to build an abstraction MA that is equivalent (e.g. bisimilar [48]) to M . This sometimes leads to
performance advantages if the state space of MA is smaller than M . This type of abstraction would more likely be used in model comparison (e.g. as in [38]). Typically, the behaviour of an
abstraction is not equivalent...
- ACM Transactions on Software Engineering and Methodology , 2000
"... In recent years, there has been a surge of progress in automated verification methods based on state exploration. In areas like hardware design, these technologies are rapidly augmenting key
phases of testing and validation. To date, one of the most successful of these methods has been symbolic mode ..."
Cited by 24 (7 self)
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In recent years, there has been a surge of progress in automated verification methods based on state exploration. In areas like hardware design, these technologies are rapidly augmenting key phases
of testing and validation. To date, one of the most successful of these methods has been symbolic model checking, in which large finite-state machines are encoded into compact data structures such as
binary decision diagrams (BDDs) -- and are then checked for safety and liveness properties. However, these techniques have not realized the same success on software systems. One limitation is their
inability to deal with infinite-state programs -- even those with a single unbounded integer. A second problem is that of finding efficient representations for various variable types. We recently
proposed a model checker for integer-based systems that uses arithmetic constraints as the underlying state representation. While this approach easily verified some subtle, infinite-state concurrency | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=115446","timestamp":"2014-04-19T13:12:14Z","content_type":null,"content_length":"41610","record_id":"<urn:uuid:7f1efd20-1649-47d2-878a-14d538dc81fa>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00600-ip-10-147-4-33.ec2.internal.warc.gz"} |
Edgewater, MD Math Tutor
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The Path-Planning Problem
Next: EPB/PDO Implementation.3 Up: Reasoning About Path-Planning with Previous: Reasoning About Path-Planning with
Path planning problems are a far more common concern in robotics than the simple sliding problem presented in the previous section. The typical problem formulation in path planning is the ``(piano)
mover's problem'', or ``findpath problem''. This problem, like my sliding problem, involves a number of obstacles and a single moving object. The difference between the mover's problem and the
sliding problem is that in the mover's problem, the goal is to find a path through the obstacles while avoiding touching any of them (rather than finding what can happen when maintaining contact with
The obstacles in my examples are all right angled polygons, with a mixture of concave and convex vertices, and with multiple levels of detail on the boundary. This is also true of the moving object.
Most of the implemented code allows for angles which are not right angles, and for curved edges, but examples with these properties were not tested, for reasons explained later.
The reasoning performed in solving this problem was intended to complement that performed in the sliding problem. It involves motion of the moving object through free space, and discovery of
non-contact paths between obstacles. The goal state is a position for the moving object which is on the other side of the obstacles initially surrounding it, and provides unbounded movement away from
those obstacles. The solution to the problem also includes a suggested path which will enable the moving object to reach that goal state.
The remainder of this section first discusses the implementation of the extended polygon boundary/partial distance ordering representation, and then describes the reasoning strategies used to solve
path planning problems. The overall stages in solving a problem are discussed first, and the method of solution for each stage is then described.
Next: EPB/PDO Implementation.3 Up: Reasoning About Path-Planning with Previous: Reasoning About Path-Planning with Alan Blackwell | {"url":"http://www.cl.cam.ac.uk/~afb21/publications/masters/node76.html","timestamp":"2014-04-21T02:36:33Z","content_type":null,"content_length":"5461","record_id":"<urn:uuid:35bf33fa-0e61-423b-8882-bc17b64aad5c>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00306-ip-10-147-4-33.ec2.internal.warc.gz"} |
I was asked how to calculate the power to a 3 phase delta connected heater if the load is not balanced. We all know the formula W = LV x LA x 1.732 x PF for 3 phase power but I can't remember how to
do the same for an un-balanced load. My intuition tells me that using the average current for LA would be close but that seems too simple.
Anyone remember the formula?
Responding to Roy Matson's 21-Dec (00:34) question... yes, you are correct. Averaging is not the answer. But first, please clarify your need:
Given the three line currents, Ia, Ib, and Ic, as well as the three line-to-line voltages, Vab, Vbc, and Vca, you want to determine:
1) The heater's total power consumption in Watts?
2) The power consumption of the three heating element?
The answer, like the your previous qustion, can be found in the Control List archives. If a search proves unfruitful, then contact me... on or off list!
Regards, Phil Corso (cepsicon@aol.com)
Roy, have you resolved the calculation question asked in your 21-Dec-07 (00:34) post?
Phil Corso (cepsicon@aol.com)
Not exactly, I asked several engineer buddies, they couldn't remember either however I concluded that 2 out of 6 elements were toast.
Roy, if you still want the formula, please provide (if available) the three ph-ph voltages, and the three line currents measured before discovering that 2 elements were gone!
Regards, Phil Corso (cepsicon@aol.com)
Roy, the unsymmetrical loss of elements can be easily explained if the load was wye-connected, with two-paralleled elements per phase.
If the above connection has no neutral wire the remaining resistances are exposed to voltages that could be much higher than expected line-neutral voltage!
Regards, Phil Corso (cepsicon@aol.com)
Yes, even I could calculate a Y configuration but the elements were in delta. When site finally got back to me the phase currents were as follows.
Working element
3.3, 3.3, 3.3
Faulty element
1.9, 1.9, 0
So I concluded there must be 2 elements out. I am still curious as to how to calculate unbalance kW.
It seems from the above results if you subtract the lowest reading 3.3 and calculate that
3.3 x 600 x 1.732 = 3429
then add
1.9 x 600 = 1140, end result 4569 will be correct.
But that's easy, what if all 3 are different?
I thought there would be a simple formula.
It's not that important now, so don't stress over it.
Roy, it depends on what parameter you have chosen for current!
1) The general formula for a balanced delta-connected resistive-load is:
a) If 3.3A repesents line-current, Ia, then total power is:
Pt = Sqrt(3)(Vab)(Ia).
b) If 3.3A represents phase-current thru an element of the delta-connected load, Iab, then total power is:
Pt = (3)x(Vab)x(Iab).
2) The general formula for an unbalanced delta-
connected resistive load is:
a) If one has access to the individual elements then total power is:
Pt = (Vab)(Iab)+(Vbc)(Ib)+(Vca)(Ic).
b) For the case you cite, it would be helpful to know what parameter the two 1.9A measurements represent!
I apologize for misunderstanding your original question. I thought you wanted to determine the total power for a three-phase load (delta or wye) given the 3 phase-phase voltage magnitudes, |Vab|, |
Vbc|, & |Vca|, as well as the 3-line currents magnitudes, |Ia|, |Ib|, & |Ic|.
The above problem requires the use of a mathematical approach known as Symmetrical Components!
Regards, Phil
I work in smelting furnaces, which are basically resistance furnaces. It is a three phase AC system feeding large magnitudes of current (in kiloamperes) to raw material (charge). There is always
difference in the current drawn by each phase as well as voltage between them. I want to calculate power consumed what formula should be used.
Another point is whether I should consider load as star or delta for power calculations and how it will be affecting calculations, which one will be more practical? I have a facility to measure
voltage between electrode-hearth. But due to unbalance load nature I cannot measure the correct voltage. Any suggestions?
Roy, I forgot to mention in my earlier post that the 1.9A measurement is indicative of one blown element!
That is, current magnitude, 1.9A, for the damaged case, is 57.7% of the current magnitude, 3.3A, representing the undamaged case. An additional clue is the fact that two of the line-currents are
equal, and the 3rd is zero!
Regards, Phil Corso
In keeping with one of Control.com's goals, i.e., education of List members (admittedly some acrimoniously), the following Table compares three cases illustrating current distribution and power in a
3-phase, delta-connected system! They are: 1) the balanced case with equal resistive branch-elements; 2) open-circuiting of one branch-element; and 3) open-circuiting of two branch-elements:
TABLE A: Per-Unit A & Pwr, D-Load
Unit Balanced 1-Res Opn 2-Res Opn
----- -------- --------- ----------
Ia 1.00 1.00 0.00
Ib 1.00 0.58 0.58
Ic 1.00 0.58 0.58
----- -------- --------- ----------
Iab 0.58 0.58 0.00
Ibc 0.58 0.00 0.58
Ica 0.58 0.58 0.00
----- -------- --------- ----------
Pwr 1.00 0.67 0.33
The above was derived from the fact that ANY three-phase power problem, configured as wye or delta, balanced or unbalanced, can be represented as a two-loop circuit consisting of just two sources of
voltage, and three resistive or complex loads! If additional detail is required, contact me!
Phil Corso (cepsicon@aol.com)
Unbalanced systems can be evaluated by computing phase current and then apply Kirchoff's law (vector form)
I neutral=-(Ia+Ib+Ic)
First, calculate Ia=Vab/Zab(Zab=R for resister)
similarly calculate Ib=Vbc/Zbc and Ic=Vac/Zbc
Total power dissipated=total I^2*R
(Calculate I using vector form.)
It will be easier to see if we draw 3 phase, delta connected, and then calculate Ia, Ib, Ic separately, also calculate real power for each phase separately and then total it up.
Note: Real current value will be calculated by vector form.
I hope this helps.
Dilip... the system you describe is somewhat like the classic "unbalanced 3-phase resistive" circuit. However, apparently information in this thread is not quite what you are looking for.
To get idea of the unbalance magnitude you face, please provide:
A) the 3 line-line voltages.
B) the 3 phase-neutral voltages.
C) the 3 line currents.
D) the transformer nameplate data.
Regards, Phil
We work with resistive heating elements in our products. At times we have unbalanced numbers of heating elements per phase. (i.e., 12 elements divided between three phases)
Could you provide an example calculation of leg current using heater elements arranged 1-1-2 across a three phase delta supply?
L. Burton... your post is the same as Eric L's in Control.Com Thread
I sent him the general solution covering n-elements/phase arranged as 3-phase, 3-wire delta-load
If you want a copy, contact me at cepsicon@aol.com
Regards, Phil Corso
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On spaces having the homotopy type of a CW complex
Results 1 - 10 of 47
- J. Pure Appl. Algebra
"... We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the G-CW-version EF(G) and the numerable G-space version JF(G). They agree
if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact su ..."
Cited by 55 (28 self)
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We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the G-CW-version EF(G) and the numerable G-space version JF(G). They agree if G
is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups
in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also
discuss the relevance of these spaces for the Baum-Connes Conjecture about the topological K-theory of the reduced group C ∗-algebra, for the Farrell-Jones Conjecture about the algebraic K-and
L-theory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.
, 1995
"... Riemann-Roch theorems assert that certain algebraically defined wrong way maps (transfers) in algebraic K–theory agree with topologically defined ones [BaDo]. Bismut and Lott [BiLo] proved such
a Riemann–Roch theorem where the wrong way maps are induced by the projection of a smooth fiber bundle, an ..."
Cited by 22 (3 self)
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Riemann-Roch theorems assert that certain algebraically defined wrong way maps (transfers) in algebraic K–theory agree with topologically defined ones [BaDo]. Bismut and Lott [BiLo] proved such a
Riemann–Roch theorem where the wrong way maps are induced by the projection of a smooth fiber bundle, and the topologically defined transfer map is the Becker–Gottlieb transfer. We generalize and
refine their theorem, and prove a converse stating that the Riemann–Roch condition is equivalent to the existence of a fiberwise smooth structure. In the process, we prove a family index theorem
where the K–theory used is algebraic K–theory, and the fiber bundles have topological (not necessarily smooth) manifolds as fibers.
, 1997
"... We provide a "toolkit " of basic lemmas for the comparison of homotopy types of homotopy colimits of diagrams of spaces over small categories. We show how this toolkit can be used on quite
different fields of applications. We demonstrate this with respect to 1. Bjorner's "Generalized Homot ..."
Cited by 18 (2 self)
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We provide a "toolkit " of basic lemmas for the comparison of homotopy types of homotopy colimits of diagrams of spaces over small categories. We show how this toolkit can be used on quite
different fields of applications. We demonstrate this with respect to 1. Bjorner's "Generalized Homotopy Complementation Formula" [4], 2. the topology of toric varieties, 3. the study of homotopy
types of arrangements of subspaces, 4. the analysis of homotopy types of subgroup complexes.
- Proc. London Math. Soc. 88 (2004) 632–658, arXiv: math.AG/0003093. T. Hausel, N. Proudfoot / Topology 44 , 2002
"... This paper will show that, in the rank 2 case, the cohomology ring of this noncompact space is again generated by universal classes. A companion paper [23] gives a complete set of explicit
relations between these generators ..."
Cited by 15 (6 self)
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This paper will show that, in the rank 2 case, the cohomology ring of this noncompact space is again generated by universal classes. A companion paper [23] gives a complete set of explicit relations
between these generators
- Theory Appl. Categ
"... We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space.
From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the ..."
Cited by 10 (0 self)
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We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From
this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module G. We use this interpretation to define a twisting of
Yetter’s Invariant by cohomology classes of crossed modules, defined
- Math. Zeitschrift , 1999
"... Abstract. We prove the Arnold conjecture for closed symplectic manifolds with π2(M) = 0 and cat M = dim M. Furthermore, we prove an analog of the Lusternik– Schnirelmann theorem for functions
with “generalized hyperbolicity ” property. ..."
Cited by 8 (0 self)
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Abstract. We prove the Arnold conjecture for closed symplectic manifolds with π2(M) = 0 and cat M = dim M. Furthermore, we prove an analog of the Lusternik– Schnirelmann theorem for functions with
“generalized hyperbolicity ” property.
"... this paper we explore the relationship between combinatorial vector bundles and real vector bundles. As a consequence of our results we get theorems relating the topology of the combinatorial
Grassmannians to that of their real analogs. The theory of oriented matroids gives a combinatorial abstract ..."
Cited by 6 (3 self)
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this paper we explore the relationship between combinatorial vector bundles and real vector bundles. As a consequence of our results we get theorems relating the topology of the combinatorial
Grassmannians to that of their real analogs. The theory of oriented matroids gives a combinatorial abstraction of linear algebra; a k-dimensional subspace of R | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=2175671","timestamp":"2014-04-20T19:33:27Z","content_type":null,"content_length":"32141","record_id":"<urn:uuid:1b957bb5-93e7-4389-8834-4eb1b0a6b383>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00012-ip-10-147-4-33.ec2.internal.warc.gz"} |
gamma distribution
October 30th 2008, 03:57 AM
gamma distribution
If X is a gamma(2,lambda) distribution
lambda(^2)xe^(-lambda x) x >0
0 otherwise
How do i obtain the mean and variance of X?
October 30th 2008, 04:18 AM
mr fantastic
$f(x) = \lambda^2 \, x \, e^{-\lambda x}$ for x > 0 and zero elsewhere.
$E(X) = \lambda^2 \int_{0}^{+\infty} x^2 \, e^{-\lambda x} \, dx$.
$Var(X) = E(X^2) - [E(X)]^2$.
$E(X^2) = \lambda^2 \int_{0}^{+\infty} x^3 \, e^{-\lambda x} \, dx$.
The integrals can be evaluated by repeated application of integration by parts.
October 30th 2008, 08:28 AM
Chris L T521
$f(x) = \lambda^2 \, x \, e^{-\lambda x}$ for x > 0 and zero elsewhere.
$E(X) = \lambda^2 \int_{0}^{+\infty} x^2 \, e^{-\lambda x} \, dx$.
$Var(X) = E(X^2) - [E(X)]^2$.
$E(X^2) = \lambda^2 \int_{0}^{+\infty} x^3 \, e^{-\lambda x} \, dx$.
The integrals can be evaluated by repeated application of integration by parts.
Or by doing it in a way where integration by parts is not required ;)
I'll quickly do $E(X^2)$. A similar thing can be done with $E(X)$.
Let $u=\lambda x\implies x=\frac{u}{\lambda}$. Thus, $\,dx=\frac{\,du}{\lambda}$
The integral transforms into $\lambda^2\int_0^{\infty}\frac{1}{\lambda}\left(\fr ac{u}{\lambda}\right)^3e^{-u}\,du\implies\frac{1}{\lambda^2}\int_0^{\infty}u^ 3e^{-u}\,du$.
But note that $\int_0^{\infty}e^{-u}u^3\,du=\Gamma(4)$.
So we have $\frac{1}{\lambda^2}\int_0^{\infty}u^3e^{-u}\,du=\frac{\Gamma(4)}{\lambda^2}$.
Since $\Gamma(4)=3!=6$, we now see that $\color{red}\boxed{E(X^2)=\frac{6}{\lambda^2}}$ | {"url":"http://mathhelpforum.com/advanced-statistics/56565-gamma-distribution-print.html","timestamp":"2014-04-16T05:06:10Z","content_type":null,"content_length":"9823","record_id":"<urn:uuid:47ab9b9d-a22e-4bd4-bf77-cbb452556076>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00631-ip-10-147-4-33.ec2.internal.warc.gz"} |
relational beta-module
Basic concepts
Relational $\beta$-modules
One of my early Honours students at Macquarie University baffled his proposed Queensland graduate studies supervisor who asked whether the student knew the definition of a topological space. The
aspiring researcher on dynamical systems answered positively: “Yes, it is a relational $\beta$-module!” I received quite a bit of flak from colleagues concerning that one; but the student Peter
Kloeden went on to become a full professor of mathematics in Australia then Germany.
—Ross Street, in An Australian conspectus of higher categories?
In 1970, Michael Barr gave an abstract definition of topological space based on a notion of convergence between ultrafilters (building on work by Ernest Manes on compact Hausdorff spaces).
Succinctly, Barr defined topological spaces as ‘relational $\beta$-modules’. It was subsequently realized that this was a special case of the notion of generalized multicategory. Here we unpack this
definition and examine its properties.
The correctness of this definition (in the sense of matching Bourbaki's definition) is equivalent to the ultrafilter principle ($UP$). However, the definition can be treated on its own, even in a
context without $UP$. So we also consider the properties of relational $\beta$-modules when these might not match Bourbaki spaces.
Abstract description
If $S$ is a set, let $\beta{S}$ be the set of ultrafilters on $S$. This set is canonically identified with the set of Boolean algebra homomorphisms
$P(S) \to \mathbf{2},$
from the power set of $S$ to $\mathbf{2}$, the unique Boolean algebra with two elements. The 2-element set carries a dualizing object structure that induces an evident adjoint pair
$(Set \stackrel{P}{\to} Bool^{op}) \; \dashv \; (Bool^{op} \stackrel{\hom(-, \mathbf{2})}{\to} Set)$
so that the composite functor $\beta = \hom(P-, \mathbf{2}): Set \to Set$ carries a monad structure.
The functor $\beta : Set \to Set$ extends to Rel as follows: given a binary relation $r\colon X \to Y$, written as a subobject in $Set$
$R \stackrel{\langle \pi_1, \pi_2 \rangle}{\to} X \times Y,$
we define $\beta(r): \beta(X) \to \beta(Y)$ to be the relation obtained by taking the image of $\langle \beta(\pi_1), \beta(\pi_2) \rangle: \beta(R) \to \beta(X) \times \beta(Y)$. It turns out,
although it is by no means obvious, that $\beta$ is according to this definition a strict functor on $Rel$.
The monad structure on $\beta: Set \to Set$, given by a unit $u: 1 \to \beta$ and multiplication $m: \beta \beta \to \beta$, extends not to a strict monad on $Rel$, but rather one where the
transformations $u, m$ are op-lax in the sense of there being inequalities
$\array{ X & \stackrel{u_X}{\to} & \beta X & & & & & & \beta \beta X & \stackrel{m_X}{\to} & \beta X \\ \mathllap{r} \downarrow & \leq & \downarrow \mathrlap{\beta(r)} & & & & & & \mathllap{\beta \
beta (r)} \downarrow & \leq & \downarrow \mathrlap{\beta(r)} \\ Y & \underset{u_Y}{\to} & \beta Y & & & & & & \beta \beta Y & \underset{m_Y}{\to} & \beta Y }$
(while of course the monad associativity and unit conditions remain as equations: hold on the nose).
Then a relational $\beta$-module is a lax algebra? (module) of $\beta$ on the 2-poset $Rel$. In other words, a set $S$ equipped with a relation $\xi: \beta S \to S$ such that the following
inequalities hold:
(1)$\array{ S & \stackrel{u_S}{\to} & \beta S & & & & & & \beta \beta S & \stackrel{m_S}{\to} & \beta S \\ & \mathllap{1_S} \searrow \; \leq & \downarrow \mathrlap{\xi} & & & & & & \mathllap{\beta(\
xi)} \downarrow & \leq & \downarrow \mathrlap{\xi} \\ & & S & & & & & & \beta S & \underset{\xi}{\to} & S }$
Arguably, it is better to consider $Rel$ as a proarrow equipment in this construction, in order to accommodate continuous functions between topological spaces (not continuous relations!) as the
appropriate abstract notion of morphism between relational $\beta$-modules. We touch on this below, but for a much wider context, see generalized multicategory.
Bridge to a concrete description
A relational $\beta$-module is a set $S$ and a binary relation $\xi: \beta S \to S$ between ultrafilters on $S$ and elements of $S$ that satisfy the conditions (1) . For $F \in \beta S$ and $x \in S$
, we write $F \rightsquigarrow_\xi x$ if $(F, x)$ satisfies the relation $\xi$, or often just $F \rightsquigarrow x$ if the relation is clear. We pronounce this by saying “the ultrafilter $F$
converges to the point $x$”, so that $\xi$ plays the role of “notion of convergence”.
Preliminary to explaining the conditions (1), we first set up a Galois connection between $\xi \in Rel(\beta S, S)$ and subsets $\mathcal{C} \in P P(S)$, so that fixed points on the $P P(S)$ side are
exactly topologies on $S$, and fixed points on the other side are (as we show below) lax $\beta$-module structures on $S$. The Galois connection would then of course restrict to a Galois
correspondence between topologies and lax module structures.
Recall that each topology $\mathcal{O} \subseteq P(S)$ induces a notion of convergence where $F \rightsquigarrow x$ means $N_x \subseteq F$ ($F$ contains the filter of neighborhoods of $x$).
Accordingly, for general $\mathcal{C} \subseteq P(S)$, define the relation $conv(\mathcal{C}) = \xi: \beta S \to S$ by
$F \rightsquigarrow_\xi x \;\;\; \Leftrightarrow \;\;\; (\forall_{U: P(S)})\; U \in \mathcal{C} \; \wedge \; x \in U \; \Rightarrow \; U \in F.$
Conversely, a topology $\mathcal{O}$ can be retrieved from its notion of convergence: under the ultrafilter principle, the neighborhood filter of a point $x$ is just the intersection of all
ultrafilters containing it (hence all $F$ such that $F \rightsquigarrow x$), and then a set is open if it is a neighborhood of all of its elements. Accordingly, for general “notions of convergence” $
\xi \in Rel(\beta S, S)$, we define a collection $\tau(\xi) \subseteq P(S)$ by
$\tau(\xi) \coloneqq \{U \subseteq S: \; (\forall_{F: \beta S, x: S})\; (x \in U \; \wedge\; F \rightsquigarrow_\xi x) \Rightarrow U \in F\}.$
$\tau(\xi)$ is a topology on $S$, for any $\xi: Rel(\beta S, S)$.
It is trivial that $S \in \tau(\xi)$. If $U, V \in \tau(\xi)$, and if $x \in U \cap V$ and $F \rightsquigarrow x$, then also $x \in U$ and $x \in V$ and we conclude $U, V \in F$, whence $U \cap V \in
F$ since $F$ is an ultrafilter, so that $U \cap V$ satisfies the condition of belonging to $\tau(\xi)$. Given a collection of elements $U_i \in \tau(\xi)$, if $x \in \cup_i U_i$ and $F \
rightsquigarrow x$, then $x \in U_i$ for some $i$ and we conclude $U_i \in F$, whence $\cup_i U_i \in F$ since $F$ is upward closed. Therefore $\cup_i U_i$ satisfies the condition of belonging to $\
tau(\xi)$ (vacuously so if the collection is empty).
There is a Galois connection between notions of convergence on $S$ and subsets of $P(S)$, according to the bi-implication
$\mathcal{C} \subseteq \tau(\xi) \; \Leftrightarrow \; \xi \subseteq conv(\mathcal{C}).$
To establish the bi-implication, it suffices to observe that both containments $\mathcal{C} \subseteq \tau(\xi)$ and $\xi \subseteq conv(\mathcal{C})$ are equivalent to the condition
$\forall_{F: \beta S} \forall_{U: P S} \forall_{x: S} (F \rightsquigarrow_\xi x) \; \wedge \; (U \in \mathcal{C}) \; \wedge \; (x \in U) \; \; \Rightarrow \; \; (U \in F).$
If $\mathcal{O}$ is a topology on $S$, then $\mathcal{O} = \tau(conv(\mathcal{O}))$ (i.e., topologies are fixed points of the closure operator $\tau \circ \conv$).
We already have $\mathcal{O} \subseteq \tau(conv(\mathcal{O}))$ from Proposition 2. For the other direction, we must show that any $V$ belonging to $\tau(conv(\mathcal{O}))$ is an $\mathcal{O}$
-neighborhood of each of its points. Suppose the contrary: that $x \in V$ but $V$ is not an $\mathcal{O}$-neighborhood of $x$. Then for every $\mathcal{O}$-neighborhood $U \in N_x$, we have $U \cap
eg V eq \emptyset$, so that sets of this form generate a filter. By the ultrafilter principle, we may extend this filter to an ultrafilter $F$; clearly we have $F \rightsquigarrow x$ and $eg V \in F$
, but since $F \rightsquigarrow x$ and $V \in \tau(conv(\mathcal{O}))$ and $x \in V$, we also have $V \in F$, which is inconsistent with $eg V \in F$.
Propositions 1, 2, and 3 more or less show that a topological space $(S, \mathcal{O})$ is a particular type of pseudotopological space:
A pseudotopological space is set $S$ equipped with a relation $\xi: \beta S \to S$ such that $1_S \leq \xi \circ u_S$.
All that remains is to check is:
The lax unit condition $1_S \leq \xi \circ u_S$ holds if $\xi = conv(\mathcal{O})$, for a topology $\mathcal{O}$.
The unit $u_S: S \to \beta S$ may also be denoted $prin_S$, as it takes an element $x \in S$ to the principal ultrafilter
$prin_S(x) = \{U \subseteq S: x \in U\}$
and now the unit condition says $prin_S(x) \rightsquigarrow_\xi x$ for all $x$. For $\xi = conv(\mathcal{O})$, this says $N_x \subseteq prin_S(x)$, or that $x \in V$ for all neighborhoods $V \in N_x$
, which is a tautology.
One of our goals is to prove the following theorem:
(Main Theorem) An arrow $\xi: \beta(S) \to S$ in $Rel$ is of the form $conv(\tau(\xi))$ if and only if the following inequalities are satisfied:
$1_S \leq \xi \circ prin_S, \qquad \xi \circ \beta(\xi) \leq \xi \circ m_S$
where $m_S: \beta \beta(S) \to \beta(S)$ is the multiplication on the ultrafilter monad.
Ultrafilter monad on $Rel$
Before rolling up our sleeves and proving the main theorem, we pause to consider some more abstract contexts in which to place the concept of lax $\beta$-module, leading up to the context of
generalized multicategories.
Extending the ultrafilter functor to $Rel$
First we examine more closely the extension of the ultrafilter functor $\beta: Set \to Set$ to $Rel$, showing in particular that the extension is a strict functor. First we slightly rephrase our
earlier definition:
For a relation $r: X \to Y$ between sets, given by a subobject $R \hookrightarrow X \times Y$ in $Set$ with projections $f: R \to X$ and $g: R \to Y$, define $\bar{\beta}(r)$ to be the composite
$\beta(X) \stackrel{\beta(f)^{o}}{\to} \beta(R) \stackrel{\beta(g)}{\to} \beta(Y)$
in the bicategory of relations.
Any span of functions $(h: S \to X, k: S \to Y)$ that represents $r$ (in the sense that $r = k h^{o}$ in the bicategory of relations) would serve in place of $(f, g)$, since for any such span there
is an epi $s: S \to R$ with $h = f s$, $k = g s$, whence $\beta(s)$ is epi (because the epi $s$ splits in $Set$) and we have
$\array{ \beta(g)\beta(f)^{o} & = & \beta(g)\beta(s)\beta(s)^{o}\beta(f)^{o} \\ & = & \beta(g)\beta(s)(\beta(f)\beta(s))^{o} \\ & = & \beta(g s)\beta(f s)^{o} \\ & = & \beta(k)\beta(h)^{o}. }$
In particular, $\bar{\beta}$ is well-defined. Since $\bar{\beta}$ extends $\beta: Set \to Set$, there is no harm in writing $\beta(r)$ in place of $\bar{\beta}(r)$. If $r \leq r': X \to Y$, then $\
beta(r) \leq \beta(r')$ (as can be seen from the calculation displayed above, but replacing the epi $s$ by a general map $t$, and the first equation by an inequality $\geq$).
The functor $\beta: Set \to Set$ satisfies the Beck-Chevalley condition (and therefore the extension $\beta: Rel \to Rel$ is a strict functor).
Referring to the pullback diagram in Remark 1, let $Q = R \times_Y S$ be the pullback. We must show that the canonical map
$\beta(R \times_Y S) \to \beta(R) \times_{\beta(Y)} \beta(S)$
is epic. Viewing this as a continuous map between compact Hausdorff spaces (see this section of the article on compacta), it suffices to show that the canonical map
$R \times_Y S \to \beta(R) \times_{\beta(Y)} \beta(S)$
has a dense image. Let $(G, H) \in \beta(R) \times_{\beta(Y)} \beta(S)$, so that $\beta(g)(G) = \beta(h)(H)$ are the same ultrafilter $J \in \beta(Y)$. Let $\hat{A}$ and $\hat{B}$ be basic open
neighborhoods of $G$ and $H$ in $\beta(R)$ and $\beta(S)$ respectively; we must show that there is $(r, s) \in R \times_Y S$ such that
$(prin(r), prin(s)) \in \hat{A} \times \hat{B}$
or in other words such that $r \in A$ and $s \in B$. We have $g^{-1}(g(A)) \in G$ since $A \in G$ and $A \subseteq g^{-1}(g(A))$, so that $g(A)$ belongs to
$J = \beta(g)(G) \coloneqq \{C \subseteq Y: g^{-1}(C) \in G\}$
and similarly $h(B) \in J$. It follows that $g(A) \cap h(B) \in J$ so that $g(A) \cap h(B) eq \emptyset$. Any element $y \in g(A) \cap h(B)$ can be written as $y = g(r)$ and $y = h(s)$ for some $r \
in A$ and $s \in B$, and this completes the proof.
Ultrafilter monad on the equipment $\mathbf{Rel}$
As mentioned in an earlier section, the natural transformations $u = prin: 1_{Set} \to \beta$, $m: \beta\beta \to \beta$do not extend to (strict) natural transformations on the locally posetal
bicategory $Rel$, but only to transformations that are op-lax in the sense of inequalities
$prin_Y \circ r \leq \beta(r) \circ prin_X, \qquad m_Y \circ \beta\beta(r) \leq \beta(r) \circ m_X$
for every relation $r: X \to Y$. These are equivalent to inequalities
$r \leq prin_Y^o \circ \beta(r) \circ prin_X, \qquad \beta\beta(r) \leq m_Y^o \circ \beta(r) \circ m_X$
and they may be deduced simply by staring at naturality diagrams in $Set$, in which we represent or tabulate $r$ by $\pi_2 \circ \pi_1^o$:
$\array{ & & R & & & & & & & & & & \beta\beta (R) & & \\ & \mathllap{\pi_1} \swarrow & \downarrow_\mathrlap{prin_R} & \searrow \mathrlap{\pi_2} & & & & & & & & _\mathllap{\beta\beta\pi_1} \swarrow &
\downarrow_\mathrlap{m_R} & \searrow_\mathrlap{\beta\beta\pi_2} & \\ X & & \beta (R) & & Y & & & & & & \beta \beta (X) & & \beta (R) & & \beta \beta (Y) \\ _\mathllap{prin_X} \downarrow & \swarrow_\
mathrlap{\beta \pi_1} & & _\mathllap{\beta \pi_2} \searrow & \downarrow_\mathrlap{prin_Y} & & & & & & \mathllap{m_X} \downarrow & \swarrow_\mathrlap{\beta \pi_1} & & _\mathllap{\beta \pi_2} \searrow
& \downarrow \mathrlap{m_Y} \\ \beta (X) & & & & \beta (Y) & & & & & & \beta (X) & & & & \beta (Y) }$
To get an actual monad, it is more satisfactory in this context to consider not the bicategory $Rel$, but rather the equipment or framed bicategory $\mathbf{Rel}$. That is, there is a 2-category
$Equip$ of equipments (as a sub-2-category of a 2-category of double categories), so that the notion of monad makes sense therein, and it turns out the data to hand induces such a monad $\bar{\beta}:
\mathbf{Rel} \to \mathbf{Rel}$.
In more detail: the 0-cells of $\mathbf{Rel}$ are sets, and the horizontal arrows are relations between sets. Vertical arrows are functions between sets, and a 2-cell of shape
$\array{ A & \stackrel{r}{\to} & B \\ \mathllap{f} \downarrow & \Downarrow & \downarrow \mathrlap{g} \\ C & \stackrel{s}{\to} & D; }$
is an inequality $g \circ r \leq s \circ f$. We straightforwardly get a double category $\mathbf{Rel}$, and the ultrafilter functor on $Set$ extends to a functor $\bar{\beta}: \mathbf{Rel} \to \
mathbf{Rel}$ between double categories (or in this case, equipments), preserving all structure in sight.
Some attention must be paid to the notion of transformation between functors $F, G: \mathbf{B} \to \mathbf{C}$ between equipments. A transformation $\eta: F \to G$ assigns to each 0-cell $b$ of $\
mathbf{B}$ a vertical arrow $\eta b: F b \to G b$, and to each horizontal arrow $r: b \to b'$ a 2-cell $\eta r$ of the form
$\array{ F b & \stackrel{F r}{\to} & F b' \\ \mathllap{\eta b} \downarrow & \Downarrow \mathrlap{\eta r} & \downarrow \mathrlap{\eta b'} \\ G b & \stackrel{G r}{\to} & G b'; }$
suitably compatible with the double category structures.
We thus find that the op-lax structures of the transformations $prin: 1 \to \bar{\beta}$, $m: \bar{\beta} \bar{\beta} \to \bar{\beta}$ on $Rel$ qua bicategory are exactly what we need to produce
honest transformations $u: 1 \to \bar{\beta}$, $m: \bar{\beta} \bar{\beta} \to \bar{\beta}$ on $\mathbf{Rel}$ qua equipment, and the result is an ultrafunctor monad on the equipment $\mathbf{Rel}$.
Given a monad $T$ on an equipment $\mathbf{B}$, one may proceed to construct a horizontal Kleisli equipment $HKl(\mathbf{B}, T)$ with the same 0-cells and vertical arrows as $\mathbf{B}$, but whose
horizontal arrows are of the form $r: b \to T b'$. A 2-cell in $HKl(\mathbf{B}, T)$ (with vertical source $f$ and vertical target $g$) is a 2-cell in $\mathbf{B}$ of the form
$\array{ b & \stackrel{r}{\to} & T b' \\ \mathllap{f} \downarrow & \Downarrow \mathrlap{\alpha} & \downarrow \mathrlap{T g} \\ c & \stackrel{s}{\to} & T c'; }$
with horizontal compositions being performed in familiar Kleisli fashion. (When we say “familiar Kleisli fashion”, we are using the fact that an equipment allows one to “translate” vertical arrows,
in particular the map $m_b: T T b \to T b$, into horizontal arrows, which are then composed horizontally. Similarly, the unit of the monad is translated into a horizontal arrow, where it plays the
role of an identity in the Kleisli construction.)
In an equipment, there is a notion of monoid and monoid homomorphism. A monoid consists of a horizontal arrow $\xi: b \to b$ together with unit and multiplication 2-cells
$\array{ b & \stackrel{1_b}{\to} & b & & & & & & b & \stackrel{\xi}{\to}\;\;\; b \;\;\; \stackrel{\xi}{\to} & b\\ \mathllap{1} \downarrow & \Downarrow \mathrlap{\eta} & \downarrow \mathrlap{1} & & &
& & & \mathllap{1} \downarrow & \Downarrow \mathrlap{\mu} & \downarrow \mathrlap{1}\\ b & \stackrel{\xi}{\to} & b & & & & & & b & \stackrel{\xi}{\to} & b }$
satisfying evident identities. A monoid homomorphism from $(b, \xi)$ to $(c, \theta)$ consists of a vertical arrow and 2-cell $(f, \phi)$ of the form
$\array{ b & \stackrel{\xi}{\to} & b \\ \mathllap{f} \downarrow & \Downarrow \mathrlap{\phi} & \downarrow \mathrlap{f} \\ c & \underset{\theta}{\to} & c }$
that is suitably compatible with the unit and multiplication cells.
The following notion gives an interim notion of generalized multicategory that applies in particular to relational $\beta$-modules.
Given a monad $T$ on an equipment $\mathbf{B}$, a $T$-monoid is a monoid in the horizontal Kleisli equipment $HKl(\mathbf{B}, T)$. A map of $T$-monoids is a homomorphism between monoids in $HKl(\
mathbf{B}, T)$.
For the ultrafilter monad $\beta$ on the equipment $\mathbf{Rel}$, a structure of $\beta$-monoid is equivalent to a structure of relational $\beta$-module, and a homomorphism of $\beta$-monoids is
the same as a lax map of relational $\beta$-modules in the bicategory $Rel$.
This is really just a matter of unwinding definitions. The data of a $\beta$-monoid in the equipment $\mathbf{Rel}$ amounts to a set $X$ together with a horizontal arrow in the Kleisli construction,
that is to say a relation $c: X \to \beta X$ (opposite to our conventional direction, i.e., $c = \xi^o$). The unit and multiplication cells for $c$ are inequalities $1_X \leq c$ and $c \circ_{Kl} c \
leq c$ (the vertical source and target being identity maps), where the identity $1_X$ in the Kleisli construction uses the unit for $\beta$ and the Kleisli composition uses the multiplication. Back
in the bicategory $Rel$ these translate to relational inequalities
$prin_X \leq c, \qquad m_X \circ \beta c \circ c \leq c$
or, with $c = \xi^o$,
$prin_X \leq \xi^o, \qquad m_X \circ (\beta (\xi))^o \circ \xi^o \leq \xi^o.$
These boil down to relational inequalities
$1_X \leq prin_X^o \circ \xi^o, \qquad (\beta (\xi))^o \circ \xi^o \leq m_X^o \circ \xi^o$
or to
$1_X \leq \xi \circ prin_X, \qquad \xi \circ \beta (\xi) \leq \xi \circ m_X,$
as in the axioms on relational beta-modules. Similarly, a $\beta$-monoid homomorphism $(X, c) \to (Y, d)$ is a vertical arrow $f: X \to Y$ in $HKl(\mathbf{Rel}, \beta)$ together with a suitable
2-cell, which after some unraveling comes down to a relational inequality
$\beta (f) \circ c \leq d \circ f$
or to an inequality $\beta (f) \circ \xi_X^o \leq \xi_Y^o \circ f$, which may be further massaged into the form $\xi_X^o \circ f^o \leq (\beta (f))^o \circ \xi_Y^o$, or simply to
$f \circ \xi_X \leq \xi_Y \circ \beta (f)$
as advertised in the notion of lax morphism of relational $\beta$-modules (cf. theorem 4 below).
Proof of Main Theorem
We now return to the task of proving theorem 1.
We now break up our Main Theorem 1 into the following two theorems.
If $\xi = conv(\mathcal{O})$ for a topology $\mathcal{O}$, then the two inequalities of (1) are satisfied.
The first inequality (lax unit condition) was already verified in proposition 4. For the second (lax associativity), let us represent the relation $\xi$ by a span $\beta S \stackrel{\pi_1}{\
leftarrow} R \stackrel{\pi_2}{\to} S$, so that $\beta(\xi) = \beta(\pi_2) \beta(\pi_1)^o$. The lax associativity condition becomes
$\pi_2 \pi_1^o \beta(\pi_2) \beta(\pi_1)^o \leq \pi_2 \pi_1^o m_S$
which (using $\beta(\pi_1) \dashv \beta(\pi_1)^o$) is equivalent to
(2)$\pi_2 \pi_1^o \beta(\pi_2) \leq \pi_2 \pi_1^o m_S \beta(\pi_1)$
or in other words that for all $\mathcal{G}: \beta(R)$, $x: S$
(3)$\beta(\pi_2)(\mathcal{G}) \rightsquigarrow_\xi x \;\; \vdash \;\; m_S \beta(\pi_1)(\mathcal{G}) \rightsquigarrow_\xi x.$
Here $\beta(\pi_2)(\mathcal{G})$ is, by definition,
$\{U \subseteq S: \pi_2^{-1}(U) \in \mathcal{G}\},$
with $\beta(\pi_1)(\mathcal{G})$ defined similarly. The monad multiplication $m_S: \beta \beta S \to \beta S$ is by definition
$(\mathcal{U}: \beta\beta S) \;\; m_S(\mathcal{U}) \coloneqq \{A \subseteq S: \hat{A} \in \mathcal{U}\}$
where $\hat{A} = \{F \in \beta S: A \in F\}$ (see also the previous section).
Thus, (3) translates into the following entailment (using remark 3):
$\array{ & & \mathcal{O}_x \subseteq \{A \subseteq S: \pi_2^{-1}(A) \in \mathcal{G}\} \\ & \vdash & \forall_{U: P S} U \in \mathcal{O}_x \Rightarrow \pi_1^{-1}(\hat{U}) \in \mathcal{G}. }$
This would naturally follow if
$\forall_{U \in \mathcal{O}_x} \pi_2^{-1}(U) \subseteq \pi_1^{-1}(\hat{U}).$
But a pair $(F, y)$ belongs to $\pi_2^{-1}(U)$ if $F \rightsquigarrow_\xi y$ and $y \in U$; we want to show this implies $F = \pi_1(F, y)$ belongs to $\hat{U}$, or in other words that $U \in F$. But
this is tautological, given how $conv(\mathcal{O})$ is defined in terms of a topology $\mathcal{O}$.
The next theorem establishes the converse of the preceding theorem; the two theorems together establish the Main Theorem. First we need a lemma.
Given any relation $\xi: Rel(\beta S, S)$ and $x: S$, $A \subseteq S$, we have that $x$ belongs to the closure $\bar{A}$ wrt the topology $\tau(\xi)$ if and only if $\exists_{F: \beta S} A \in F \; \
wedge \; F \rightsquigarrow_\xi x$.
As usual, let $eg A$ denote the complement of a subset $A$. By definition of the topology $\tau(\xi)$, we have that $eg A$ is a neighborhood of $x$ if $\forall_{F: \beta S} F \rightsquigarrow_\xi x
\; \Rightarrow \; eg A \in F$. In other words,
$\array{ x \in int(eg A) & \Leftrightarrow & \forall_{F: \beta S} \; F \rightsquigarrow_\xi x \; \Rightarrow \; eg A \in F \\ & \Leftrightarrow & \forall_{F: \beta S} \; eg ((A \in F) \; \wedge \; (F
\rightsquigarrow_\xi x)) }$
since in an ultrafilter $F$, we have $eg (A \in F)$ iff $(eg A) \in F$. Negating both sides of this bi-implication gives
$\array{ x \in \bar{A} & \Leftrightarrow & \exists_{F: \beta S} \; A \in F \; \wedge \; F \rightsquigarrow_\xi x }$
as desired.
If $\xi: \beta S \to S$ in $Rel$ satisfies the inequalities of (1), then $\xi = conv(\tau(\xi))$.
We have $\xi \leq conv(\tau(\xi))$ from the Galois connection (proposition 2), so we just need to prove $conv(\tau(\xi)) \leq \xi$, or that $F \rightsquigarrow_{conv(\tau(\xi))} x$ (henceforth
abbreviated as $F \rightsquigarrow_{\tau(\xi)} x$) implies $F \rightsquigarrow_\xi x$ under the conditions (1).
If $F \rightsquigarrow_{\tau(\xi)} x$, then every neighborhood $V$ of $x$ belongs to $F$, so that for every $U \in F$, every neighborhood $V$ of $x$ intersects $U$ in a nonempty set. But this just
means $x \in \bar{U}$ for every $U \in F$, or in other words (using lemma 1) that
$U \in F \; \; \vdash \; \; \exists_{G: \beta S} U \in G \; \wedge \; G \rightsquigarrow_\xi x.$
Representing the relation $\xi$ as usual by a subset $\langle \pi_1, \pi_2 \rangle: R \hookrightarrow \beta S \times S$, another way of expressing the existential formula on the right of this
entailment is:
$\exists_{(G, x): \beta S \times S} \; (G, x) \in R \; \wedge \; G \in \hat{U}$
$\exists_{\gamma: R} \pi_1(\gamma) \in \hat{U} \wedge \pi_2(\gamma) = x$
or even just
(4)$\pi_1^{-1}(\hat{U}) \wedge \pi_2^{-1}(\{x\}) eq \emptyset$
as subsets of $R$, as $U$ ranges over all elements of $F$. We therefore have that subsets of the form (4) generate a proper filter of $R$. By the ultrafilter principle, we may extend this filter to
an ultrafilter $\mathcal{G} \in \beta R$.
By construction, we have
$F \subseteq \{B \subseteq X: \pi_1^{-1}(\hat{B})) \in \mathcal{G}\} \qquad prin_S(x) \subseteq \{A \subseteq X: \pi_2^{-1}(A) \in \mathcal{G}\}$
but in fact these inclusions are equalities since the left sides and right sides are ultrafilters. Put differently, we have established
$F = (m_S \circ \beta(\pi_1))(\mathcal{G}), \qquad prin_S(x) = \beta(\pi_2)(\mathcal{G}).$
Notice that the lax unit condition of (1) implies that $prin_S(x) = \beta(\pi_2)(\mathcal{G}) \rightsquigarrow_\xi x$, or that $(\mathcal{G}, x)$ belongs to the relation $\xi \circ \beta(\pi_2)$.
Recall also that the lax associativity condition is equivalent to (2), which says
$\xi \circ \beta(\pi_2) \leq \xi \circ m_S \circ \beta(\pi_1);$
in other words $(\mathcal{G}, x)$ belongs to $\xi \circ m_S \circ \beta(\pi_1)$, i.e., $F = (m_S \circ \beta(\pi_1))(\mathcal{G}) \rightsquigarrow_\xi x$, as was to be shown.
This completes the proof of the Main Theorem (theorem 1).
Continuous maps
A function between two topological spaces $f: X \to Y$ is continuous if and only if $f \circ \xi \leq \theta \circ \beta(f)$ for their respective topological notions of convergence $\xi, \theta$.
Suppose first that $f$ is continuous, and that $(F, y) \in \beta(X) \times Y$ belongs to $f \circ \xi$, i.e., there is $x$ such that $F \rightsquigarrow x$ and $f(x) = y$. We want to show $\beta(f)
(F) \rightsquigarrow y = f(x)$, or that any open set $V$ containing $f(x)$ belongs to $\beta(f)(F)$. The latter means $f^{-1}(V) \in F$, which is true since $f^{-1}(V)$ is an open set containing $x$
and $F \rightsquigarrow x$.
Now suppose $f \circ \xi \leq \theta \circ \beta(f)$. To show $f$ is continuous, it suffices to show that
$f(\bar{A}) \subseteq \widebar{f(A)}$
for any $A \subseteq X$ (easy exercise). For $x \in \bar{A}$, lemma 1 shows there is $F: \beta(X)$ with $A \in F$ and $F \rightsquigarrow x$. Under the supposition we have $\beta(f)(F) \
rightsquigarrow f(x)$, and we also have $f(A) \in \beta(f)(F)$, because $A \subseteq f^{-1}(f(A))$ and $F$ is upward closed and $A \in F$ implies $f^{-1}(f(A)) \in F$. Then again by lemma 1, $f(A) \
in \beta(f)(F)$ and $\beta(f)(F) \rightsquigarrow f(x)$ implies $f(x) \in \widebar{f(A)}$, as desired.
Corollary (Barr)
The category of topological spaces is equivalent (even isomorphic to) the category of lax $\beta$-modules and lax morphisms between them.
As above, a subset $A$ of $S$ is open if $A \in \mathcal{U}$ whenever $\mathcal{U} \to x \in A$. On the other hand, by lemma 1, $A$ is closed if $x \in A$ whenever $A \in \mathcal{U} \to x$.
A relational $\beta$-module is compact if every ultrafilter converges to at least one point. It is Hausdorff if every ultrafilter converges to at most one point. Thus, a compactum is (assuming $UF$)
precisely a relational $\beta$-module in which every ultrafilter converges to exactly one point, that is in which the action of the monad $\beta$ lives in $Set$ rather than in $Rel$. Full proofs may
be found at compactum; see also ultrafilter monad.
A continuous map $f$ from $(X, \xi: \beta X \to X)$ to $(Y, \theta: \beta Y \to Y)$ is proper if the square
$\array{ \beta X & \stackrel{\xi}{\to} & X \\ \mathllap{\beta (f)} \downarrow & & \downarrow \mathrlap{f} \\ \beta Y & \underset{\theta}{\to} & Y }$
commutes (strictly) in $Rel$, and $f$ is open if the square
$\array{ \beta X & \stackrel{\xi}{\to} & X \\ \mathllap{\beta (f)^o} \uparrow & & \uparrow \mathrlap{f^o} \\ \beta Y & \underset{\theta}{\to} & Y }$
commutes in $Rel$. From this point of view, a space $X$ is Hausdorff if the diagonal map $\delta: X \to X \times X$ is proper, and compact if $\epsilon: X \to 1$ is proper (and these facts remain
true even for pseudotopological spaces). See Clementino, Hofmann, and Janelidze, infra corollary 2.5.
The following ultrafilter interpolation result is due to Pisani:
A topological space $(X, \xi)$ is exponentiable if, whenever $m_X(\mathcal{U}) \rightsquigarrow_\xi x$ for $\mathcal{U} \in \beta\beta X$ and $x \in X$, there exists $F \in \beta X$ with $\mathcal{U}
\rightsquigarrow_{\beta(\xi)} F$ and $F \rightsquigarrow_\xi x$.
For an convergence-approach extension of this result to exponentiable maps in $Top$, see Clementino, Hofmann, and Tholen.
A continuous map $f: X \to Y$ is a discrete fibration if, whenever $G \rightsquigarrow y$ in $Y$ and $f(x) = y$, there exists a unique $F \in \beta(X)$ such that $\beta(f)(F) = G$ and $F \
rightsquigarrow x$ in $X$.
A continuous map $f: X \to Y$ is étale (a local homeomorphism) if $f$ and $\delta_f: X \to X \times_Y X$ are both discrete fibrations.
For more on this, see Clementino, Hofmann, and Janelidze.
Relation to nonstandard analysis
In nonstandard analysis (which implicitly relies throughout on $UF$), one may define a topological space using a relation between hyperpoints (elements of $S^*$) and standard points (elements of $S$
). If $u$ is a hyperpoint and $x$ is a standard point, then we write $u \approx x$ and say that $x$ is a standard part? of $u$ or that $u$ belongs to the halo? (or monad, but not the
category-theoretic kind) of $x$. This relation must satisfy a condition analogous to the condition in the definition of a relational $\beta$-module. The nonstandard defintions of open set, compact
space, etc are also analogous. (Accordingly, one can speak of the standard part of $u$ only for Hausdorff spaces.)
So ultrafilters behave very much like hyperpoints. This is not to say that ultrafilters are (or even can be) hyperpoints, as they don't obey the transfer principle?. Nevertheless, one does use
ultrafilters to construct the models of nonstandard analysis in which hyperpoints actually live. Intuitions developed for nonstandard analysis can profitably be applied to ultrafilters, but the
transfer principle is not valid in proofs.
Relation to other topological concepts
If $\beta$ is treated as a monad on $Set$ instead of on $Rel$, then its algebras are the compacta (the compact Hausdorff spaces), again assuming $UP$; see ultrafilter monad, and more especially
One might hope that there would be an analogous treatment of uniform spaces based on an equivalence relation between ultrafilters. (In nonstandard analysis, this becomes a relation $\approx$ of
infinite closeness between arbitrary hyperpoints, instead of only a relation between hyperpoints and standard points.) The description in terms of generalized multicategories is known to generalize
to a description of uniform spaces, but rather than using relations between ultrafilters, this description uses pro-relations between points.
For more on relations between Barr’s approach to topological spaces, Lawvere’s approach to metric spaces, as well as uniform structures, prometric spaces, and approach structures?, see Clementino,
Hofmann, and Tholen.
• Michael Barr, Relational algebras, Springer Lecture Notes in Math. 137 (1970), 39-55.
• Maria Manuel Clementino, Dirk Hofmann, and George Janelidze, On exponentiability of étale algebraic homomorphisms, Preprint 11-35, University of Coimbra. (pdf)
• Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen, The convergence approach to exponentiable maps, Portugaliae Mathematica (Nova Series), Vol. 60 Issue 2 (2003), 139-160. (web) (pdf)
• Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen, One Setting for All: Metric, Topology, Uniformity, Approach Structure. (pdf)
• Gavin J. Seal, Canonical and op-canonical lax algebras, Theory and Applications of Categories, 14 (2005), 221–243. (web)
• Claudio Pisani, Convergence in exponentiable spaces, Theory Appl. Categories 5 (1999), 148-162. (web) | {"url":"http://ncatlab.org/nlab/show/relational+beta-module","timestamp":"2014-04-16T04:47:48Z","content_type":null,"content_length":"230447","record_id":"<urn:uuid:2f1f1e66-fb8f-4f75-a198-4e961ed4af53>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00204-ip-10-147-4-33.ec2.internal.warc.gz"} |
This problem is a game where n objects are arranged in a circle and every mth object is deleted until 1 object is left.
The original problem is said to derive from a suicide pact among 41 rebels trapped by the Romans in the first century. The rebels decided to kill every third person until one is left and can escape
in a boat that can take only one person. Josephus, the historian who lived to tell the story quickly figured out where he should position himself from the starting point in the circle so that he
would survive.
Suppose you were in that situation with the only difference that you have your palm PC to quickly find out the lucky position to survive. Your program should be able to handle the following
n people (n > 0 ) are initially arranged in a circle, facing inwards, and numbered from 1 to n. The numbering from 1 to n proceeds consecutively in a clockwise direction.
Starting with person number 1, counting continues in a clockwise direction, until we get to person number k (k > 0), who is promptly killed.
Counting then proceeds from the person to his immediate left, to kill the kth person, and so on, until only one person remains.
For example, when n = 5 and k = 2, the order of execution is 2, 4, 1, and 5. The survivor is 3.
Hint you have to implement the circular list for your program to be efficient."how that could be !"
Input data is to be read from a file roulette.in. Each line in this file contains values for n and k (in that order). A line containing values of 0 for n and k will terminate input. Your program does
know not the maximum number of people taking part in this tragic event.
For each input line, output in a file roulette.out the position of the sole survivor.
Sample Input
Sample Output
- to be serious with you my friends i did not understand this question very well !
and am not here to just get the solution without learning!
.. i know here a lot of people who have excellent background they can help
me step by step. so please will you help me !! to learn and solve this problem ! | {"url":"http://www.dreamincode.net/forums/topic/224965-find-out-the-lucky-position-to-survive-will-you-help-me-please/","timestamp":"2014-04-21T09:11:53Z","content_type":null,"content_length":"155329","record_id":"<urn:uuid:57e3123f-ac10-416e-ba8d-c55c7ff9fd6c>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00177-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Herding Cats: Managing in the presence of all things
Came across this Twitter from Dave Snowden
The notion of trying to manage risk, chaos, or anything with an underlying probabilistic driving process is futile. But attempts to reduce complexity or chaos. Or reduce risk, without understanding
the underlying probabilistic driving processes is also futile.
Blanket statement about reducing complexity that don't state how this complexity will be reduced, or how the underlying probabilistic drivers will be altered is futile, or worse nonsense. The naive
and populist notion that complexity simply emerges in the absence of the equations of motion for that complexity is the start is the problem of managing in the presence of complexity and managing in
the presence of uncertainty.
So when you hear someone say the world is too complex, we need to reduce the complexity, ask what are the units of measure of the complexity, what are the driving functions for this complexity, what
are the equations of motion exhibited by these driving functions, and how would these be changed to reduce the complexity.
Recent Comments | {"url":"http://herdingcats.typepad.com/my_weblog/2012/06/managing-in-the-presence-of-all-things.html","timestamp":"2014-04-17T06:41:31Z","content_type":null,"content_length":"46887","record_id":"<urn:uuid:b910ccf2-36b7-4a53-a685-8b3d45543ba6>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00124-ip-10-147-4-33.ec2.internal.warc.gz"} |
Post a reply
We can approximate by simulation (J again):
which is close to the exact answer of 425.
Brief explanation:
1) Get random uniform values in the range for x and y.
2) Get the maximum of 10x+25y among the values where x+y<=20
From the LPP:
= 425
We can easily obtain the tacit definition using
Typing in sim gives the tacit definition. We can then use that output to build more complicated definitions. | {"url":"http://www.mathisfunforum.com/post.php?tid=20012&qid=284367","timestamp":"2014-04-17T18:47:33Z","content_type":null,"content_length":"21433","record_id":"<urn:uuid:88434beb-fb94-418d-b25c-6cea5fb455c2>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00253-ip-10-147-4-33.ec2.internal.warc.gz"} |
Deepen your R experience with Rcpp
July 17, 2013
By Joseph Rickert
by Joseph Rickert
It is very likely that even a very casual observer what is happening in the world of R these past few months would have come across some mention of Rcpp, the R package that greatly facilitates R and
C++ integration. Rcpp is hot! Over 130 R packages now depend on Rcpp and it is likely to keep growing. The following plot built using code Tal Galili posted to examine the log files from RStudio’s
CRAN mirror shows the number of downloads of Rcpp around the time R 3.0.0 was released.
The intense activity over the first three days and relatively slow tapering off is noteworthy, especially for what might be considered an “advanced” package that takes some expertise to use. So, it
is not surprising that here have been quite a few conference presentations this year about some aspect or another of Rcpp, and Dirk Eddelbuettel, Romain Francois and other experts seem to be hard
pressed to keep up with the demand for training. I had the opportunity last week at the useR 2013 conference in Spain to attend the tutorial on Rcpp given by Hadley Wickham and Romain. And, at
roughly the same time that on the other side of the world, Dirk gave a similar tutorial to the Sydney Users of R Forum (SURF).
Romain and Hadley's tutorial was geared to people with some R skills, but not necessarily any C++ experience. It was very well done; exceptionally well done. Hadley and Romain are two experienced
trainers who are so good at what they do that they can quickly get a diverse group to a comfortable place where they can begin dealing with the material. The class was positive, challenging and very
While I was sitting there, probably hallucinating in the Albacete heat, I had the thought the Rcpp phenomenon probably says something about the future of R. No, I don’t mean that Rcpp or C++ is the
future. It occurred to me though that I was seening the results of how a small but committed group of R experts cooperated to deal with a potential threat to R’s continued success. To my way of
thinking, this kind of sustained, creative effort and the willingness of R developers to connect R to the rest of the computational world indicates that R is likely to be the platform of choice for
statistical computing some time to come.
So what is the threat? It is not big not big news that R can be slow. The following code from Hadley and Romain's tutorial shows a straightforward C++ function to compute a simple weighted mean, a
naïve implementation of this same function in R, and the built in weighted.mean() function from the base stats package.
# Script to compare C++ and R
# C++ Function in Rcpp wrapper
double wmean(NumericVector x, NumericVector w) {
int n = x.size();
double total = 0, total_w = 0;
for(int i = 0; i < n; ++i) {
total += x[i] * w[i];
total_w += w[i];
return total / total_w;
# Naive R function
wmeanR <- function(x, w) {
total <- 0
total_w <- 0
for (i in seq_along(x)) {
total <- total + x[i] * w[i]
total_w <- total_w + w[i]
total / total_w
x <- rnorm(100000000)
w <- rnorm(100000000)
# The proper way to compute a simple weighted mean in R
# using a built in function from the base stats package
Created by Pretty R at inside-R.org
On my laptop, the naïve R function took 229.47 seconds to run, the built in R function ran in 4.52 seconds, and the C++ function took only 0.28 seconds to execute. Yes, C++ is a lot faster. But, this
is a somewhat contrived example and it is not unreasonable to expect that a statistician could spend her entire career running weighted.mean() on vectors of reasonable size and never even consider
that R might be slower that something else. (For vectors of length 1,000,000, weighted.mean() took 0.06 seconds to run on my PC). Speed of execution needs to be evaluated in context. I can't imagine
any statistician interuppting the flow of an R session to save a few seconds on a once-in-a-while calculation. However, it is nice to know that there is a reasonable way to proceed in R if the
calculation needs to be done 100,000 times.
My three main take-aways from my tutorial were;
1. For garden variety programming (no objects or classes) C++ is not only accessible, but might also be fun.
2. Rcpp along with RTools does an incredible amount of “heavy lifting”, hiding the details of working with a compiled language from the R user and providing a big league environment for writing high
performance, R based code.
3. Even if you have some considerable experience with R, it may turn out that R is even richer than you thought.
It was a delightful surprise to realize that gaining some experience C++ might enhance one’s motivation to learn even more about R. Yes, it is important to know that one can attempt serious work in R
that might have critical execution time constraints and that there are tools such as Rcpp available to help one power through bottlenecks. However, the richer experience of the tutorial was to
consider the rewards of learning more about the structure of R and all R it has to offer.
for the author, please follow the link and comment on his blog:
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Play with the controls! Use the "Shell" slider to pick another shell, then move the "Section" slider around. Click the "New Colors" button. If you have red-blue 3D glasses, change the "Stereo Mode"
to "Anaglyph" and click the "Edges" checkbox. If you have a ColorCodeViewerTM (see below), change the "Stereo Mode" to "ColorCode" and click the "Edges" checkbox.
Atom Bomb
Cosmos is the Universe regarded as an ordered system.[1] The philosopher Pythagoras is regarded as the first person to apply the term cosmos (Greek κόσμος) to the order of the Universe.[2] Cosmology
[edit] Cosmology is the study of the cosmos in several of the above meanings, depending on context. All cosmologies have in common an attempt to understand the implicit order within the whole of
This is not a "Global warming" website. Although it has played a role in the melting of the polar regions, this website addresses the scientific data & information about other serious global changes
to our planet and how it is affecting our weather, seismic & volcanic stability. The menu contains links to articles, data, images and information from NASA, SOHO, ESA, NOAA, NWS, USGS, Smithsonian
Institution, Goddard Space Center, FEMA & other taxpayer funded Agencies. All information contained on this website has been based on verifiable information from official sources to ensure you can
access and verify the authenticity of this information. Global Changes
Subject Information Notes for Teachers This Vedic mathematics exercise can provide practice in basic arithmetic manipulation (addition and multiplication), as well as providing a context in which
students can explore mathematical connections. The task set out on the sheet is suitable for a wide age and ability range. At one level, it can generate an artistically pleasing set of mathematical
designs; more advanced work can go on to explore and explain some of the patterns which are developed.
Fibonacci, de Gulden Snede en het Zonnebloemmotief A.h.v. onderstaande figuur laat ik nog eens zien hoe cellen zich telkens opnieuw ordenen, wanneer deze zich naar buiten toe over een steeds grotere
omtrek verspreiden. Dat is hier weergegeven als een uitrekking in de horizontale ( tangentiële ) richting. Omdat het totale oppervlak, dat deze cellen gezamenlijk innemen, hierbij in principe niet
verandert - even afgezien dan van eventuele groei of krimp - gaat dat in verticale ( radiale ) richting gepaard met een samentrekking. Zijn aanvankelijk cellen A, B, C en D nog geordend in een
vierkant, na samentrekking in verticale en uitrekking in horizontale richting zijn dit vervolgens de cellen B, C, D en E. .In eerste instantie vallen de twee actuele spiralen nog samen met de zijden
van het oorspronkelijke vierkant en verlopen derhalve in de richtingen BA en BC. .Dit vierkant vervormt echter geleidelijk aan tot een ruit met een maximale hoek van 120 graden.
On 6th June 2006, I added optional satellite dish pointing angles - azimuth, elevation and feed rotation polarisation angles. These are based on the latitude and longitude of your site location and
the orbit position longitude of the geostationary satellite selected which is above the equator. Not all the bugs may be out of this yet. Lat - Long Finder: This page helps you find Latitude and
UFOs, ETs, Crop Circles and Close Encounters
UFO Top Secret: The Bob Lazar Interview pt.1
New Science, Holographic Universe, and Supranormal Phenomena
FAQ on Vedic Mathematics (From Georges Ifrah's The Universal History of Numbers) FAQ in Ancient Indian Mathematics If you have questions on Vedic Mathematics or related topics, don't be bashful, send
them to Aryabhatta@indicethos.org. No question will be deemed too simple or obvious.
The Hitchhiker's Guide to the Galaxy The Hitchhiker's Guide to the Galaxy is a comic science fiction series created by Douglas Adams. Originally a radio comedy broadcast on BBC Radio 4 in 1978, later
it was adapted to other formats, and over several years it gradually became an international multi-media phenomenon. Adaptations have included stage shows, a "trilogy" of five books published between
1979 and 1992, a sixth novel penned by Eoin Colfer in 2009, a 1981 TV series, a 1984 computer game, and three series of three-part comic book adaptations of the first three novels published by DC
Comics between 1993 and 1996.
Cosmology - UR
Gravitation, or gravity, is a natural phenomenon by which all physical bodies attract each other. It is most commonly recognized and experienced as the agent that gives weight to physical objects,
and causes physical objects to fall toward the ground when dropped from a height. During the grand unification epoch, gravity separated from the electronuclear force. Gravity is the weakest of the
four fundamental forces, and appears to have unlimited range (unlike the strong or weak force). The gravitational force is approximately 10-38 times the strength of the strong force (i.e., gravity is
38 orders of magnitude weaker), 10-36 times the strength of the electromagnetic force, and 10-29 times the strength of the weak force.
I was at a talk the other night listening to a nervous grad student present a paper about knots when, for whatever reason – I was thinking about Michael Heizer – wham!An idea came to me: The
eccentric son of a billionaire takes his share of the family fortune and moves to Utah. He buys loads and loads of land, tens of thousands of acres – and, with it, huge digging machines. This
includes state of the art tunnelers, taken straight from the oil industry.
Tesla komt na 100 jaar toch weer om de hoek
Life in the Universe #2: Where is Everybody? (FINAL CUT)
Life in the Universe #1: Just on Earth, or Everywhere?
Diamagnetic Gravity Vortexes by Richard LeFors Clark Waves of gravity pour from an extraordinary (and imaginary) cosmic accident —a head-on collision of two black holes approaching from left and
There are many competing theories about the ultimate fate of the universe. Physicists remain unsure about what, if anything, preceded the Big Bang. Many refuse to speculate, doubting that any
information from any such prior state could ever be accessible. There are various multiverse hypotheses, in which some physicists have suggested that the Universe might be one among many universes
that likewise exist.[11][12]
Segment of the macrocosm showing the elemental spheres of terra (earth), aqua (water), aer (air), and ignis (fire). Robert Fludd. 1617. Many philosophies and worldviews have a set of classical
elements believed to reflect the simplest essential parts and principles of which anything can consist or upon which the constitution and fundamental powers of everything are based. | {"url":"http://www.pearltrees.com/t/science/id4194345","timestamp":"2014-04-17T13:40:50Z","content_type":null,"content_length":"28510","record_id":"<urn:uuid:4380fa6e-92c3-4c65-8598-d2897fc569e5>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00118-ip-10-147-4-33.ec2.internal.warc.gz"} |
Provides a type for natural numbers.
data Natural Source
The type of natural numbers.
Note that matching a natural number against a negative pattern might not work as you expect. For example, evaluating the following expression results in a run-time error, instead of the result "plus
case 5 :: Natural of
-5 -> "minus five"
5 -> "plus five"
The reason is that the == operator of Natural is used for checking if the patterns match, making it necessary to convert -5 to Natural.
Enum Natural
Eq Natural
Integral Natural
Num Natural
Ord Natural
Read Natural
Real Natural
Show Natural | {"url":"http://hackage.haskell.org/package/natural-numbers-0.1.0.1/docs/Data-Natural.html","timestamp":"2014-04-17T22:48:37Z","content_type":null,"content_length":"4302","record_id":"<urn:uuid:639ff018-9cac-4710-b00c-d3a20c6cd4da>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00140-ip-10-147-4-33.ec2.internal.warc.gz"} |
Students are often asked to find the equation of a line that is perpendicular to another line and that passes through a point. Watch the video tutorial below to understand how to do these prolems
and, if you want,
download this free worksheet
if you want some extra practice. | {"url":"http://www.mathwarehouse.com/algebra/linear_equation/write-equation/equation-of-line-perpendicular-through-point.php","timestamp":"2014-04-19T02:44:03Z","content_type":null,"content_length":"17540","record_id":"<urn:uuid:84191bc2-6c78-4ab6-99d9-65f505d8f3f6>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00399-ip-10-147-4-33.ec2.internal.warc.gz"} |
Rewrite Systems, Pattern Matching, and Code Generation
Eduardo Pelegri-Llopart
EECS Department
University of California, Berkeley
Technical Report No. UCB/CSD-88-423
June 1988
Trees are convenient representations because of their hierarchical structure, which models many situations, and the ease with which they can be manipulated. A rewrite system is a collection of
rewrite rules of the form Alpha->Beta where Alpha and Beta are tree patterns. A rewrite system defines a transformation between trees by the repeated application of its rewrite rules.
Two research directions are pursued in this dissertation: augmenting the expressive power of individual rewrite rules by using new types of patterns, and analyzing the interaction of the rewrite
rules. The dissertation contains new algorithms for linear and non-linear patterns, for a new type of non-local pattern, and for typed patterns in which the variables are restricted to tree
The REACHABILITY problem for a rewrite system R is, given an input tree T and a fixed goal tree G, to determine whether there exists a rewrite sequence in R, rewriting T into G and, if so, to obtain
one such sequence. REACHABILITY can be used to solve problems related to the mapping between concrete and abstract syntax trees, to construct a pattern matching algorithm for typed non-local
patterns, and to provide algorithms for compiler code generation. A new class of rewrite system called finite bottom-up rewrite system (finite-BURS) is introduced for which the REACHABILITY problem
can be solved efficiently with a table-driven algorithm.
The C-REACHABILITY problem is similar to REACHABILITY except that rewrite sequences are assigned costs, and the obtained sequence is required to have minimum cost over all candidates. If the cost of
a rewrite sequence is defined as the sum of the costs of its rewrite rules, the algorithm for REACHABILITY can be modified for a subclass of finite-BURS to solve C-REACHABILITY in such a way that all
cost manipulation is done at table-creation time. The subclass extends the machine grammars used by Graham and Glanville for code generation. A code generator based on this approach has been
implemented and tested with several machine descriptions. The code generators obtained produce locally optimal code, are faster than comparable ones based on Graham-Glanville techniques, and are
significantly faster than other recent proposals that manipulate costs explicitly at code generation time. Table size is comparable to the Graham-Glanville code generator.
Advisor: Susan L. Graham
BibTeX citation:
Author = {Pelegri-Llopart, Eduardo},
Title = {Rewrite Systems, Pattern Matching, and Code Generation},
School = {EECS Department, University of California, Berkeley},
Year = {1988},
Month = {Jun},
URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/5507.html},
Number = {UCB/CSD-88-423},
Abstract = {Trees are convenient representations because of their hierarchical structure, which models many situations, and the ease with which they can be manipulated. A rewrite system is a collection of rewrite rules of the form Alpha->Beta where Alpha and Beta are tree patterns. A rewrite system defines a transformation between trees by the repeated application of its rewrite rules. <p>Two research directions are pursued in this dissertation: augmenting the expressive power of individual rewrite rules by using new types of patterns, and analyzing the interaction of the rewrite rules. The dissertation contains new algorithms for linear and non-linear patterns, for a new type of non-local pattern, and for typed patterns in which the variables are restricted to tree languages. <p>The REACHABILITY problem for a rewrite system <i><b>R</b></i> is, given an input tree <i>T</i> and a fixed goal tree <i>G</i>, to determine whether there exists a rewrite sequence in <i><b>R</b></i>, rewriting <i>T</i> into <i>G</i> and, if so, to obtain one such sequence. REACHABILITY can be used to solve problems related to the mapping between concrete and abstract syntax trees, to construct a pattern matching algorithm for typed non-local patterns, and to provide algorithms for compiler code generation. A new class of rewrite system called finite bottom-up rewrite system (finite-BURS) is introduced for which the REACHABILITY problem can be solved efficiently with a table-driven algorithm. <p>The C-REACHABILITY problem is similar to REACHABILITY except that rewrite sequences are assigned costs, and the obtained sequence is required to have minimum cost over all candidates. If the cost of a rewrite sequence is defined as the sum of the costs of its rewrite rules, the algorithm for REACHABILITY can be modified for a subclass of finite-BURS to solve C-REACHABILITY in such a way that all cost manipulation is done at table-creation time. The subclass extends the machine grammars used by Graham and Glanville for code generation. A code generator based on this approach has been implemented and tested with several machine descriptions. The code generators obtained produce locally optimal code, are faster than comparable ones based on Graham-Glanville techniques, and are significantly faster than other recent proposals that manipulate costs explicitly at code generation time. Table size is comparable to the Graham-Glanville code generator.}
EndNote citation:
%0 Thesis
%A Pelegri-Llopart, Eduardo
%T Rewrite Systems, Pattern Matching, and Code Generation
%I EECS Department, University of California, Berkeley
%D 1988
%@ UCB/CSD-88-423
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/5507.html
%F Pelegri-Llopart:CSD-88-423 | {"url":"http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/5507.html","timestamp":"2014-04-19T19:42:13Z","content_type":null,"content_length":"9634","record_id":"<urn:uuid:efdce6ff-cf4a-462e-bcc9-494e112d1e36>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00016-ip-10-147-4-33.ec2.internal.warc.gz"} |
Student Support Forum: 'Converging Summations' topic
Author Comment/Response
Mike Willhide
Hi. Newbie here. I'm working on a mechanics problem that requires me to find the
limit of a sum as the iterator goes to infinity. Let's see if I can type
Summation from n=0 to n=infinity of:
{ e^(2n) * h } + { e^(2n+1) * h}
This limit exists if e < 1. Otherwise, it diverges. Here is the meat of my
How do I tell Mathematica that e will always be less than 1 such that it
produces the limit as a function of e and h (analytical solution)?
I can obtain numerical solutions by using:
Limit[ theSummationPresentedAbove /. {e->NumberLessThanOne, h->AnyValue},
n->infinity ]
Does an analytical solution exist, or will it be different with each value of e
and h?
Also, this doesn't work, can you tell me why?
f[MyH_,MyE_] := Limit[ Sum[ {MyE^(2n) * MyH + MyE^(2n+1) * MyH} ], {n, 0,
It just returns f[ a, b].
Sorry about the length of this post, but it's hard to get these things across
without the luxury of Mathematica's notation. Thanks a bundl
URL: , | {"url":"http://forums.wolfram.com/student-support/topics/2330","timestamp":"2014-04-20T08:42:59Z","content_type":null,"content_length":"24195","record_id":"<urn:uuid:1ea06dca-052b-4c10-b531-0f8eb1bd6a38>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00148-ip-10-147-4-33.ec2.internal.warc.gz"} |
Translation to and from Polish notation
By C. L. Hamblin
"Reverse Polish" notation is embodied in the instruction languages of two recent machines, and
"Forward Polish" notation is of use in mechanized algebra. This article illustrates, using a simple
language without detail, some methods of translating between these notations and an "orthodox"
one of the kind used in FORTRAN and ALGOL.
The question of efficient translation between an is in terms of the order of the number-denoting symbols
"orthodox" mathematical notation of the kind ordinarily (numbers and number variables): in "pure" translation
used in writing algebraic formulae (and copied as closely these symbols remain unaltered and in the same order
as is practicable in FORTRAN and ALGOL) and in the translated formula as they were in the original.
"Polish" notation has come to prominence as a result Thus we shall say that the transformation of
Downloaded from comjnl.oxfordjournals.org by guest on August 31, 2010
of the use of what is in effect Polish notation as the "a +(b X c)" into Forward Polish " + a x J c " is a
basic instruction language of two recent computers.* case of pure translation, whereas its transformation into
Polish notation is so-called because of its extensive " + X b c a" though this is an equivalent form, involves
use in Polish logical writings since its invention by manipulation as well as translation, since the order
-Lukasiewicz (1921, 1929). -Lukasiewicz demonstrated "b c a" of the number variables is different. We confine
that if operators are written always in front of their ourselves to pure translation, so defined, in what follows.
operands, instead of (as in the case of the diadic operators This restriction, however, does not yet entirely remove
of arithmetic, " + ", "—", " x " and so on) between them, the possibility that a formula in a given notation should
there is never any need for brackets to indicate associa- have alternative forms. This is because of the asso-
tion of terms. Thus if in place of "a -f- b" we write ciativity of some arithmetical operators. Thus in
" + a b", and so on, the brackets in an expression such orthodox notation "(a + b) + c" is equivalent to
as "(a + b) X c" may be dispensed with in translation, "a +{b + c)," and the brackets are usually omitted;
since " X + a b c" indicates unambiguously the result but to these formulae correspond in Forward Polish
of operating with " x " on "-\-ab" and "c": for the formulae "++abc" and "+a + bc" respec-
"a + (b X c)" we should instead write " + a X b c." tively. To resolve ambiguity we distinguish two special
The resulting notation, in the case of long formulae, is cases, the early-operator and late-operator forms respec-
a little harder to read, since brackets aid the eye, but it tively of Polish formulae. A Polish formula is in early-
has some other advantages. In particular Reverse operator (late-operator) form if all operator symbols
Polish—the notation which results if operators are occur as early (late) in it as possible. Thus "a+b-\-c+d"
placed after operands, as in "a b +"—has the property becomes "+ + + a b c d" in early-operator Forward
that the operators appear in the order in which they are Polish, " + a + b + c d" in late-operator Forward
required in computation. Reverse Polish is hence in Polish, "a b + c + d + " in early-operator Reverse
some sense a natural notation for an instruction language, Polish, and "a b c d + + + " in late-operator Reverse
each symbol being interpretable as an instruction. Polish. There are of course intermediate forms such as
(Number variables are "fetch" instructions.) The "++ab+cd" and "a b + c d + + " which, though
absence of brackets further makes Polish notation (— valid Forward and Reverse Polish respectively, are
either Forward or Reverse, but probably preferably neither early-operator nor late-operator.
Forward—) useful in mechanized algebra, since it eli-
minates a continual source of complication in algebraic In the case of Reverse Polish for use as an instruction
manipulations. language it is usually the early-operator form that is
desirable, since this uses the minimum number of
Machine translation from one notation to another is locations in the push-down store.
needed in writing compilers for the new machines, and
it is possible to foresee a variety of future uses for it. By Orthodox A I shall mean a language constructed
This article illustrates, using a simple language without with orthodox symbol-order out of the following
detail, some translation methods. In general, translation symbols.
is extremely simple if done in the right way (i) Number-variables a, b, c, d, . . . (The use of
It is convenient to distinguish "pure" translation from actual numerals raises no essential new issues; we
translation which involves manipulation or rearrange- need not consider it here.)
ment. A simple way of characterizing this distinction (ii) Operators + , —, neg, x, f. Of these, "neg"
* The English Electric KDF9 and the Burroughs B5000. Each (representing "negative") is monadic, i.e. operates on a
of these uses a "push-down" (or "nesting") type of store for single number, and is placed in front of its operand, as
arithmetic operands and results, following a scheme suggested by in "neg a": the others are diadic and stand between
the present author (Hamblin, 1957, 1957, 1960; see also Hamblin,
Humphreys, Karoly and Parker, 1960). their operands, as in "a + b". Symbol " f " denotes
Translation of Polish notation
exponentiation: thus for "a " we write "a | b." (There one time: if the list has entries £,, E2,. . ., £„, it is
is, of course, no difficulty in arbitrarily extending the necessary to remove £„ before £„_! can be inspected,
range of permitted operators, but these are enough for and so on.)
our present purpose.) In detail, the following are the operations to be
(iii) Brackets (, ). We assume that " - r " and "—" carried out when symbol Sj of the Orthodox A formula
are weaker (that is, more weakly associative) than "neg," is examined.
which is in turn weaker than " x , " which is in turn (a) If Sj is a number variable a, b, c, d, . . . it is
weaker than " f." Hence the absence of brackets will transcribed directly to output.
never actually lead to any ambiguity. For example (b) If Sj is a L.H. bracket symbol, it is transcribed
to list N.
neg axb+c^dxe +/ x g (c) If Sj is an operator symbol, the last entry—call
will mean —ab + c?e +fg. it E—of list N is examined: if E is an operator not
weaker than Sj, E is transcribed to output and the next
Brackets are used to associate symbols into a group last entry similarly examined; and so on until Nis empty
when they are not automatically so associated by these or its last entry is a L.H. bracket or an operator weaker
rules. (There is, of course, no penalty if brackets are than Sj. Then Sj is transcribed to list N.
used superfluously.) (d) If Sj is a R.H. bracket symbol, entries are tran-
It is a trivial matter to convert formulae in a fully
Downloaded from comjnl.oxfordjournals.org by guest on August 31, 2010
scribed from list N to output until a L.H. bracket symbol
orthodox notation to Orthodox A, provided, of course, is reached: this is deleted.
that they use only the permitted range of mathematical (e) After the last symbol of the Orthodox A formula
notions. The essential rules are as follows. has been dealt with, the remaining entries of N are
(i) Alter "—" to "neg" wherever it occurs at the transcribed to output.
beginning of a formula or immediately following a As described, this procedure gives as output the
L.H. bracket. early-operator form of Reverse Polish. An alteration
(ii) Insert " f" wherever there is a change from of detail yields a procedure which gives the late-
normal type-face to that used for exponents, and put operator form: paragraph (c) is replaced by:
brackets round the exponent which follows if it contains (c') If Sj is an operator symbol the last entry—call
any operator. Then use the same type-face throughout. it E—of list N is examined: if E is an operator and Sj
(iii) Insert " x " wherever a number variable or R.H. is weaker than E, or if JT is " - " and S; is " + " or " - , "
bracket is followed by a number variable or L.H. E is transcribed to output and the next last entry
bracket. similarly examined; and so on until N is empty or its
The Polish notations considered here will have exactly last entry is a L.H. bracket or an operator not as
the same range of symbols as Orthodox A, except, of described. Then Sj is transcribed to list N.
course, the brackets. The special provisions regarding " 4 . " and " —" here
The following cases of translation will be considered guard against error owing to the incomplete asso-
in detail: ciativity of "—": thus, for example, "a — b + c" does
not have separate early-operator and late-operator
I. Orthodox A to Reverse Polish. forms, becoming "ab — c + " in either case. Actually,
II. Orthodox A to Forward Polish. "—" in orthodox notation is associative to the left: this
III. Forward Polish to Orthodox A. corresponds with early-operator Polish directly, but will
IV. Forward Polish to Reverse Polish. always lead to a special rule in other cases.
These cases provide a survey of the relevant techniques.
As will appear, only minor modifications are needed to
II. Orthodox A to Forward Polish
give the other cases of interest.
Translation to or from early-operator (late-operator)
Forward Polish is closely the same as translation to or
I. Orthodox A to Reverse Polish from late-operator (early-operator) Reverse Polish back-
This is the simplest of the cases. Let StS2 . . . Sm be wards, i.e. from right to left. In fact the only fore-
the Orthodox A formula. The symbols of this formula and-aft asymmetry that occurs is not in the Polish
are examined one by one in order from left to right, notations, but in the Orthodox A, and then only refers
and the translated formula is written out symbol-by- to "neg" which appears in front of its operands when
symbol directly. Number variables are transcribed as the formula is read forwards and after them when the
soon as they are encountered. Operator-symbols, which formula is read backwards, and to the associativity
can never occur earlier in the sequence of number properties of " —." Consequently, under this heading
variables in Reverse Polish than they do in Orthodox A, two translation methods will be considered, of which
are held in a "nesting list" N until conditions for their the first, which is by far the simpler, is a modification
transcription are satisfied. (A "nesting list" is a list of that described above, used backwards. Circumstances
operated on the "last-in-first-out" principle. That is, might arise, however, in which it was not desirable to
of the entries in the list only the last is available at any be forced to write and read formulae backwards, and
Translation of Polish notation
in such cases a method such as the second must be Polish. Let Ax, A2, • . . , Am be the addresses of the
resorted to. The extra complexity of this method is a symbols in the Orthodox A formula. Against each,
considerable penalty, but it is unavoidable since in unless it is a bracket or the final symbol S'n, we write
translation from Orthodox A to Forward Polish the an address, Bt, B2,. . ., Bm. The final output will then
operators must be moved forward in the formula, not be taken as follows: given that Syi is the starting symbol,
back; and this cannot be done on-the-run. The alter- it is sent to output and the symbol at address Bn is
native of "queueing" the number-variables until the fetched—let this be Sj2: this is sent to output and the
operators are sorted out is not as simple as it sounds, symbol at address Bj2 is fetched—let this be SJ3: and so
since in most cases all the number-variables need to be on until a blank address is reached.
placed in the queue before a single one is taken out and Let list L consist at any time of p entries Eu E2, .. ., Ep
sent to output, and one might as well have no queue (where p may, of course, be zero). Each entry Ej con-
but simply resort to more than one run-through of the sists of a symbol 7} and two addresses Cj and D}. 7}
formula; for example, to a translation first to Reverse is one of the symbols a, (,+,—, neg, x , \ . Every entry
Polish, followed by a translation from Reverse to stands for a sequence of symbols in the final (Polish)
Forward as described in IV. Method 2, below, would formula: if 7} is a there is a number-denoting expression
usually be faster than this. which can be found in the Orthodox A formula by
starting with the symbol at address C}—call it Ski:
Downloaded from comjnl.oxfordjournals.org by guest on August 31, 2010
Method 1 taking next the symbol at Bkl—call it Sk2: and so on
Let SXS2 . . . Sm be the Orthodox A formula, and let until the symbol at address Dj has been taken. If 7}
it contain p bracket symbols: after translation let the is a diadic operator there is a similar sequence consisting
resulting Forward Polish formula be S& . . . Si,, where of that symbol followed by a number-denoting expression,
of course n = m — p. Symbols of the Orthodox A its first operand. If 7} is a monadic operator we always
formula are taken one by one in the reverse order have Cj = Dj (there is, as it were, a one-symbol sequence).
Sm, Sm_t,. . . and the translated formula is written out If Tj is a bracket symbol the entries that follow it are
symbol-by-symbol in the reverse order S'n, S'n^\,. . . all contained within a bracket-pair in the Orthodox A
The procedure each time a symbol Sj is examined is the formula: here C, and Dj are left blank and are not relevant.
same as in I above, except that if Sj is the symbol "neg" At various stages, to be specified, an entry which is a
it is transcribed to output directly in the same way as a merger of a succession of existing entries is formed. To
number variable; and that under (c) in I, for "not weaker merge £,(= T&D,), £}(= TJCJDJ), and £,(= TkCkDk)
than" it is necessary to read "weaker than", and for we replace these entries by a single one, namely by
"weaker than" it is necessary to read "not weaker than". aC,Dk if Tk is a, otherwise by TjCjDk: at the same time
This gives the early-operator form. For the late- against the symbol (in the Orthodox A formula) at
operator form a comparable, if slightly more complicated, address D, we write the address Cj\ and against the
modification of (c') is substituted. symbol at address Dj we write Ck. Similarly for the
merger of a longer or shorter sequence of entries.
Method 2 The procedure for the writing-in of addresses against
Here we first effect a "virtual" reordering of the the symbols of the Orthodox A formula can now be
Orthodox A symbols without rewriting them, by placing fully specified. The symbols Su S2,. .., Sm are examined
against each (other than brackets) the present address of in order and for each Sj the following action is taken.
the symbol which is to follow it in the revised order. A (a) If Sj is a number variable an entry aAjAj is added
separate indication gives the starting-symbol. For to the list L.
example, if symbols "ABCDEF" were stored at addresses (b) If Sj is a L.H. bracket an entry "( 0 0" is added to
18-23 respectively, we could indicate our intention of the list L.
reordering them "CBDFEA" by noting the address (20) (c) If Sj is a monadic operator symbol an entry
of C as starting-point, writing against C at address 20 SjAjAj is added to the list L.
the address (19) of B, against B at address 19 the (<•/) If Sj is a diadic operator symbol list L is examined
address (21) of D, and so on; thus: backwards from the last entry (without removing any
entries) until either a weaker operator symbol, or a
Start bracket, or the beginning of the list is encountered. Then
Address 18 19 20 21 22 23
A B
(i) if what is encountered (say at Ek) is a weaker
C D E F
operator symbol, Ek+, is replaced by the merger of SJAJAJ,
(next address) 21 19 23 18 22 Ek+U Ek+2,. .., Ep; and Ek+2,. . ., Ep are deleted.
This can be done in a single run-through of the formula (ii) If what is encountered (say at Ek) is a bracket
with the aid of a subsidiary list L: each entry of L symbol, Ek is replaced by the merger of SJAJAJ, Ek+i,
consists of a symbol and two addresses, indicating a Ek+2,. . ., Ep; and Ek+U . . ., Ep are deleted.
subsequence of the finished formula. L reduces at the (iii) If what is encountered is the beginning of the list,
end of the process to a single entry. Ei is replaced by the merger of SjAjAj, Eu E2, • • •, Ep;
Let SlS2 . . . Sm be the Orthodox A formula, and and E2, . . ., Ep are deleted.
S{S2 . . . Si, the corresponding formula in Forward (e) If Sj is a R.H. bracket list L is examined back-
Translation of Polish notation
wards vwunuut removing entries) until a bracket symbol this is sent to output but left in the list, marked "written."
is encountered (say at Ek); and Ek is replaced by the If no such operator is reached, the translation is complete.
merger of Ek+X, Ek+2, • • •, Ep, which are deleted. A closely similar method can naturally be applied,
When all the symbols of the Orthodox A formula used backwards, to the translation of Reverse Polish to
have been dealt with, if p > 1, Ex is replaced by the Orthodox A: compare method 1 of II.
merger of Ex, E2,. . ., Ep\ and E2,.. ., Ep are deleted. IV. Forward Polish to Reverse Polish
Now C, gives the address of the first symbol in the
Polish formula (and Dx that of the last). The simplest of all methods of converting from
To yield late-operator Forward Polish instead of Forward Polish to Reverse or vice versa is simply to
early-operator, only a minor modification is needed: read the pertinent formula backwards: this is not quite
under (d) above, and under (d)(i), in place of "a weaker accurate as it stands, since certain operators such as
operator symbol" write "a weaker or equally weak "—" and " f" are asymmetrical (Forward Polish
operator symbol (other than the symbol '—'in case "— a b" means "a — b" whereas Reverse Polish
Sj is ' + ' or ' - ' ) . " "b a —" means "b — a"), but it may be possible to
allow for this in interpretation. The order of the
m . Forward Polish to Orthodox A number-denoting symbols is of course reversed. But
where this procedure is unacceptable the following
This is a relatively simple case, not unlike I: the
Downloaded from comjnl.oxfordjournals.org by guest on August 31, 2010
method is appropriate.
operators may similarly be stored up in a nesting list. The Forward Polish formula SXS2 . . . Sm is taken
However, provision must, of course, be made for symbol-by-symbol as before, using a nesting list with
inserting brackets where necessary; and since the asso- provision as in III for placing a mark against each
ciative influence of an operator extends in the result entry. In this case a mark placed against an entry
after it as well as before it the writing of an operator in indicates that only one operand of the operator con-
the output does not mean that it can be cancelled imme- cerned remains to be completely written. As each symbol
diately from the nesting list. Hence an extra provision Sj is examined, operations are carried out as follows.
must be made in the nesting list for putting a mark
against entries to indicate that they have been "written." (a) If Sj is a diadic operator it is transcribed to the
nesting list.
The symbols of the Forward Polish formula SiS2 • • Sm (b) If S} is the monadic operator "neg" it is transcribed
are examined in order and the following operations are to the nesting list with a "mark" against it.
carried out. (c) If Sj is a number variable it is transcribed to out-
(a) If Sj is a diadic operator it is transcribed to the put : then the last entry of the nesting list is transcribed
nesting list; but a R.H. bracket is written in the nesting to output if it is "marked," and similarly the next last,
list first if Sj is weaker than the operator which is the and so on until an unmarked entry is reached: this is
current last entry in the list, if any: in this case also a "marked." If there is no unmarked entry translation
L.H. bracket is sent to output. is complete.
(b) If Sj is the monadic operator "neg," and if this is This procedure, perhaps somewhat surprisingly, trans-
weaker than the operator which is the current last entry lates early-operator Forward Polish into early-operator
in the nesting list, a L.H. bracket is sent to output and Reverse, and late-operator Forward Polish into late-
a R.H. bracket is written in the nesting list: then, and operator Reverse; and intermediate forms into inter-
in any case whether this is so or not, "neg" is transcribed mediate forms. A procedure which would translate,
to output and also "neg" is added to the nesting list, say, early-operator Forward into late-operator Reverse,
marked "written." or which would always give early-operator Reverse
(c) If Sj is a number variable, it is transcribed to out- whatever the form of the original Forward, would
put: then if the last entry in the nesting list is an operator need to be rather more complicated.
marked "written," it is cancelled; if it is a R.H. bracket It is immediate from considerations of symmetry that
it is transcribed to output. The next last entry is taken an identical procedure used backwards—that is, reading
from the nesting list and treated in the same way, and and writing the relevant formulae from right to left—
so on until an operator not marked "written" is reached: translates Reverse Polish to Forward Polish.
HAMBLIN, C. L. (1957). "An Addressless Coding Scheme based on Mathematical Notation," W.R.E. Conference on Computing,
Proceedings, Weapons Research Establishment, Salisbury, South Australia.
HAMBLIN, C. L. (1957). "Computer Languages," Australian Journal of Science, Vol. 20, p. 135.
HAMBLIN, C. L. (1960). "GEORGE, an Addressless Coding Scheme for DEUCE," Australian National Committee on Com-
putation and Automatic Control, Summarised Proceedings of First Conference, paper C6.1.
HAMBLIN, C. L., HUMPHREYS, H. L., KAROLY, G., and PARKER, G. J. (1960). "Considerations of a Computer with an Addressless
Order Code" and "Logical Design for ADM, an Addressless Digital Machine," Australian National Committee on Com-
putation and Automatic Control, Summarised Proceedings of First Conference, papers C6.2 and C6.3.
LUKASIEWICZ, J. (1921). "Logika dwuwartosciowa" (Two-valued logic), Przeglqd Filozoficzny, Vol. 23, p. 189.
LUKASIEWICZ, J. (1929). Elementy logiki matematyczny (Elements of mathematical logic), Warsaw. | {"url":"http://www.docstoc.com/docs/52733285/Translation-to-and-from-Polish-notation","timestamp":"2014-04-20T19:08:59Z","content_type":null,"content_length":"84361","record_id":"<urn:uuid:d3185cfc-f6ae-4ec4-a535-a232480d3ce3>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00290-ip-10-147-4-33.ec2.internal.warc.gz"} |
Simplex method - algebraic vs tabular form
Thank you very much for your help!
Iīm now learning the big M method and learning about the radiation therapy problem where the ≤ isnīt always that way but itīs also ≥.
See picture
I was wondering if the big M method was the only way to solve it when it looks like this or could i solve it by using the tabular form?
And if I use the tabular form, would I also put -x5 +x6 in the bottom line where it looks like this: ≥ ?
In introductory textbooks there are two methods discussed: (1) the 2-phase method; and (2) the big-M method. The two-phase method deals with the following two questions: (a) is the problem feasible
at all, and if it is, what is a basic feasible solution; then (b) if feasible, go to optimality. (a) is Phase I and (b) [if needed] is Phase II. Here are two little related examples:
(I) max x1 + x2 + x3
2*x1 + 4*x2 + x3 <= 8
2*x1 + x2 + 2*x3 >= 8
2*x1 + 3*x2 + 4*x3 >= 30
x1,x2,x3 >= 0
(II) Same as (I), except change 30 on RHS of third constraint to 34.
Question: is (I) a feasible problem? One way is to solve a Phase I problem:
min a2 + a3 <----Phase I objective.
2*x1 + 4*x2 + x3 + s1 = 8
2*x1 + x2 + 2*x3 -s2 + a2 = 8
2*x1 + 3*x2 + 4*x3 - s3 + a3 = 30
Here, s1 is a slack variable, s2 and s3 are surplus variables and a2, a3 are artificial variables. If the optimal phase I objective is > 0 that means that it is impossible to have both artificial
variables = 0, so it is impossible to satisfy all of the original constraints; that is, the original problem is infeasible. However, if the optimal Phase I objective = 0, it is possible to have all
artificial vars = 0, so the original problem is feasible; furthermore, the output solution to Phase I is a basic feasible solution to the original problem, which can the be used to start Phase II
(the optimization phase). Phase I just forgets about the original objective until it has been determined whether or not the problem is even feasible; after all, if it is infeasible, the objective
does not matter.
The Phase I problem for (II) is similar, except it has 34 on the right of the 3rd equation. If we solve both problems, we find: (I) is feasible, but (II) is infeasible. Neither statement is "obvious"
when you first look at these problems.
The big-M method tries to combine Phases I and II in a single problem.
Real-world, commercial-scale LP codes do not actually use either of these methods; some of them start with infeasible solutions (having one or more negative vars which are supposed to be >= 0) and
use modified simplex steps to try to reduce the amount of infeasiblilty. But, those are details best left to more advanced treatments. | {"url":"http://www.physicsforums.com/showthread.php?p=3805368","timestamp":"2014-04-18T21:24:32Z","content_type":null,"content_length":"42522","record_id":"<urn:uuid:acf43951-21b0-422d-aeb3-e9e1db5e9f36>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00427-ip-10-147-4-33.ec2.internal.warc.gz"} |
physics(I'm taking a different way to solve)
Posted by phys on Saturday, April 10, 2010 at 12:02am.
In the 1968 Olympic Games, University of Oregon jumper Dick Fosbury
introduced a new technique of high jumping called the "Fosbury flop."
It contributed to raising the world record by about 30 cm and is
presently used by nearly every world-class jumper. In this technique,
the jumper goes over the bar face up while arching his back as much as
possible, as shown below. This action places his center of mass outside
his body, below his back. As his body goes over the bar, his center of
mass passes below the bar. Because a given energy input implies a
certain elevation for his center of mass, the action of arching his
back means his body is higher than if his back were straight. As a
model, consider the jumper as a thin, uniform rod of length L.
When the rod is straight, its center of mass is at its center. Now bend
the rod in a circular arc so that it subtends an angle of θ = 81.5°
at the center of the arc, as shown in Figure (b) below. In this
configuration, how far outside the rod is the center of mass? Report
your answer as a multiple of the rod length L.
y=r^2/L times integral of sin(theta)dtheta
I'm just confused about the angle that i should use, i used angle from 49.25 to 130.75, but i got it wrong.
• physics(I'm taking a different way to solve) - drwls, Saturday, April 10, 2010 at 9:25am
This is basically an integration problem. You want the CM of a uniform circular arc that faces downward and subtends 81.5 degrees, which is 1.422 radians. Your integral should extand from 49.25
to 130.75 degrees, which is 0.860 to 2.282 radians.
Your integrand appears to be wrong. The difference between the CM and the top of the arc is the integral of
(1 - cos theta)*r*(d theta), divided by the subtended arc angle, 1.42 radians.
the arc length is L = 1.422*r
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GHAPACK: A Library for the Generalized Hebbian Algorithm - Kurt Grandis
GHAPACK: A Library for the Generalized Hebbian Algorithm
I recently joined a new open source project called GHAPACK. The project currently provides the functionality and the means to use the Generalized Hebbian Algorithm. I came across this project after
banging my head against some of the practical limitations of Singular Value Decomposition (SVD). GHA is a Hebbian-based neural network-like algorithm that approximates SVD’s ability to perform eigen
decomposition. Its added bonus is that it allows for incremental training so you can refine your model with new data without having to recompute using the entire dataset.
Your Trusty SVD Tool
SVD is one of those tools that every machine learning practitioner and computational geek will pull out at some time or another. It’s a powerful matrix factorization technique that allows you to get
at the matrix’s eigenvectors and eigenvalues. One of reason it tends to be used so often is the fact that it can be used on those pesky M x N matrices, which us data junkies tend to generate.
For most small problems I can just use scipy and numpy’s svd and never give it a second thought. LAPACK’s suite of SVD routines power the svd functions of scipy, numpy, and MATLAB among others. It is
developed for dense matrices and processes them in their entirety. What happens when you start dealing with problems in high-dimensional space? Those dense representations and full processing are
expensive. So, when your problem space is better suited for sparse matrices you tend to run into not enough memory, non-convergence…no SVD.
At the time I was considering a problem that would be well-suited for incremental training, meaning I did not want to have to rerun the entire dataset through SVD after adding a small set of new
data; GHA lets you avoid that sort of inconvenience and approximates the same outcome (as far as my problem was concerned).
GHAPACK is written by Genevieve Gorrell and based on her work using GHAs to perform Latent Semantic Analysis (LSA).
“Offline” Calculations
My first order of business upon joining the project was to get the offline training working. This allows you to compute a pseudo-SVD based on a massive matrix without having to load the whole thing
into memory. No more out-of-memory segfaults. Now, you’re just limited by the resiliency of your hardware. This is now working.
Memory Management
I addressed a few memory leaks, but will likely do some restructuring to optimize memory management.
Resource Library
I would like the core of the GHA magic to be extracted into a library that others could embed in their own projects. So, I intend to move core functionality into a library and restructure the
existing apps into commandline tools that utilize those libs.
GHA, off-the-bat, is not known for its speed compared to some other eigen decomposition approaches. Besides that, there is room for some major gains in performance. Let’s see what we can squeeze out
of GHAPACK and perhaps lean on things like BLAS.
Testing Framework
A testing framework that objectively keeps track of performance gains, while ensuring computational integrity through unit testing always makes refactoring work that much less stressful.
Lots of work to do and a hearty thanks to Professor Gorrell for letting me join her efforts.
Other SVD Resources
There are other SVD libraries out there that will carry you farther if SVD is what you really want and not necessarily the means to an end.
ScaLAPACK has parallel SVD code, which creams LAPACK’s performance when you have access to multiple cores and/or MPI. ARPACK and SVDPACK both offer Lanczos-based SVD solutions for sparse matrices
with ARPACK being well-suited for parallel processing.
Stochastic algorithms, like Gorrell’s GHA approach to SVD, are much faster at finding solutions quickly to a few decimal places of accuracy. With fixed floating-point arithmetic, it’s all
approximate, so it’s just a matter of how much accuracy you need from your SVD. Just tweak the convergence parameters.
I discuss this in a blog entry on SVMs and logistic regression, but the ideas also apply to SVD.
I reimplemented the GHA approach as part of LingPipe. There’s an SVD Tutorial which goes over some of the classic demos.
Categories: Data, Machine Learning, Software Engineering / Tags: | {"url":"http://kurtgrandis.com/blog/2009/02/08/ghapack-a-library-for-the-generalized-hebbian-algorithm/","timestamp":"2014-04-17T03:48:23Z","content_type":null,"content_length":"30890","record_id":"<urn:uuid:499806b8-2603-4754-96d6-3248548cf365>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00235-ip-10-147-4-33.ec2.internal.warc.gz"} |
Partial Differential Equations: An Introduction, 2nd Edition
Partial Differential Equations: An Introduction, 2nd Edition
December 2007, ©2008
Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of
Partial Differential Equations
provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject,
illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations.
In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least
possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.
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Chapter 1: Where PDEs Come From
1.1 What is a Partial Differential Equation?
1.2 First-Order Linear Equations
1.3 Flows, Vibrations, and Diffusions
1.4 Initial and Boundary Conditions
1.5 Well-Posed Problems
1.6 Types of Second-Order Equations
Chapter 2: Waves and Diffusions
2.1 The Wave Equation
2.2 Causality and Energy
2.3 The Diffusion Equation
2.4 Diffusion on the Whole Line
2.5 Comparison of Waves and Diffusions
Chapter 3: Reflections and Sources
3.1 Diffusion on the Half-Line
3.2 Reflections of Waves
3.3 Diffusion with a Source
3.4 Waves with a Source
3.5 Diffusion Revisited
Chapter 4: Boundary Problems
4.1 Separation of Variables, The Dirichlet Condition
4.2 The Neumann Condition
4.3 The Robin Condition
Chapter 5: Fourier Series
5.1 The Coefficients
5.2 Even, Odd, Periodic, and Complex Functions
5.3 Orthogonality and the General Fourier Series
5.4 Completeness
5.5 Completeness and the Gibbs Phenomenon
5.6 Inhomogeneous Boundary Conditions
Chapter 6: Harmonic Functions
6.1 Laplace’s Equation
6.2 Rectangles and Cubes
6.3 Poisson’s Formula
6.4 Circles, Wedges, and Annuli
Chapter 7: Green’s Identities and Green’s Functions
7.1 Green’s First Identity
7.2 Green’s Second Identity
7.3 Green’s Functions
7.4 Half-Space and Sphere
Chapter 8: Computation of Solutions
8.1 Opportunities and Dangers
8.2 Approximations of Diffusions
8.3 Approximations of Waves
8.4 Approximations of Laplace’s Equation
8.5 Finite Element Method
Chapter 9: Waves in Space
9.1 Energy and Causality
9.2 The Wave Equation in Space-Time
9.3 Rays, Singularities, and Sources
9.4 The Diffusion and Schrodinger Equations
9.5 The Hydrogen Atom
Chapter 10: Boundaries in the Plane and in Space
10.1 Fourier’s Method, Revisited
10.2 Vibrations of a Drumhead
10.3 Solid Vibrations in a Ball
10.4 Nodes
10.5 Bessel Functions
10.6 Legendre Functions
10.7 Angular Momentum in Quantum Mechanics
Chapter 11: General Eigenvalue Problems
11.1 The Eigenvalues Are Minima of the Potential Energy
11.2 Computation of Eigenvalues
11.3 Completeness
11.4 Symmetric Differential Operators
11.5 Completeness and Separation of Variables
11.6 Asymptotics of the Eigenvalues
Chapter 12: Distributions and Transforms
12.1 Distributions
12.2 Green’s Functions, Revisited
12.3 Fourier Transforms
12.4 Source Functions
12.5 Laplace Transform Techniques
Chapter 13: PDE Problems for Physics
13.1 Electromagnetism
13.2 Fluids and Acoustics
13.3 Scattering
13.4 Continuous Spectrum
13.5 Equations of Elementary Particles
Chapter 14: Nonlinear PDEs
14.1 Shock Waves
14.2 Solitions
14.3 Calculus of Variations
14.4 Bifurcation Theory
14.5 Water Waves
A.1 Continuous and Differentiable Functions
A.2 Infinite Sets of Functions
A.3 Differentiation and Integration
A.4 Differential Equations
A.5 The Gamma Function
Answers and Hints to Selected Exercises
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• Eigenvalue problems (Chapters 10 and 11) are covered in appropriate depth.
• Treatment of distributions and Green’s functions eliminates student confusion by giving instructors the option of going directly from Green's functions in Chapter 7 to distributions and Fourier
transforms in Chapter 12.
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• Numerous exercises, varying in difficulty.
• Frequent mention of wave propagation, heat and diffusion, electrostatics, and quantum mechanics puts PDE into context, which is especially important for engineering and science majors.
• Rational organization of material: from science to mathematics, from one dimension to multidimensions, from full-line to half-line to finite interval, etc.
• Companion solutions manual allows students to see detailed worked out solutions.
• Introduction to nonlinear PDEs (Chapter 14) provides the student with a taste of the most important problems studied today by researchers in mathematics and science.
• Provides appropriate introduction to numerical analysis (Chapter 8).
• Organization gives instructors flexibility in chapter coverage; for example, one can go from Chapter 6 to Chapters 7 and 12 (Green’s functions and distributions), Chapters 13 and 14 (science and
nonlinear PDEs), Chapter 8 (numerical), Chapter 9 (waves), or to Chapter 10 (disks, spheres).
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Students Resources
Wiley Student Companion Site
Coming Soon!
View Sample content below:
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Partial Differential Equations: An Introduction, 2nd Edition
ISBN : 978-0-470-47318-4
464 pages
October 2008, ©2009
Partial Differential Equations: An Introduction, 2nd Edition
ISBN : 978-0-470-05456-7
464 pages
December 2007, ©2008
Partial Differential Equations: An Introduction, Textbook and Student Solutions Manual, 2nd Edition
ISBN : 978-0-470-38553-1
March 2008, ©2008
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How do I add the same number/value for an entire column in Excel?
Enter the number 10 into any blank cell
Copy that cell
Select your range of data
Paste Special
Check the box for "Add"
Click OK
Delete the 10 that you entered originally
10 has now been added to all of your data.
"Gregorio" wrote:
> What I am doing here is this for an example:
> Let's Say:
> A1 is 4
> A2 is 9
> A3 is 14
> A4 is 56
> A5 is 99
> And what I want to do is add the number 10 to each one of these numbers in
> this column. How would I go about doing this? Is there an equation? I hope
> someone can help... | {"url":"http://www.excelbanter.com/showthread.php?t=90651","timestamp":"2014-04-18T21:26:52Z","content_type":null,"content_length":"42383","record_id":"<urn:uuid:f708aa8d-2134-4619-b5ad-12094b3bb23f>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00359-ip-10-147-4-33.ec2.internal.warc.gz"} |
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19 Cards in this Set
An object goes from one point in space to another. After it
arrives at its destination, its displacement is: 3. either less than or equal to
1. either greater than or equal to Displacement can be negative, and when it is, it is less than
2. always equal to distance, which is the absolute value of displacement!
3. either less than or equal to
4. not related to
the distance it traveled.
A skydiver is falling straight down, along the negative v<0, a<0
y direction. During the initial part of the fall, her speed
increases from 16 to 28 m/s in 1.5 s. What sign are
the velocity and acceleration?
During a later part of the fall, after the parachute has
opened, her speed decreases from 48 to 26 m/s in 11 s. 3) v<0, a>0
Which of the following is correct? If speed is increasing, v and a are in same direction.
1) v>0, a>0 If speed is decreasing, v and a are in opposite direction.
2) v>0, a<0
3) v<0, a>0
4) v<0, a<0
A ball is thrown straight up in the air and returns to its initial position. 3 - Average velocity is zero but average acceleration is not zero.
For the time the ball is in the air, which of the following statements is Free fall: acceleration is constant (-g)
true? Initial position = final position: Δx=0
1 - Both average acceleration and average velocity are zero. ⇒ ave vel = Δx/ Δt = 0
2 - Average acceleration is zero but average velocity is not zero.
3 - Average velocity is zero but average acceleration is not zero.
4 - Neither average acceleration nor average velocity are zero.
An object is dropped from rest. If it falls a distance D in time t
then how far will if fall in a time 2t ? 4. 2D
1. D/4 Correct x=1/2 at2
2. D/2
3. D
4. 2D
5. 4D
An object is dropped from rest. If the object has speed v at
time t then what is the speed at time 2t ? 4. 2v
1. v/4
2. v/2 Correct v=at
3. v
4. 2v
5. 4v
I am going to roll the ball down the inclined plane. If the ball
reaches mark at distance 1 ft at time t1, when will the ball reach 3. t9 = 3t1
the mark at distance 9 ft? x=1/2VoT +1/2at2
1. t9 = 9t1
2. t9 = √18 t1
3. t9 = 3t1
A ball is thrown downward (not dropped) from the top of a
tower. After being released, its downward acceleration will 2. exactly g
be: after the ball is released, it is only being accelerated by gravity, the throw just sets the initial velocity
1. greater than g
2. exactly g
3. smaller than g
3. velocity is zero and acceleration is not zero
A ball is thrown vertically upward. At the very top of its At the top of the path, the velocity of the ball is
trajectory, which of the following statements is true: zero,but the acceleration is not zero. The velocity at
1. velocity is zero and acceleration is zero the top is changing, and the acceleration is the rate at
2. velocity is not zero and acceleration is zero which velocity changes.
3. velocity is zero and acceleration is not zero
4. velocity is not zero and acceleration is not zero
Dennis and Carmen are standing on the edge of a
cliff. Dennis throws a basketball vertically upward,
and at the same time Carmen throws a basketball same, when denis's ball returns to it's initial position it's velocity equals the initial velocity, the same as carmen's
vertically downward with the same initial speed.
You are standing below the cliff observing this
strange behavior. Whose ball is moving fastest
when it hits the ground?
2. Remain vertically under the plane.
There is no acceleration along
Without air resistance, an object dropped from a plane horizontal - object continues to
flying at constant speed in a straight line will travel at constant speed (same as
1. Quickly lag behind the plane. that of the plane) along horizontal.
2. Remain vertically under the plane. Due to gravitational acceleration
3. Move ahead of the plane the object’s speed downwards
A seagull flies through the air with a velocity of 9 m/s if there 2. -4 m/s
were no wind. However, it is making the same effort and flying Seagull’s velocity relative to the wind = 9 m/s
in a headwind. If it takes the bird 20 minutes to travel 6 km as • i. e., in the frame relative to the wind, wind velocity is zero
measured on the earth, what is the velocity of the wind? • Seagull travels at 6000/1200 = 5 m/s relative to earth. Therefore,
1. 4 m/s the wind velocity relative to earth is 5-9=-4 m/s.
2. -4 m/s
3. 13 m/s
4. -13 m/s
A seagull is flying at 9m/s and covers 6.00km to an island and back. 2. The round trip time is always larger with the wind
How are the rounds-trip times with and without a 4.00m/s wind related Time taken for the round trip without wind is: 12000 m / 9 m/s
if the seagull always goes at 9.00 m/s relative to the wind? = 1333 s = 22.2 minutes
1. The round-trip time is the same with/without the wind • Time taken for the round trip with wind is: 27.7 minutes
2. The round trip time is always larger with the wind
3. It is not possible to calculate this
Beth will reach the shore first because the vertical component of
Three swimmers can swim equally fast relative to the water. They have a race her velocity is greater than that of the other swimmers.
to see who can swim across a river in the least time. Relative to the water, The key here is how fast the vector in the vertical direction is. "B"
Beth (B) swims perpendicular to the flow, Ann (A) swims upstream, and Carly focuses all of its speed on the vertical vector, while the others
(C) swims downstream. Which swimmer wins the race? divert some of their speed to the horizontal vectors.
A) Ann Time to get across =
B) Beth width of river/vertical component of velocity
C) Carly
3. speeds up
An object is held in place by friction on an inclined surface. The When the object is at rest net force on it is zero.
angle of inclination is increased until the object starts moving. If When the object starts to move there is change in velocity - i.e.,
the surface is kept at this angle, the object there is acceleration or a net force due to gravity
1. slows down The force remains constant when inclination is kept at that angle
2. moves at uniform speed leading to constant acceleration - continuous speed up.
3. speeds up
4. none of the above
You are a passenger in a car and are not wearing your seat belt. Starting at the time of collision, the door exerts a leftward
Without increasing or decreasing its speed, the car makes a sharp force on you
left turn, and you find yourself colliding with the right-hand side Law of inertia: your body tends to move in a straight line forward.
door. Which of the following is correct analysis of the situation? It collides with the door which being part of the car is beginning to
Before and after the collision there is a rightward force curve leftward. When the contact happens you feel the door’s
pushing you into the door force on you.
Starting at the time of collision, the door exerts a leftward
force on you
You are pushing a wooden crate across the floor at
constant speed. You decide to turn the crate on end,
reducing by half the surface area in contact with the equally as great
floor. In the new orientation, to push the same crate as the force required before you changed the crate
across the same floor with the same speed, the force Fricotrioiennalt faotricoen d.
that you apply must be about oes not depend on the area of contact. It depends only on
a) four times as great the normal force and the coefficient of friction for the contact.
b) twice as great
c) equally as great
d) half as great
e) one-fourth as great
Consider a person standing in an elevator that is accelerating
upward. The upward normal force N exerted by the elevator floor a) larger than
on the person is Person is accelerating upwards - net upwards force is non zero
a) larger than
b) identical to
c) less than
the downward weight W of the person.
You are driving a car up a hill with constant velocity. On a piece of
paper, draw a Free Body Diagram (FBD) for the car.
How many forces are acting on the car? Name them 3
1 normal force, weight, road on car | {"url":"http://www.cram.com/flashcards/physics-midterm-1-conceptual-questions-887350","timestamp":"2014-04-17T22:05:23Z","content_type":null,"content_length":"99145","record_id":"<urn:uuid:ab2523af-ce7f-4432-badf-11b32f25b991>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00358-ip-10-147-4-33.ec2.internal.warc.gz"} |
Baylor University || Mathematics Department || Markus Hunziker
Markus Hunziker
Associate Professor of Mathematics
Ph. D., University of California, San Diego, 1997 (Advisor: N. Wallach)
Dipl. Phil. II, University of Basel, 1993 (Advisor: H. Kraft)
Dr. Hunziker joined the Baylor faculty in 2004. Prior to coming to Baylor he was teaching at Brandeis University (1997-2000) and at the University of Georgia (2000-2004). Dr. Hunziker is originally
from Basel, Switzerland. He came to the USA for graduate school in 1993. He is married to Kyunglim Nam and has a two children: Tobias, who was born in March 2007, and Hana, who was born in December
2008. He enjoys traveling, cooking, and spending time with his family and friends.
Academic Interests and Research:
Dr. Hunziker's research area is the representation theory of Lie groups and related algebraic geometry and combinatorics.
Selected Publications:
(with M. Sepanski and R. Stanke) The minimal representation of the conformal group and classical solutions to the wave equation, J. Lie Theory 22 (2012), 301-360.
(with M. Sepanski) Distinguished orbits and the L-S category of simply connected compact Lie groups, Topology Appl. 156 (2009), 2443-2451.
(with W. Graham) Multiplication of polynomials on Hermitian symmetric spaces and Littlewood-Richardson coefficients, Canad. J. Math. 61 (2009), 351-372.
(with B. Boe) Kostant modules in blocks of category O_S, Comm. in Alg. 37 (2009), 323-356.
(with T. Enright) Hilbert series and resolutions of determinantal varieties and unitary highest weight representations, J. Algebra 273 (2004), 608-639.
Ph.D. Students:
Jordan Alexander
Gail Hartsock
Drew Pruett (completed degree in 2010)
Teaching Interests:
Dr. Hunziker's teaching interests range from introductory calculus classes for undergraduates to specialized courses for Ph.D. students. He has also taught teacher preparation courses. In fact, one
of his favorite courses he taught was a geometry course for prospective elementary teachers at the University of Georgia.
Courses taught at Baylor:
MTH 1321 - Calculus I
MTH 1322 - Calculus II
MTH 2321 - Calculus III
MTH 3312 - Foundations of Combinatorics and Algebra
MTH 3325 - Ordinary Differential Equations
MTH 3350 - Structure of Modern Geometry
MTH 4V90 - Topics in Mathematics
MTH 5310 - Advanced Abstract Algebra I
MTH 5311 - Advanced Abstract Algebra II
MTH 5340 - Differential Geometry
MTH 6V13 - Advanced Topics in Algebra
MTH 6V23 - Advanced Topics in Analysis
MTH 6V43 - Advanced Topics in Representation Theory
MTH 6V99 - Dissertation | {"url":"http://www.baylor.edu/math/index.php?id=54010","timestamp":"2014-04-20T01:48:36Z","content_type":null,"content_length":"19545","record_id":"<urn:uuid:b4bea584-12f6-477b-a2d7-f93bbc05b827>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00342-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: extrapolate to find the intercept
Replies: 4 Last Post: Oct 18, 2013 10:29 AM
Messages: [ Previous | Next ]
Torsten Re: extrapolate to find the intercept
Posted: Oct 18, 2013 7:16 AM
Posts: 1,439
Registered: "Linda" wrote in message <l3r34b$7j9$1@newscl01ah.mathworks.com>...
11/8/10 > hi all,
> i'm sure this is a simple one but i'm not sure how to do this the most efficient way in matlab: i have a vector x=[2.5;3.456];. how do i extrapolate this vector so that it hits the
intercept? or find the intercept in a different way? is there a simple function for that?
> any help is appreciated.
What do you mean by "hitting the intercept" ?
Best wishes
Date Subject Author
10/18/13 extrapolate to find the intercept Linda
10/18/13 Re: extrapolate to find the intercept Torsten
10/18/13 Re: extrapolate to find the intercept Steven Lord
10/18/13 Re: extrapolate to find the intercept Jos
10/18/13 Re: extrapolate to find the intercept Curious | {"url":"http://mathforum.org/kb/thread.jspa?threadID=2602509&messageID=9309667","timestamp":"2014-04-20T13:29:48Z","content_type":null,"content_length":"21052","record_id":"<urn:uuid:69850891-3bea-425f-807f-813174d00d50>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00507-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Topic-models] LDA-based document retrieval
Stephen Thomas sthomas at cs.queensu.ca
Fri Aug 26 21:03:07 EDT 2011
Topic Modelers,
I want to perform document retrieval using LDA. I need to compute the
similarity of a query and each document in the corpus. According to
Wei and Croft 2006 (Equation 5), as well as Steyvers and Griffiths
2007 (Equation 21.9), I should compute this conditional probability
P(Q | D) = \prod_{q \in Q} P(q | D)
Isn't this a bit harsh? Since this is a product, we are saying that if
any one of the query's words gets a score of 0 (i.e., P(q | D) = 0 for
some q), then the entire similarity score is 0. If the query had 10
words, 9 of which were highly relevant and 1 of which was irrelevant,
we would assign the query a score of 0? Am I understanding this
Why isn't this a summation instead of a product? (We can normalize by
query length.)
Thanks in advance,
title={LDA-based document models for ad-hoc retrieval},
author={Wei, X. and Croft, W.B.},
booktitle={Proceedings of the 29th annual international ACM SIGIR
conference on Research and development in information retrieval},
title={Probabilistic topic models},
author={Steyvers, M. and Griffiths, T.},
journal={Handbook of latent semantic analysis},
publisher={Lawrence Erlbaum Associates}
More information about the Topic-models mailing list | {"url":"https://lists.cs.princeton.edu/pipermail/topic-models/2011-August/001542.html","timestamp":"2014-04-18T02:58:02Z","content_type":null,"content_length":"4190","record_id":"<urn:uuid:0276907e-2877-4b78-898d-d6bfaf4bd58e>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00561-ip-10-147-4-33.ec2.internal.warc.gz"} |
Maxima and Minima of a Function of Several Variables.
June 17th 2010, 06:24 PM
Maxima and Minima of a Function of Several Variables.
Hello all.
Please have a look at the function below. Here A,B and C are constants while x, y and z are independent variables. Please note that this is a general representation and the variables do not refer
to space and time. I want to find the critical points for determination of maxima and minima of this function. Can anyone help me on this one? Thanks.
June 18th 2010, 05:28 AM
Are you trying to find the extrema over some region of R^3?
June 18th 2010, 07:02 AM
Take the partial derivatives and set them equal to 0. Solve those equations for x, y, and z. Can you find the partial derivatives?
June 18th 2010, 11:24 AM
Hello all.
Please have a look at the function below. Here A,B and C are constants while x, y and z are independent variables. Please note that this is a general representation and the variables do not refer
to space and time. I want to find the critical points for determination of maxima and minima of this function. Can anyone help me on this one? Thanks.
Okay, to find the critical points of a function of multiple variables you have to find the zeros of the partial derivitves like the poster above me said. To take a partial derivitive with respect
to a specific variable you treat all the other variables as if they were constants. So if we had:
The partial derivite with respect to x is written as http://latex.codecogs.com/gif.latex?...rtial&space;x} and is taken by treating y as a constant. Since y^2 is simply a "constant" squared, it
is also a constant, and will be treated as such. This is what the partial derivitive comes out to:
Simmilarly with respect to y we have:
For my exapmle, to find the critical points I simply find the zeros of these functions. This apllies to your 3 varible equation also. Hope this helps.
June 18th 2010, 04:47 PM
Thank you mfetch22 for the example.
For Ackbeet's question, my variable x lie in the range of 0.4 and 1 while f lies in the range of 0.5 and 1.
In my case, if I take derivatives w.r.t. x, y and z. Then I set them equal to zero. So can I get any value of any of the variable by comparison of any two derivative. Say if I compare derivatives
w.r.t. x and z with each other, and I get the value of z in terms of x and y. So does that value apply to the original function as well in the ranges specified.
June 18th 2010, 05:32 PM
Usually, for a problem like this, there are intervals for all three variables x, y, and z. It's a bit unusual to have an interval for f, although I suppose you could. That might impose a
constraint on the independent variables.
As for comparing derivatives, I wouldn't do that. It merely confuses the issue. Set the derivatives equal to zero and find the set of points (x,y,z) that satisfy those conditions. In addition,
you'll need to evaluate the function f on the boundaries of the regions in which you're looking. | {"url":"http://mathhelpforum.com/calculus/148748-maxima-minima-function-several-variables-print.html","timestamp":"2014-04-19T20:32:03Z","content_type":null,"content_length":"8582","record_id":"<urn:uuid:a15c3462-5ceb-4ac7-877a-9543baa0a1ee>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00579-ip-10-147-4-33.ec2.internal.warc.gz"} |
the first resource for mathematics
Uniform asymptotic smoothing of Stokes’ discontinuities.
(English) Zbl 0683.33004
Stokes’ discontinuities occur in the asymptotic approximation of functions defined by integrals or differential equations and dependent on a large complex parameter
. Usually several asymptotic estimates are available in different sectors of the complex
-plane. At the border lines of the sectors, usually called Stokes lines, the change in the approximations is not always smooth; the jumps may be exponentially large. If the expansion is truncated
near its least term, the change may be smooth, and can be described in terms of an error function. This new interpretation introduces an interesting development in the theory of asymptotic
expansions. The author uses the formalism of Dingle to describe the phenomenon, and he gives numerical illustrations for Dawson’s integral and Airy functions.
33B20 Incomplete beta and gamma functions
30E15 Asymptotic representations in the complex domain
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) | {"url":"http://zbmath.org/?q=an:0683.33004","timestamp":"2014-04-19T02:06:15Z","content_type":null,"content_length":"21461","record_id":"<urn:uuid:ddbd0df3-ab36-452b-be61-0a525be385d8>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00265-ip-10-147-4-33.ec2.internal.warc.gz"} |
The derp package
A parser based on derivatives of parser combinators (Might and Darais). Our paper on Arxiv details the theory of parsing with derivatives: http://arxiv.org/abs/1010.5023. This implementation uses my
latest work on the theory that brings the O(n*|G|^2) complexity bound to O(n) for parsing most not-painfully-ambiguous grammars. (|G| would be the size of the initial grammar, n would be size of the
input. These bounds are based off of observation and intuition; they are not proven yet.) This implementation will not terminate if the resulting parse forest is infinite. We know how to extend the
implementation to work for infinite parse forests with little effort. If this is something you would like to see, send me an email.
Maintainers' corner
For package maintainers and hackage trustees | {"url":"http://hackage.haskell.org/package/derp-0.1.4","timestamp":"2014-04-16T17:36:49Z","content_type":null,"content_length":"4146","record_id":"<urn:uuid:93ce3efc-2ddd-4383-a209-73611544b6ea>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00594-ip-10-147-4-33.ec2.internal.warc.gz"} |
Conversion help
September 7th 2008, 11:25 AM
Conversion help
here is the problem i m stuck on.. can someone help me get the answer..
i m really more interested then the work then the answer.
Convert 8.2mg/mL to (micro)g/L
September 7th 2008, 11:53 AM
Matt Westwood
Ooh delightful, a question in metric! That doesn't often happen round here. Full marks to the teacher.
There's 8.2 mg in every mL. There are 1000 mL in each L. So there's 1000 times as many mg in every L as there are in every mL. So multiply the 8.2 by 1000 to get the number of mg in every L.
Then there are 1000 microg in every mg. So there's 8.2 x 1000 microg in every mL. So theres 8.2 x1000 x1000 microg in every L.
Does that help?
September 7th 2008, 12:13 PM
Ooh delightful, a question in metric! That doesn't often happen round here. Full marks to the teacher.
There's 8.2 mg in every mL. There are 1000 mL in each L. So there's 1000 times as many mg in every L as there are in every mL. So multiply the 8.2 by 1000 to get the number of mg in every L.
Then there are 1000 microg in every mg. So there's 8.2 x 1000 microg in every mL. So theres 8.2 x1000 x1000 microg in every L.
Does that help?
not really.. its hard for me to understand it without math.. this is how i do it and i m getting the way wrong answer:
(8.2mg/1ml) x (10^-3 g / 1mg) = .0082g
.0082g/1 ) x ( 1 microg / 10^-6) = 8200
i m way wrong, but u see the way i do it.. can u show me that way how u got ur answeR??
September 7th 2008, 12:15 PM
not really.. its hard for me to understand it without math.. this is how i do it and i m getting the way wrong answer:
(8.2mg/1ml) x (10^-3 g / 1mg) = .0082g
.0082g/1 ) x ( 1 microg / 10^-6) = 8200
i m way wrong, but u see the way i do it.. can u show me that way how u got ur answeR??
see if either of the methods employed here can help you
one of them is doing exactly as Matt pointed out, but it is set out to look familiar to the method you are using
September 7th 2008, 12:19 PM
see if either of the methods employed here can help you
one of them is doing exactly as Matt pointed out, but it is set out to look familiar to the method you are using
yah i understand those easier problems with 1 conversion.. but these problems have a dual conversion it seems to me and i m absolutly lost.
September 7th 2008, 12:28 PM
September 7th 2008, 12:52 PM
Convert 8.2mg/mL to (micro)g/L
ok so convert 82.mg/ml into g/l to mcg/l i duno. can u show me the steps
September 7th 2008, 12:57 PM
September 7th 2008, 01:07 PM
1 mg = .001 g
1 g = 1000mcg
1 ml = .001 L
September 7th 2008, 01:11 PM
Matt Westwood
not really.. its hard for me to understand it without math.. this is how i do it and i m getting the way wrong answer:
(8.2mg/1ml) x (10^-3 g / 1mg) = .0082g
.0082g/1 ) x ( 1 microg / 10^-6) = 8200
i m way wrong, but u see the way i do it.. can u show me that way how u got ur answeR??
You've successfully calculated the concentration in microg per mL, but the question calls for microg per L.
September 7th 2008, 01:15 PM
no, 1 g = 1000000 micrograms
so that means, wherever you see mg, you can change it to 0.001g. then simplify. then change the g to 1000000 mcg then simplify, then change the ml to 0.001L and simplify, and you are done.
can you show your steps now?
September 7th 2008, 01:21 PM
Convert 8.2mg/mL to (micro)g/L
write it like a grid to solve the conversion problems here.
8.2 mg |
1 mL |
Then since there is 1000 (mL) in 1 (L) and you need to cancel the (mL), put it on top
8.2 mg | 1000 mL |
1 mL | 1 L |
Same for mg to (micro)g. There are 1000 (micro)g in 1 mg. mg goes on bottom to cancel the one already on top.
8.2 mg | 1000 mL | 1000 (micro)g |
1 mL | 1 L | 1 mg |
The horizontal line is devide and vertical lines become multiply. The mg and mL cancel and the proper units remain in the proper place.
September 7th 2008, 01:44 PM
yes thank you every1! i got it ! ty alot.
here is what i did:
8.2mg/1mL (1mL / 10^-3 L)(10^-3 g/1 mcg)= 8.2 g/L
8.2g/L (10^6 mcg/1g) = 8.2 x 10^6
is what i did logical? i got the right answer though!
September 7th 2008, 02:50 PM | {"url":"http://mathhelpforum.com/math-topics/48057-conversion-help-print.html","timestamp":"2014-04-18T13:35:28Z","content_type":null,"content_length":"15879","record_id":"<urn:uuid:2841e8d6-979d-41be-9e21-3a865f097626>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00099-ip-10-147-4-33.ec2.internal.warc.gz"} |
Bridge Oscillator
Wien Bridge Oscillator
OPWIEN.CIR (Closed-Loop Circuit) Download the SPICE file
The opamp Wien-bridge oscillator provides a nice view into classic oscillator design using feedback analysis. Feedback analysis reveals if your circuit is stable (well behaved) or unstable (may
oscillate). When designing amplifiers (especially high-speed ones), the trick is to avoid the conditions that make the circuit oscillate. When designing oscillators, you strive to achieve those
conditions in a predictable way.
Feedback analysis simply means opening the circuit and injecting an AC signal VTEST at one end of the circuit. Then, by looking at the magnitude (gain) and phase (time-shift) of signal as it travels
around the opened loop, you can tell whether you’ve got an amplifier or an oscillator on your hands.
When wearing your oscillator design hat, the idea is to pick components that will make the open-loop analysis meet the conditions for oscillation at your chosen frequency: The conditions are as
If the AC gain around the opened loop is 1 V/V and the total phase shift is -360 or 0 deg, the circuit will oscillate at that frequency.
Open the loop of the Wien Bridge Oscillator at the op amp’s output. This is a good point because of the relatively low impedance of the output terminal. Likewise, VTEST also has a low output
impedance. Therefore, opening the loop here does not significantly alter the circuit’s behavior.
OPWIEN_OL.CIR (Open-Loop Circuit) Download the SPICE file
Two basic sections form the Wien-bridge oscillator: an RC tuning network and an amplifier. Both are necessary to achieve the conditions for oscillations. The RC network is characterized by a center
where R=R1=R2 and C=C1=C2. At the center frequency, two interesting things occur at V(3). First, the phase shift goes through 0 degrees. And second, the magnitude reaches a peak of 1/3 V/V.
CIRCUIT ANALYSIS Run a SPICE simulation of the open-loop circuit OPWEIN_OL.CIR. Add trace VM(3) to see what the Magnitude looks like at the center frequency. For R1=R2=10k and C1=C2=16nF, the center
frequency should be near 1 kHz. What does the phase look like? Add another plot window and then add trace VP(3) to see the Phase shift. The RC network does a nice job of meeting one of the conditions
for oscillation: the phase is 0 degrees at the design frequency.
The RC network falls short of the oscillation conditions in that the gain is only 1/3 V/V. How is the gain of 1V/V around the loop to be achieved? As you might have guessed, the non-inverting
amplifier provides the needed gain. How much? A gain of 3 V/V makes the total gain 1/3 x 3 = 1 V/V. Setting the correct op amp gain is critical. Not enough - oscillations will cease. Too much –
oscillation amplitude will grow until the output saturates.
What’s needed is a mechanism to guarantee oscillations will start (GAIN > 3), yet, limit the gain (GAIN=3) at steady state. Enter our heros - D1, D2 and R12. The circuit adjusts its gain depending on
the signal level. For small signals, the diodes do not conduct and the gain is set by
For larger signals, the voltage across R12 is big enough to make D1 and D2 conduct. The shunt resistance of the conducting diodes effectively reduces the R12 resistance, consequently, reducing the
overall gain to GAIN=3.
CIRCUIT ANALYSIS Run a simulation of OPWIEN_OL.CIR. View the AC output of the op amp VM(4). For R10=10k, R11=18k and R12=5k, the op amp gain is (1 + (18+5)/10) = 3.3 V/V. This should make the overall
open-loop gain equal to 1/3 x 3.3 = 1.1 V/V. Does the peak at VM(4) reach this expected gain?
It’s time to close the loop and try out the Wien-Bridge Oscillator. Run a simulation of closed-loop circuit OPWIEN.CIR and plot the Transient Analysis at V(4). How much time does it take for the
amplitude to stabilize?
HANDS-ON DESIGN Design the circuit with a different oscillation frequency. Calculate the values for R and C. (Example: For fo = 10kHz, choose R1=R2=10k and calculate C1=C2=1/(2π x R1 x fo) = 1.6nF.)
Test drive your oscillator. If there’s too little or too many sinewaves on the plot, adjust the total time of the Transient Analysis to another value like 5 ms by modifying the .TRAN statement to
look like
.TRAN 0.05MS 5MS
If there’s no input signal to an oscillator, what starts the oscillations? Current source IS injects a pulse into the RC network to jump start the oscillations. In a real circuit, the large transient
at power up will kick the circuit into action.
CIRCUIT INSIGHT What happens if there’s not enough gain around the loop? Reduce R12 to 1k making the total loop gain less than 1. Run a simulation. The circuit rings briefly, but there’s not enough
gain to sustain oscillations.
Download the file or copy this netlist into a text file with the *.cir extention.
OPWIEN_OL.CIR - OPAMP WIEN-BRIDGE OSC, OPEN-LOOP ANALYSIS
VTEST 40 0 AC 1
* RC TUNING
R2 40 6 10K
C2 6 3 16NF
R1 3 0 10K
C1 3 0 16NF
R10 0 2 10K
R11 2 5 18K
XOP 3 2 4 OPAMP1
R12 5 4 5K
*D1 5 4 D1N914
*D2 4 5 D1N914
.model D1N914 D(Is=0.1p Rs=16 CJO=2p Tt=12n Bv=100 Ibv=0.1p)
* OPAMP MACRO MODEL, SINGLE-POLE
* connections: non-inverting input
* | inverting input
* | | output
* | | |
.SUBCKT OPAMP1 1 2 6
RIN 1 2 10MEG
* DC GAIN (100K) AND POLE 1 (100HZ)
EGAIN 3 0 1 2 100K
RP1 3 4 1K
CP1 4 0 1.5915UF
EBUFFER 5 0 4 0 1
ROUT 5 6 10
.AC DEC 10 10 10MEG
* VIEW RESULTS
.PRINT AC VM(3) VP(3)
.PLOT AC VM(3) VP(3)
Download the file or copy this netlist into a text file with the *.cir extention.
IS 0 3 PWL(0US 0MA 10US 0.1MA 40US 0.1MA 50US 0MA 10MS 0MA)
* RC TUNING
R2 4 6 10K
C2 6 3 16NF
R1 3 0 10K
C1 3 0 16NF
R10 0 2 10K
R11 2 5 18K
XOP 3 2 4 OPAMP1
R12 5 4 5K
D1 5 4 D1N914
D2 4 5 D1N914
.model D1N914 D(Is=0.1p Rs=16 CJO=2p Tt=12n Bv=100 Ibv=0.1p)
* connections: non-inverting input
* | inverting input
* | | output
* | | |
.SUBCKT OPAMP1 1 2 6
RIN 1 2 10MEG
* DC GAIN (100K) AND POLE 1 (100HZ)
EGAIN 3 0 1 2 100K
RP1 3 4 1K
CP1 4 0 1.5915UF
EBUFFER 5 0 4 0 1
ROUT 5 6 10
.TRAN 0.05MS 10MS
* VIEW RESULTS
.PRINT TRAN V(4)
.PLOT TRAN V(4)
© 2002 eCircuit Center | {"url":"http://www.ecircuitcenter.com/Circuits/opwien/opwien.htm","timestamp":"2014-04-17T03:49:01Z","content_type":null,"content_length":"13549","record_id":"<urn:uuid:72251a8f-3028-4fff-aa67-448405cb4d92>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00136-ip-10-147-4-33.ec2.internal.warc.gz"} |
Subactivity Theory
The PSL Ontology uses the subactivity relation to capture the basic intuitions for the composition of activities.
The core theory subactivity.th alone does not specify any relationship between the occurrence of an activity and occurrences of its subactivities. For example, the specification of subactivities
alone does not allow us to distinguish between a nondeterministic activity and a deterministic activity.
The basic ontological commitments of the Subactivity Theory are based on the following intuitions:
Intuition 1:
The composition relation is a discrete partial ordering, in which primitive activities are the minimal elements.
Informal Semantics for the Subactivity Theory
(subactivity ?a1 ?a2) is TRUE in an interpretation of Subactivity Theory if and only if activity ?a1 is a subactivity of activity ?a2.
(primitive ?a) is TRUE in an interpretation of Subactivity Theory if and only if the activity ?a has no proper subactivities.
Last Updated: Wednesday, 15-December-2003 11:42:40
Return to the PSL homepage | {"url":"http://www.mel.nist.gov/psl/psl-ontology/part12/subactivity_page.html","timestamp":"2014-04-19T01:10:55Z","content_type":null,"content_length":"1604","record_id":"<urn:uuid:a653260c-8731-431c-a2cc-cd3f404c8be5>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00116-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: relations aren't types?
From: John Jacob <jingleheimerschmitt_at_hotmail.com> Date: 31 Dec 2003 11:23:07 -0800 Message-ID: <72f08f6c.0312311123.321a9da6@posting.google.com>
> > Why is this a requirement for a useful definition of scalar?
> Scalar is not a useful concept so I fail to understand your question.
So you advocate an untyped language?
> With all due respect, if everything one can represent with a finite number
> of bits is a scalar, everything one can represent with a computer system is
> a scalar, which is rather too encompassing to have any utility.
First, the fact that any scalar can be represented with a finite number of bits (true of every value, scalar or otherwise) does not imply that any finite number of bits is a scalar. Second, and
perhaps more importantly, the physical representation of the value in the computer system has no bearing on the logical model. This is physical data independence which you so voraciously defended
previously. If we start using the physical representations of values to define the logical boundaries for type categories, we have taken a *huge* step backwards.
> > You can invoke "A join B" without knowing the attributes, but what
> > does the result mean?
> You can invoke A * B without knowing the exact types or dimensions, but what
> does the result mean? You can invoke A AND B without knowing the bits, but
> what does the result mean? You can invoke Distance(P1,P2) without knowing
> any coordinate systems, but what does the result mean? You can invoke P2 -
> P1 without knowing any coordinate systems, but what does the result mean?
> What relevance does your question have?
The primary reason for types is to imbue our code with meaning that the compiler can understand and verify. The primary reason for scalar types is to allow the user to invoke the very operators you
are describing without having to know anything about the possible representations of those values, and still be able to understand the meaning of the invocation. The primary reason for non-scalar
types is to allow the description of values which do have some visible structure in the logical model, like relations, lists, tuples, etc.,.
At this point I am forced to ask what is your overall point? Do you disagree with the characterization of some values as scalar, and others as non-scalar? Do you disagree that we need some logical
construct to describe relation values? What would a relational language that did not distinguish between scalar and relation values look like? How would it be more useful than one that did? Received
on Wed Dec 31 2003 - 13:23:07 CST | {"url":"http://www.orafaq.com/usenet/comp.databases.theory/2003/12/31/0309.htm","timestamp":"2014-04-18T06:12:34Z","content_type":null,"content_length":"9418","record_id":"<urn:uuid:0d637008-abbb-4f6d-a98c-2891b6900358>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00134-ip-10-147-4-33.ec2.internal.warc.gz"} |
chance of rain , probability
July 30th 2009, 09:56 AM #1
Junior Member
Jul 2009
chance of rain , probability
can anyone help with this?
chance of rain
the weather report claims that the chance of rain for certain days
next week are
Monday: 50% chance of rain
Tuesday: 20% chance of rain
Wednesday: 30% chance of rain
Thursday: 30% chance of rain
Friday: 10% chance of rain
what is the chance of rain
for the entire week as a whole ?
The chance of not raining is
$0.5 \cdot 0.8 \cdot 0.7 \cdot 0.7 \cdot 0.9 =0.1764$
The complement to the event "it is not raining" is "it is raining".
A="It is not raining".
So $P(A^{C})=1-0.1764= 0.8236$
can anyone help with this?
chance of rain
the weather report claims that the chance of rain for certain days
next week are
Monday: 50% chance of rain
Tuesday: 20% chance of rain
Wednesday: 30% chance of rain
Thursday: 30% chance of rain
Friday: 10% chance of rain
what is the chance of rain
for the entire week as a whole ?
Do you mean raining every day during the week?
Or at least once during the whole week?
If it's the former it's just the product of the probabilities for each given day.
If it's the latter, think of the probability of something happening "at least once", as 1 - Probability of it not happening at all...
July 30th 2009, 10:28 AM #2
July 30th 2009, 12:55 PM #3
Mar 2008 | {"url":"http://mathhelpforum.com/statistics/96536-chance-rain-probability.html","timestamp":"2014-04-18T17:23:26Z","content_type":null,"content_length":"35875","record_id":"<urn:uuid:040da4d6-8e6c-4e5d-82a9-fbfc1b72048f>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00517-ip-10-147-4-33.ec2.internal.warc.gz"} |
[racket] Multiplying by 0
From: Vincent St-Amour (stamourv at ccs.neu.edu)
Date: Sun Feb 13 13:13:48 EST 2011
At Sun, 13 Feb 2011 16:56:13 +0000,
José Lopes wrote:
> I understand. However, not only that disregards type promotion but also
> is incoherent since (+ 0 0.0) evaluates to 0.0.
It is somewhat inconsistent, and it makes it possible to shoot
yourself in the foot (if you expect the result of the multiplication
to be inexact), but there are reasons why this behavior is useful:
- As Carl said, that makes mathematical sense; anything multiplied by
0 is 0.
- A complex number with an imaginary part of exact 0 can be demoted to
a real number (which use less space and are processed more
efficiently). Therefore, if an imaginary part of exact 0 arises as
the result of a multiplication, you are back to real numbers. This
comes in handy for some computations.
That said, I'm not sure this is worth the inconsistency, but there are
reasons for this decision.
In the meantime, if you have code that depends on the result of a
multiplication of exact and inexact reals being inexact, the safe
thing to do is to wrap the exact number in a call to exact->inexact.
Posted on the users mailing list. | {"url":"http://lists.racket-lang.org/users/archive/2011-February/044288.html","timestamp":"2014-04-20T01:13:27Z","content_type":null,"content_length":"6142","record_id":"<urn:uuid:9ab33f7a-8c49-4cd8-bbd5-8f312d4c6218>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00183-ip-10-147-4-33.ec2.internal.warc.gz"} |
denumerable vs uncountable
December 13th 2008, 03:32 PM #1
Dec 2008
denumerable vs uncountable
why is the set of all integer powers of 2 {2^x|x is in z} is denumerable
Then why is the set of all prime numbers denumeralbe
and explain why the number of points on a circle is denumerable or uncountable
we can find a bijective function from the naturals to the set. so it has the same cardinality as the naturals.
or better yet, since the integers are denumerable, it suffices to find a bijection from the integers to the set, which would be a bit easier to describe. the desired result follows by
Then why is the set of all prime numbers denumeralbe
you can list and enumerate them using the natural numbers. since there are an infinite number of primes, it follows the set is denumerable.
and explain why the number of points on a circle is denumerable or uncountable
you can think of a circle as a section of the real line bent into a circle, right? is a segment of the real line denumerable or uncountable?
There exist the obvious mapping x->2^x and there is a well known mapping from N to Z: f(n)= n/2 if n is even, -(n+1)/2 if n is odd.
Then why is the set of all prime numbers denumeralbe
The set of prime numbers is infinite and a subset of N.
and explain why the number of points on a circle is denumerable or uncountable
Denumerable or UNCOUNTABLE? That's like asking you to explain why every integer must be either even or odd! It has to be one or the other. (In fact, since the function $f(x)= (R cos(2\pi x), R
sin(2\pi x))$ maps the interval [0, 1) onto the circle with radius R, it is easy to prove that it is uncountable.
Last edited by HallsofIvy; December 31st 2008 at 05:14 AM.
December 13th 2008, 03:39 PM #2
December 23rd 2008, 06:02 PM #3
MHF Contributor
Apr 2005 | {"url":"http://mathhelpforum.com/advanced-math-topics/64833-denumerable-vs-uncountable.html","timestamp":"2014-04-19T09:15:16Z","content_type":null,"content_length":"39564","record_id":"<urn:uuid:bf490514-a31f-4519-bf57-40515b04e8cb>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00377-ip-10-147-4-33.ec2.internal.warc.gz"} |
Odds Calculator
Odds Probability Calculator
Calculator Use
Convert stated odds to a decimal value of probability and a percentage value of success.
Odds Formulas:
This calculator will convert "odds of winning" an event into a percentage chance of winning. If odds are stated as an A to B chance of success then the probability of success is given as P = B / (A +
B). The percent chance of winning is calculated as P x 100. | {"url":"http://www.calculatorsoup.com/calculators/games/odds.php","timestamp":"2014-04-19T19:55:03Z","content_type":null,"content_length":"19357","record_id":"<urn:uuid:d5538f4c-93e1-4cec-8623-54a81572f158>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00405-ip-10-147-4-33.ec2.internal.warc.gz"} |
On the Locality of the Prüfer Code
The Prüfer code is a bijection between trees on the vertex set $[n]$ and strings on the set $[n]$ of length $n-2$ (Prüfer strings of order $n$). In this paper we examine the 'locality' properties of
the Prüfer code, i.e. the effect of changing an element of the Prüfer string on the structure of the corresponding tree.
Our measure for the distance between two trees $T$ and $T^*$ is $\Delta(T,T^*)=n-1-\vert E(T)\cap E(T^*)\vert$. We randomly mutate the $\mu$th element of the Prüfer string of the tree $T$, changing
it to the tree $T^*$, and we asymptotically estimate the probability that this results in a change of $\ell$ edges, i.e. $P(\Delta=\ell\, \vert \, \mu).$ We find that $P(\Delta=\ell\, \vert \, \mu)$
is on the order of $ n^{-1/3+o(1)}$ for any integer $\ell>1,$ and that $P(\Delta=1\, \vert \, \mu)=(1-\mu/n)^2+o(1).$
This result implies that the probability of a 'perfect' mutation in the Prüfer code (one for which $\Delta(T,T^*)=1$) is $1/3.$
Full Text: | {"url":"http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r10/0","timestamp":"2014-04-16T19:57:25Z","content_type":null,"content_length":"15280","record_id":"<urn:uuid:1e94020e-92ce-4646-85d2-1a14ef17aa5d>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00087-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: st: Meta-analysis of rates greater than 1 (when the event number is
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Re: st: Meta-analysis of rates greater than 1 (when the event number is greater than the sample size)
From Nick Cox <njcoxstata@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: Meta-analysis of rates greater than 1 (when the event number is greater than the sample size)
Date Wed, 20 Apr 2011 09:40:06 +0100
Your rate is manifestly not a proportion bounded by 0 and 1, so it can
be confirmed that Freeman-Tukey transformations and the method you
cite do not apply. It is not just that they "do not seem to work";
they are quite wrong. For example, any number greater than 1 has a
square root that is also greater than one, so the arcsine of that is
undefined. Similarly if p, supposedly a proportion, is greater than 1,
then 1 - p < 0 and sqrt(p (1 - p)) is a complex number without
statistical interpretation here
A more positive answer to your question might be forthcoming from
people who use meta-analysis routinely. Such people might be helped by
your indicating which Stata program or programs you imagine using.
On Wed, Apr 20, 2011 at 8:55 AM, Wang, Alberta L
<Alberta.L.Wang@uth.tmc.edu> wrote:
> Is anyone familiar with how to perform a meta-analysis of rates in Stata when the number of events is greater than the sample size (i.e., when the rate is greater than 1)?
> The data I'm working with has one group per study. To further explain, I am performing a meta-analysis of partner notification outcome rates--for example, the number of sex partners notified per index patient. The index patient is the patient diagnosed with a STI who is notifying his/her sex partners of possible exposure. The number of partners notified (number of events) can be higher than the number of index patients (sample size) because one index patient can notify more than one sex partner.
> I tried using the Freeman-Tukey arcsin transformation and back transformation method for meta-analysis of proportions as described in the archives (http://www.stata.com/statalist/archive/2004-09/msg00386.html). However, this method does not seem to work when the rate is greater than 1.
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Re: st: reliability with -icc- and -estat icc-
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Re: st: reliability with -icc- and -estat icc-
From "JVerkuilen (Gmail)" <jvverkuilen@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: reliability with -icc- and -estat icc-
Date Wed, 27 Feb 2013 15:30:22 -0500
On Wed, Feb 27, 2013 at 11:57 AM, Rebecca Pope <rebecca.a.pope@gmail.com> wrote:
> Jay, I don't know which Rabe-Hesketh & Skrondal text you are referring
> to, but if it is MLM Using Stata (2012) you'll want Ch 9.
Yes, that's it. I'm not near my copies at the moment to look at them,
but I've read it. I'll look later.
> In the interest of full disclosure, like Nick, psychometrics is not my
> specialty.
Lots of folks compute ICCs.
The following is as much for my edification as to add to
> the group discussion. Joseph has used two random effects rather than
> one (leaving aside the whole target/rater issue for now). This
> corresponds to crossed effects (advised by R-H & S, so I should have
> read the book yesterday instead of adapting UCLA's code to match
> -icc-) and will reduce the ICC. This differs by design from what is
> implemented with -icc-, which treats the target as fixed, as does the
> code I posted originally. In short Jay, while _all: R isn't wrong, my
> use of fixed effects for part of the model was. Does that sum it up
> appropriately?
I'm not sure but I think so.
If the apps are the ones of interest, then that's fixed effects. If,
on the other hand, you want to generalize to other apps then that's
random effects both ways. In psychlinguistics this is referred to as
the "language as fixed effects" fallacy because they should be using
crossed models due to the desire to generalize to both the corpus and
the population of respondents.
Anyway, I was trying to figure out why the model was specified that
way, though it turns out they were the same, which is not unusual in
mixed models.
> If dramatic disagreement is a mark of unreliable raters, what does
> that say for elections, user reviews, or for that matter faculty
> search committees? Don't take this to mean I don't understand the
> general concept that you want raters to concur. However, we're talking
> about smartphone apps, not e.g. radiology where there is a "true"
> answer (yes/no lung cancer). Hypothetically at least, you could
> legitimately have strong and divergent opinions.
Totally agree. Modeling agreement can have many motives. So for
instance, folks who study romantic couples often use ICC-type
coefficients as measures of similarity. It's not obvious that they
should be high, medium, or low.
> I would argue here that the issue of rater reliability is not an issue
> of disagreement but rather Rater 2's utter inability to distinguish
> between applications. Now, perhaps this means that the applications
> really aren't substantively different from each other and you found 3
> people who just wanted to accomodate you by completing the ranking
> task and Rater 2 happened to be honest. Who knows. I'd say it's
> unlikely, but I've seen some pretty unlikely things happen...
Yes, totally agree. The reason to model something is to determine
this. I think the general view I take is that a meaningful consensus
wasn't arrived at and so the notion of an ICC is probably not
meaningful here. Why that is true needs to be considered further, for
instance by interviewing the raters.
JVVerkuilen, PhD
"It is like a finger pointing away to the moon. Do not concentrate on
the finger or you will miss all that heavenly glory." --Bruce Lee,
Enter the Dragon (1973)
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Applications - Boundless Open Textbook
Given the algebraic equation for a quadratic function, one can calculate any point on the function, including critical values like minimum/maximum and x- and y-intercepts.
These calculations can be more tedious than is necessary, however. A graph contains all the above critical points and more, and is essentially a clear and concise representation of a function. If one
needs to determine several values on a quadratic function, glancing at a graph is quicker than calculating several points.
Consider the function:
$F(x)=-\frac {x^2}{10}+50x-750$
Suppose this models the profit (f(x)) in dollars that a company earns as a function of the number of products (x) of a given type that are sold, and is valid for values of x greater than or equal to
0 and less than or equal to 500.
If one wanted to find the number of sales required to break even, the maximum possible loss (and the number of sales required for this loss), and the maximum profit (and the number of sales required
for this profit), one could calculate algebraically or simply reference a graph.
Interactive Graph: Profit vs. Sales
Graph of quadratic equation $y=\frac{x^2}{10}+50x-750$. Graphical representation of profits versus sales can help one visualize, and thus strategize, accordingly. This can be less time-consuming than
performing several calculations. Based on this graph, what price point is most effective for maximum sales?
By inspection, we can find that the maximum loss is $750, which is lost at both 0 and 500 sales. Maximum profit is $5500, which is achieved at 250 sales. The break-even points are between 15 and 16
sales, and between 484 and 485 sales.
The above example pertained to business sales and profits, but a similar model can be used for many other relationships in finance, science and otherwise. For example, the reproduction rate of a
strand of bacteria can be modeled as a function of differing temperature or pH using a quadratic functionality. | {"url":"https://www.boundless.com/algebra/functions-equations-and-inequalities/graphs-of-quadratic-functions/applications/","timestamp":"2014-04-19T22:07:19Z","content_type":null,"content_length":"62974","record_id":"<urn:uuid:feacd809-d0fd-4bb6-9ff5-2aafbc261219>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00180-ip-10-147-4-33.ec2.internal.warc.gz"} |
Can anybody give me give me some idea how to tackle this: 1/ (2/5) - 1/ (2/3)
When you have an expression like $x=\frac a{\dfrac bc}$ it's the same if we say $x=a:\dfrac bc$ Does that make sense?
OK. These are completely new to me but here goes: 1/ (2/5) - 1/ (2/3) = 1 : 2/5 - 1: 2/3 1/1 divide by 2/5 flip to get 1/1 multiply 5/2 1/1 divide by 2/3 flip to get 1/1 multiply 3/2 5/2 - 3/2 = 2/2
= 1 I have a feeling this is totally wrong....
It is correct! Trust in your answers, if you apply correctly the learnt, there's nothing to worry about.
There is a way to avoid the ":" stuff. Consider the fraction: $\frac{1}{\frac{2}{5}}$ We need to remove the fraction in the denominator of the "overall" fraction. How would you do this? Well, note
that if we multiply $\frac{2}{5}$ by 5 the result is an integer: 2. But what we multiply in the denominator we must also multiply in the numerator. So: $\frac{1}{\frac{2}{5}} = \frac{1}{\frac{2}{5}}
\cdot \frac{5}{5} = \frac{1 \cdot 5}{\frac{2}{5} \cdot 5} = \frac{5}{2}$ (No offense Krizalid. I hate ratio notation with a passion. -Dan
Doesn't matter. If it's necessary, I try to explain pedagogically so that the user understands the problem well. | {"url":"http://mathhelpforum.com/math-topics/19375-fractions.html","timestamp":"2014-04-18T20:14:33Z","content_type":null,"content_length":"57884","record_id":"<urn:uuid:8bbffa69-3092-4261-9e28-45d18b4f086b>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00599-ip-10-147-4-33.ec2.internal.warc.gz"} |
Is there a proof that OEIS-A002387 is $[ e^{n-\gamma} ]$ ?
up vote 8 down vote favorite
Based on the comments on OEIS-A002387:
$a_{n}$ = 1, 2, 4, 11, 31, 83, 227, 616,...
it is likely, that the sequence $a_{n}$ coincides with $[ e^{n-\gamma} ]$ , where $\gamma$ is the Euler-Mascheroni constant and $[\cdot]$ is the rounding function (remark made by Dean Hickerson).
My Question: Is there a formal proof, that OEIS-A002387 is $[ e^{n-\gamma} ]$ ?
Equivalently, there exist no integers $k \ge 1$ and $n$ with $\log(k+1/2) + \gamma < n < \log(k+1/2) + \gamma + 1/(24k^2)$. – Gerald Edgar Jun 23 '13 at 12:00
As $a_{n}$ essentially grows as a geometric progression of common ratio $e$, maybe there is a way to derive the analogue of Binet's formula for the Fibonacci sequence. – Sylvain JULIEN Jun 23 '13
at 12:09
@Gerald I suppose there might exists integers in your equality, yet the value of $H_k$ to be the floor of the lower bound and the conjecture will be still true. – joro Jun 23 '13 at 12:45
2 I doubt a proof is known or can be obtained using any known techniques. Basically, it comes down to whether a bizarre numerical coincidence occurs with Euler's contant $\gamma$. The approximation
properties of $\gamma$ seem intractable, which makes the problem feel hopeless, but I guess you never know where a nice argument might be hiding (or I might be overlooking some aspect of this). –
Henry Cohn Jun 23 '13 at 14:28
Let me summarize. $k=a(n)$ means $$n\in I_k:=[H(k-1),H(k) )$$ $k=\lfloor e^{n-\gamma} +0.5\rfloor$ is equivalent to $$n\in J_k:=[L(k-1), L(k) ),$$ for $L(k):=\log(k+1/2)+\gamma$. Asymptotics for
$H$ and $L$ imply that $I_k$ and $J_k$ are quite close intervals, which makes more and more likely that $I_k$ meets $\mathbb{N}$ iff $J_k$ does (the thesis being that this is true for all $k$).
Moreover, it is encouraging that $H(k)$ is never an integer for $k > 1$ (a consequence of the Bertrand's postulate), that is, if an integer $n$ is in $I_k$, it is not on its boundary. – Pietro
Majer Jun 23 '13 at 21:12
show 10 more comments
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How to Add and Multiply Fractions
Edit Article
Adding FractionsMultiplying Fractions
Edited by Lojjik Braughler, Elyne, Kalpit, Teresa and 1 other
There are many ways to write numbers. Different types of numbers require different steps for completing operations. Once you learn how to add and multiply fractions, you will be able to combine parts
of numbers that are less than 1.
Method 1 of 2: Adding Fractions
1. 1
Look at the denominators of the fractions you want to add. The basic principle in adding fractions is to make the denominators the same, because you cannot add when the denominators are
2. 2
Find the Lowest Common Multiple of the Denominators (often referred to as the LCD or the Lowest Common Denominator). In order to add fractions with different denominators, we must find the LCD.
Note that if the denominators contain same number, then the LCD is also just the same number.
3. 3
Figure out how many times the original denominator goes into the LCD. Multiply the numerator and denominator of each fraction by that number to get the fraction with new numerator and the
previously found LCD. Multiplying the numerator and denominator of any fraction by the same number does not change its value.
4. 4
Write the original problem with transformed fractions (having the same denominator)
5. 5
Add the numerators. Once you have written the new numerators over the LCD, you are ready to add. Simply add the numerators; do not do anything to the denominator.
6. 6
Simplify your answer. Once you have your answer, check to see if you can simplify it.
Method 2 of 2: Multiplying Fractions
1. 1
Multiply the top numbers in the fractions as regular whole numbers and ignore the bottom numbers.
2. 2
Conduct multiplication with the bottom numbers of the fractions in the same manner as you carried out with the top numbers.
3. 3
Write the answer as a fraction with the answer from multiplying the top numbers over the answer you got when you multiplied the bottom numbers
4. 4
Inspect the answer to determine if the top and bottom number can be divided by the same number and reduce, if possible.
• Your answers can be checked by using a calculator or website that offers math calculators.
Article Info
Categories: Fractions
Recent edits by: Teresa, Kalpit, Elyne
Thanks to all authors for creating a page that has been read 7,407 times.
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Do teachers deserve these holidays??
31 03 2013, 12:35 #31
Established Poster
Join Date
Jul 2008
With holidays taken into account that roughly equates to a 40 hour working week, which is average.
(48 hours over 39 weeks = 1872 hours, divide by 46 (the number of weeks an average worker works) = 40.7)
So maybe the answer is to get rid of the long holidays and replace them with two 2 week breaks and two 1 week breaks spread evenly through the year, with the length of the school day reduced
accordingly. This would mean that the same number of hours are worked but in a more sensible pattern and would be a lot better for the children as well as we wouldn't have the problem of them
forgetting half of what they learnt over the holidays.
With holidays taken into account that roughly equates to a 40 hour working week, which is average.
(48 hours over 39 weeks = 1872 hours, divide by 46 (the number of weeks an average worker works) = 40.7)
So maybe the answer is to get rid of the long holidays and replace them with two 2 week breaks and two 1 week breaks spread evenly through the year, with the length of the school day reduced
accordingly. This would mean that the same number of hours are worked but in a more sensible pattern and would be a lot better for the children as well as we wouldn't have the problem of them
forgetting half of what they learnt over the holidays.
This figure already accounted for holidays.
With holidays taken into account that roughly equates to a 40 hour working week, which is average.
(48 hours over 39 weeks = 1872 hours, divide by 46 (the number of weeks an average worker works) = 40.7)
So maybe the answer is to get rid of the long holidays and replace them with two 2 week breaks and two 1 week breaks spread evenly through the year, with the length of the school day reduced
accordingly. This would mean that the same number of hours are worked but in a more sensible pattern and would be a lot better for the children as well as we wouldn't have the problem of them
forgetting half of what they learnt over the holidays.
But the report quoted that this 48 hours only applied to three quarters of teachers and only for a three month period. I don't think there's enough evidence there to conclude a 48 hour week is
the norm. We need to know what the lazy quarter are doing and also why choose only a 3 month period. Another flaw is that it seems the evidence, if it is such, is based solely on what the
teachers say. Can we believe them? One thing is certain, they have a vested interest in making their working hours appear long, in order to get some justification for their long holidays and fat
pensions now under Goves magnifying glass. I fear that all is not what it is stated to be in this report, and tat an independent assessment would be better.
So maybe the answer is to get rid of the long holidays and replace them with two 2 week breaks and two 1 week breaks spread evenly through the year, with the length of the school day reduced
accordingly. This would mean that the same number of hours are worked but in a more sensible pattern and would be a lot better for the children as well as we wouldn't have the problem of them
forgetting half of what they learnt over the holidays.
That may be sensible but there are many disadvantages
1. can you expect pupils to be at school for 47/48 weeks per year?
2. Would the teachers teaching time be reduced: if it takes 48 hours to plan, prepare, deliver and assess lesson for a week, then this will still be the case if pupils are in school for more
3. What about the work done during the holidays? For example, take report writing, if I don't get 2 weeks at Easter, when could I put in the extra work to write the reports?
Also, I dream of only working 48 hours per week in term time!
With holidays taken into account that roughly equates to a 40 hour working week, which is average.
(48 hours over 39 weeks = 1872 hours, divide by 46 (the number of weeks an average worker works) = 40.7)
So maybe the answer is to get rid of the long holidays and replace them with two 2 week breaks and two 1 week breaks spread evenly through the year, with the length of the school day reduced
accordingly. This would mean that the same number of hours are worked but in a more sensible pattern and would be a lot better for the children as well as we wouldn't have the problem of them
forgetting half of what they learnt over the holidays.
Does it take into account the fact that most teachers to my knowledge work on at least some of their holidays?
it IS a silly argument. I criticise the government all the time, but I don't need to become an MP. Lots of people find that the job they start doing doesn't suit them or is beyond their
capabilities. They then quite rightly move on to something else. Teachers are not unique in this.
What you are saying is that in order to express an opinion about a particular worker or trade or profession you have to be prepared to apply for the job. By any rational standards that is a silly
argument. I had every basis to attack his argument. It is a ridiculous argument.
Nope, it isn't a silly argument at all. I don't criticise physicists as I don't understand physics. For me to do so would be asinine.
As you clearly don't understand education or the educational establishment, it's exactly the same. With no pedagogical qualifications, what right do you feel you have to "have your say?"
I mean, no one is going to stop you of course, but to have a "say" based on ignorance and complete lack of understanding, discounting evidence like the fact that 1/4 of new teachers quit the
profession due to not being able to handle it, kind of shows that your say is largely like banging a hollow drum.
Lots of noise, no substance.
But the report quoted that this 48 hours only applied to three quarters of teachers and only for a three month period. I don't think there's enough evidence there to conclude a 48 hour week is
the norm. We need to know what the lazy quarter are doing and also why choose only a 3 month period. Another flaw is that it seems the evidence, if it is such, is based solely on what the
teachers say. Can we believe them? One thing is certain, they have a vested interest in making their working hours appear long, in order to get some justification for their long holidays and fat
pensions now under Goves magnifying glass. I fear that all is not what it is stated to be in this report, and tat an independent assessment would be better.
Based on what? You're presented with evidence and your response is "THEY ARE LYING FOR THEIR GREED". How ridiculous.
This is why your criticisms fall on deaf ears really - you again prove your complete lack of understanding.
This is why there's going to be some more significant strikes soon. Terrible attitudes based on the politics of envy - again flembo, if teachers have it so good why do so many quit early in their
careers, and why do some schools have serious trouble retaining staff?
Ridiculous, disgusting, envious rantings once more from someone completely ignorant and without the first clue concerning the profession.
Thankfully these kind of comments tend to be restricted to "have your say" forum areas, where they cause little damage aside from hurting the morale of an already disaffected bunch of
But the report quoted that this 48 hours only applied to three quarters of teachers and only for a three month period. I don't think there's enough evidence there to conclude a 48 hour week is
the norm. We need to know what the lazy quarter are doing and also why choose only a 3 month period. Another flaw is that it seems the evidence, if it is such, is based solely on what the
teachers say. Can we believe them? One thing is certain, they have a vested interest in making their working hours appear long, in order to get some justification for their long holidays and fat
pensions now under Goves magnifying glass. I fear that all is not what it is stated to be in this report, and tat an independent assessment would be better.
No. Once again you have not troubled to do you research. Was comprehension not on the "very well educated" menu either?
The 3 months and 3/4 applies only to the number who referenced working at least one "all-nighter".
The hours worked was simply a mean of the whole sample. (And why on earth one would imagine that anyone would average only 3/4 of a sample is beyond me).
And of course it is perfectly possible that all, or a substatial portion, of the 1,600 respondents lied. But given that the alternative sample is you and your hunch, the former looks more
persuasive on the face of it.
can't we just agree that teachers are marxist communist subversive lazy PC do gooders and they should all be forced to fight in a war?
No of course not. In any case they wouldn't make very good soldiers. Just as a battle starts they would declare a training day,or say they couldn't get to the front because there were rumours of
a snowflake being expected.
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MATHEMATICA BOHEMICA, Vol. 125, No. 3, pp. 365-370 (2000)
On the extension of exponential polynomials
Laszlo Székelyhidi
Laszlo Székelyhidi, Institute of Mathematics and Informatics, Kossuth Lajos University, H-4010 Debrecen, Pf. 12., Hungary
Abstract: Exponential polynomials are the building bricks of spectral synthesis. In some cases it happens that exponential polynomials should be extended from subgroups to whole groups. To achieve
this aim we prove an extension theorem for exponential polynomials which is based on a classical theorem on the extension of homomorphisms.
Keywords: exponential polynomial, extension
Classification (MSC2000): 39B05, 39B99
Full text of the article:
[Previous Article] [Next Article] [Contents of this Number] © 2005 ELibM and FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition | {"url":"http://www.kurims.kyoto-u.ac.jp/EMIS/journals/MB/125.3/9.html","timestamp":"2014-04-16T19:20:51Z","content_type":null,"content_length":"2107","record_id":"<urn:uuid:39508d24-4894-4b8d-815f-0994a50d65ea>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00092-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calculus III problem
July 28th 2010, 08:03 PM #1
Jul 2010
Hi, so tomorrow's pretty much the last day of the summer semester for me and I have a homework assignment due tomorrow that's confusing me sooo much..
Im here asking for some help in direction on how to do these problems so that I can find the solution by myself... I would really really appreciate any help anyone could provide ...
I am not sure how to represent the domain as a function, and if anyone could point me in the right direction, I'm sure I could get the answer.
The Problem -
I would really really really appreciate any help at all. I need to know how to represent the domain as a function.
Last edited by mr fantastic; July 28th 2010 at 10:03 PM.
Don't think of the domain as one big function. Think of it as 3 parts as in the descrption:
C1: line from (0,0) to (1,0)
C2: circular arc from (1,0) to the line x=y. What is x in this case?
C3: line from (x,x) to (0,0)
If you don't know how to parametrize lines, look in your book somewhere around parametric equations of a line.
The circular arc you can just parametrize using sines and cosines.
I hope this helps!
I think we should go for Green's Theorem.
$\displaystyle{\int_{\partial R} P dx + Q dy = \int\int_{R}\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dxdy}$
So, lets do it:
Let have $P = \sin x - 6x^2y$ and $Q=3xy^2-x^3$.
$\frac{\partial Q}{\partial x} = 3y^2 - 3x^2$
$\frac{\partial P}{\partial y} = - 6x^2$
$\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} = 3y^2 + 3x^2$ (How convenient!!)
Now we have to determine the region bounded by the curve C. I've made a sketch:
You see that we're "walking" counter-clockwise - thus we should have $\int\int_{R} 3y^2 + 3x^2~dxdy$. There would be a negative signal if we walked clockwise.
Well. We should determine the limits of integration. I mentioned above the convenience because this integral is asking so much that we change it for polar coordinates - therefore we can easily
find the limits of integration (I think the sketch speaks by itself). So we put:
$x = r cos t$ and $y=r sin t$.
Our angle will vary from 0 to $\pi/2 + \pi/4 = 3\pi/4$ and the radius will vary from 0 to 1 (unit circle). This will lead us to:
Ah, before, we can't forget the Jacobian (I will not show this, but you should know how its done), so $dx dy = r dr dt$. Finally, the integral becomes:
$\displaystyle{\int\int_{R} 3 (y^2 + x^2)~dxdy = \int_{0}^{3\pi/4}\int_{0}^{1} 3 r^2\cdot r~drdt$ because $x^2+y^2 = r^2$ by our change.
Last edited by bondesan; July 28th 2010 at 08:45 PM. Reason: Wrong figure
It says counter-clockwise in the question!
Oh, my mistake, sorry. I'll correct it.
Thank you so much for the quick replies. I appreciate it a looooot! I will try it out and see what I can come up with. And keep you guys updated.
Already done, I guess you can follow now.
July 28th 2010, 08:19 PM #2
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On the Anthropic Trilemma
Eliezer’s Anthropic Trilemma:
So here’s a simple algorithm for winning the lottery: Buy a ticket. Suspend your computer program just before the lottery drawing – which should of course be a quantum lottery, so that every
ticket wins somewhere. Program your computational environment to, if you win, make a trillion copies of yourself, and wake them up for ten seconds, long enough to experience winning the
lottery. Then suspend the programs, merge them again, and start the result. If you don’t win the lottery, then just wake up automatically. The odds of winning the lottery are ordinarily a
billion to one. But now the branch in which you win has your “measure”, your “amount of experience”, temporarily multiplied by a trillion. So with the brief expenditure of a little extra
computing power, you can subjectively win the lottery – be reasonably sure that when next you open your eyes, you will see a computer screen flashing “You won!” As for what happens ten seconds
after that, you have no way of knowing how many processors you run on, so you shouldn’t feel a thing.
See the original post for assumptions, what merging minds entails etc. He proposes three alternative bullets to bite: accepting that this would work, denying that there is “any meaningful sense in
which I can anticipate waking up as myself tomorrow, rather than Britney Spears” so undermining any question about what you should anticipate, and Nick Bostrom’s response, paraphrased by Eliezer:
…you should anticipate winning the lottery after five seconds, but anticipate losing the lottery after fifteen seconds. To bite this bullet, you have to throw away the idea that your joint
subjective probabilities are the product of your conditional subjective probabilities. If you win the lottery, the subjective probability of having still won the lottery, ten seconds later, is
~1. And if you lose the lottery, the subjective probability of having lost the lottery, ten seconds later, is ~1. But we don’t have p(“experience win after 15s”) = p(“experience win after 15s”|
”experience win after 5s”)*p(“experience win after 5s”) + p(“experience win after 15s”|”experience not-win after 5s”)*p(“experience not-win after 5s”).
I think I already bit the bullet about there not being a meaningful sense in which I won’t wake up as Britney Spears. However I would like to offer a better, relatively bullet biting free solution.
First notice that you will have to bite Bostrom’s bullet if you even accept Eliezer’s premise that arranging to multiply your ‘amount of experience’ in one branch in the future makes you more likely
to experience that branch. Call this principle ‘follow-the-crowd’ (FTC). And let’s give the name ‘blatantly obvious principle’ (BOP) to the notion that P(I win at time 2) is equal to P(I win at time
2|I win at time 1)P(I win at time 1)+P(I win at time 2|I lose at time 1)P(I lose at time 1). Bostrom’s bullet is to deny BOP.
We can set aside the bit about merging brains together for now; that isn’t causing our problem. Consider a simpler and smaller (for the sake of easy diagramming) lottery setup where after you win or
lose you are woken for ten seconds as a single person, then put back to sleep and woken as four copies in the winning branch or one in the losing branch. See the diagram below. You are at Time 0 (T
[0]). Before Time 1 (T[1]) the lottery is run, so at T[1] the winner is W[1] and the loser is L[1]. W[1] is then copied to give the multitude of winning experiences at T[2], while L[2] remains
Now using the same reasoning as you would to win the lottery before, FTC, you should anticipate an 80% chance of winning the lottery at T[2]. There is four times as much of your experience winning
the lottery as not then. But BOP says you still only have a fifty percent chance of being a lottery winner at T2:
P(win at T[2]) = P(win at T[2]|win at T[1]).P(win at T[1])+P(win at T[2]|lose at T[1]).P(lose at T[1]) = 1 x 1/2 + 0 x 1/2 = 1/2
FTC and BOP conflict. If you accept that you should generally anticipate futures where there are more of you more strongly, it looks like you accept that P(a) does not always equal P(a|b)P(b)+P(a|-b)
P(-b). How sad.
Looking at the diagram above, it is easy to see why these two methods of calculating anticipations disagree. There are two times in the diagram that your future branches, once in a probabilistic
event and once in being copied. FTC and BOP both treat the probabilistic event the same: they divide your expectations between the outcomes according to their objective probability. At the other
branching the two principles do different things. BOP treats it the same as a probabilistic event, dividing your expectation of reaching that point between the many branches you could continue on.
FTC treats it as a multiplication of your experience, giving each new branch the full measure of the incoming branch. Which method is correct?
Neither. FTC and BOP are both approximations of better principles. Both of the better principles are probably true, and they do not conflict.
To see this, first we should be precise about what we mean by ‘anticipate’. There is more than one resolution to the conflict, depending on your theory of what to anticipate: where the purported
thread of personal experience goes, if anywhere. (Nope, resolving the trilemma does not seem to answer this question).
Resolution 1: the single thread
The most natural assumption seems to be that your future takes one branch at every intersection. It does this based on objective probability at probabilistic events, or equiprobably at copying
events. It follows BOP. This means we can keep the present version of BOP, so I shall explain how we can do without FTC.
Consider diagram 2. If your future takes one branch at every intersection, and you happen to win the lottery, there are still many T[2] lottery winners who will not be your future. They are your
copies, but they are not where your thread of experience goes. They and your real future self can’t distinguish who is actually in your future, but there is some truth of the matter. It is shown in
Now while there are only two objective possible worlds, when we consider possible paths for the green thread there are five possible worlds (one shown in diagram 2). In each one your experience
follows a different path up the tree. Since your future is now distinguished from other similar experiences, we can see the weight of your experience at T[2 ]in a world where you win is no greater
than the weight in a world where you lose, though there are always more copies who are not you in the world where you win.
The four worlds where your future is in a winning branch are each only a quarter as likely as one where you lose, because there is a fifty percent chance of you reaching W1, and after that a twenty
five percent chance of reaching a given W2. By the original FTC reasoning then, you are equally likely to win or lose. More copies just makes you less certain exactly where it will be.
I am treating the invisible green thread like any other hidden characteristic. Suppose you know that you are and will continue to be the person with the red underpants, though many copies will be
made of you with green underpants. However many extra copies are made, a world with more of them in future should not get more of your credence, even if you don’t know which future person actually
has the red pants. If you think of yourself as having only one future, then you can’t also consider there to be a greater amount of your experience when there are a lot of copies. If you did
anticipate experiences based on the probability that many people other than you were scheduled for that experience, you would greatly increase the minuscule credence you have in experiencing being
Britney Spears when you wake up tomorrow.
Doesn’t this conflict with the use of FTC to avoid the Bolzmann brain problem, Eliezer’s original motivation for accepting it? No. The above reasoning means there is a difference between where you
should anticipate going when you are at T[0], and where you should think you are if you are at T[2].
If you are at T[0] you should anticipate a 50% chance of winning, but if you are at T[2] you have an 80% chance of being a winner. Sound silly? That’s because you’ve forgotten that you are
potentially talking about different people. If you are at T[2], you are probably not the future of the person who was at T[0], and you have no way to tell. You are a copy of them, but their future
thread is unlikely to wend through you. If you knew that you were their future, then you would agree with their calculations.
That is, anyone who only knows they are at T[2] should consider themselves likely to have won, because there are many more winners than losers. Anyone who knows they are at T[2] and are your future,
should give even odds to winning. At T[0], you know that the future person whose measure you are interested in is at T[2] and is your future, so you also give even odds to winning.
Avoiding the Bolzmann brain problem requires a principle similar to FTC which says you are presently more likely to be in a world where there are more people like you. SIA says just that for
instance, and there are other anthropic principles that imply similar things. Avoiding the Bolzmann brain problem does not require inferring from this that your future lies in worlds where there are
more such people. And such an inference is invalid.
This is exactly the same as how it is invalid to infer that you will have many children from the fact that you are more likely to be from a family with many children. Probability theory doesn’t
distinguish between the relationship between you and your children and the relationship between you and your future selves.
Resolution 2
You could instead consider all copies to be your futures. Your thread is duplicated when you are. In that case you should treat the two kinds of branching differently, unlike BOP, but still not in
the way FTC does. It appears you should anticipate a 50% chance of becoming four people, rather than an 80% chance of becoming one of those people. There is no sense in which you will become one of
the winners rather than another. Like in the last case, it is true that if you are presently one of the copies in the future, you should think yourself 80% likely to be a winner. But again ‘you’
refers to a different entity in this case to the one it referred to before the lottery. It refers to a single future copy. It can’t usefully refer to a whole set of winners, because the one
considering it does not know if they are part of that set or if they are a loser. As in the last case, your anticipations at T[0] should be different from your expectations for yourself if you know
only that you are in the future already.
In this case BOP gives us the right answer for the anticipated chances of winning at T[0]. However it says you have a 25% chance of becoming each winner at T[2] given you win at T[1], instead of 100%
chance of becoming all of them.
Resolution 3:
Suppose that you want to equate becoming four people in one branch as being more likely to be there. More of your future weight is there, so for some notions of expectation perhaps you expect to be
there. You take ‘what is the probability that I win the lottery at T1?’ to mean something like ‘what proportion of my future selves are winning at T1?’. FTC gives the correct answer to this question
– you aren’t especially likely to win at T1, but you probably will at T2. Or in the original problem, you should expect to win after 5 seconds and lose after 15 seconds, as Nick Bostrom suggested. If
FTC is true, then we must scrap BOP. This is easier than it looks because BOP is not what it seems.
Here is BOP again:
P(I win at T2) is equal to P(I win at T2|I win at T1)P(I win at T1)+P(I win at T2|I lose at T1)P(I lose at T1)
It looks like a simple application of
P(a) = P(a|b)P(b)+P(a|-b)P(-b)
But here is a more extended version:
P(win at 15|at 15) = P(win an 15|at 15 and came from win at 5)P(win at 5|at 5)+P(win at 15|at 15 and came from loss at 5)P(lose at 5|at 5)
This is only equal to BOP if the probability of having a win at 5 in your past when you are at 15 is equal to the probability of winning at 5 when you are at 5. To accept FTC is to deny that. FTC
says you are more likely to find the win in your past than to experience it because many copies are descended from the same past. So accepting FTC doesn’t conflict with P(a) being equal to P(a|b)P(b)
+P(a|-b)P(-b), it just makes BOP an inaccurate application of this true principle.
In summary:
1. If your future is (by definitional choice or underlying reality) a continuous non-splitting thread, then something like SIA should be used instead of FTC, and BOP holds. Who you anticipate being
differs from who you should think you are when you get there. Who you should think you are when you get there remains as something like SIA and avoids the Bolzmann brain problem.
2. If all your future copies are equally your future, you should anticipate becoming a large number of people with the same probability as that you would have become one person if there were no extra
copies. In which case FTC does not hold, because you expect to become many people with a small probability instead of one of those many people with a large probability. BOP holds in a modified form
where it doesn’t treat being copied as being sent down a random path. But if you want to know what a random moment from your future will hold, a random moment from T1 is more likely to include losing
than a random moment from T2. For working out what a random T2 moment will hold, BOP is a false application of a correct principle.
3. If for whatever reason you conceptualise yourself as being more likely to go into future worlds based on the number of copies of you there are in those worlds, then FTC does hold, but BOP becomes
I think the most important point is that the question of where you should anticipate going need not have the same answer as where a future copy of you should expect to be (if they don’t know for some
reason). A future copy who doesn’t know where they are should think they are more likely to be in world where there are many people like themselves, but you should not necessarily think you are
likely to go into such a world. If you don’t think you are as likely to go into such a world, then FTC doesn’t hold. If you do, then BOP doesn’t hold.
It seems to me the original problem uses FTC while assuming there will be a single thread, thereby making BOP look inevitable. If the thread is kept, FTC should not be, which can be conceptualised as
in either of resolutions 1 or 2. If FTC is kept, BOP need not be, as in resolution 3. Whether you keep FTC or BOP will give you different expectations about the future, but which expectations are
warranted is a question for another time.
6 responses to “On the Anthropic Trilemma”
1. Brilliant
2. Looks quite good. Typo: in the first declaration of BOP “P(I win at time 1) is equal” should be “P(I win at time 2) is equal”.
□ Fixed.
3. Nice analysis.
Personally I think your resolution 2 is correct, but then I also disagree with you on the nature of identity, believing that the future mes who are causally derived from the present me are all
me, and since Brittany Spears is not, she is not and never will be me.
4. This seems to be a very tortured and confused argument for the obvious. If I understand the problem correctly, this “Eliezer” seem to not be able to distinguish the analytical fact that copies
are not identical to their original. If x is a copy of y then necessarily, x and y are not the same objects (not identical). This simple fact would seem to obviate many of the confusions of the
original problem (really a pseudo-problem). Once that is understood, BOP, as you have called it, takes care of the rest.
This entry was posted in 1 and tagged Anthropics. Bookmark the permalink. | {"url":"http://meteuphoric.wordpress.com/2011/05/19/on-the-anthropic-trilemma/","timestamp":"2014-04-19T06:52:16Z","content_type":null,"content_length":"110034","record_id":"<urn:uuid:3333b0c9-85c1-4ac4-b0c5-c28706e9ed54>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00268-ip-10-147-4-33.ec2.internal.warc.gz"} |
Course Catalog Descriptions
1A -- Calculus [4 units]
Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
Prerequisites: Three and one-half years of high school math, including trigonometry and analytic geometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the UC/CSU
math diagnostic test, or 32. Consult the mathematics department for details. Students with AP credit should consider choosing a course more advanced than 1A.
Credit option: Students will receive no credit for 1A after taking 16B and 2 units after taking 16A.
Description: This sequence is intended for majors in engineering and the physical sciences. An introduction to differential and integral calculus of functions of one variable, with applications and
an introduction to transcendental functions.
1B -- Calculus [4 units]
Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
Prerequisites: 1A.
Credit option: Students will receive 2 units of credit for 1B after taking 16B.
Description: Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary
differential equations; oscillation and damping; series solutions of ordinary differential equations.
H1B -- Honors Calculus [4 units]
Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
Prerequisites: 1A.
Credit option: Students will receive 2 units of credit for H1B after taking 16B.
Description: Honors version of 1B. Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations.
Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations.
16A -- Analytic Geometry and Calculus [3 units]
Course Format: Two hours of lecture and one hour of discussion/workshop per week; at the discretion of the instructor, an additional one hour to one and one-half hours of lecture or discussion/
workshop per week.
Prerequisites: Three years of high school math, including trigonometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the UC/CSU math diagnostic exam, or 32. Consult
the mathematics department for details.
Credit option: Students will receive no credit for 16A after taking 1A. Two units of 16A may be used to remove a deficient grade in 1A.
Description: This sequence is intended for majors in the life and social sciences. Calculus of one variable; derivatives, definite integrals and applications, maxima and minima, and applications of
the exponential and logarithmic functions.
16B -- Analytic Geometry and Calculus [3 units]
Course Format: Two hours of lecture and one hour of discussion/workshop per week; at the discretion of the instructor, an additional hour of lecture or discussion/workshop per week.
Prerequisites: 16A.
Credit option: Students will receive no credit for 16B after 1B, 2 units after 1A. Two units of 16B may be used to remove a deficient grade in 1A.
Description: Continuation of 16A. Application of integration of economics and life sciences. Differential equations. Functions of many variables. Partial derivatives, constrained and unconstrained
24 -- Freshman Seminars [1 unit]
Course Format: One hour of seminar per week.
Credit option: Course may be repeated for credit as topic varies.
Grading option: Sections 1-2 to be graded on a letter-grade basis. Sections 3-4 to be graded on a passed/not passed basis.
Description: The Berkeley Seminar Program has been designed to provide new students with the opportunity to explore an intellectual topic with a faculty member in a small-seminar setting. Berkeley
Seminars are offered in all campus departments, and topics vary from department to department and semester to semester.
32 -- Precalculus [4 units]
Course Format: Two hours of lecture and two hours of discussion per week, plus, at the instructor's option, an extra hour of lecture/discussion per week.
Prerequisites: Three years of high school mathematics, plus satisfactory score on one of the following: CEEB MAT test, math SAT, or UC/CSU diagnostic examination.
Credit option: Students will receive no credit for 32 after taking 1A-1B or 16A-16B and will receive 3 units after taking 96.
Description: Polynomial and rational functions, exponential and logarithmic functions, trigonometry and trigonometric functions. Complex numbers, fundamental theorem of algebra, mathematical
induction, binomial theorem, series, and sequences.
(2-4) Two to four hours of seminar per week.
49 -- Supplementary Work in Lower Division Mathematics [1-3 units]
Course Format: Meetings to be arranged.
Prerequisites: Some units in a lower division Mathematics class.
Credit option: Course may be repeated for credit.
Description: Meetings to be arranged. Students with partial credit in lower division mathematics courses may, with consent of instructor, complete the credit under this heading.
53 -- Multivariable Calculus [4 units]
Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
Prerequisites: 1B.
Credit option: Students will receive 1 unit of credit for 53 after taking 50B and 3 units of credit after taking 50A.
Description: Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and
H53 -- Honors Multivariable Calculus [4 units]
Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
Prerequisites: 1B.
Credit option: Students will receive 1 unit for H53 after taking 50B and 3 units after taking 50A.
Description: Honors version of 53. Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems
of Green, Gauss, and Stokes.
54 -- Linear Algebra and Differential Equations [4 units]
Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
Prerequisites: 1B.
Credit option: Students will receive 1 unit of credit for 54 after taking 50A and 3 units of credit after taking Math 50B.
Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary
differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.
H54 -- Honors Linear Algebra and Differential Equations [4 units]
Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
Prerequisites: 1B.
Credit option: Students will receive 1 unit for H54 after taking 50A and 3 units after taking 50B.
Description: Honors version of 54. Basic linear algebra: matrix arithmetic and determinants. Vectors spaces; inner product spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous
ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.
55 -- Discrete Mathematics [4 units]
Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
Prerequisites: Mathematical maturity appropriate to a sophomore math class. 1A-1B recommended.
Credit option: Students will receive no credit for 55 after taking Computer Science 70.
Description: Logic, mathematical induction sets, relations, and functions. Introduction to graphs, elementary number theory, combinatorics, algebraic structures, discrete probability, theory, and
74 -- Transitions to Upper Division Mathematics [3 units]
Course Format: Three hours of lecture and two hours of discussion per week.
Grading option:
Prerequisites: 53 and 54.
Description: The course will focus on reading and understanding mathematical proofs. It will emphasize precise thinking and the presentation of mathematical results, both orally and in written
form. The course is intended for students who are considering majoring in mathematics but wish additional training.
H90 -- Honors Undergraduate Seminar in Mathematical Problem Solving [1 units]
Course Format: Two hours of seminar per week.
Grading option: Course may be repeated for credit.
Prerequisites: Consent of instructor; undergraduate standing.
Description: This seminar is designed especially, but not exclusively, to prepare students for the annual national Putnam Mathematical Competition in December. Students will develop problem solving
skills and experience by attempting the solution of challenging mathematical problems that require insight more than knowledge.
91 -- Special Topics in Mathematics [4 units]
Course Format: Three hours of lecture/discussion per week.
Grading option: Course may be repeated for credit.
Description: Topics to be covered and the method of instruction to be used will be announced at the beginning of each semester that such courses are offered. See department bulletins.
(F,SP) Staff
98 -- Supervised Group Study [1-4 units]
Course Format:
Grading option: Must be taken on a passed/not passed basis.
Description: Directed Group Study, topics vary with instructor.
99 -- Supervised Independent Study [1-4 units]
Course Format: Enrollment is restricted; see the Introduction to Courses and Curricula section of the catalog. Independent study, weekly meeting with faculty.
Grading option: Course may be repeated for credit. Must be taken on a passed/not passed basis.
Prerequisites: Restricted to freshmen and sophomores only. Consent of instructor.
Description: Supervised independent study by academically superior, lower division students. 3.3 GPA required and prior consent of instructor who is to supervise the study. A written proposal must
be submitted to the department chair for pre-approval.
(F,SP) Staff
C103 -- Introduction to Mathematical Economics [4 units]
Course Format: Three hours of lecture per week.
Description: Selected topics illustrating the application of mathematics to economic theory. This course is intended for upper-division students in Mathematics, Statistics, the Physical Sciences,
and Engineering, and for economics majors with adequate mathematical preparation. No economic background is required. Also listed as Economics C103.
104 -- Introduction to Analysis [4 units]
Course Format: Three hours of lecture per week; at the discretion of the instructor, an additional two hours of discussion per week.
Prerequisites: 53 and 54.
Description: The real number system. Sequences, limits, and continuous functions in R and R^n. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series.
Mean value theorem and applications. The Riemann integral.
(F,SP) Staff
H104 -- Introduction to Analysis [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 53 and 54.
Description: Honors section corresponding to 104. Recommended for students who enjoy mathematics and are good at it. Greater emphasis on theory and challenging problems.
105 -- Second Course in Analysis [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 104.
Description: Differential calculus in R^n: the derivative as a linear map; the chain rule; inverse and implicit function theorems. Lebesgue integration on the line; comparison of Lebesgue and
Riemann integrals. Convergence theorems. Fourier series, L^2 theory. Fubini's theorem, change of variable.
110 -- Linear Algebra [4 units]
Course Format: Three hours of lecture per week and an additional two hours of discussion at the discretion of the instructor.
Prerequisites: 54 or a course with equivalent linear algebra content.
Credit option: No credit allowed after completion of Math 112 or 113B.
Description: Matrices, vector spaces, linear transformations, inner products, determinants. Eigenvectors. QR factorization. Quadratic forms and Rayleigh's principle. Jordan canonical form,
applications. Linear functionals.
(F,SP) Staff
H110 -- Linear Algebra [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 54 or a course with equivalent linear algebra content.
Credit option: No credit allowed after completion of Math 112 or 113B.
Description: Honors section corresponding to course 110 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.
113 -- Introduction to Abstract Algebra [4 units]
Course Format: Three hours of lecture per week; at the discretion of the instructor, an additional two hours of discussion per week.
Prerequisites: 54 or a course with equivalent linear algebra content.
Description: Sets and relations. The integers, congruences and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals and quotient fields. The theory of
polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.
(F,SP) Staff
H113 -- Introduction to Abstract Algebra [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 54 or a course with equivalent linear algebra content.
Description: Honors section corresponding to 113. Recommended for students who enjoy mathematics and are good at it. Greater emphasis on theory and challenging problems.
114 -- Second Course in Abstract Algebra [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 110 and 113, or consent of instructor.
Description: Further topics on groups, rings, and fields not covered in Math 113. Possible topics include the Sylow Theorems and their applications to group theory; classical groups; abelian groups
and modules over a principal ideal domain; algebraic field extensions; splitting fields and Galois theory; construction and classification of finite fields.
(SP) Staff
115 -- Introduction to Number Theory [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 53 and 54.
Description: Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions,
additive problems.
116 -- Cryptography [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 55.
Description: Construction and analysis of simple cryptosystems, public key cryptography, RSA, signature schemes, key distribution, hash functions, elliptic curves, and applications.
118 -- Wavelets and Signal Processing [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 53 and 54.
Description: Introduction to signal processing including Fourier analysis and wavelets. Theory, algorithms, and applications to one-dimensional signals and multidimensional images.
121A -- Mathematical Tools for the Physical Sciences [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 53 and 54.
Description: Intended for students in the physical sciences who are not planning to take more advanced mathematics courses. Rapid review of series and partial differentiation, complex variables and
analytic functions, integral transforms, calculus of variations.
121B -- Mathematical Tools for the Physical Sciences [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 53 and 54.
Description: Intended for students in the physical sciences who are not planning to take more advanced mathematics courses. Special functions, series solutions of ordinary differential equations,
partial differential equations arising in mathematical physics, probability theory.
123 -- Ordinary Differential Equations [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 104.
Description: Existence and uniqueness of solutions, linear systems, regular singular points. Other topics selected from analytic systems, autonomous systems, Sturm-Liouville Theory.
125A -- Mathematical Logic [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 113 or consent of instructor.
Description: Sentential and quantificational logic. Formal grammar, semantical interpretation, formal deduction, and their interrelation. Applications to formalized mathematical theories. Selected
topics from model theory or proof theory.
126 -- Introduction to Partial Differential Equations [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 53 and 54.
Description: Waves and diffusion, initial value problems for hyperbolic and parabolic equations, boundary value problems for elliptic equations, Green's functions, maximum principles, a priori
bounds, Fourier transform.
127 -- Mathematical and Computational Methods in Molecular Biology [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 53, 54, and 55; Statistics 20 recommended.
Description: Introduction to mathematical and computational problems arising in the context of molecular biology. Theory and applications of combinatorics, probability, statistics, geometry, and
topology to problems ranging from sequence determination to structure analysis.
128A -- Numerical Analysis [4 units]
Course Format: Three hours of lecture and one hour of discussion per week. At the discretion of instructor, an additional hour of discussion/computer laboratory per week.
Prerequisites: 53 and 54.
Description: Programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations. Practice on the
128B -- Numerical Analysis [4 units]
Course Format: Three hours of lecture and one hour of discussion per week. At the discretion of the instructor, an additional hour of discussion/computer laboratory per week.
Prerequisites: 110 and 128A.
Description: Iterative solution of systems of nonlinear equations, evaluation of eigenvalues and eigenvectors of matrices, applications to simple partial differential equations. Practice on the
130 -- The Classical Geometries [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 110 and 113.
Description: A critical examination of Euclid's Elements; ruler and compass constructions; connections with Galois theory; Hilbert's axioms for geometry, theory of areas, introduction of
coordinates, non-Euclidean geometry, regular solids, projective geometry.
135 -- Introduction to the Theory of Sets [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 113 and 104.
Description: Set-theoretical paradoxes and means of avoiding them. Sets, relations, functions, order and well-order. Proof by transfinite induction and definitions by transfinite recursion.
Cardinal and ordinal numbers and their arithmetic. Construction of the real numbers. Axiom of choice and its consequences.
136 -- Incompleteness and Undecidability [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 53, 54, and 55.
Description: Functions computable by algorithm, Turing machines, Church's thesis. Unsolvability of the halting problem, Rice's theorem. Recursively enumerable sets, creative sets, many-one
reductions. Self-referential programs. Godel's incompleteness theorems, undecidability of validity, decidable and undecidable theories.
140 -- Metric Differential Geometry [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 104.
Description: Frenet formulas, isoperimetric inequality, local theory of surfaces in Euclidean space, first and second fundamental forms. Gaussian and mean curvature, isometries, geodesics,
parallelism, the Gauss-Bonnet-Von Dyck Theorem.
141 -- Elementary Differential Topology [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 104 or equivalent and linear algebra.
Description: Manifolds in n-dimensional Euclidean space and smooth maps, Sard's Theorem, classification of compact one-manifolds, transversality and intersection modulo 2.
142 -- Elementary Algebraic Topology [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 104 and 113.
Description: The topology of one and two dimensional spaces: manifolds and triangulation, classification of surfaces, Euler characteristic, fundamental groups, plus further topics at the discretion
of the instructor.
143 -- Elementary Algebraic Geometry [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 113.
Description: Introduction to basic commutative algebra, algebraic geometry, and computational techniques. Main focus on curves, surfaces and Grassmannian varieties.
151 -- Mathematics of the Secondary School Curriculum I [4 units]
Course Format: Three hours of lecture and zero to one hour of discussion per week.
Prerequisites: 1A-1B, 53, or equivalent.
Description: Theory of rational numbers based on the number line, the Euclidean algorithm and fractions in lowest terms. The concepts of congruence and similarity, equation of a line, functions,
and quadratic functions.
(F,SP) Staff
152 -- Mathematics of the Secondary School Curriculum II [4 units]
Course Format: Three hours of lecture and zero to one hour of discussion per week.
Prerequisites: 151; 54, 113, or equivalent.
Description: Complex numbers and Fundamental Theorem of Algebra, roots and factorizations of polynomials, Euclidean geometry and axiomatic systems, basic trigonometry.
(F,SP) Staff
153 -- Mathematics of the Secondary School Curriculum III [4 units]
Course Format: Three hours of lecture and zero to one hour of discussion per week.
Prerequisites: 151, 152.
Description: The real line and least upper bound, limit and decimal expansion of a number, differentiation and integration, Fundamental Theorem of Calculus, characterizations of sine, cosine, exp,
and log.
(F,SP) Staff
160 -- History of Mathematics [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 53, 54, and 113.
Description: History of algebra, geometry, analytic geometry, and calculus from ancient times through the seventeenth century and selected topics from more recent mathematical history.
170 -- Mathematical Methods for Optimization [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 53and 54.
Description: Linear programming and a selection of topics from among the following: matrix games, integer programming, semidefinite programming, nonlinear programming, convex analysis and geometry,
polyhedral geometry, the calculus of variations, and control theory.
(F,SP) Staff
172 -- Combinatorics [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 55.
Description: Basic combinatorial principles, graphs, partially ordered sets, generating functions, asymptotic methods, combinatorics of permutations and partitions, designs and codes. Additional
topics at the discretion of the instructor.
(F,SP) Staff
185 -- Introduction to Complex Analysis [4 units]
Course Format: Three hours of lecture per week; at the discretion of the instructor, an additional two hours of discussion per week.
Prerequisites: 104.
Description: Analytic functions of a complex variable. Cauchy's integral theorem, power series, Laurent series, singularities of analytic functions, the residue theorem with application to definite
integrals. Some additional topics such as conformal mapping.
(F,SP) Staff
H185 -- Introduction to Complex Analysis [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 104.
Description: Honors section corresponding to Math 185 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.
189 -- Mathematical Methods in Classical and Quantum Mechanics [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 104, 110, 2 semesters lower division Physics.
Credit option: Course may be repeated for credit.
Description: Topics in mechanics presented from a mathematical viewpoint: e.g., hamiltonian mechanics and symplectic geometry, differential equations for fluids, spectral theory in quantum
mechanics, probability theory and statistical mechanics. See department bulletins for specific topics each semester course is offered.
191 -- Experimental Courses in Mathematics [1-4 units]
Course Format: Hours to be arranged.
Prerequisites: Consent of instructor.
Credit option: Course may be repeated for credit.
Description: The topics to be covered and the method of instruction to be used will be announced at the beginning of each semester that such courses are offered. See departmental bulletins.
195 -- Special Topics in Mathematics [4 units]
Course Format: Hours to be arranged.
Prerequisites: Consent of instructor.
Credit option: Course may be repeated for credit.
Description: Lectures on special topics, which will be announced at the beginning of each semester that the course is offered.
196 -- Honors Thesis [4 units]
Course Format: Hours to be arranged.
Prerequisites: Admission to the Honors Program; an overall GPA of 3.3 and a GPA of 3.5 in the major.
Credit option: Course may be repeated for credit.
Description: Independent study of an advanced topic leading to an honors thesis.
197 -- Field Study [1-4 units]
Course Format: Three hours of work per week per unit.
Prerequisites: Upper division standing. Written proposal signed by faculty sponsor and approved by department chair.
Credit option: Must be taken on a passed/not passed basis.
Description: For Math/Applied math majors. Supervised experience relevant to specific aspects of their mathematical emphasis of study in off-campus organizations. Regular individual meetings with
faculty sponsor and written reports required. Units will be awarded on the basis of three hours/week/unit.
(F,SP) Staff
198 -- Directed Group Study [1-4 units]
Course Format: Group study.
Prerequisites: Must have completed 60 units and be in good standing.
Credit option: Must be taken on a passed/not passed basis.
Description: Topics will vary with instructor.
(F,SP) Staff
199 -- Supervised Independent Study and Research [1-4 units]
Course Format: Hours to be arranged.
Prerequisites: The standard college regulations for all 199 courses.
Grading option: Must be taken on a passed/not passed basis.
202A -- Introduction to Topology and Analysis [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 104.
Description: Metric spaces and general topological spaces. Compactness and connectedness. Characterization of compact metric spaces. Theorems of Tychonoff, Urysohn, Tietze. Complete spaces and the
Baire category theorem. Function spaces; Arzela-Ascoli and Stone-Weierstrass theorems. Partitions of unity. Locally compact spaces; one-point compactification. Introduction to measure and
integration. Sigma algebras of sets. Measures and outer measures. Lebesgue measure on the line and R^n. Construction of the integral. Dominated convergence theorem.
202B -- Introduction to Topology and Analysis [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 202A and 110.
Description: Measure and integration. Product measures and Fubini-type theorems. Signed measures; Hahn and Jordan decompositions. Radon-Nikodym theorem. Integration on the line and in R^n.
Differentiation of the integral. Hausdorff measures. Fourier transform. Introduction to linear topological spaces, Banach spaces and Hilbert spaces. Banach-Steinhaus theorem; closed graph theorem.
Hahn-Banach theorem. Duality; the dual of L^p. Measures on locally compact spaces; the dual of C(X). Weak and weak-* topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem.
Additional topics chosen may include compact operators, spectral theory of compact operators, and applications to integral equations.
203 -- Asymptotic Analysis in Applied Mathematics [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 104.
Description: Asymptotic methods for differential equations, with emphasis upon many physical examples. Topics will include matched asymptotic expansions, Laplace's method, stationary phase,
boundary layers, multiple scales, WKB approximations, asymptotic Lagrangians, bifurcation theory.
204 -- Ordinary Differential Equations [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 104.
Description: Rigorous theory of ordinary differential equations. Fundamental existence theorems for initial and boundary value problems, variational equilibria, periodic coefficients and Floquet
Theory, Green's functions, eigenvalue problems, Sturm-Liouville theory, phase plane analysis, Poincare-Bendixon Theorem, bifurcation, chaos.
205 -- Theory of Functions of a Complex Variable [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 185.
Description: Normal families. Riemann Mapping Theorem. Picard's theorem and related theorems. Multiple-valued analytic functions and Riemann surfaces. Further topics selected by the instructor may
include: harmonic functions, elliptic and algebraic functions, boundary behavior of analytic functions and HP spaces, the Riemann zeta functions, prime number theorem.
206 -- Banach Algebras and Spectral Theory [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 202A-202B.
Description: Banach algebras. Spectrum of a Banach algebra element. Gelfand theory of commutative Banach algebras. Analytic functional calculus. Hilbert space operators. C*-algebras of operators.
Commutative C*-algebras. Spectral theorem for bounded self-adjoint and normal operators (both forms: the spectral integral and the "multiplication operator" formulation). Riesz theory of compact
operators. Hilbert-Schmidt operators. Fredholm operators. The Fredholm index. Selected additional topics.
208 -- C*-algebras [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 206.
Description: Basic theory of C*-algebras. Positivity, spectrum, GNS construction. Group C*-algebras and connection with group representations. Additional topics, for example, C*-dynamical systems,
209 -- Von Neumann Algebras [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 206.
Description: Basic theory of von Neumann algebras. Density theorems, topologies and normal maps, traces, comparison of projections, type classification, examples of factors. Additional topics, for
example, Tomita Takasaki theory, subfactors, group actions, and noncommutative probability.
212 -- Several Complex Variables [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 185 and 202A-202B or their equivalents.
Description: Power series developments, domains of holomorphy, Hartogs' phenomenon, pseudo convexity and plurisubharmonicity. The remainder of the course may treat either sheaf cohomology and Stein
manifolds, or the theory of analytic subvarieties and spaces.
214 -- Differentiable Manifolds [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 202A.
Description: Smooth manifolds and maps, tangent and normal bundles. Sard's theorem and transversality, Whitney embedding theorem. Morse functions, differential forms, Stokes' theorem, Frobenius
theorem. Basic degree theory. Flows, Lie derivative, Lie groups and algebras. Additional topics selected by instructor.
215A-215B -- Algebraic Topology [4;4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 113 and point-set topology (e.g. 202A).
Description: Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. Homotopy theory, fibrations, relations between
homotopy and homology, obstruction theory, and topics from spectral sequences, cohomology operations, and characteristic classes. Sequence begins fall.
C218A -- Probability Theory [4 units]
Course Format: Three hours of lecture per week.
Description: Some knowledge of real analysis and metric spaces, including compactness, Riemann integral. Knowledge of Lebesgue integral and/or elementary probability is helpful, but not essential,
given otherwise strong mathematical background. Measure theory concepts needed for probability. Expectation, distributions. Laws of large numbers and central limit theorems for independent random
variables. Characteristic function methods. Conditional expectations; martingales and theory convergence. Markov chains. Stationary processes. Also listed as Statistics C205A.
C218B -- Probability Theory [4 units]
Course Format: Three hours of lecture per week.
Description: Some knowledge of real analysis and metric spaces, including compactness, Riemann integral. Knowledge of Lebesgue integral and/or elementary probability is helpful, but not essential,
given otherwise strong mathematical background. Measure theory concepts needed for probability. Expectation, distributions. Laws of large numbers and central limit theorems for independent random
variables. Characteristic function methods. Conditional expectations; martingales and theory convergence. Markov chains. Stationary processes. Also listed as Statistics C205B.
219 -- Dynamical Systems [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: 214.
Description: Diffeomorphisms and flows on manifolds. Ergodic theory. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor.
220 -- Introduction to Probabilistic Methods in Mathematics and the Sciences [4 units]
Course Format: Three hours of lecture per week.
Prerequisites: Some familiarity with differential equations and their applications.
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278 -- Topics in Analysis [4 units]
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290 -- Seminars [1-6 units]
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295 -- Individual Research [1-12 units]
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299 -- Reading Course for Graduate Students [1-6 units]
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300 -- Teaching Workshop [3 units]
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Description: Individual study for the comprehensive or language requirements in consultation with the field adviser.
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Description: Individual study in consultation with the major field adviser intended to provide an opportunity for qualified students to prepare themselves for the various examinations required for
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Diophantine approximations and quadratic polynomials
up vote 1 down vote favorite
I am working on a problem these days and the following issue came up. I am not sure yet that I understand it's depth very well, so I would like to discuss a simple case. For those interested, the
problem has applications in coding theory.
Consider a quadratic polynomial $f \left( x_1, x_2, x_3, x_4 \right)$ with real roots and coefficients drawn from a continuous distribution (and therefore irrationals with probability 1). Is there a
strictly positive lower bound on $|f \left( x_1, x_2, x_3, x_4 \right)|$ if we constrain all $x_1, x_2, x_3, x_4$ to lie in $\mathbb{Z}$ ? In other words, is there a $\gamma > 0$ such that $\
displaystyle |f \left( x_1, x_2, x_3, x_4 \right)| \geq \gamma~~ \text{for all} ~~ x_1, x_2, x_3, x_4 \in \mathbb{Z}$ ?
It seems to me that this is a Diophantine approximation - type problem. Note that one can show through a simple application of Khintchine-Groshev theorem that $|f \left( x_1, x_2, x_3, x_4 \right)|$
will be strictly positive for all $x_1, x_2, x_3, x_4 \in \mathbb{Z}$, in that case with probability 1. This is relatively straightforward.
1 What does it mean for a polynomial in more than one variable to have real roots? – Qiaochu Yuan Mar 15 '12 at 22:42
It means that there exist real numbers $x_1^*, x_2^*, x_3^*, x_4^*$ such that $f(x_1^*, x_2^*, x_3^*, x_4^*) = 0$. – Stewart Mar 15 '12 at 22:45
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1 Answer
active oldest votes
I think the answer is no. For example, the Oppenheim conjecture (proved by Margulis in 1987) states that if an indefinite nondegenerate quadratic form has at least 3 variables and it is
up vote 3 not proportional to a rational quadratic form, then its set of values taken at integers are dense in the real line.
down vote
another related theorem - the fractional parts of $\alpha n^2$ are equidistributed in the unit interval (by a refinemnet of the Kronecker lemma, proved by Weyl/Hardy, and by
Frustenberg). I'm guessing that if you take the coeff. if $f$ to be drwan in some iid manner and to be say diophantine generic, you can even get a quantitative version of those
results, that is to get estimate on the maximal sizes of $x_1,\ldots, x_4$ which are needed in order to get $f(x_{1},\ldots,x_{4})$ to be less than some epsilon. – Asaf Mar 15 '12 at
Thanks for the pointer GH, I am already looking into it! Do you maybe know if there is an analogue for higher degree polynomials as well? – Stewart Mar 15 '12 at 23:40
@Stewart: I think there is an analogue for higher degree polynomials as long as the number of variables is large (in terms of the degree). Traditionally these problems are treated by
the Hardy-Littlewood circle method (even when the variables are restricted to special integers such as primes). I am sure MathSciNet or the web helps you find some interesting
literature. Margulis' achievement in the quadratic case was to bring down the number of variables to 3. – GH from MO Mar 15 '12 at 23:43
@GH: My problem in its full generality involves $n$-degree polynomials with $n^2$ variables. So I think I will be able to show something there. Your suggestions seem to be critical,
thank you very much. – Stewart Mar 15 '12 at 23:57
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ion Co
Calculating Intraclass Correlation with AgreeStat 2011.1
Posted : October 24, 2011
AgreeStat 2011.1 for Excel Windows provides the simplest way for researchers to compute the Intraclass Correlation Coefficient (ICC). It is a self-automated workbook containing a Visual Basic for
Applications program, and requires no installation. You simply need to have MS Excel 2007 or 2010 for Windows. The Mac version of AgreeStat 2011.1 will be released in the beginning of 2012. Download
the trial version of AgreeStat 2011.1 here, and you will be surprised to see how easy and intuitive it is.
Numerous versions of the intraclass correlation coefficient are offered for the purpose of evaluating the extent of agreement among multiple raters based on interval or ratio data. Here are 3 key
advantages for using AgreeStat 2011.1 to compute the intraclass correlation coefficients:
• AgreeStat 2011.1 is based on the familiar user-friendly Excel environment within which you can manipulate your data with ease.
• AgreeStat 2011.1 implements the most popular intraclass correlation coefficients introduced by Shrout and Fleiss (1979).
• Unlike the existing statistical packages (including SPSS, SAS, and others), AgreeStat 2011.1 can handle missing values with no problem. Records with a few missing scores are not removed from
analysis. Instead, non-missing from those records are used in the calculation if the intraclass correlation coefficient as suggested by Searle (1997). This results in a more efficient use of your
statical data, and a more accurate evaluation of the extent of agreement among raters.
• In addition to computing inter-rater reliability coefficients based on intraclass correlation, AgreeStat 2011.1 also allows you to compute intra-rater reliability coefficients, which are commonly
used by researchers.
Figure 1 shows AgreeStat 2011.1's main menu, which allows to specify a 2-rater or 3-rater analysis. Although this post focuses on intraclass correlation, AgreeStat 2011.1 allows you to compute
chancecorrected agreement coefficients such as Cohen's kappa, Fleiss' generalized kappa, Gwet's AC[1], Scott's π, and many more, along with their standard errors.
Figure 2 shows how you would capture your data for the purpose of calculating the intraclass correlation coefficient of your choice. You would first select your workbook, then the worksheet within
your workbook, before specifying the data itself.
Figure 3 shows the "Options (ICC)" form that allows you to specify the model you want to use for calculating the intraclass correlation coefficient. Depending on the model of your choice, you would
calculate Shrout-Fleiss' ICC(1,1), ICC(1,k), ICC(2,1), ICC(2,k), ICC(3,1), or ICC(3,k) coefficients (see Shrout and Fleiss, 1979 for more information regarding these coefficients.
Figure 1: AgreeStat's Main Menu
Figure 2: Capturing Data for the Calculation of Intraclass Correlation Coefficients
Figure 3: Specifying Inter-Rater Reliability Model
• Shrout, P.E., and Fleiss, J.L. (1979). "Intraclass Correlations: Uses in Assessing Rater Reliability." Psychological Bulletin, 86, 420-428.
• Searle, S.R. (1997). Linear Models (Wiley Classics Library), Wiley-Interscience: John Wiley & Sons, Inc. (see pages 473- 493)
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Crossover Slopes 24dB per Octave
Frequency Response +/-1dB from 20Hz to 20kHz
Signal to Noise Ratio >100dB
Input Sensitivity 200 millivolts to 2 volts
Input Impedance >10k ohms
Input Voltage Range 10.5 volts to 15.0 volts
Bass EQ 0 to + 12dB @ 45Hz
Typical current draw at idle 750 milliamps
Minimum Impedance 2 ohm bridged/1ohm stereo
Dimensions, XS2200, XS2300 8.85"L x 9.25"W x 2.1"H
Dimensions, XS2500 11.0"L x 9.25"W x 2.1"H
Dimensions, XS4300 12.25"L x 9.25"W x 2.1"H
Dimensions, XS4600 14.25"L x 9.25"W x 2.1"H
Dimensions, XS6600 14.25"L x 11.25"W x 2.1"H
What years where the XS series amplifiers produced?
1997 - Current
Are the XS series amplifiers stable to 2 ohms bridged?
Yes, but there are a couple of things you MUST consider when running an XS amp at 2 ohms bridged. The amplifier must have a steady supply of current from the car's electrical system so you'll want to
run a minimum of 4 gauge power cable from front to rear. If the main cable run is longer than about 10 feet (most are), then you'll want to step up to 2 gauge. A capacitor is HIGHLY recommended for 2
ohm bridged operation. The capacitor should have at least 1 farad of capacitance. The only other consideration is heat. With more power comes more heat so you may have to have some form of cooling
fan system to prevent thermal shut down.
How much power does an XS series amplifier provide at 2 ohms bridged?
The continuous power at 2 ohms or 1 ohm bridged remains about the same.
When designing the XS series amps, we wanted a design that made its best power at normal loads. By normal, we mean 4 ohms bridged. We know that no matter what we say, or how loud we say it - People
will still hook the amp up to lower impedances trying to get more power. So, our number one goal is that the amp continue to operate at lower impedances. The only way to do that is to limit the
continuous power output of the amp when it's hooked up to lower impedances. Here's why -
Wattage is equal to voltage times current. Running the amp at 2 ohms bridged means more output current. Current makes heat. We can't allow the amp to make more heat than the heatsink can reasonably
dissipate. With the addition of cooling fans, the heatsink can manage a more heat than what's generated from a 4 ohm bridged load so there's some room for a bit more power. Here's how we limit the
power increase to what's manageable:
The XS series amps have a circuit that detects the amount of current leaving the speaker terminals. If the current is high enough, AND lasts long enough (longer than 50 milliseconds), the circuit
limits the amount of voltage that the power supply can produce. This in turn limits the power output of the amp. So continuous power output remains about the same with the power composed of more
current and less voltage.
There is a bright side to all this. Remember I said that the excess current demand had to last longer than 50ms. The dynamic peaks in most music last less than 20ms. Therefore, the circuit never has
a chance to affect the power supply voltage for musical peaks. All this means that the amp is allowed to make more power with musical peaks while continuous power remains about the same. You could
say that the "headroom" of the amp is increased.
If you have to run the amp 2 ohms bridged, you can. Just understand that you must follow the above suggestions to make sure your amplifier isn't damaged or not operating correctly. I wouldn't
recommend running it 1 ohm bridged. The amp will shut down (red LED) if the output current gets too high (looks like a shorted speaker). | {"url":"http://webfaq.phoenixphorum.com/XS.htm","timestamp":"2014-04-20T21:12:26Z","content_type":null,"content_length":"31394","record_id":"<urn:uuid:4fc2dd13-e790-4d12-9e14-3cd5bdf7ea95>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00531-ip-10-147-4-33.ec2.internal.warc.gz"} |
Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry
Has Miller restored faith in Euclidean geometry by revealing a natural foundation upon which it rests? This is a question that could be asked after reading his book. Miller assesses his work in this
Our formalization of Euclid’s proof methods is useful for several reasons: it lets us better understand his proofs; it allows us to prove metamathematical results about these kinds of proofs; and
it shows that there is no inherent reason that modern foundations of geometry must look completely different from the ancient foundations found in the Elements. (p. 4)
In the title of his book, Miller alludes to Charles L. Dodgson’s (i.e., Lewis Carroll’s)
Euclid and his Modern Rivals
(1885). The famous author contended that none of Euclid’s rivals of Dodgson’s time were, in Miller’s paraphrase, “a more appropriate basis for the study of a beginning geometry student” (p. 1). Of
his own work, Miller remarks:
[T]he aims of this book are not far removed from Dodgson’s aims in 1879: to show that, while modern developments of logic and geometry may require changes in Euclid’s development, his basic ideas
are neither outdated nor obsolete. (p. 4)
I provide a very rough indication of Miller’s formalization (which is too extensive to present in this review). Miller’s goal is ‘to give…a formal system in which Euclid’s proofs can be duplicated’
(p. 13), in this way demonstrating their correctness. He provides a way of proving Euclid’s propositions by capturing formally the topology of Euclid’s geometry, by generating topological cases, and
then by eliminating cases from consideration, leaving a solution. This formalization, Miller claims, brings Euclid into the realm of completely formal systems. (‘The simplest possible definition that
we can adopt here is that a system is completely formal if it can be completely implemented on a computer’ (p. 13).) Miller remarks:
A crucial idea will be that all crucial information given by a diagram is contained in its topology, on the general arrangement of its points and lines in the plane. Another way of saying this is
that if one diagram can be transformed into another by stretching, then the two diagrams are essentially the same. This is typical of diagrammatic reasoning, and, although it has not been
previously treated formally, this idea has a long tradition in geometry. Proclus, the fifth century commentator on Euclid, writes that each case in a geometric proof ‘announces different ways of
construction and alternative positions due to the transposition of points or lines or planes or solids.’ (This is Sir Thomas Heath’s translation, given in Euclid (1956).) Thus, we see that the
idea of considering cases to be different when the arrangements of the geometric object being considered are topologically different is an ancient one. (p. 4)
With regard to SAS (the side-angle-side criterion for triangle congruence),
[t]he proof is essentially identical to Euclid’s proof, with a lot of tedious extra cases showing all of the ways that the triangles could possibly intersect. The idea is to move the two
triangles together using the symmetry transformations and to then check that they must be completely superimposed. (p. 44)
Figure 1
More specifically,
[t]he first step [of the proof] is to apply rule S1 to the diagram in [Figure 1], moving triangle ABC so that the imageA'B' of AB lies along DE. The possible cases that result are shown in the
diagram arrays in [Figure 2] and [Figure 3]. For the sake of readability, many markings have been left off these diagrams, although all markings that are later needed have been left. (p. 25)
Note that curved lines are needed to depict many of the topological cases. (Does this distance Miller’s treatment from Euclid? — a worry raised in my mind.) Also note that the generation of cases
takes place within Miller’s formalism (which he has implemented with a computer algorithm). The software-generated diagrams assist human understanding. Figure 4 magnifies four diagrams in the
upper-left corner of Figure 2.
Figure 2
Figure 3
Figure 4
Norman’s second step ‘remove[s] all of the extra cases using the rules of inference CA and CS’ (p. 45). Informally, a diagram is removed by inference CA if in the diagram there is a line properly
contained in another line of equal length. Similarly, informally, a diagram is removed by inference CS if in the diagram there is an angle properly contained in another angle of equal degree. Norman
observes that ‘[a]ll the diagrams shown in [Figure 2] except the very first one can be eliminated by applying CS to A'B'and DE’ (p. 45). Furthermore, all the cases in Figures 2 and 3 can be
eliminated using CA or CS except the first one in Figure 2. The first diagram in Figure 2, then, is ‘provable’ from the diagram in Figure 1. Miller gives a formal definition of provable. Indeed, the
mathematical work is done by the formalism, implemented by a computer algorithm.
Miller’s book seems to have thrown new light — philosophical and mathematical — on an old classic.
Dodgson, Charles. 1885. Euclid and his Modern Rivals. London: MacMillan and Co., 2^nd edition.
Euclid. 1956. The Elements . Translated with an introduction by Thomas L. Heath.
Dennis Lomas (dlomas@upei.ca) has studied computer science (MSc), mathematics (half dozen, or so, graduate courses), and philosophy (PhD). Retired, he does some part-time teaching at the University
of Prince Edward Island. | {"url":"http://www.maa.org/publications/maa-reviews/euclid-and-his-twentieth-century-rivals-diagrams-in-the-logic-of-euclidean-geometry","timestamp":"2014-04-17T22:06:21Z","content_type":null,"content_length":"101736","record_id":"<urn:uuid:a886888f-ee22-4a53-babf-9ca5e54a4dd9>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00405-ip-10-147-4-33.ec2.internal.warc.gz"} |
Finding the Solution Set for an Inequality
1. I need to find the solution set for |(3x+2)/(x+3)|>3.
3. When I solve the inequality (3x+2)/(x+3)>3, I get 2>9 which is clearly false. When I solve the inequality (3x+2)/(x+3)> -3, I come with the solution set (-inf. -11/6). My teacher is saying that
there the solutotion set is (-inf. -3)U(-3, -11/6).
I just can't figure out how to get to that solution. I can't figure out where that -3 is coming from. In his sparse notes on my assignment, he says there are two subcases for each of the two cases in
number 1. Those are when x < -3 and when x > -3. I just cant' figure out how to use these cases. | {"url":"http://www.physicsforums.com/showthread.php?t=361924","timestamp":"2014-04-18T18:23:34Z","content_type":null,"content_length":"25658","record_id":"<urn:uuid:f9f0f634-8d81-4bd0-80b2-2fa4a0a84b9b>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00433-ip-10-147-4-33.ec2.internal.warc.gz"} |
Geometric Transformations - Lesson Plan
• Given a figure (pre-image) and a transformation specify the figure resulting from the transformation (image).
• Given a transformed figure (image) and a transformation, specify the original figure (pre-image).
• Given a figure (pre-image) and a transformation draw the result of the transformation.
• Given two figures, and a transformation recognize whether or not the second figure is the result of applying the transformation to the first figure.
• Given a figure (pre-image) and another figure that is a transformation of the first (image), describe the transformation that generates the second figure.
• Given a figure, recognize whether or not it has line and point symmetry and specify points and lines of symmetry.
• Recognize and produce instances of the concepts: pre-image, image,line of reflection, center of rotation, rotation angle,translation vector, center of dilation, dilation factor.
Types of transformations addressed are: translations, rotations, reflections, and dilations.Figures are referred to using both vertex names and cartesian coordinates.
Teaching Plan (1 week)
DAY 1 (50 Minutes)
Mappings (10 minutes)
The objectives of this lesson are to help students identify images and preimages of a mapping and recognize an isometry.
This lesson could also be taught as a presentation with the whole class working together.
1. Using a computer projector, demonstrate how to go to the eNLVM website, go to your school and class, and login. (5 minutes)
2. (2 activities) Direct students to Mappings: Introduction. Use the first two pages to demonstrate to students how to enter their answer and move between activites. Present some background on
Esher and tesselations. Ask students to enter their observations and continue through the lesson on their own.
3. (4 activities) Students are presented with the definition of preimage and image. They are also introduced to the 4 transformations taught in the eModule and are asked to identify characteristics
of each.
4. (1 activity) Students are presented with the definition of a mapping and a transformation.
5. (1 activity) Students identify different transformations.
6. (2 activities) Students are introduced to mapping notation.
7. (2 activities) Students are presented with a definition of congruence, isometry, and similarity. They are asked to identify if a translation is congruent or similar and justify their answer.
8. (1 activity) Students are presented with how to prove that a mapping is an isometry
9. (1 activity) Questions check for the students' understanding of concepts and terms.
10. When students have completed the Introduction, bring the class together and discuss the concepts they learned. Asess their understanding and clarify questions before moving onto the practice.
1. (5 activities) Direct students to the Practice section. Students will name and identify images and preimages.
2. (1 activity) Students create figures by rotating, translating, and flipping (reflecting) blocks, then identify the number of flips needed.
3. (1 activity) Students identify isometries.
4. (1 activity) Students identify a rotation and create a tessalation.
5. (1 activity) Students describe a scene with transformations.
6. (1 activity) Students pick a transformation and create a scene.
Reflections (40 - 50 minutes)
The objective of the second lesson is to name a reflection image, to recognize line and point symmetry, and to draw a reflection image.
1. Direct students to comple the section on reflections. Observe the students working and provide help and clarifcation when needed.
2. (3 activities) Students are presented with examples and definitions of reflection, line of reflection, and point of reflection.
3. (3 activities) Students are presented with the following properties of reflections: reflections preserve collinearity, betweeness, angle and distance measures.
4. (1 activity) Students are shown how to construct a reflection image over a line.
5. (1 activity) Students are presented with the definition of line of symmetry.
6. (2 activities) Students are taught how to identify a line of symmetry.
7. (2 activities) Students practice identifying lines of symmetry.
8. (2 activities) Students are presented with the definitintion of point of symmetry.
9. (3 activities) Students are presented with examples of points of symmetry.
10. (1 activities) Students are given questions that check their understanding of concepts. Bring the class together if you want to provide any clarification.
1. (1 activity) Students name reflections.
2. (2 activities) Students draw the reflection image of figures across a line.
3. (2 activities) Students identify lines of symmetry.
4. (1 activity) Students draw lines of symmetry.
5. (1 activity) Students identify lines and points of symmetry.
6. (2 activities) Students draw reflection images through a point.
7. (2 activities) Students describe how to check for a line of symmetry using a reflection and a point symmetry using rotation.
8. (2 activities) Students create a figure with point symmetry or line symmetry.
9. (17 activities) If you identify a need or if a student has extra time, a section of Extra Practice is available.
DAY 2 (50 Minutes)
Translations (50 minutes)
The objective of this lesson is to name a translation image with respect to parallel lines and to draw a such a translation image.
1. Review the concepts of the previous day and address any problems you observed. Direct students to complete the Translations section. Continue to observe their work and correct and clarify as
2. (1 activitiy) Students are presented with an example and a definition of a translation.
3. (2 activities) Students are shown how to construct a translation as a reflection over two parallel lines.
4. (1 activity) Students are presented with properties preserved by reflections. Students are asked to recall the difference between an isometry and a similarity transform.
5. (1 activity) Students are shown how to build 3D images with translations.
6. (1 activity) Students work with an example of how translations can be used in tessalation wallpaper.
1. (2 activities) Students identify translations.
2. (2 activities) Students draw the translation images of figures across parallel lines.
3. (2 activities) Students identify translation images.
4. (1 activity) Students name reflection and translation images.
5. (2 activities) Students draw translation images with respect to two parallel lines.
6. (1 activity) Students plan a proof that translations preserve certain qualities.
7. (1 activitiy) Students create 3D images with translations.
8. (1 activity) Students create a translation.
9. (10 activities) If you identify a need or if a student has extra time, a section of Extra Practice is available.
DAY 3 (50 Minutes)
Rotations (50 minutes)
The objective of this lesson is to help students name a rotation image with respect to intersecting lines and to draw a such a rotation image.
1. Review the concepts of the previous day and address any problems you observed. Direct students to complete the Rotations section. Continue to observe their work and correct and clarify as
2. (2 activities) Students are presented with the definitions of rotation and angle of rotation.
3. (2 activities) Students identify angle of rotation and construct rotation images.
4. (1 activity) Students identify the relationship between angle of rotation and an angle between intersecting lines.
5. (1 activity) Students identify angles of rotation.
6. (1 activity) Students are presented questions that check their understanting of the definitions and concepts. Bring the class together if you wish to provide any clarification.
1. (1 activity) Students name reflection and rotation images.
2. (2 activities) Students draw the rotation images of figures across intersecting lines.
3. (2 activities) Students identify rotations and angles of rotation.
4. (2 activities) Students determine angle of rotation.
5. (1 activity) Students identify which properties are preserved by rotation.
6. (1 activity) Students identify rotations and reflections.
7. (1 activitiy) Students create 3D images with translations.
8. (7activities) If you identify a need, or if a student has extra time, a section of Extra Practice is available.
DAY 4 (50 Minutes)
Dilations (50 minutes)
The objective of this lesson is to help students understand the meaning of scale factors, identify the center and scale factor of a dilation, and draw a dilation given a center and scale factors.
1. Review the concepts of the previous day and address any problems you observed. Direct students to complete the Dilations section. Continue to observe their work and correct and clarify as
2. (2 activities) Students are presented with the definition of a dilation or similarity transform and the scale factor of a dilation.
3. (3 activities) Students explore the effect of the scale factor on a dilation.
4. (1 activity) Students are shown how to construct a dilation image on a geoboard.
5. (2 activities) Students are shown how to identify the center of dilation and the scale factor.
6. (1 activity) Students practice finding a center and scale factors
1. (2 activities) Students find scale factors given a dilation.
2. (2 activities) Given a dilation, students find the center of dilation and scale factor.
3. (4 activities) Students construct a dilation image of a figure given the center and scale factor.
4. (1 activity) Students identify which transformations will move the ball to win a game of golf..
Review the content periodically as misunderstandings become apparent. Spend time after all the lessons are completed to review properties of each transformation and how each transformation is
constructed before having students complete the quiz.
Ask students to complete the online Quiz. If they have time, ask them to complete the Student Feedback Questionaire as well.
│ Lesson Design │ Joel Duffin, Jean Culbertson │
│ Web Development │ Joel Duffin, Liz Hart │
│ Mathlets │ Argyll Home Education Centre, MyMaths, NLVM, Shodor Foundation Inc., Stephen Webber │
│ Images │ Denise Chandler, US Library of Congress │
Correlation to Standards
Correlation to NCTM Standards
Geometry - Standard 3. Apply transformations and use symmetry to analyze mathematical situations.
• Grades PreK-2
□ Recognize and apply slides, flips, and turns.
□ Recognize and create shapes that have symmetry.
• Grades 3-5
□ Predict and describe the results of sliding, flipping, and turning two-dimensional shapes.
□ Describe a motion or a series of motions that will show that two shapes are congruent.
□ Identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs.
• Grades 6-8
□ Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.
□ Examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
• Grades 9-12
□ Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices.
□ Use various representations to help understand the effects of simple transformations and their compositions.
Correlation to Utah Standards
• Math 3 - 3.3 Visualize and identify geometric shapes after applying transformations.
□ Demonstrate the effect of a slide (translation) or flip (reflection) on a figure, using manipulatives.
□ Determine whether two polygons are congruent by sliding, flipping, or turning to physically fit one object on top of the other.
• Math 4 - 3.3 Visualize and identify geometric shapes after applying transformations.
□ Identify a slide (translation) or a flip (reflection) of a geometric shape using manipulatives.
• Math 5 - 3.3 Visualize and identify geometric shapes after applying transformations.
□ Identify a slide (translation) or a flip (reflection) of a shape across a line.
□ Demonstrate the effect of a turn (rotation) on a figure using manipulatives.
• Math 6 - 3.3 Visualize and identify geometric shapes after applying transformations.
□ Turn (rotate) a shape around a point and identify the location of the new vertices.
□ Slide (translate) a polygon either horizontally or vertically on a coordinate grid and identify the location of the new vertices.
□ Flip (reflect) a shape across either the x- or y-axis and identify the location of the new vertices.
• Math 7 - 3.3 Visualize and identify geometric shapes after applying transformations, and identify lines of symmetry.
□ Identify line(s) of symmetry in plane figures.
□ Transform geometric shapes using translations (slides), rotations (turns), and reflections (flips).
• Geometry - 3.2.4. Perform and analyze transformations (translations, rotations, reflections, and dilations) using coordinate geometry.
Correlation to Arizona Standards
Strand 4. Concept 2. Apply spatial reasoning to create transformations and use symmetry to analyze mathematical situations.
• Grade 4
□ Demonstrate translation using geometric figures.
• Grade 5
□ Demonstrate reflections using geometric figures.
□ Describe the transformations that created a tessellation.
• Grade 6
□ Identify reflections, and translations using pictures.
□ Perform elementary transformations to create a tessellation.
• Grade 7
□ Identify rotations about a point, using pictorial models.
□ Recognize simple single rotations, translations or reflections on a coordinate grid.
• Grade 8
□ Identify the planar geometric figure that is the result of a given rigid transformation.
□ Model a simple transformation on a coordinate grid
• High School
□ Identify the properties of the planar figure that is the result of two or more transformations.
□ Determine whether a given pair of figures on a coordinate plane represents a translation, reflection, rotation, or dilation.
□ Classify transformations based on whether they produce congruent or similar figures.
□ Determine the effects of a single transformation on linear or area measurements of a planar geometric figure. | {"url":"http://www.enlvm.usu.edu/ma/classes/__shared/emready@transformations/info/lessonplan.html","timestamp":"2014-04-19T10:24:53Z","content_type":null,"content_length":"28467","record_id":"<urn:uuid:793e7f1b-b83d-4ed2-aab6-04257d50815d>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00219-ip-10-147-4-33.ec2.internal.warc.gz"} |
Auto-insert row based on cell input.
Once again, you have a posted a unclear requirement.
i.e. "A1 row: =A2-A1. the inserted A2 row: =A3-A2"
First, A1 and A2 are not rows, A1 and A2 are cells. I don't know what you mean by "A1 row" and "A2 row"
If you mean the cells A1 and A2, then you have another problem. You can't have =A2-A1 in A1 or =A3-A2 in A2.
That will cause a circular reference since you are asking Excel to calculate a value using a value from the same cell in which you want the answer. That can't be done.
Click Here Before Posting Data or VBA Code ---> How To Post Data or Code. | {"url":"http://www.computing.net/answers/office/autoinsert-row-based-on-cell-input/17586.html","timestamp":"2014-04-21T14:55:34Z","content_type":null,"content_length":"41929","record_id":"<urn:uuid:4b1e2a28-9903-4782-b1cd-6f0cfbb18299>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00515-ip-10-147-4-33.ec2.internal.warc.gz"} |
Engineering 107 > Hassan > Notes > Lec_15_S07.ppt | StudyBlue
ENGR 107: Engineering Fundamentals Lecture 15 Chap 6: Dimensions, Units, and Conversions Department of Electrical and Computer Engineering George Mason University References http://go.hrw.com/
resources/go_sc/hst/HP1PE729.PDF http://www.dot.state.ak.us/stwddes/research/assets/documents/rfps/examples/metric.doc http://www.hazelwood.k12.mo.us/~grichert/sciweb/convert.htm http://www.epa.gov/
ORD/NRMRL/pubs/600r01106/600R01106appA.pdf SI Unit Conversion Tables Paper & on-line Physical Quantities Fundamental Dimensions: For example: Length (L) = 20 meters Units: magnitude of physical
quantities only be understood when compared with pre-determined reference amounts called units Time Force Length Temperature Mass These are abstract qualities of measurement without scale, (i.e.
length) Dimensions Units are used to describe physical quantities can be conveniently and usefully manipulated when expressing all physical quantities of a particular field of science or engineering
e.g. length (meter) Fundamental Dimensions: Precision today: Inch: Cubit: tip of middle finger to the elbow distance covered by three barley corns, round and dry, laid end to end Length of a meter is
determined by the distance traveled by light in a vacuum during a specified amount of time Derived Demensions a combination of fundamental dimensions e.g. velocity: length/time (mps) i.e., a
combination of two fundamental dimensions e.g. In this case, length and time Derived Dimensions: Units Two fundamental systems of units are used in the world today: The US has adopted this standard
and is in the very slow process of implementing this system The metric system,modified over the years now called the System International (SI). SI is now considered the new international standard
system of units 2. Engineering System used in the United States ? is a system based on the foot, foot-pound, and second 1. Metric System used in almost every industrial country in the world today ?
it?s a decimal, absolute, system based on the meter, kilogram, and second (MKS) International System Divided into three classes of units Supplementary units Base units there are seven base units in
the SI system Derived units ? combination of units formed by combining base, supplementary, and other derived units The International System of Units (SI) was developed and maintained by the General
Conference on Weights and Measures and intended as a basis for worldwide standardization of measurements. The key element is that it?s a decimal system Base Units Quantity Name Symbol Length meter m
Mass kilogram kg Time time s Electric current ampere A Thermodynamic temp kelvin K Amount of substance mole mol Luminous intensity candela cd Supplementary Units Quantity Name Symbol Plane angle
radian rad Solid angle steradian sr Plane angle: a rad is the plane angle between two radii of a circle that cuts off a circumference of an arc equal in length to the radius. A circle subtends two pi
radians about the origin. Steradian: measure of the angular ?area? subtended by a two dimensional surface about the origin in three dimensional space. A sphere subtends 4 pi steradians about the
origin. Derived Units Frequency Hz hertz s-1 Force N newton kg*m*s-2 Pressure stress Pa pascal kg*m-1*s-2 Energy or work J joule kg*m2**s-2 Quantity of heat J joule kg*m2*s-2 Power radiant flux W
watt kg*m2*s-3 Electric charge C coulomb A*s Electric Potential V volt kg*m2*s-3*A-1 Potential difference V volt kg*m2*s-3*-1 Electromotive force V volt kg*m2*s-3*A-1 Capacitance F farad A2*s4*kg-1*m
-2 Quantity SI Unit Name Base Units Symbol Derived Units Electric resistance ? ohm kg*m2*s-3*A-2 Conductance S siemens kg-1*m2*s3*A2 Magnetic flux Wb weber kg-1*m*s-2*A-1 Magnetic flux density T
tesla kg*s2*A-1 Inductance H henry kg*m2*s-2*A-2 Luminous flux lm lumen cd*sn Illuminance lx lux cd*sn*m-2 Celsius temperature C degree Celsius K Activity (radionuclides) Bq becqueret s -1 Adsorbed
dose Gy gray m2*s -2 Dose equivalent S sievert m 2*s -2 Multiplier SI Unit Name Base Units Symbol Decimal Multiples 1018 exa E 1015 peta P 1012 tera T 109 giga G 106 mega M 103 kilo k 102 hecto h 101
deka da 10-1 deci d 10-2 centi c 10-3 milli m 10-6 micro U 10-9 nano N 10-12 pico p Quantity Prefix name Symbol Rarely used Engineering Notation Engineering Notation is expressed in powers that are
multiples of 103 For example: 12 345 678* Ordinary Notation Scientific Notation Engineering Notation 1.234 567 8 x 107 12.345 678 x 106 * An exact number 0.039 A 3.9 x 10-2 A 0.39 x 10-3 A 12 M? = ?
39 mA Rules for Using SI Units No periods after symbols except at the end of a sentence Unit symbols written in lowercase unless the symbols are derived from a proper name Lowercase Uppercase m, kg,
s, mol, cd A, K, Hz, Pa, C Symbols rather than self-styled abbreviations should be used to represent units Correct Not Correct A amp s sec SI Rules (continued) ?s? never added to the symbol (plural)
Space between numerical values and the unit symbol Correct Not Correct 43.7 km 43.7km 0.25 Pa 0.25Pa Exception: no space between the numerical values and the symbols for: degree, minute, and second
of angles and for degree Celsius No space between the prefix and the unit symbols Correct Not Correct mm, Mohm k m , ? F SI Rules (continued) Unit names: lowercase all letters except at the beginning
of a sentence, even if the unit is derived from a proper name Plurals: required when writing unit names. Example Singular Plural henry henries exceptions are noted: Singular Plural lux lux hertz
hertz siemens siemens No hyphen or space between a prefix and the unit name. In three cases the final vowel is omitted: megohm, kilohm, and hectare SI Rules (continued) Symbols used in preference to
unit names; symbols are standardized An exception is made when a number is written, in that case the unit name should be used. For example: ten meters not ten m and 10 m not 10 meters Multiplication
and Division When writing unit names as a product, always use a space (preferred) or a hyphen Correct Usage Newton meter or newton-meter When expressing a quotient using unit names, always use the
word per and not a solidus (/). The solidus or slash is reserved for use with symbols. Correct Not Correct meter per second meter / second mps or m/s Multiplication and Division When writing a unit
name that requires a power, use a modifier, such as squared or cubed, after the unit name. For an area or volume the modifier can be placed before the unit name. Correct Not Correct degrees squared
square degrees When expressing products using unit symbols, the center dot is preferred. Preferred N?m for newton meter or newton-meter Multiplication and Division SI Rules (Cont) Example: m/s or m?
s-1 or m s In more complicated cases, use negative powers or parentheses, e.g. m/s2 or m·s-2 but not m/s/s for acceleration When denoting a quotient by unit symbols, any of the following methods are
accepted form. Numbers To denote a decimal point: period When expressing numbers less than one: zero before the decimal Examples: 15.6 or 0.93 Comma is used in many countries to denote a decimal
point Avoid using commas to separate data into groups of three, instead, counting from the decimal to the right or left insert a space Correct and Recommended Procedure 6.513 824 76 851 7 434 296
0.187 62 $5649733250 Try reading these numbers: $5 649 733 250 or Calculating with SI Units One unit is used to represent each physical quantity, e.g. meter for length. SI unit are coherent; that is,
each new derived unit is a product or quotient of the fundamental and supplemental units without any numerical factors. A newton is the force required to impart an acceleration of one meter per
second squared (1.0 m/s2) to a mass of one kilogram (1.0 kg) 1 N = (1.0 kg) (1.0 m/s2) Applying Newton?s second law F = ma/gc, where gc = ma/F or gc = 1.0 kg · 1.0 m N · s2 Calculations Using SI
Fundamental relationships are simple and easier to use because of coherence. Recognize how to manipulate units. e.g. Since Watt = J/s = N·m/s you should realize that N·m/s = (n/m2)(m3/s) = (pressure)
(volume flow rate) Understand the advantage of adjusting all variables to base units such as, replacing N with kg·m/s2 or Pa with kg·m-1·s-2 and so on. Develop a proficiency with exponential notation
to be used with unit prefixes. (1 mm)3 = (10-3 m)3 = 10-9 m3 1 ns-1 = (10-9 s)-1 = s-1 109 Non-SI Units Accepted for Use in the US Time minute hour day min h d 60 s 3 600 s 86 400 s Plane angle
degree minute second º ' " ?/180 rad ?/10 800 rad ?/648 000 rad Volume Liter L or l 10-3 m3 Mass metric ton unified atomic mass unit t m 103 kg 1.660 57x10-27 kg L is recommended Non-SI Units
Accepted for Use in the US Land area Hectare ha 104 m2 Energy electronvolt eV 1.602x10-19 J (approx) SI Rules Use the letter ?L? not ?l? to represent Liters. What does this mean? 16.007 l m Does this
mean 16.007 liter meters? Does it mean (16.007 l) meters ? There is no way you can tell without using L for liters. 16.007 L leaves no doubt. Example Problem 6.1 Problem: A weight of 100 kg (the unit
itself indicates mass) is suspended by a rope. Calculate the tension in the rope in newtons to hold the mass stationary when the local gravitational acceleration is 9.087 m/s2 1.63 m/s2 Theory:
Tension in the rope or the force required to hold the object when the mass is at rest or moving at constant velocity is: F= mgL/gc Example Problem 6.1 Continued F= mgL/gc Where gL replaces a and is
the local acceleration of gravity, gc is the proportionality constant, and m is the mass of the object. Due to coherence, gc = 1.0 kg?m N ?s2 Assumptions: Neglect the mass of the rope Example Problem
6.1 Continued 1.000 x 102 kg Solution a) For gL = 9.807 m/s2 F = mgL/gC = (100 kg )(9.807m/s2)/ 1.0 kg ? m/N ? s2 = 980.7 N F = (100 kg) (9.807 m) x 1 (N ? s2) = 980.7 N s2 kg ? m Example Problem 6.1
Continued 1.00 x 102 kg F = (100 kg) (1.63 m) x 1 (N ? s2) = 163 ? s2 kg ? m b) For gL = 1.63 m/s2 F = mgL/gC = (100 kg )(1.63 m/s2)/ 1.0 kg ? m/N ? s2 = .163 kN Common Conversion Factors C x 9/5+32
= F (F- 32) x 5/9 = C C x 9/5 +32 = F C +273 = K F +460 = K See conversion tables, Eide Text beginning page 501 References at the beginning of the lesson, most complete is ref. 4 No parentheses
required, apply PEMDAS Parentheses required, apply PEMDAS No parentheses required, apply PEMDAS Conversion of Units Remember: a conversion factor relates two equal physical quantities Example: 1 m =
3.208 8 ft Convert 2.97 ft to m 2.97 ft 1m 1 3.208 8 ft Pretty straight forward, but what if I said the conversion factor between feet and meters is 0.3048? Convert 4.8 feet to meters. 4.8 ft 1 m ft
Are there 0.3048 ft in a m? Are there 0.3048 m in a ft? 0.3048 1 = 1.463 m = 0.926 m = 1.5 m The conversion relates two equal physical quantities. Conversion of Units Using the tables of Unit
Conversions beginning on page 501 of the Eide text, convert: Convert 4.80 ft to m Convert 4.80 ft Multiply: By: To Obtain 3.048 x 10-1 m From page 502 = 1.463 m 4.80 ft 3.048 x 10-1 m 1 ft = 1.46 m
Conversion of Temperatures Convert 60.0° F to degrees C (F- 32) x 5/9 = C (60.0 ? 32.0)(5/9) = Convert 15.6° C to degrees K C +273 = K = 15.6 ° C 15.5556 15.6 ° C +273 = 288.6 K 15.6 ° C +273 = = 289
K Note: no degree symbol used with Kelvin Engineering System Units Quality Unit Symbol Mass pound-mass lbm Length foot ft Time second s Force pound-force lbf Conversion of Units Example problem 6.2
Convert 6.7 in to millimeters Solution: Write the identity - 6.7 in = 6.7 in 1 Then multiply by the appropriate conversion factor. 6.7 in = 6.7 in * 25.4 mm = 1.7 x 102 mm 1 1 in Conversion of Units
Example Problem. 6.3 Convert 85.0 lbm/ft3 to kilograms per cubic meter. Solution: 85.0 lbm/ft3 = 85.0 lbm (1 ft)3 0.453 6 kg 1 ft3 (0.3048m)3 lbm = 1.36 x 103 kg/m3 Note errors in text, p219: 85.0
lbm not 850 lbm and 0.304 8 not 6.304 8 m /ft Conversion of Units Example Problem 6.4 Determine the gravitational force (in newtons) on an automobile with a mass of 3 645 lbm. The acceleration of
gravity is known to be 32.2 ft/s2. Theory: F = mgL/gC Solution: m = 3 645 lbm 1 kg = 1 653 kg 1 2.204 6 lbm Conversion of Units Example Problem 6.4 continued gL = 32.2 ft 0.304 8 m = 9.814 6m/s2 1 s
2 1 ft = 1.62 x104 N F = (1 653 kg)(9.814 6 m/s2) N ?s2 1 kg?m F = (1 653.36 kg)(9.814 6 m/s2) / (1.0 kg?m/N?s2) F = mgL/gC where m = 1.653 kg Alternate Solution Example Problem 6.4 Solution: F = mgL
/ gC = 3 645 lbm 32.2 ft 1 kg 0.304 8 m 1N?s2 1 1 s2 2.204 6 lbm 1 ft 1.0 kg?m F = 16.2 kN Conversion of Units Example Problem 6.5 Convert a mass flow rate of 195 kg/s (typical of the airflow through
a turbofan) to slugs per minute. 782 slugs min 1 slug = 14.594 kg 60 s 1 min Solution: 195 kg/s = 195 kg 1 s From conversion tables of Eide text Page 505: Multiply by To Obtain Slugs 1.459 4 x 101 kg
Conversion of Units Example 6.6 Compute the power output of a 225 hp engine in a) Btu and b) kilowatts Solution 225 hp = 225 hp 2.546 1 x 103 Btu 1 h = 9.55x103 Btu/min 1 1hp?h 60 min b) 225hp = 225
hp 0.745 70 kW = 168 kW 1 1 hp Homework Assignment Study: Data Analysis and Statistics (Chapter 8, Eide, et al; pages 251 ? 268 Problems: 5.1a,d,g; 5.2 b, i; 5.4b; 6.1b,f; 6.2 d, f; 6.3b,c; 6.7b | {"url":"http://www.studyblue.com/notes/note/n/lec_15_s07ppt/file/323331","timestamp":"2014-04-18T20:43:57Z","content_type":null,"content_length":"43327","record_id":"<urn:uuid:7de02c85-789c-42b1-874f-5d193f9aaedd>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00154-ip-10-147-4-33.ec2.internal.warc.gz"} |
The exponential eigenmodes of the carbon-climate system, and their implications for ratios of responses to forcingsCSIRO Marine and Atmospheric Research, Canberra, ACT 2601, Australia
Abstract. Several basic ratios of responses to forcings in the carbon-climate system are observed to be relatively steady. Examples include the CO[2] airborne fraction (the fraction of
the total anthropogenic CO[2] emission flux that accumulates in the atmosphere) and the ratio T/Q[E] of warming (T) to cumulative total CO[2] emissions (Q[E]). This paper explores the
reason for such near-constancy in the past, and its likely limitations in future.
The contemporary carbon-climate system is often approximated as a set of first-order linear systems, for example in response-function descriptions. All such linear systems have
exponential eigenfunctions in time (an eigenfunction being one that, if applied to the system as a forcing, produces a response of the same shape). This implies that, if the
carbon-climate system is idealised as a linear system (Lin) forced by exponentially growing CO[2] emissions (Exp), then all ratios of responses to forcings are constant. Important cases
Definitions are the CO[2] airborne fraction (AF), the cumulative airborne fraction (CAF), other CO[2] partition fractions and cumulative partition fractions into land and ocean stores, the CO[2]
sink uptake rate (k[S], the combined land and ocean CO[2] sink flux per unit excess atmospheric CO[2]), and the ratio T/Q[E]. Further, the AF and the CAF are equal. Since the Lin and Exp
idealisations apply approximately to the carbon-climate system over the past two centuries, the theory explains the observed near-constancy of the AF, CAF and T/Q[E] in this period.
A nonlinear carbon-climate model is used to explore how future breakdown of both the Lin and Exp idealisations will cause the AF, CAF and k[S] to depart significantly from constancy, in
ways that depend on CO[2] emissions scenarios. However, T/Q[E] remains approximately constant in typical scenarios, because of compensating interactions between CO[2] emissions
trajectories, carbon-climate nonlinearities (in land–air and ocean–air carbon exchanges and CO[2] radiative forcing), and emissions trajectories for non-CO[2] gases. This theory
establishes a basis for the widely assumed proportionality between T and Q[E], and identifies the limits of this relationship.
Citation: Raupach, M. R.: The exponential eigenmodes of the carbon-climate system, and their implications for ratios of responses to forcings, Earth Syst. Dynam., 4, 31-49, doi:10.5194/
esd-4-31-2013, 2013. | {"url":"http://www.earth-syst-dynam.net/4/31/2013/esd-4-31-2013.html","timestamp":"2014-04-17T06:45:02Z","content_type":null,"content_length":"25133","record_id":"<urn:uuid:dca66952-d538-4f6b-9ed6-cf9ad9e194d4>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00209-ip-10-147-4-33.ec2.internal.warc.gz"} |