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Beauty in mathematics
June 2009
This article is the winner of the schools category of the Plus new writers award 2009.
Often when reading a good maths book, the author will get to the end of an explanation of a particularly complicated proof, theorem, or idea, and mention the "beauty" of the maths involved. I always
wonder what, exactly, this means. Did I miss a particularly neat diagram? Or, as seems to be the case, is mathematical beauty something buried deep: something that, perhaps, I need a PhD to get to
grips with?
I used to think that it was the latter — maybe one day, after years of studying maths at its highest level, I'd suddenly gain a glimpse of some incomprehensibly deep truth and realise the incredible
beauty of things which now seem boring and trivial.
Maths can be like a dense jungle — it's hard to penetrate but you never know whom you might might.
But actually, I think you can get a glimpse of what mathematicians mean by beauty without too much effort at all. That's what I'm going to try and convince you of in the rest of this article.
Mathematics can be a bit like a dense, never-ending jungle. It can feel like you're hacking away and away at it and never getting anywhere, but if you stop and look around yourself, every once in a
while you see incredible, exotic plants and animals to marvel at — and ever so often you find large new swathes of jungle to explore.
The particular thing that I want to introduce you to, that I think is so beautiful, is something that was mentioned in passing on a television programme I was watching. I hardly knew what it meant,
and I certainly had no idea how it came about, but I knew I had to find out more.
I am talking about Euler’s identity
Now you probably think I'm crazy. What's beautiful about that? Well, I ought to warn you, I'm not alone — Mathematical Intelligencer readers voted the identity the "most beautiful theorem in
mathematics". The physicist Richard Feynman called the formula it is derived from "one of the most remarkable, almost astounding, formulas in all of mathematics".
But what is so special about it? Well, first I ought to explain what the symbols actually mean.
You're probably familiar with irrational numbers – they have an infinite number of decimal places and you can't write them down as one integer divided by another.
Probably the strangest of these three numbers is imaginary number, and you can't find it anywhere along the normal number line, as none of the ordinary real numbers give a negative number when
Are you starting to get an idea of the beauty of Euler's identity? If you take the constant
Euler's identity is named after Leonhard Euler, one of the most prolific mathematicians of all times.
So, why does this happen? You might think that it is down to some really complex idea — how do we even take a number to the power of formula, which leads to his beautiful identity, in full
Doesn't look quite as nice and neat now, does it? But don't be put off. To understand how this formula comes about, we need something called Taylor series. These are just a way of expressing
functions such as Brook Taylor (who was also part of the committee which adjudicated the argument between Isaac Newton and Gottfried Leibniz about who first invented the calculus).
The Taylor series for the function
You can verify this Taylor series using a calculator: choose a number
for as many terms as you like, that is for a number
The Taylor series for the other two functions appearing in Euler's formular are
Again you can check this using your calculator, bearing in mind that the angle
Now let's multiply the variable
But certain powers of
We can gather the terms involving
Now notice that these two series are the same as the series for
which is Euler's formula!
All we have to do now is substitute
So you see, after a sequence of fairly complex mathematics we arrive back where we started — at the (seemingly) simple numbers 1 and 0. That is what I think is so beautiful about this identity: it
links very strange numbers with very ordinary and fundamental ones. Seeing why it works feels a bit like treading a little-known path through the mathematical jungle to reach a secret destination
somewhere in the thick undergrowth.
About the author
Surein Aziz is 17 years old and currently in year 12 at Farnborough Sixth Form College. He thinks maths is very interesting (and beautiful!) because of the incredible truths and interconnections you
can uncover simply by following a sequence of logical steps and identifying patterns. He loves to spend his time thinking about (and sometimes, in simple cases, solving) interesting maths problems,
and is hoping to read mathematics at university after he gets his A-levels. He also enjoys playing the violin and fencing. He first encountered Euler's Identity and the idea of its beauty on a TV
program, after which he knew he had to research the subject further.
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October 20
Mt Aspiring
Carl Brannen
is currently enjoying
where he will be speaking about his derivation of the lepton masses. It is a large conference with a
String theory
session, which has some interesting sounding talks such as
on the
geometric Langlands
program. Amongst bloggers, at least
Gordon Watts
appears to be there. I guess we're heading into a busy conference season!
The Euler characteristic for a Coxeter complex based on A_n goes like
chi = (-1)^n.2n.(n - 2)!!(n - 2)!!
which grows quickly but is only -6 for the case of A_3. Goodness me, that number does have a tendency to crop up, doesn't it now?
Weinberg described a theory of electroweak forces in 1967. He shared the Nobel prize for this unification with Glashow and Salam in 1979. Another gauge theory, quantum chromodynamics, took much
longer to be accepted as experimental verification slowly came in. Gluons were only discovered at PETRA II at DESY in 1979.
The electroweak theory required a Higgs boson to explain the aquisition of mass of particles. It is a shame that these events occurred in the order that they did, although of course it had to be. For
a long time many physicists took the Higgs mechanism seriously and failed to investigate clues from QCD. QCD is, after all, a theory for quarks which participate in the weak interactions.
Replacing the Higgs mechanism within the framework of rigorous QFT has proven to be a daunting task. It was, however, quite clearly never an explanation for mass quantum numbers, which by definition
must arise in a quantum gravitational theory.
On Monday January 22 2007 the NSF Distinguished Lecture will be given by the respected cosmologist Sean Carroll. The lecture has the title: Dark Energy, or Worse: Was Einstein Wrong?
From October 29 to November 3 2006 the Joint Meeting of the Pacific region Particle Physics communities will be held in Honolulu. Make sure you hear Carl Brannen's talk if you are lucky enough to be
in Hawaii.
Leonhard Euler lived from 1707 til 1783. He published such an astonishing amount of mathematics that the St. Petersburg Academy continued publishing his work for more than 30 years after his death.
Eventually he went blind, but continued doing enormous calculations in his head. He could recite the entire Aeneid of Virgil. One thing he did was study the multiple zeta values. He proved the two
argument (depth 2) version of the result that the value of the Riemann zeta function at the 1-ordinal n was the sum over (depth k , weight n) MZVs such that the first argument was greater than 1. The
depth 3 case was proved in 1996.
Euler's MZVs were largely forgotten until recent times, but since their appearance in QFT structures they have arisen in many contexts. Multiple polylogarithms are a natural generalisation. Now we
know that the MZVs are algebraic integrals for the cohomology of moduli of punctured spheres.
The details of the new Standard Model of Connes, Marcolli and Chamseddine is now out. Recall that John Barrett also has a recent paper out on a Lorentzian version of the Connes model. These ideas
bring neutrino mass generation into the SM in a natural way, but the number of generations is really put in by hand. Should we be focusing on the NCG language in order to interpret this new SM?
The physical problems of QG and Yang-Mills and precise mass values are closely related to the Riemann hypothesis, which is naturally what Connes and Marcolli are trying to solve, as is well known.
One important ingredient in this program is the notion of Grothendieck-Teichmuller group, as discussed in this lecture by Schneps. Note the pretty tree diagram on page 11. Many people, such as
Kontsevich and Cartier, have thought about this structure from different angles. Kontsevich said we should think of the GT group as the quotient of the motivic Galois group by its action on the
spectrum of an algebra generated by (2 pi i), its formal inverse and all the MZVs. Apparently there are some problems with this idea. But the question is, do we really need to define this GT group?
Perhaps we can get physical parameters much more directly.
In AQFT one prefers to think about Tomita-Takesaki modular theory. I went to an interesting NCG seminar on this yesterday by Paolo Bertozzini. He has been trying to understand the basic NCG geometry/
algebra correspondence from a more categorical point of view. In fact, he made it clear that the categorical duality is far from understood. The correspondence usually works only on the level of
objects: take a spin manifold and get a spectral triple, or take a spectral triple and find its spectrum. But to describe the correspondence properly the adjunction natural transformations need to be
fully described. The categories need morphisms. Now one can do this, but it leads to the question that Paolo is thinking about: the spectral triples appear to be approximations to something higher
categorical. They are recovered as endofunctors of some kind in a richer structure. What is this structure? Paolo was thinking along the lines of a quantum topos theory. Funny thing was that just
after I was introduced to Paolo we realised that we had met on the internet, in a discussion on the fqxi funds on Woit's blog. Paolo was one of those rejected for his over enthusiastic category
theoretic proposal.
It was a nice day yesterday. There was a Feynman film night run by the Physics group. I was talking to a quantum optics guy and he told me they are getting six new staff in Quantum Information next
year. Given the small size of the department at present, this is just flabbergasting. I stood there like a stunned mullet and he pointed out that there was an awful lot of money in this game. None of
these guys seemed to have the least interest in what the Category Theory group are doing, even though they haunt the same corridors. One of the new guys will be Terno who has been at Perimeter, so
I'm looking forward to meeting him in a few weeks when he arrives.
The pizza was yummy, too.
I continue to be alarmed at the disrespect that many of my colleagues pay to their climate science colleagues, even now, as the once green pastures of New South Wales turn to dust and atmospheric CO2
levels rise above anything they would believe possible.
On the news yesterday I heard an interview with a local astronomer, who felt it necessary to defend funding for science at a time when food and commodity prices were rising. The unfortunate reality
is that we cannot expect this problem to go away. You might think it unlikely. Look at the data yourself. Water shortages and forced migrations have always caused economic and political tension. They
have never happened on the scale that they soon will. It's a pretty simple story, really. It's time to think about what you take for granted: the fresh water, long showers, luxury items, enough food
to eat.
Yes, people like me are called alarmist. I've been hearing that for a long time. That's why it's all so depressing. We live at a time when pretty well everybody on earth needs to change their life.
And they're just not doing it.
Tony Smith, who likes octonions and Clifford algebras a lot, has a nice page on the surreal numbers. The further one moves up the tree, the more rational numbers one gets!
We've also seen trees in phylogenetics and knot theory, but most importantly in Batanin's operads. Recall that 1-level trees represented the Stasheff associahedra. These turn up everywhere, such as
in tiling the real moduli M(0,n) of genus zero surfaces.
There is an amazing series of papers by Connes, Marcolli and others on From Physics To Number Theory. See for example here or here or here. This goes back to work of Kreimer and Broadhurst, which is
now very well known. Some of the older papers are here. I particularly recommend the paper: Broadhurst and Kreimer, Association of Multiple Zeta Values with Positive Knots via Feynman Diagrams up to
9 Loops, Phys. Lett. 393 B (1997) 403-412.
Its about turning knots into simple Feynman diagrams into Multiple Zeta Values. These MZVs satisfy all sorts of crazy relations, which the mathematicans have been studying like crazy. But really
they're quite simple. They act on a set of k ordinals (yes, that's right, you should be thinking 1-ordinals) and are characterised by two numbers, namely the weight n, which is the sum of these, and
k itself, the so called depth. Of course these naturally show up as special integrals of something called Mixed Tate Motives (don't even ask), so we know that the weight n is the same n of M(0,n+3).
Goodness, me. The Yang-Mills problem and the Riemann hypothesis seem to be related. Well, well.
The real question, however, is how to go beyond scalars to other entities in QFT. Any guesses?
The gallant kneemo gave me a link to some great slides by Zvi Bern who works on perturbative quantum gravity. Before twistor strings came along he was thinking about the KLT relations between gravity
amplitudes and colour free diagrams, such as MHV tree level diagrams for n gluons.
These amplitudes are surprisingly simple, and apparently people don't really understand why! For example, the MHV tree amplitude for 4 gluons in QCD looks like A4 = (k1 + k2)^2 / (k2 + k3)^2.
It does make one wonder about the modelling of Witten's gluon spaces by projective twistor geometry in the context of M theory. Remember that there are three complex moduli of real dimension six,
namely M(0,6), M(1,4) and M(2,0). We already know the first one is interesting because it has an orbifold Euler characteristic of minus six. What if we needed to draw little loops on representative
Riemann surfaces? There is a nice mathematical way to think about this, but just imagine cutting up the two holed surface from end to end, straight through the two holes and at 90 deg to the correct
way to cut a bagel. That cut marks six points on the two holed surface. It turns out that special loops on this surface map to ones on the M(0,6) by taking the six points to the six punctures. Zvi
Bern would say that the graviton polarisation tensor is written as a square of gluon polarisation vectors.
Of course one can consider any number of punctures on the sphere to get tree amplitudes for n gluons. All one needs to know is that the compactified real moduli are all tiled by 1-operad Stasheff
associahedra, as shown in the work of Devadoss. For example, the three dimensional case of six points is tiled by the 3D 14 vertex polytope.
From both a physical and mathematical point of view, we would like to better understand Yang-Mills theory in 4D. To quote Jaffe and Witten: we would like to prove that for any compact simple gauge
group G, a non-trivial quantum Yang–Mills theory exists on R4 and has a mass gap d > 0. One of the ATLAS people recently said: our field must get some serious profit from LHC start-up and first
data, and we better teach ourselves right now how to explain Higgs, SUSY and extra dimensions to the public and the media. Oops. This statement needs a little revision.
The volume The Physicist's Conception of Nature, edited by Mehra, is a collection of lectures given at the 70th birthday celebrations for Dirac in 1972. The list of contributors is impressive:
Chandrasekhar, Dirac, Wheeler, Heisenberg, Wigner and Schwinger, to name a few.
Pascual Jordan's contribution is entitled The Expanding Earth. He explains that, having been deeply impressed with Dirac's 1937 idea of a varying G/c^2, he spent time investigating the possibility
that the Earth had been expanding over time. The lecture includes some beautiful geological diagrams regarding mid-ocean drift, and he talks about the difficulty that Wegener had with geologists
accepting the theory of continental drift. Now we understand that the value of G/c^2 is decreasing as we go back in time, and consequences of this should indeed be measurable on Earth. Jordan says:
There exists a great diversity between the mentalities of physicist's and of geologists. Physicist's are eager to learn about new facts and new ideas caused by new facts.
Pascual Jordan is one of the founders of Jordan algebras, which appear in M theory.
For anyone who happens to be around Sydney next week: come to the Feynman fest at the University of Macquarie at 5.30 pm on Wednesday Oct 25 for free pizza!
The LHC schedule for 2007 means that we won't be looking at new physics until 2008. But there is a lot of work to do before then! The 450 Gev calibration run is now planned for November 2007. The LHC
will be interesting for heavy ion physics. One important process is gg fusion, but it is said that final state effects such as energy loss will make quantification difficult unless we can better
understand such processes.
Fortunately twistor string theory has made great advances with the MHV diagram technique, which creates Feynman trees from maximal helicity violating vertices. Maybe we have time to improve on this a
This is really just another boring post about the speed of light rather than some comments on mathematical M theory. The string blogger MathPhys made an interesting comment on Woit's blog recently:
you all missed c < 1. Silly me...I wasn't sure whether he was referring to the speed of light or central charge!
In rational CFT one considers a deformation parameter q which is a root of unity in the complex plane. For q = exp(2.pi.i/N), the basic case, this depends only on the positive integer N. The same N
labels a triple of points (0,q,oo) on the Riemann sphere, which can be used to cover moduli, described by the q=1 case. And before one knows it there are modular tensor categories, Galois groups and
all sorts of other goodies floating around, which might explain why Terence Tao has been interested in physical distance scales recently.
Brannen has looked at different scales in the Standard Model with such a varying c. If c was supposed to be the speed of light one might equally ask about the domain c > 1, which has of course been
considered by Riofrio. So c could be very, very big, or it could be very, very small.
World climate change, environmental degradation, poverty, violence and more...and now we have to worry about problems at NASA. Where is that renewable resource of political Will that Al Gore spoke
about? Apparently Lee Smolin's book The Trouble with Physics discusses such serious issues. I haven't seen a copy of it yet, but I'm looking forward to reading it. (Anyone feel like sending me a
When I was plotting the snow level data from the Snowy Mountain hydroelectric scheme twenty years ago, I observed that seasonal snow levels at 1800m had fallen 30% in 50 years. Everybody told me it
was due to the formation of Jindabyne dam. They don't say that anymore.
To cheer us up, I thought I'd show a pretty and colourful picture, from Huterer, of the CMBR and its coincidence with the ecliptic
People tell me that this is just because of the way photons interact with stuff on their way here. Oh, really. If there were local future horizons defining an ecliptic, then by T-duality their
signature might appear in the cosmic CMBR. Apparently this is a more radical interpretation.
Have a nice day!
Carl Brannen has reminded me of Cartier's classic paper, A Mad Day's Work. He discusses everything, from Grothendieck's biography to symmetry groups for a point. In particular, he points out that a
sensible notion of symmetry group for a point comes from considering points as functors between toposes. Since there are natural transformations between functors, one might find a group of invertible
natural transformations between a functor and itself.
The really cool thing about all this is that the group is not fundamental. Eat your heart out Gauge Theory!
Which reminds me that I meant to say something about Grothendieck's motives. As Cartier explains, motives are a part of Grothendieck's dream, a vision of unifying number theory and modern topology,
and hence almost everything else as well. The theory of motives is still mysterious, although an impressive amount of progress in the related physics and mathematics has been made in the last 30
years. Consider for example the work of Kontsevich on motives and operads in deformation quantization. It's kind of funny that the mathematicians have chosen a word (motives) that starts with M. It's
their version of M-theory!
An important intuition behind motives is that of projective geometry. Motives obey powerful relations, an example of which is the equation
M(projective plane) = M(plane) + M(line) + M(point)
which expresses the usual grading of a projective plane (over any field) into an affine space with a line and point at infinity. This feature of a grading in dimension is typical of motives, as it is
for categorical dimension.
It is said that Grothendieck, one of the greatest mathematicians of the 20th century, is now mad. A piece of evidence often cited in support of this hypothesis is his fixation with the speed of light
, a mental exercise that might be recommended to many of the critics.
The arbitrary local numerical value of this quantity depends on the arbitrary old definition of the metre from Napolean's time. After some international political wrangling, some French guys measured
the meridian from Dunkerque to Barcelona in the years 1792 to 1798. If they had chosen a different geographical location the platinum metre bar would no doubt have come out slightly differently and
maybe, with a little stretch of the imagination, we would not be plagued with awkward values for c today. As Einstein said in a lecture in 1921:
In order to complete the definition of time we may employ the principle of the constancy of the velocity of light in a vacuum.
With emphasis on the word may. The constancy of c was not to be taken as a fundamental consideration, but as a convenient means of defining clocks for observers in uniform motion. To assume that the
constancy of c should suffice for quantum gravitational clocks is rather stupid. Fortunately people have considered alternatives. Louise Riofrio has some very pretty pictures and graphs which use a
varying c to explain away the magical Dark Energy.
The biology blogger Dcase complimented me recently on my knowledge of biomathematics. Now, whether talking about the biology or the fancy String mathematics, either way my knowledge is actually very
poor. But the point is that we both recognise a direction here, which I allude to in many of my posts. The application of trees, networks or categories to genetics, linguistics, computer science,
physics, physiology or whatever else is not merely a coincidental appearance of a new type of calculus. Certainly this is one way to see things, because this combinatorics does open vast new vistas,
mathematically speaking. But the biologists are not just talking about modelling systems. They are talking about a unified theory for understanding systems; something they have never had before.
Physicists are quite used to the idea of unifying laws of nature. Ever since the ancient Greeks they have worked with the unreasonable effectiveness of mathematics (to quote Weyl). Most physicists
are therefore convinced that a theory of Quantum Gravity (a loose term for something that unifies QFT and GR) exists. Moreover, this theory must be predictive. The idea of a Landscape is outrageous
and, since we already have better ideas anyway, one wonders why people persist with such investigations.
A theory of Quantum Gravity will say some radical things. Many physicists are now happy with the idea that spacetime disappears and is in some sense generated by the matter degrees of freedom.
Einstein could not, in the end, incorporate a Machian inertia into GR, but we expect Quantum Gravity to be able to achieve this. After all, it only fell over with Einstein's commitment to a classical
differential geometry. However, to believe that any old background independent description of quantum covariance which yields roughly the standard cosmology would be radical enough is, perhaps, to
underestimate the meaning of the word radical.
For starters, some of us are now fully convinced that Quantum Gravity will do for biology what QM did for chemistry. Of course, this is an arrogant physicist's point of view. A biologist might say
that the unified theory of biology happens to provide Quantum Gravity as well. Whatever. It's the same theory.
A little while back we were talking about the number of generations in the Standard Model. It was pointed out that this follows from the orbifold Euler characteristic of the moduli of the six
punctured sphere. Actually, the higher n-operads of Batanin highlight the fact that something special happens when one considers the statistics on six objects. Tamarkin showed that for n > 1 the
polytopes that are usually considered cannot stabilise moduli. But Batanin's can! The simplest Tamarkin example is for six points as branches of a two level nine edged tree. Since in this setting
2-operads are used to study points in the real plane, this enters into the correct combinatorics for the six punctured sphere. Look out for a paper on this soon!
On the phylogenetic tree of life on earth, the oldest of the three main branches is that of the Archaea, a class of prokaryote including extremophiles, organisms that inhabit environments far outside
the range which is comfortable for humans. For example, in the submarine volcanic environment of Loihi, which erupted violently in 1996, microbial mats have since been found (see picture). The
evidence for life on earth dates back to the oldest rocks on earth, namely the Akilia island sediments of West Greenland. Although the evidence in carbon isotopes for life in these particular
sediments has been open to question, there is plenty of evidence in other ancient rock sediments which typically date life back 3.55 billion years.
The theory that life originated in space, and was transported to the Earth's surface from space, is known as panspermia. But if the oldest life on Earth, perhaps as old as Earth, likes hot
environments, is there perhaps a different explanation? Astrobiologists such as Lawrence Krauss think that extremophiles will radically alter our understanding of the origins of life.
If there was a Black Hole at the centre of the Earth, would it have anything to tell us about the evolution of life?
Hearty congratulations to John C. Mather and George F. Smoot, who have just received the 2006 Nobel Prize for their leading roles in the COBE experiment measuring the CMBR.
Yesterday I was reading a new book by Joel R. Primack and Nancy Ellen Abrams, The View from the Center of the Universe. This book has a commendable grand vision: to look at how the current revolution
in cosmology can benefit humanity as a whole by altering its conception of Nature itself. Primack was apparently one of the physicists who predicted the anisotropy of the CMBR, based on the existence
of Dark Matter. Despite the book's relatively conservative, and hence quite erroneous, view of current cosmology, it offers brilliant physical insights in a very accessible way. I would like to quote
a little:
We don't normally think of reality as funnelling from great galaxy clusters into us and spreading cell to cell, then soaring inward to the molecular level, the atomic, the quantum levels - and our
humanness the fulcrum at the centre of the entire process. But we need to. We need to experience the universe from the inside. We have to imagine ourselves in our proper place, inside the symbols,
part of the symbols, the point of the symbols.
There is also a reasonable discussion about the celestial sphere, the badly named surface of last scattering, and how this returns us in some sense to a cosmology with Earth at the centre, but in a
way that the Greeks could never have imagined. In the standard modern cosmology (the one that is purported to be revolutionary) this celestial sphere is a fixed surface in a concrete reality that,
despite Primack's promises, the mathematics has not escaped. We continuously receive light from this primordial sphere. Compare the COBE results to those of the more recent WMAP satellite. The
blotches look roughly the same. Over great lengths of time on Earth it is supposed that the light reaching us will become more and more redshifted as the concrete universal spacetime itself expands.
There are other, even more profound, possibilities. If we accept that the temperature of the CMBR is an indicator of cosmic epoch, then its measurement is a kind of clock. The consequences of this
simple observation are not considered in the standard cosmology.
The local standard for time is now the cesium clock, which is accurate to an incredible 2 nanoseconds per day or, equivalently, one second in 1400000 years. A second, by definition, is precisely the
time it takes for 9192631770 cycles of microwave light (of a particular wavelength) from cesium133 atoms in their ground state to be absorbed or emitted. Observe that this definition requires only
counting (of cycles) and an understanding of the measurement of wavelength.
In the overzealous use of the basic equations of GR people have forgotten that it was the mathematician Minkowski who packaged spacetime neatly into a box all tied up with string. It has always been
physically clear that time and space, though necessarily related, are conceptually distinct.
In The Combinatorics of Iterated Loop Spaces, Batanin describes an operad based on the poset of faces of the nth Stasheff associahedron. The case of the pentagon looks like this:
It is a broken pentagon, but the top side is an identity if we want the sequence to form a 1-operad in the usual sense. Otherwise, the sequence of permutohedra form a kind of non-commutative operad.
Moreover, there is a map of operads from these permutohedra to the diagrams with collapsing identities. This was outlined concretely by Loday.
This is an example of a low dimensional operadic map which we might otherwise have viewed as a broken parity cube in a tetracategorical context.
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Yahoo Groups
Calculate first 3.2 trillion primes in 39 hours on standard PC
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Forgive the intrusion as I'm a physicist with an interest in efficient computing rather than a mathematician who has studied number theory.
In order to benchmark system performance I've been playing with an algorithm I created to calculate primes and was surprised by the results and wondered how it compared to other methods and
I calculated the first 3.2 trillion primes in 38 hours and 32 minutes on standard intel home PC (4-cores, 4GHz, Linux, mean memory usage using mean of ~300MB of memory). In this pass I merely
calculated pi(x) rather than output the data as I don't have 25 terabytes of disk space to hand. My results for pi(x) are correct according to other published data for the number of primes up to
A little research after the fact shows my method is a multi-threaded variant on a sieve of Eratosthenes with an additional sieve to filter multiples of the first 8 integers coprime with 2.
A bit of tinkering shows the algorithm scales reasonably well; calculation time increases by around 15% as x increases by 10-fold but memory usage creaps up to keep the algorithm efficient at
high values of x. Also there is a uint64 limit as I haven't used a big number library.
Since this is my first stab I wondered what I could realistically aim for in terms of cycle time for finding primes with realtively low values of x (<1.8x10^19)?
View Source
Take a look here:
Looks like you can extrapolate that implementation to get the first
3.2 trillion primes in about 5-6 hours on an Intel i7-920, assuming
sufficient RAM available.
I'm not sure RAM is that much of a limiting factor though; I've
written fast sieves that sieved in what I called "slices", and
could easily do sieves of this size with much less RAM than today's
machines commonly have available.
Generally, as you chase speed, you end up tuning the algorithm
parameters to minimize cache misses -- cache misses are the biggest
impediment to speed.
On 3/4/2013 2:20 PM, James Firth wrote:
> Hi,
> Forgive the intrusion as I'm a physicist with an interest in efficient computing rather than a mathematician who has studied number theory.
> In order to benchmark system performance I've been playing with an algorithm I created to calculate primes and was surprised by the results and wondered how it compared to other methods and
> I calculated the first 3.2 trillion primes in 38 hours and 32 minutes on standard intel home PC (4-cores, 4GHz, Linux, mean memory usage using mean of ~300MB of memory). In this pass I
merely calculated pi(x) rather than output the data as I don't have 25 terabytes of disk space to hand. My results for pi(x) are correct according to other published data for the number of
primes up to 1x10^14.
> A little research after the fact shows my method is a multi-threaded variant on a sieve of Eratosthenes with an additional sieve to filter multiples of the first 8 integers coprime with 2.
> A bit of tinkering shows the algorithm scales reasonably well; calculation time increases by around 15% as x increases by 10-fold but memory usage creaps up to keep the algorithm efficient
at high values of x. Also there is a uint64 limit as I haven't used a big number library.
> Since this is my first stab I wondered what I could realistically aim for in terms of cycle time for finding primes with realtively low values of x (<1.8x10^19)?
> fdj
> ------------------------------------
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://primes.utm.edu/
> Yahoo! Groups Links
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Thanks Jack, I'll take a look.
RAM becomes an issue in my implementation at least because as x gets large then if the sieve 'slice' is too small one wastes cycles looping through values of iy where there are no values of ix in
range (ix, iy being cartesians on the sieve grid).
Increase sieve slice size removes waste and improves efficiency by manyfold.
I created a Javascript visualisation here:
[Non-text portions of this message have been removed]
View Source
Jack Brennen wrote:
> Looks like you can extrapolate that implementation to get the first
> 3.2 trillion primes in about 5-6 hours on an Intel i7-920, assuming
> sufficient RAM available.
I think the results of a willy-waving contest nearly a decade ago between Terje Mathisen and James Van Buskirk on c.l.a.x, concluded that about an order of magnitude faster than that speed should
be possible. If you're James, that is. Tomas e Silva's sieve uses pretty much the same algorithm, but he hasn't optimised it quite as much. (James is an extreme optimiser, very few people can
optimise as much!)
View Source
--- In
, Phil Carmody <thefatphil@...> wrote:
> Jack Brennen wrote:
> > Looks like you can extrapolate that implementation to get the first
> > 3.2 trillion primes in about 5-6 hours on an Intel i7-920, assuming
> > sufficient RAM available.
> I think the results of a willy-waving contest nearly a decade ago between Terje Mathisen and James Van Buskirk on c.l.a.x, concluded that about an order of magnitude faster than that speed
should be possible. If you're James, that is. Tomas e Silva's sieve uses pretty much the same algorithm, but he hasn't optimised it quite as much. (James is an extreme optimiser, very few
people can optimise as much!)
> http://www.ieeta.pt/~tos/software/prime_sieve.html
> Phil
Nearly two orders of magnitude is possible. YAFU and primesieve both compute pi(x) to 3.2e12 using the sieve of Eratosthenes in about 30 minutes on my machine (a pretty fast server). They are
both threadable too - with 4 threads YAFU takes about 8 minutes.
- ben.
View Source
--- In
, "Ben Buhrow" <bbuhrow@...> wrote:
> Nearly two orders of magnitude is possible. YAFU and primesieve both compute pi(x) to 3.2e12 using the sieve of Eratosthenes in about 30 minutes on my machine (a pretty fast server). They
are both threadable too - with 4 threads YAFU takes about 8 minutes.
> - ben.
Apologies, I was confusing pi(1e14) ~= 3.2e12 with pi(3.2e12). After running some experiments, it looks like the two programs I mentioned will take about 18 hrs for pi(1e14), so more like a
factor of 2.
View Source
Thank you Ben.
I have already optimised down to 18 hours for pi(1e14) however primesieve is consistently running at least 4 times faster on my machine.
I am aware that I might be using an odd method to stitch together my sieve segments. Some Javascript here (when you click on the demo link) explains how I have calculated my boundaries:
I am not seeing the efficiency improvements the primesieve author gets when the sieve memory fits into the L2 cache; I will probably have to switch to Tomás Oliveira e Silva's bucket method.
I had a hunch my method, which allows starting on a prime's square for starting values greater than seg_start/start_val, might be efficient, but alas...
I might try improving the storage compression, I'm currently using modulo 30, but I doubt I'll find another 4x optimisation in my code.
--- In primenumbers@yahoogroups.com, "Ben Buhrow" wrote:
> Nearly two orders of magnitude is possible. YAFU and primesieve both compute pi(x) to 3.2e12 using the sieve of Eratosthenes in about 30 minutes on my machine (a pretty fast server). They
are both threadable too - with 4 threads YAFU takes about 8 minutes.
> - ben.
Apologies, I was confusing pi(1e14) ~= 3.2e12 with pi(3.2e12). After running some experiments, it looks like the two programs I mentioned will take about 18 hrs for pi(1e14), so more like a
factor of 2.
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Needville Math Tutor
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Fitting a mesh to a density function
up vote 7 down vote favorite
Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). For large $N$, is
it possible to place $N$ points $X_1,\dots,X_N$ in the region so that the points $X_i$ are distributed according to $f(x)$, and also form a mesh of (approximately) equilateral triangles? This is
clearly trivial when $f(x)$ is uniform (just put the $X_i$ in a uniform triangular lattice).
For the non-uniform case, obviously some triangles will be larger than others, but I want each individual triangle to be approximately equilateral (e.g. maximum side length and minimum side length
are within 1% of each other, etc.). One possibility for the non-uniform case would be to sample $N$ points independently at random from $f(x)$ and then take their Delaunay triangulation, but I don't
think there is a guarantee that the triangles will be roughly equilateral (i.e. some will be long and skinny) as $N$ becomes large.
The picture below is along these lines, if you ignore the big ugly hole in the center; each triangle is roughly equilateral, but points are not uniformly distributed.
geometry pr.probability geometric-probability euclidean-lattices
add comment
2 Answers
active oldest votes
Here is one possible interpretation of your question.
Assume a probability density function $f$ is given. Is there a sequence of triangulations $T_n$ with $\varepsilon_n$-equilateral triangles such that counting probability measure on
nodes converges to $f$ and $\varepsilon_n\to 0$ as $n\to\infty$.
up vote 5
down vote (Say a triangle is $\varepsilon_n$-equilateral if the ratio of maximum side length and minimum side length is $\le 1+\varepsilon$.)
I am almost sure that the answer is "YES" if and only if $f$ is conformal factor of a flat metric; i.e., if and only if $f=e^{2{\cdot}\phi}$ and $\Delta \phi\equiv 0$.
2 Thanks a lot, Anton -- I hadn't made the connection to conformal maps, but that's clearly the right way to think about things. I gather that, in my particular case with $f(x) = \alpha
\|x\|^{-\beta}$ on the unit disk, the answer is therefore no? – John Gunnar Carlsson Jan 15 '12 at 2:40
@John: Sorry my last comment was not correct, so I delete it. My answer is OK in the case if $f$ has no zeros and the domain is simply connected. In particular $f=\alpha{\cdot}|x|^{-\
2 beta}$ is OK once the domain is simply connected. $$ $$ If the domain is not simply connected then in addition the holonomy group should be $\mathbb{Z}_6$ in $\mathbb S^1$. For
example $f=|x|^{-2/7}$ should be OK for the annulus (I might make a mistake). (Constructing a triangulation near zero of $f$ seems to be impossible, so you need to cut it from your
domain.) – Anton Petrunin Jan 15 '12 at 6:25
1 Ah, interesting; many thanks for the follow-up. When you say "$f=\alpha{\cdot}|x|^{-\beta}$ is OK once the domain is simply connected", do you mean that such a triangulation DOES NOT
exist for this case (I inferred this from your next example with $|x|^{-2/7}$ on an annulus)? Also, did the $-2/7$ come from anywhere in particular, or would, say, $-2/3$ work as
well? (I will admit that I do not know what a holonomy group is, and clearly have quite a bit of reading to do) – John Gunnar Carlsson Jan 15 '12 at 9:05
2 Take a cone $C_n$ with angle $n{\cdot}\tfrac\pi3$. (We need angle proportional to $\tfrac\pi3$ so $C_n$ admits a triangulation in equilateral triangles) The map $C_n\to \mathbb C$,
defined as $z\mapsto z^{6/n}$ is conformal. The conformal factor is proportional to $(|z|^{6/n-1})^2$. So any $\beta= 2-12/n$ will do. (Sorry if I made a mistake in calculations.) –
Anton Petrunin Jan 15 '12 at 18:20
Got it! Many thanks! – John Gunnar Carlsson Jan 15 '12 at 20:32
add comment
There is an analogy in mechanics that might help: think of the nodes of the mesh as being connected by springs, which have tension proportional to something meaningful, e.g. the integral of
$f(x)$ along the segment $[X_i,X_j]$. Then, if you let it stabilize, you will get a mesh with nodes distributed roughly according to $f$; if you pre-process $f$ to make is smooth enough so
up vote 3 that it does not change much on every initial triangle, you should end up with roughly equilateral triangles, too.
down vote
add comment
Not the answer you're looking for? Browse other questions tagged geometry pr.probability geometric-probability euclidean-lattices or ask your own question.
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PHYS 437 / METM 305
Introduction to Solid State Physics
Fall 1997 Syllabus
Instructor: Carlo U. Segre
Office hours: 10:00-11:00 Tuesdays and Thursdays
Textbook: Elementary Solid State Physics, M. Ali Omar, (Addison-Wesley, 1993).
Material to be covered: Chapters 1-5, 8-10 of textbook plus additional topics if time permits.
Homework Assignments
Tentative Schedule
• Crystal structures and interatomic forces: Bravais lattices, symmetry
• Miller indices, interatomic forces, atomic bonding.
• Diffraction in crystals: generation of X-rays, Bragg's law, X-ray scattering from atoms and from crystals, reciprocal lattice, experimental techniques, neutron and electron diffraction.
• Lattice vibrations: elastic waves, enumeration of modes, Debye and Einstein models, phonons, density of states of a lattice, theory of specific heat, thermal conductivity, scattering by phonons
lattice optical properties in the infrared.
• Free-electron model: conduction electron, free-electron gas, electrical conductivity and resistivity, Fermi surface, thermal conductivity in metals, motion in a magnetic field, optical
properties, failure of the free-electron model.
• Energy bands in solids: energy spectra and bands, the Bloch theorem, band symmetry, Brillouin zones, nearly-free-electron model, tight-binding model, calculation of energy bands, density of
states, effective mass, electron dynamics.
• Dielectric and optical properties: dielectric constant and polarizibility, dipolar, ionic and electronic polarizibility, piezoelectricity and ferroelectricity.
• Magnetism: susceptibility, Langevin diamagnetism, paramagnetism, magnetism in metals, ferromagnetism in insulators, antiferromagnetism, ferromagnetism in metals.
• Superconductivity: zero resistance, Meissner effect and perfect diamagnetism, the critical field, thermodynamics of superconductors, electrodynamics of superconductors, theory of
superconductivity, tunnelling and the Josephson effect.
Course Organization: There will be 2 Midterm Examinations and one Final Examination. In addition homework will be assigned on a regular basis and graded. The percentage distribution for the course
will be:
Midterm Exam 1 20%
Midterm Exam 2 20%
Homework 10%
Final Exam 40%
The remaining 10% of the grade will be assessed on a term project and/or class participation.
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isnan (MATLAB Functions)
MATLAB Function Reference
TF = isnan(A) returns an array the same size as A containing logical true (1) where the elements of A are NaNs and logical false (0) where they are not. For a complex number z, isnan(z) returns 1 if
either the real or imaginary part of z is NaN, and 0 if both the real and imaginary parts are finite or Inf.
For any real A, exactly one of the three quantities isfinite(A), isinf(A), and isnan(A) is equal to one.
See Also
isfinite, isinf, is*
© 1994-2005 The MathWorks, Inc.
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Math problem solving investigation: guess and chec
Number of results: 243,954
Math problem solving investigation: guess and chec
no idea how many squares you have, but start counting. If there are n squares larger than the smallest, then the area of the largest is 3*2^n. The areas are: 3,6,12,24,48,96,... If you want to ask
questions about diagrams, you gotta describe what's going on here.
Saturday, December 29, 2012 at 8:29pm by Steve
Math problem solving investigation: guess and chec
The area of each square is twice the area of the next smaller square drawn in it. If the area of the smallest square is 3 square centimeters, what is the area of the largest square
Saturday, December 29, 2012 at 8:29pm by Alex
classroom instruction
Problem solving is a basic skill needed by today’s learners. I guess for each grade level (elementary and middle school) i would present the problem, look at the pros and cons of each potential
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Friday, March 19, 2010 at 9:18pm by scooby
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Wednesday, January 20, 2010 at 5:02pm by kim
Writeacher, can you ask a science teacher to help me with my application? (methods adopted when analyzing experiment data concerning biodiversity in water) I would be extremely grateful to you.
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Thursday, February 2, 2012 at 11:20am by Henry2
classroom instruction
Problem solving is a basic skill needed by today’s learners. I guess for each grade level (elementary and middle school) i would present the problem, look at the pros and cons of each potential
solution (come up with them first) and then select the best one. I will have to ...
Friday, March 19, 2010 at 9:18pm by scooby
I really need your help to determine if the following techniques are possible when analyzing scientific data (biodiversity in water) with pupils. I think an English science teacher can help me
determine the techniques. Approach 1) Task organization and information distribution...
Thursday, February 2, 2012 at 10:09am by Henry2
Guess and Check is an important problem-solving strategy. Use guess and check to solve each puzzle. a) A number times itself is 100. What is the number? b) A number times itself is 225. What is the
Monday, September 9, 2013 at 7:54pm by Anonymous
I really need a science teacher to check point 1 and tell me if the rest is possible. Thank you very much! 1) Task organization and information distribution The various tasks will be divided among
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Friday, February 3, 2012 at 3:14am by Henry2
8th grade pre algerbra
problem solving investigation: draw a diagram problem: kelly is using 3-inch square tiles to cover a 4-foot by 2-foot area the tiles are 0.5 inches tall if the tiles were stacked on top of each other
to create a tower, how many inches tall would the tower be?
Thursday, October 21, 2010 at 7:44pm by mojo
You will need to ask a much more specific question than that. No one here is going to guess at what your "trouble with problem solving is."
Sunday, May 1, 2011 at 12:39pm by Writeacher
Math Problem Solving
i dont know i want help to solve the problem solving
Monday, January 18, 2010 at 7:26pm by bisan
a+3d = 3a a+6d = 2(a+2d)+1 a=3 d=2 chec: 3+6 = 3*3 3+12 = 2(7)+1
Friday, February 7, 2014 at 8:31am by Steve
cultural diversity
not really sure, I guess I'm going to have to do some further investigation
Wednesday, September 21, 2011 at 9:34pm by Vanessa
beginning chemistry
The problem says indicate the the number of sig. figs. that should be present in the answers to the following problem without solving the problem. 0.70 x 7.00 x 6.880= What is a sig.fig. and how can
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Why is problem solving considered the highest level of mental activity? What makes an individual good at problem solving?
Saturday, September 10, 2011 at 11:27am by Linda
1. To come up with possible solutions to a problem is referred to as: A. decision making. B. concept mapping. C. critical thinking. D. problem solving. is it D 2. "Should I apply for a job or go for
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Are you being asked to devise an equation, a strategy for solving the problem, or ? You haven't really stated.
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Hello can you please help me with this investigation. I am having lots of problems with it can you please help. Thandeka wanted to investigate whether the number of vibrations per second depended on
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Saturday, November 5, 2011 at 4:56am by kyle
Bobpursely, thank you for helping me! Our project involving three schools is about biodiversity in water. In particular, participating students will have to carry out a chemical and microbiological
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Thursday, February 2, 2012 at 6:12pm by Henry2
Bobpursely, thank you for helping me! Our project involving three schools is about biodiversity in water. In particular, participating students will have to carry out a chemical and microbiological
analyses of water in each school's local area : river water for us and the ...
Thursday, February 2, 2012 at 6:13pm by Henry2
physics practical investigation
Investigation about the amount of heat produced in HCl and Zn
Monday, March 25, 2013 at 5:05am by Lekgothoane prudence
What is a experimental investigation? What is a descriptive investigation?
Tuesday, October 5, 2010 at 6:40pm by Emily
Please help solving this problem. Solve the following word problem: One poll reported that 48% of city residents were against building a new stadium. The polling service stated that the poll was
accurate to within 3%. What is the minimum percent of city resident that oppose ...
Tuesday, April 6, 2010 at 12:12pm by bindiya farswani
IN an investigation 3 plants each receive the same amount of sunlight, ut different aounts of water. WHat is the control in the investigation?
Wednesday, September 23, 2009 at 5:50pm by linda
Here's my common sense, but unconventional, way of solving this problem. If the original price was $10, then 75% off would mean it now sells for $2.50. 2.50 is 50% of $5.00. The original discount
must have been 50%. Perhaps a math tutor will show you a more mathematical way of...
Wednesday, April 13, 2011 at 4:47pm by Ms. Sue
One investigation conducts experiments to prove or disprove a hypothesis. The onter investigation describes what has been observed.
Tuesday, October 5, 2010 at 6:40pm by Ms. Sue
physics practical investigation
Plan,design and conduct.. Investigation about the amount of heat produced in HCl and Zn
Monday, March 25, 2013 at 5:05am by Lekgothoane prudence
Algebra 2
Helllo I have no idea how to do this problem given f(x) = x-3(x + 4)^(1/2) how to solve for x when f(x) = -5 I know I could just guess and check but I would like to know how to do this algebraically
by pluging in -5 like so -5 = x - 3 (x + 4)^(1/2) and solving for x which I ...
Tuesday, May 26, 2009 at 5:57pm by Dylan
"The area of a rectangle is 360m2. If its lengh if increased by 10m and its width is decreased by 6m, then its area does not change. Find the perimeter of the original rectangle." I need to use the
"guess and test" or the "draw a diagram" method to solve; which would be most ...
Monday, February 18, 2013 at 12:24am by john
This problem deals with Pressure, Volume, and Temperature. Sounds like a PV/T=PV/T problem. The reason you don't know pressure is - That's what you're solving for! The thing you're solving for must
always be included in the equation that you use to solve for it.
Wednesday, April 29, 2009 at 8:39pm by Bill
Discuss how, as a professional, you may be able to assist children in refining their learning and metacognitive problem-solving strategies. At what stage in the maturation process do you believe it
is most important to address a child’s learning and problem-solving strategies...
Wednesday, June 11, 2008 at 6:54pm by Kell
Can you type out the problem? Maybe then people can actually help you. By reading what you said it sounds like it's sq (x^2/x-6) = 0... but I guess that's not what the problem looks like, be sure you
indicate where the parentheses are in your problem.
Sunday, January 13, 2008 at 6:28pm by Jake1214
COULD I GET HELP SOLVING THIS PROBLEM USING THE DISTRIBUTIVE PROPERTY 5(W-5)= 10 W =
Saturday, January 21, 2012 at 4:21pm by DJ
math problem
haveing problem solving 8x -(6x+7)= -3 The solution is x =
Saturday, January 21, 2012 at 5:02pm by DJ
lattice math
Why should we estimate before solving a problem? We should estimate before solving a problem to make sure our answer is close to the real answer. Here's a simple problem. Suppose we want to find wnat
a 6% tax on a $10 purchase would be. When you do the math, the answer depends...
Tuesday, October 3, 2006 at 9:46pm by marc
math:solving for an exponent.
No. The math he did was handwritten. I guess he must have made a mistake because this is driving me nuts. Everything listed in my post is CORRECTLY copied from the piece of paper I got it on.
Tuesday, October 4, 2011 at 7:27pm by Sar
Accident Investigation: Law+Physics
Thanks Ms. Sue!!! But for "inadequate brakes", is it supposed to be only the car's problem (brakes don't work properly), or can it the driver's problem (didn't react in time) too?
Thursday, January 12, 2012 at 10:37pm by Cicilia
problem solving, math
thanks, problem solved!
Monday, September 8, 2008 at 6:19pm by Laura
Child Development
Discuss how as a professional, you may be able to assist children in refining their learning and metacognitive problem-solving strategies. At what stage in the maturation process do you believe it is
most important to address a child’s learning and problem-solving strategies?
Saturday, September 20, 2008 at 6:41pm by Melvenia
classroom instruction
• Write a 200- to 300-word response describing how you might teach this concept to students in a lower elementary grade and to students in a middle school grade. i picked problem solving i have this
so far. i guess for each grade level (elem and middle school) i would present ...
Friday, March 19, 2010 at 9:18pm by scooby
criminal investigations
do the objectives of a criminal investigation change with the type of investigation
Wednesday, December 7, 2011 at 8:58pm by Annette
probably the most famous approach to problem solving is Polya's four-step process. Describe the four-step process.
Friday, October 14, 2011 at 7:28am by alya
Solving by graphing DQ 1 Solving by substitution Solving by elimination Which method do you think would be best to solve this system? y=10x-8 y=1/2x+5
Tuesday, October 19, 2010 at 11:36pm by Anonymous
problem solving
You have a problem to solve. You may need to type the problem or question here.
Tuesday, November 4, 2008 at 12:23pm by Ms. Sue
Problem solving:for 3rd grade
write a real-world problem involving multiple steps, then solve your problem
Thursday, October 25, 2012 at 10:29pm by Phyllis
That was my question as well. I believe this is solving for similar triangles...or why else would the second triangle be part of the problem? What is the significance of the second triangle? And I am
not getting a legitimate answer solving for x.
Saturday, October 17, 2009 at 1:11pm by Anonymous
math:solving for an exponent.
Bob, thank you. I thought I was going insane there for a minute. I guess the professor made a mistake. We are all human (but it drives me crazy when I'm trying to study haha)
Tuesday, October 4, 2011 at 7:27pm by Sar
I've been having problems with solving equations that involve fractions with the following problem can I get some help for solving equations that involve fractions? x+1=2/3x Thanks for the help.
Saturday, March 14, 2009 at 4:02am by Jen
As of July 2004, the population of the United States was 293,027,571. If more than half of these people live within 80 kilometers of an ocean, about how many people live within distance? I guess I
have to look to find the conversion of kilometers to diatance. I had no luck in ...
Wednesday, December 17, 2008 at 2:14am by Nehemiah
You are playing Guess you card with (3) other players. Here is wha you see: Andy has the cars 1,5, and 7 Belle has the cards 5,4, and 7 Carol has the cards 2,4, and 6 Andy draws the question card, "
do you see two or more players whose cards sum to the same value? He answers "...
Monday, January 28, 2013 at 1:33am by Mike
English expression
Thank you! Then what is the full form of "Guess What? Can it be one of them below? 1. Guess what it is. 2. Gues what they are. 3. Guess what the pictures below are. 4. Guess what the follwing are.
Thursday, March 6, 2008 at 2:44pm by John
the prob(correct guess on first try) = 1/5 Did you leave out a second question? The interesting part of the problem does not get involved e.g. What is the probability that he will guess correctly
after either one or two tries.?
Monday, April 29, 2013 at 2:46am by Reiny
Math Problem Solving
Oh im sorry i did this problem wrong. Ms.sue did it correct :) Sorry again.
Tuesday, December 21, 2010 at 5:44pm by Anna
The problem states that "5 to 8 is as 15 to w." I'm having some trouble solving this because I am unsure of what the problem is asking me to do. Some help would be appreciated. Thank you! :)
Monday, January 13, 2014 at 10:26pm by Melisande
I need help in solving this: Problem #1 For f(x)=-6x3+10 find f(4) and f(-4) Problem #2 For g(x)=20-5x2 find f(1) and f(-1)
Wednesday, October 6, 2010 at 3:03pm by Anonymous
Need help in solving these Problem #1 For f(x)=-6x3+10 find f(4) and f(-4) Problem #2 For g(x)=20-5x2 find f(1) and f(-1)
Sunday, October 10, 2010 at 11:48pm by Anonymous
Did you and I work on this problem yesterday. If not, then let me tell you quickly where your problem(s) is/are. First you MUST use Kelvin for T1. That is 80.1 + 273.16. I think the problem asks for
T in C BUT you must solve the problem first, using Kelvin, then convert back ...
Sunday, January 25, 2009 at 8:41pm by DrBob222
7th grade pre algebra
I have a problem solving a math problem in my sons math book. We have figured out all except this one. Please Help.. I need to insert parentheses to make this xpression equial 20 7+3.3-1+7 Thank
Thursday, November 6, 2008 at 8:30pm by Roni
An object floats with half of its volume beneath the surface of the water. The weight of the displaced water is 2000N. What is the weight of the object? I'm confused by this problem. How would I go
about solving this? My guess would be that the weight of the object is 2000N. ...
Monday, December 3, 2007 at 5:27pm by Tammy
math: problem solving
That's correct.
Wednesday, April 9, 2008 at 3:02pm by DrBob222
Math-Problem Solving
pr(m,a,t,h /m,a,t,h,e,i,c,s)=4/8
Wednesday, November 4, 2009 at 5:01pm by bobpursley
math problem solving
Thursday, October 14, 2010 at 6:26pm by yUo
Problem Solving Math
Wednesday, August 23, 2006 at 9:42pm by Nicole
Problem Solving Math
Wednesday, August 23, 2006 at 9:42pm by Nicole
I am having trouble solving this problem: "Find where the tangent line is horizontal for r=1+cos(theta)" I would really appreciate the feedback, I'd like to know how to go about solving problems like
these. Thank you!
Friday, March 1, 2013 at 2:46am by Anonymous
Math: Calculus
That's the problem. I have no idea. The question is exactly like i posted it here. I did the same thing but i realized i wasn't given a value of what it should equal to. I was hoping there a
different way of solving this problem
Thursday, November 11, 2010 at 2:42pm by REALLY NEED HELP!!!!
Problem solving:for 3rd grade
Write a real-world problem that you can solve by adding or subtracting.Then give your problem to a classmate to solve.
Thursday, October 25, 2012 at 10:29pm by Jon
Explain the difference between solving a system of equations by the algebraic method and the graphical method. Someone also wants to know why there are different methods for solving the same
problem-what would you tell him? Tiffany Oakes
Monday, July 12, 2010 at 10:46pm by kiMBERLY
Math: Word Problem
I finished solving it. thank you
Tuesday, September 7, 2010 at 10:58pm by Amy~
Math Problem Solving
(12n+3)/2 or 6n+3/2
Tuesday, December 21, 2010 at 6:00pm by Damon
College Math
I need help solving this problem, s/5 - 5/s=0
Monday, April 11, 2011 at 10:08pm by Leann
math (really accounting)
This is an accounting problem rather than a math problem. Accounting is not my area. That said, I would guess that using a double declining balance method would be best. But, I am on very shaky
ground here.
Tuesday, August 4, 2009 at 11:03pm by economyst
Math-Problem Solving
Four out of eight letters, I get it. Thank you!
Wednesday, November 4, 2009 at 5:01pm by Anonymous
college math
I need help on solving this problem -(25^1/2)=
Wednesday, April 7, 2010 at 4:16pm by Anonymous
math problem solving organized list
Thursday, September 9, 2010 at 8:51pm by Anonymous
Math Problem Solving
Tuesday, December 21, 2010 at 5:44pm by Ian
Math Problem Solving
Tuesday, December 21, 2010 at 5:44pm by Ian
can anybody help me to understand solving this problem 8/x=2/5x=
Wednesday, December 25, 2013 at 1:50am by cora evans
Oh, ok! I was over thinking the problem!!! Plus a fever may not help with my problem solving abilities... Thank you!
Saturday, September 29, 2012 at 8:13pm by amanda1012
Chemistry -need help in solving problem
Didn't Devron work this problem for you earlier? I think so.
Monday, March 4, 2013 at 12:03pm by DrBob222
problem solving
What do you think? We'll be glad to discuss this problem with you after you post your answer.
Monday, December 8, 2008 at 7:36pm by Ms. Sue
Problem Solving
I was asked the same problem in my class. 88.75 left weekly
Saturday, April 18, 2009 at 4:49am by Ed
Having a problem with solving this problem. 1 + 1/6p = 2(p + 5)
Tuesday, October 4, 2011 at 1:06am by maurice
i need help solving this math problem. 15*12/4-3^2+17(9)
Tuesday, September 11, 2012 at 6:32pm by Malik
How do I go about solving this math problem. -172=-1-9p
Wednesday, January 23, 2013 at 8:36pm by Robert
I need help in solving this problem: A certain substance has a heat of vaporization of 42.46 kJ/mol. At what Kelvin temperature will the vapor pressure be 5.00 times higher than it was at 291 K? I
know that I have to use the Clausius-Clapeyron equation, but what would I be ...
Sunday, February 24, 2013 at 3:27pm by Confused
Calculus - Integrals
That's my problem. When I tried solving for A, B, C, D, and E in that first problem, it didn't really work out right...
Monday, March 24, 2008 at 12:53pm by David
Need help solving this problem. If there's an equation at the end, need help in how to solve that also. Compounded interest problem. Principal $825. Rate 4%. Compounded Annually. Time 10 years.
Wednesday, February 9, 2011 at 12:22am by J
MATH (Problem Solving)
could it be 2 cos^2-7cos +3=0 (2cosx -1)(cosx-3)=0
Saturday, September 5, 2009 at 5:52pm by bobpursley
8 / (4 / (-28)) = -56 I don't understand why? what are the steps to solving this problem?
Friday, April 2, 2010 at 10:17pm by stacey
We've showed you ways of solving this problem. What is your answer?
Sunday, October 17, 2010 at 10:24pm by Ms. Sue
how can i estimate in the first step of solving a division problem
Monday, October 24, 2011 at 5:20pm by Bailey
The best way of solving this problem is to change 6 to 5 2/2 5 2/2 - 2 1/2 = 3 1/2
Tuesday, November 27, 2012 at 4:32pm by Ms. Sue
Ms sue why arent u solving mine? Is there any problem?
Saturday, February 23, 2013 at 5:57pm by Kristien
How do problem solving using model addition and subtraction
Wednesday, September 18, 2013 at 3:00pm by Anonymous
I need help solving this problem; 7-3(2t-5)+4t=-18
Wednesday, October 16, 2013 at 9:46pm by Anonymous
Algebra II
Solving the formula for the indicated variable. I=prt,for r I need help knowing how to solve this problem step by step. its easier with numbers and a fraction in the problem but this problem just
threw me off.......HELP PLEASE!!!!
Sunday, January 30, 2011 at 6:29pm by Attalah Higgs
im having a hard time solving a problem that is 9x+9y=-18 do you think you can help me with this problem
Monday, July 20, 2009 at 1:07pm by stacey
Problem Solving
What form does Bernoulli's equation take if we use the information in the problem statement that P1=P2?
Monday, November 29, 2010 at 7:02pm by JahMan
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The Frobenius problem in a free monoid
"... words is linear ..."
"... We prove that, for the uniform distribution over all sets X of m (that is a fixed integer) non-empty words whose sum of lengths is n, D_X , one of the usual deterministic automata recognizing X^∗
, has on average O(n) states and that the average state complexity of X^∗ is Θ(n). We also show that the ..."
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We prove that, for the uniform distribution over all sets X of m (that is a fixed integer) non-empty words whose sum of lengths is n, D_X , one of the usual deterministic automata recognizing X^∗ ,
has on average O(n) states and that the average state complexity of X^∗ is Θ(n). We also show that the average time complexity of the computation of the automaton D_X is O(n log n), when the alphabet
is of size at least three.
"... A (non-circular) de Bruijn sequence w of order n is a word such that every word of length n appears exactly once in w as a factor. In this paper, we generalize the concept to different settings:
the multi-shift de Bruijn sequence and the pseudo de Bruijn sequence. An m-shift de Bruijn sequence of or ..."
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A (non-circular) de Bruijn sequence w of order n is a word such that every word of length n appears exactly once in w as a factor. In this paper, we generalize the concept to different settings: the
multi-shift de Bruijn sequence and the pseudo de Bruijn sequence. An m-shift de Bruijn sequence of order n is a word such that every word of length n appears exactly once in w as a factor that starts
at a position im + 1 for some integer i ≥ 0. A pseudo de Bruijn sequence of order n with respect to an antimorphic involution θ is a word such that for every word u of length n the total number of
appearances of u and θ(u) as a factor is one. We show that the number of m-shift de Bruijn sequences of order n is an!a(m−n)(an−1) for 1 ≤ n ≤ m and is (am!) a n−m for 1 ≤ m ≤ n, where a is the size
of the alphabet. We provide two algorithms for generating a multi-shift de Bruijn sequence. The multi-shift de Bruijn sequence is important for solving the Frobenius problem in a free monoid. We show
that the existence of pseudo de Bruijn sequences depends on the given alphabet and antimorphic involution, and obtain formulas for the number of such sequences in some particular settings. 1.
"... A (non-circular) de Bruijn sequence w of order n is a word such that every word of length n appears exactly once in w as a factor. In this paper, we generalize the concept to different settings:
the multi-shift de Bruijn sequence and the pseudo de Bruijn sequence. An m-shift de Bruijn sequence of or ..."
Add to MetaCart
A (non-circular) de Bruijn sequence w of order n is a word such that every word of length n appears exactly once in w as a factor. In this paper, we generalize the concept to different settings: the
multi-shift de Bruijn sequence and the pseudo de Bruijn sequence. An m-shift de Bruijn sequence of ordern is a word such that every word of length n appears exactly once in w as a factor that starts
at a position im + 1 for some integer i ≥ 0. A pseudo de Bruijn sequence of order n with respect to an antimorphic involutionθ is a word such that for every worduof lengthnthe total number of
appearances of u and θ(u) as a factor is one. We show that the number of m-shift de Bruijn sequences of ordernis a n!a (m−n)(an −1) for1 ≤ n ≤ m and is (am!) an−m for 1 ≤ m ≤ n, where a is the size
of the alphabet. We provide two algorithms for generating a multi-shift de Bruijn sequence. The multi-shift de Bruijn sequence is important for solving the Frobenius problem in a free monoid. We show
that the existence of pseudo de Bruijn sequences depends on the given alphabet and antimorphic involution, and obtain formulas for the number of such sequences in some particular settings. 1
, 2009
"... The computational complexity of universality problems for prefixes, suffixes, factors, and subwords of regular languages ..."
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The computational complexity of universality problems for prefixes, suffixes, factors, and subwords of regular languages
"... Considering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically
linear. ..."
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Considering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically linear.
"... Considering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically
linear. 1. ..."
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Considering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically linear. 1.
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Mplus Discussion >> LCA with known classes using training variables
Andres Cardona posted on Monday, August 17, 2009 - 11:36 am
Unusual as it may sound I’m trying to run a LCA knowing exactly both the number of latent classes and the class membership of each individual. The purpose of this already "solved" LCA is to compare
the model fit of alternative latent classes restricting them to a set of different observed classes using the same data.
Model 1, for example, groups the data into 2 known classes ("high" "low"), whereas Model 2 groups individuals into 4 known classes ("high" "middle" "middle-low" "low"). How to specify such a LCA
using Mplus? I’ve already figured out a solution but I’m not sure if it’s right und just wanted to hear your expert opinion about it (I didn’t found any comparable example on the manual).
I did the following: using the option "training" with (MEMBERSHIP) I defined, for each model, dummy variables (t1 t2 and t1,t2, t3,t4 respectively) and ran a standard LCA with the same categorical
indicators in each case. By doing so, I suppose, is the latent class membership restricted to the known classes captured by the dummy-coded t-variables defined in "training". Is that right? Below is
a shortened syntax for Model 1.
Thanks for your help!
CATEGORICAL is var1 var2;
CLASSES = c (2);
TRAINING = t1 t2 (MEMBERSHIP);
Bengt O. Muthen posted on Monday, August 17, 2009 - 5:02 pm
The training data need to have (with 2 classes as an example)
t1=1 t2=0
for a person to be a known member of class 1
t1=0 t2=1
for a person to be a known member of class 2.
The two classes would then have to have distinguishable features in the MODEL command in terms of parameter restrictions so that the classes have different meanings.
I am not sure, however, if this is what you are asking.
Andres Cardona posted on Monday, August 17, 2009 - 11:33 pm
Thanks for your answer. What I'm trying to do is to fit a LCA knowing a priori both the number of classes and the class membership of every individual. The purpose of such an analysis is just to be
able to compare the fit of different models with different observed classes.
The model I wanted to run is the one explained in Example 7.3, however I'd like to restrict the class membership of every individual to match known classes in the data. These known classes should
identify exactly the latent classes in the model. A very simple task indeed.
I thought this could be easily solved by introducing, in the case of 2 classes, "TRAINING = t1 t2 (MEMBERSHIP)" in the VARIABLE command of Example 7.3. Like in this example, the MODEL command would
be empty.
In short: I'm fitting Example 7.3 restrictig latent classes to perfectly identify known classes in the data.....is "Training" the right option to introduce this restriction or is there any other way
to fit such a model?
Thanks again for your help.
ywang posted on Tuesday, August 18, 2009 - 7:10 am
Dear Drs. Muthen:
I have a follow-up question. Can the "multiple-group LCA analysis" be used instead of "LCA with training variables" to fit "the model with known classes"? What is the difference between
'multiple-group LCA" and 'LCA with training variables" in this particular situation as Andres described?
Thank you very much!
Bengt O. Muthen posted on Tuesday, August 18, 2009 - 4:49 pm
Training data can accomplish the same as Knownclass (multiple-group analysis). Neither is available when you have several latent class variables, but then you can use an observed indicator for known
class membership (see second approach to CACE modeling in the UG).
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edHelper.com - Logarithm Word Problems
1. Amy bought a diamond ring for $6,000. If the value of the ring increases at a constant rate of 3.83% per year, how much will the ring be worth in twenty-one years?
2. Greg bought a gold coin for $9,000. If the value of the ring increases at a constant rate of 1.79% per year, how many years will it be for the ring to be worth $17,978.02?
3. A bacteria population doubles every eight minutes. If the population begins with one cell, how long will it take to grow to 512 cells?
4. A bacteria population doubles every eight minutes. If the population begins with one cell, how long will it take to grow to 2,097,152 cells?
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Straight line detection
PyPK wrote:
> Does anyone know of a simple implementation of a straight line
> detection algorithm something like hough or anything simpler.So
> something like if we have a 2D arary of pixel elements representing a
> particular Image. How can we identify lines in this Image.
> for example:
> ary =
> [[1,1,1,1,1],
> [1,1,0,0,0],
> [1,0,1,0,0],
> [1,0,0,1,0],
> [1,0,0,0,1]]
> So if 'ary' represents pxl of an image which has a horizontal line(row
> 0),a vertical line(col 0) and a diagonal line(diagonal of ary). then
> basically I want identify any horizontal or vertical or diagonal line
> anywhere in the pxl array.
> Thanks.
I would recommend using a module for computing, my choice would be
You could even write your own version of hough, should not be too complex.
A fwee things you need to consider:
1) Are all the lines through the image, or would a row with
[0,0,1 ...(a few dozen ones in here) ... 1,0] be a line?
2) Do you also need edge detection? Then you might need to convolve
the image with a Laplacian or something like that, e.g.
new[i,j] = (4*old[i,j])-old[i-1,j]-old[i+1,j]-old[i,j-1]-old[i,j+1]
3) How "full" are the images?
It is much easier if only a small fraction of your image is lines,
in your example more than half of image pixels are lines.
4) How big images are you processing? I always have at least
one million pixels, so the rest may not work for small images.
To do some quicklook checks you can of course go through each row/column
and check if the values are different enough, something like
mat = numarray.array(ima)
x = mat.mean()
dx = mat.stddev()
then check if some rows are different from others, maybe
(mat[:,i].mean() > (x + N*dx)) for "white" lines or
(mat[:,i].mean() < (x - N*dx))) for "black" lines
you probably need do a few tests to get a good value of N.
repeat for columns (mat[j,:]) and diagonals:
numarray.diagonal(mat,o) where
o is offset from mat[0,0]
and if you need non-diagonal elements, say
ima = [[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 0 1]]
would contain a line of ones, then
vect = ima.flat
gives the image as a rank-1 array and you can then take strides
(every nth element) just like with normal lists, array[a:b:n]
takes every nth element in array[a:b], so vect[::7] would be [1 1 1]
I hope this helps a bit.
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Mathematica command question - matrix operator
January 13th 2011, 05:31 AM
Mathematica command question - matrix operator
Is it possible to prepare a set of commands, which lists elements of any size matrix in such sequence like this:
for 3x3 matrix:
$<br /> \left(<br /> \begin{array}{ccc}<br /> a & b & c \\<br /> d & e & f \\<br /> g & h & i<br /> \end{array}<br /> \right)<br />$
the sequence is: $a, b, c, f, i, h, g, d, e$;
for matrix 4x4:
$<br /> \left(<br /> \begin{array}{cccc}<br /> a & b & c & d \\<br /> e & f & g & h \\<br /> i & j & k & l \\<br /> m & n & o & p<br /> \end{array}<br /> \right)<br />$
the sequence is: $a, b, c, d, h, l, p, o, n, m, i, e, f, g, k, j$,
So, the operator which forms the sequence, moves on matrix like a point drawing a spiral, independently of matrix’s size.
Thanks for all your help.
January 13th 2011, 05:43 AM
I'm sure it is possible. Mathematica is a full-blown programming language with loops, if's, increments, etc. One command that will be helpful is the Dimensions command. For example,
Dimensions[{{1, 2}, {3, 4}, {5, 6}}] returns
You can access the first component of the result by indexing, which in Mathematica looks like this: [[i]]. So, for example,
Dimensions[{{1, 2}, {3, 4}, {5, 6}}][[1]] returns
Dimensions[{{1, 2}, {3, 4}, {5, 6}}][[2]] returns
One more comment: indexing into a matrix looks like this:
A[[1]][[4]], which gives you the 1,4 element of the matrix. For example,
{{1, 2}, {3, 4}, {5, 6}}[[3]][[1]] returns
I would set up some sort of a nested looping program to do what you are asking. Does this help?
January 13th 2011, 06:42 AM
It will be great, if you send me any proposition of exact program to obtain results, which I have desired.
Thanks in advance!
January 13th 2011, 06:45 AM
No, that's not the way we work around here. The helpers here are volunteers, and those asking questions are expected to put in the main effort to solve their problems. We're here to help people
get unstuck, not to exhibit complete solutions to the problems.
So it's your turn to do something. What ideas do you have?
January 13th 2011, 07:51 AM
So, if - for example - 3x3 matrix is
A = {{a, b, c}, {d, e, f}, {g, h, i}},
I may construct a list I've desired as
X = {A[[1, 1]], A[[1, 2]], A[[1, 3]], A[[2, 3]], A[[3, 3]], A[[3, 2]],
A[[3, 1]], A[[2, 1]], A[[2, 2]]}.
Is it possible to use it, for example with 'Which' and/or 'While' commands, to generalize the procedure? I'm trying to do it...
January 13th 2011, 08:31 AM
Excellent! Yes, your X list is precisely what you want. What you've got to do now is assign X programmatically. The While command would be very useful, I think. You could either append items to
your X list, or you could pre-allocate the X list and use the ReplacePart command to replace the elements one at a time.
Couple of ideas you might think about:
1. Use East, South, West, and North as indicator directions for which direction you're traveling in the matrix.
2. Use some sort of limit to tell you when to stop going in a particular direction.
Where does all this get you?
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odd functions
Show that arctan x is an odd function, that is, arctan –x = –arctan x.
I'm not objecting to CaptainBlack's proof as much as I have a question about how to get around a problem with it. Given an angle x define y: $y = tan(x)$ Then $atn(y) = atn(tan(x)) eq x$ in general
because of the domain restriction we place on the atn function to make it bijective. I can easily see how restricting the domain of the tan function would fix this, but then we aren't really using
the tan function. The only way I can think of to get around THIS is to extend the atn function so that it's no longer 1:1. But then it isn't really the inverse of the tan function any longer. I'm
kinda going in circles here... Just wondering if it wouldn't be better to prove that atn is an odd function by using something more direct. -Dan
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Summary: On Perfect Completeness for QMA
Scott Aaronson
Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with one-
sided error, has been an open problem for years. This note helps to explain why the problem
is difficult, by using ideas from real analysis to give a "quantum oracle" relative to which
QMA = QMA1. As a byproduct, we find that there are facts about quantum complexity
classes that are classically relativizing but not quantumly relativizing, among them such "trivial"
containments as BQP ZQEXP.
1 Introduction
The complexity class MA (Merlin-Arthur) was introduced by Babai [4] in 1985. Intuitively, MA
is a probabilistic version of NP; it contains all problems for which an omniscient wizard Merlin
can convince a probabilistic polynomial-time verifier Arthur of a "yes" answer, by a one-round
protocol in which Merlin sends Arthur a purported proof z, and then Arthur checks z. In the
usual definition, if the answer to the problem is "yes" then there should exist a string z that makes
Arthur accept with probability at least 2/3 (this property is called completeness), while if the
answer is "no" then no z should make Arthur accept with probability more than 1/3 (this property
is called soundness).
One of the first questions people asked about MA was whether it can be made to have perfect
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Number of results: 20
simplify [2+(-3)]^2
Monday, October 12, 2009 at 9:05pm by jerson
[2+(-3)]^2 = [-1]^2 now what is (-1)(-1) ?
Monday, October 12, 2009 at 9:05pm by Reiny
umm whats the (9,4)?
Monday, May 12, 2008 at 10:03pm by jerson
oh yes.. but how do u plug that into a calculator?
Monday, May 12, 2008 at 10:03pm by jerson
7th grade pre-alge
Monday, December 15, 2008 at 7:52pm by maya
If I understand your question correctly that would be C(9,4)(.6)^4(.4)^5 = .167
Monday, May 12, 2008 at 10:03pm by Reiny
The slope is 5/3 and the y-intercept is 2. You will have to draw the graph yourself.
Wednesday, May 26, 2010 at 7:03pm by drwls
it is C(9,4) or "9choose4" = 9!/(4!5!) If you are studying this level of probability, then you must be familiar with that concept.
Monday, May 12, 2008 at 10:03pm by Reiny
ok i figured that out. last question where'd the (.6)^4(.4)^5 i think i know where the .6 came from but the exponent is throwing me off
Monday, May 12, 2008 at 10:03pm by jerson
7th grade pre-alge
what is the answer to -40== -5p respond asap
Monday, December 15, 2008 at 7:52pm by Anonymous
What square root property is essential to solve any radical equation involving radicals?
Saturday, August 21, 2010 at 1:03pm by shelly
The angle of elevation of the sun is 31 degrees. Find the length of the shadow, to the nearest foot, of a man that is 6 feet tall.
Friday, March 30, 2012 at 8:58am by Anonymous
7th grade pre-alge
-40 = -5p -40 / - 5 = p 8 = p
Monday, December 15, 2008 at 7:52pm by Ms. Sue
Find the probability of x=4 sucesses in n=9 trials for the probability of succes p=0.6 on each trial. Round to the nearest thousanth.
Monday, May 12, 2008 at 10:03pm by jerson
Ok, let's just consider one of the possible outcomes. S = success, F = failure SSFFSFFFS (4 successes, 5 failures) the prob of that specific event is .6 x .6 x .4 x .4 x .6 x .4 x .4 x .4 x .6 = (.6)
^4 x (.4)^5 but the SSFFSFFFS can be arranged in 126 ways, so .....
Monday, May 12, 2008 at 10:03pm by Reiny
length of shadow --- x ft tan31° = 6/x x = 6/tan31 = 9.98567... or appr 10 ft
Friday, March 30, 2012 at 8:58am by Reiny
Graph the equation using the slope and the y-intercept. y=5/3x+2 I need to show the slope and y-intercept when I graph.
Wednesday, May 26, 2010 at 7:03pm by Craig
Saturday, August 21, 2010 at 1:03pm by bobpursley
depends on the calculator. On mine, which is a Casio, it is shown as nCr On many calculators it is paired with nPr which is defined as n!/(n-r)! and is used in permutations. If you cannot find it on
your calculator, look for the factorial key ! , which is found on most ...
Monday, May 12, 2008 at 10:03pm by Reiny
Prob(success) + prob(failure) = 1 so if prob(success) = .6 then prob(failure) = 1 - .6 = .4
Monday, May 12, 2008 at 10:03pm by Reiny
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Carnival of Mathematics #33 - The rushed edition!
Carnival of Mathematics #33 – The rushed edition!
Hello and welcome to the 33rd edition of the Carnival of Mathematics. This carnival very nearly didn’t happen since I didn’t realise that no one had offered to host it until a couple of days ago! I
toyed with the idea of letting this edition of the carnival lapse and write something in a fortnights time but then that would break the carnivals unbroken run of 33 publications (well..apart from
that one time which we don’t talk about) and I simply couldn’t have that. So, with only two days to go I bent the standard carnival rules a little and started leaning on people I know in order to get
submissions. After that I started leaning on people I didn’t know and I am glad to say that everyone came through and I have a nice selection of articles for you all.
Before I get onto the articles themselves, tradition dictates that I attempt to fascinate you with some interesting facts concerning the number 33. Well how about this one:
It is known that for all numbers N below 1000 that do not have the form N as a sum of three cubes. In other words
where a,b and c can be positive or negative. What does this have to do with the number 33? Well, 33 is the smallest such number for which a,b and c have not yet been found. If you fancy having a
crack at solving this be aware that the solution for N=30 is
Anyway, enough with the trivia and on with the show!
As some of you know, I am a big fan of computer algebra systems (well most of them anyway) and so I thought I would start off with some submissions from three of the big names in the CAS world,
Wolfram Research, The Mathworks and SAGE. I use the products of all three of these groups to one degree or another and so it is great to see submissions from them all. This is one of the areas where
I bent the carnival rules slightly since I emailed the blog authors and said “Hi – please submit something to the carnival.” I thank them for humoring me and not consigning my email to the spam bin.
Loren from Loren on the Art of Matlab writes a regular blog on Matlab programming and her submission is a recent post entitled Acting on Specific Elements in a Matrix where she uses several methods
to obtain the same result. This sort of article is very instructive when thinking about how to go about developing your code. Although she did not submit it, I thought that many carnival readers
would also be interested in her post called Matlab Publishing for Teaching.
Next up from the Mathworks we have Doug whose submission is a coin tossing puzzle which he invites you to solve using Matlab. Some solutions can be found in the comments section so resist the urge to
scroll down if you want to try and solve it yourself. Solving problems like this, using any system, can be a great way of learning how to use it – much more interesting than just reading through the
manual; no matter how well written it is.
Moving over to the Wolfram Research Blog we have two posts in this edition of the carnival, the first of which is called Two Hundred Thousand New formulas on the Web which is a discussion of The
Wolfram Functions Site. At the time of writing the site has over 307,00 formulas on it which is, quite frankly, astonishing! Pretty useful too!
Next up from Wolfram we have a blog post called Making Photo Mosaics. It never ceases to amaze me how much you can achieve with so little code – I will be having a play with this code using photos
from my recent vacation :) Check out the video that Theodore has produced as part of this post as I think it’s fascinating.
Moving over to the world of open source we have a submission from William Stein – Can There be a Viable Free Open Source Alternative to Magma, Maple, Mathematica and Matlab? where he discusses the
SAGE project. I have recently been looking at SAGE myself and have been very impressed with it.
This edition of the carnival isn’t just about computer algebra packages though – we also have lots of non-CAS submissions. The first of which is one from Maria over at the TCM Technology Blog where
she writes about her talk, Exploring Online Calculus, at the Michigan MAA meeting. Gotta love those graphs :)
John of jd2718 asks Can we find the area of a quadrilateral from just it’s co-ordinates?, with some interesting answers in the comments section. I reckon a nice Wolfram Demonstration could be made
from this idea.
Sam Shah thinks that algebraic manipulation is overrated – head over to his blog to see why. In another post, Sam also writes about some interesting calculus projects that he has assigned to his
students. When I was at school I used to love open-ending projects as it used to give me a sense of ‘owning the material’. I distinctly remember doing a project on the Fibonacci sequence when I was
11 years old and spending ages on it. To this day I still have a fascination for the topic and probably always will. I wonder how often such projects can be done by school children in todays
test-centric environment?
Moving on, we have Math for the Very Patient from Vlorbik on Math Ed. Vlorbik has already demonstrated his patience in the past since my blog looks horrible on his browser and yet he still reads what
I have to say – thanks Vlorbik! I seem to have a problem with IE 6 that I have no idea how to fix. Just look at this blog in IE 6 compared to firefox to see what we mean. One hexadecimal pound (thats
two pounds and fifty six pence) to the first person who can diagnose and fix the problem for me.
Over at blinkdagger (among other things, a great source of Matlab tutorials) they have a competition where you can win prizes from the people at the art of problem solving. There is still time to
enter so take a look at BlinkDagger burgers and have a go.
If you like the level of your mathematics to be a bit higher and median graphs are your thing then you will be interested in David Eppstein’s submission Median graphs and binary majorization over at
Denise of Let’s Play Math sent me the details of her latest post, The Function Machine Game. This is another one I remember doing when I was at school. As she suggests it’s probably best to limit the
functions one can choose from – “Waddya mean you couldn’t get it – BesselJ(x) is simple!” I feel yet another Wolfram Demonstration coming on :)
Next we have a post from a blog that writes posts on the all time classic combination of subjects, cats and maths – Catsynth.com. The post is about how to calculate without having to calculate all of
the primes up to x. I wonder how the various CAS systems calculate this function? Anyone care to enlighten me?
Finally, in another bending of the rules, I’d like to present Five Open Problems Regarding Convex Polytopes from Gil Kalai’s blog, Combinatorics and more. He didn’t submit this post himself but it
comes highly recommended and so I hope he will not mind having it included here.
And…that’s it for this 33rd edition of the carnival. Thank you to everyone who submitted something – without you the carnival would be..well..just me posting a load of links! Finally, would someone
please volunteer to host the 34th edition of the carnival? I think it really is a lovely tradition that has been kept going by maths bloggers for almost 18 months now, which is like an eternity in
internet years and it would be a shame to see it go. I think that it’s a great way of finding new math blogs and also of generating a sense of community in the maths blogsphere.
Update: As it says in the comments, the next Carnival will be hosted over at 360 on May 30th so please head over there and submit a post. Making a submission is as easy as saying “Hi, what about this
one…< insert link here>” 9 times out of 10 your post will be accepted so its an easy way to promote your blog.
May 16th, 2008 at 19:58
Reply | Quote | #1
outstanding. i won’t be trying for the 2.56, though …
May 17th, 2008 at 00:19
Reply | Quote | #2
Thanks for hosting this!
This carnival (or rather one of the links in it — I’m leaving it a mystery which one) inspired me to write a short post about similar triangles: http://11011110.livejournal.com/139750.html
May 18th, 2008 at 12:43
Reply | Quote | #3
Thanks for hosting this, especially with only a couple days notice!
We just got confirmation from Alon that we’ll be hosting the next Carnival of Mathematics on May 30 over at 360 (http://threesixty360.wordpress.com).
May 18th, 2008 at 20:07
Reply | Quote | #4
Thanks for mentioning my post -Gil
May 19th, 2008 at 02:28
Reply | Quote | #5
Not exactly a diagnosis and fix (I don’t have IE6 around anywhere, so I can’t see what its doing), but… a quick glance at the stylesheet shows that the sidebar class is floated, and I don’t see any
IE6-specific code to account for its buggy float rendering. The Microsoft standard way of fixing this is to put conditional comments around the extra CSS statements needed to get IE6 to behave
May 19th, 2008 at 09:40
Reply | Quote | #6
vlorbik – clearly you have too much money then ;)
Michael – thanks for the info. Not quite enough for the hex pound though ;)
I might fix it eventually but I will first have to learn about stylesheets and I figure it would be much easier to just wait until IE6 is obsolete!
Gil – You are welcome. Feel free to submit a post to next weeks carnival over at http://threesixty360.wordpress.com
David – Its good to see that the carnival can inspire blog posts as well as link to them.
Heather – Hosting the carnival is always a pleasure :)
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Department of Mathematics
Course Info
MATH 3210: Abstract Linear Algebra
Description: Linear algebra is one of the most productive branches of mathematics. Almost no science can survive without a serious use of linear algebra. Moreover, ideas throughout higher mathematics
are often at some point related to "simple" linear algebra manipulations. The key idea is "linearization," which deals with the attempt at describing the information one wants to study in terms of
linear algebra objects (vector spaces, operators, etc). We will try to understand such notions and make use of them in studying problems which at first glance may not seem to be "linear". Examples we
will look at include explicit formulas for the famous Fibonacci and Lucas numbers, polynomial interpolation, factoring integers, solving difference and differential equations, and Hurwitz's
celebrated 1,2,4,8 theorem.
Prerequisites: MATH 2210(227) and a grade of C or better in either MATH 2142(244) or 2710(213).
Offered: Spring
Credits: 3
Sections: Spring 2011 on Storrs Campus
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Combining quantum information communication and storage
The latest news from academia, regulators research labs and other things of interest
Posted: Feb 14, 2013
Combining quantum information communication and storage
(Nanowerk News) "This work represents the first step towards creating exotic mechanical
quantum states. For example, the transfer makes it possible to create a state in which the
resonator simultaneously vibrates and doesn’t vibrate," says Mika Sillanpää, professor at
Aalto University, who runs the research group.
Combining quantum information communication and storage
A qubit is the quantum-mechanical equivalent of the bits we know from computers. A
traditional bit can be in a state of 0 or 1, while a qubit can be in both states at the
same time. In theory, this inconceivable situation allows for a quantum calculation in
which the operations are performed simultaneously for all possible numbers. In the case of
a single qubit, this means zero and one, but as the number of qubits increases, the amount
of possible numbers and simultaneous calculations grows exponentially.
Merger of three quantum systems: superconducting quantum qubit, or, qubit (spheres)
interacting with two different resonant cavities. (Image: Juha Juvonen)
The quantum state of a qubit is very fragile and easily disturbed between and during the
operations. The key to successful quantum calculation is being able to protect the qubit
state from disturbances in the environment.
Combining quantum information communication and storage
"In this case, the qubit state can be stored as vibration, thus preserving the state for
much longer than the qubit itself. The resonator also functions as a mechanical quantum
memory, which is something that an ordinary memory can't do," explains Juha Pirkkalainen,
who is doing his dissertation on the topic.
Combining quantum information communication and storage
The novel work combines two Nobel Winner's achievements
Combining quantum information communication and storage
The work ("Hybrid circuit cavity quantum electrodynamics with a micromechanical resonator"
) combined the achievements of both winners of this year’s Nobel Prize for Physics. The
qubit state was measured using a superconducting cavity in the same way that Serge Haroche
measured atoms, and the qubit state was also linked to mechanical movement as in David
Wineland’s experiments. In contrast to these larger-scale measurement arrangements, the
experiment at the O.V. Lounasmaa Laboratory was prepared for a tiny silicon microchip.
This made it possible to cool the sample to near absolute zero temperatures and then use
Subscribe to a free copy of one of our daily
Nanowerk Newsletter Email Digests
with a compilation of all of the day's news.
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A Metal Bar AB, Of Length 4m, Is Held Perpendicular ... | Chegg.com
Calculate the value of the value of M, the mass suspended from B and the magnitude and direction of the net force exerted by the wall on the rod. (Hint: consider all the forces acting on the rod
alone -- they must sum to zero in any direction, and their net torque about any point must be zero)
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Beethoven (5th Symphony, Appassionata, Waldstein)
Beethoven (5th Symphony, Appassionata, Waldstein)
The use of mathematical devices is deeply embedded in Beethoven's music. Therefore, this is one of the best places to dig for information on the relationship between mathematics and music. I'm not
saying that other composers do not use mathematical devices. Practically every musical composition has mathematical underpinnings. However, Beethoven was able to extend these mathematical devices to
the extreme. It is by analyzing these extreme cases that we can find more convincing evidence on what types of devices he used.
We all know that Beethoven never really studied advanced mathematics. Yet he incorporates a surprising amount of math in his music, at very high levels. The beginning of his Fifth Symphony is a prime
case, but examples such as this are legion. He "used" group theory type concepts to compose this famous symphony. In fact, he used what crystallographers call the Space Group of symmetry
transformations! This Group governs many advanced technologies, such as quantum mechanics, nuclear physics, and crystallography that are the foundations of today's technological revolution. At this
level of abstraction, a crystal of diamond and Beethoven's 5th symphony are one and the same! I will explain this remarkable observation below.
The Space Group that Beethoven "used" (he certainly had a different name for it) has been applied to characterize crystals, such as silicon and diamond. It is the properties of the Space Group that
allow crystals to grow defect free and therefore, the Space Group is the very basis for the existence of crystals. Since crystals are characterized by the Space Group, an understanding of the Space
Group provides a basic understanding of crystals. This was neat for materials scientists working to solve communications problems because the Space Group provided the framework from which to launch
their studies. It's like the physicists needed to drive from New York to San Francisco and the mathematicians handed them a map! That is how we perfected the silicon transistor, which led to
integrated circuits and the computer revolution. So, what is the Space Group? And why was this Group so useful for composing this symphony?
Groups are defined by a set of properties. Mathematicians found that groups defined in this way can be mathematically manipulated and physicists found them to be useful: that is, these particular
groups that interested mathematicians and scientists provide us with a pathway to reality. One of the properties of groups is that they consist of Members and Operations. Another property is that if
you perform an Operation on a Member, you get another Member of the same Group. A familiar group is the group of integers: -1, 0, 1, 2, 3, etc. An Operation for this group is addition: 2 + 3 = 5.
Note that the application of the operation + to Members 2 and 3 yields another Member of the group, 5. Since Operations transform one member into another, they are also called Transformations. A
Member of the Space Group can be anything in any space: an atom, a frog, or a note in any musical space dimension such as pitch, speed, or loudness. The Operations of the Space Group relevant to
crystallography are Translation, Rotation, Mirror, Inversion, and the Unitary operation. These are almost self explanatory (Translation means you move the Member some distance in that space) except
for the Unitary operation which basically leaves the Member unchanged. However, it is somewhat subtle because it is not the same as the equality transformation, and is therefore always listed last in
textbooks. Unitary operations are generally associated with the most special member of the group, which we might call the Unitary Member. In the integer group noted above, this Member would be 0 for
addition and 1 for multiplication (5+0 = 5x1 = 5).
Let me demonstrate how you might use this Space Group, in ordinary everyday life. Can you explain why, when you look into a mirror, the left hand goes around to the right (and vice versa), but your
head doesn't rotate down to your feet? The Space Group tells us that you can't rotate the right hand and get a left hand because left-right is a mirror operation, not a rotation. Note that this is a
strange transformation: your right hand becomes your left hand in the mirror; therefore, the wart on your right hand will be on your left hand image in the mirror. This can become confusing for a
symmetric object such as a face because a wart on one side of the face will look strangely out of place in a photograph, compared to your familiar image in a mirror. The mirror operation is why, when
you look into a flat mirror, the right hand becomes a left hand; however, a mirror cannot perform a rotation, so your head stays up and the feet stay down. Curved mirrors that play optical tricks
(such as reversing the positions of the head and feet) are more complex mirrors that can perform additional Space Group operations, and group theory will be just as helpful in analyzing images in a
curved mirror. The solution to the flat mirror image problem appeared to be rather easy because we had a mirror to help us, and we are so familiar with mirrors. The same problem can be restated in a
different way, and it immediately becomes much more difficult, so that the need for group theory to help solve the problem becomes more obvious. If you turned a right hand glove inside out, will it
stay right hand or will it become a left hand glove? I will leave it to you to figure that one out (hint: use a mirror).
Let's see how Beethoven used his intuitive understanding of spatial symmetry to compose his 5th Symphony. That famous first movement is constructed largely by using a single short musical theme
consisting of four notes, of which the first three are repetitions of the same note. Since the fourth note is different, it is called the surprise note, and carries the beat. This musical theme can
be represented schematically by the sequence 5553, where 3 is the surprise note. This is a pitch based space group; Beethoven used a space with 3 dimensions, pitch, time, and volume. I will consider
only the pitch and time dimensions in the following discussions. Beethoven starts his Fifth Symphony by first introducing a Member of his Group: three repeat notes and a surprise note, 5553. After a
momentary pause to give us time to recognize his Member, he performs a Translation operation: 4442. Every note is translated down. The result is another Member of the same Group. After another pause
so that we can recognize his Translation operator, he says, "Isn't this interesting? Let's have fun!" and demonstrates the potential of this Operator with a series of translations that creates music.
In order to make sure that we understand his construct, he does not mix other, more complicated, operators at this time. In the ensuing series of bars, he then successively incorporates the Rotation
operator, creating 3555, and the Mirror operator, creating 7555. Somewhere near the middle of the 1st movement, he finally introduces what might be interpreted as the Unitary Member: 5555. Note that
these groups of 5 identical notes are simply repeated, which is the Unitary operation.
In the final fast movements, he returns to the same group, but uses only the Unitary Member, and in a way that is one level more complex. It is always repeated three times. What is curious is that
this is followed by a fourth sequence -- a surprise sequence 7654, which is not a Member. Together with the thrice repeated Unitary Member, the surprise sequence forms a Supergroup of the original
Group. He has generalized his Group concept! The supergroup now consists of three members and a non-member of the initial group, which satisfies the conditions of the initial group (three repeats and
a surprise).
Thus, the beginning of Beethoven's Fifth symphony, when translated into mathematical language, reads just like the first chapter of a textbook on group theory, almost sentence for sentence! Remember,
group theory is one of the highest forms of mathematics. The material is even presented in the correct order as they appear in textbooks, from the introduction of the Member to the use of the
Operators, starting with the simplest, Translation, and ending with the most subtle, the Unitary operator. He even demonstrates the generality of the concept by creating a supergroup from the
original group.
Beethoven was particularly fond of this four-note theme, and used it in many of his compositions, such as the first movement of the Appassionata piano sonata, see bar 10, LH. Being the master that he
is, he carefully avoids the pitch based Space Group for the Appassionata and uses different spaces -- he transforms them in tempo space and volume space (bars 234 to 238). This is further support for
the idea that he must have had an intuitive grasp of group theory and consciously distinguished between these spaces. It seems to be a mathematical impossibility that this many agreements of his
constructs with group theory just happened by accident, and is virtual proof that he was somehow playing around with these concepts.
Why was this construct so useful in this introduction? It certainly provides a uniform platform on which to hang his music. The simplicity and uniformity allow the audience to concentrate only on the
music without distraction. It also has an addictive effect. These subliminal repetitions (the audience is not supposed to know that he used this particular device) can produce a large emotional
effect. It is like a magician's trick -- it has a much larger effect if we do not know how the magician does it. It is a way of controlling the audience without their knowledge. Just as Beethoven had
an intuitive understanding of this group type concept, we may all feel that some kind of pattern exists, without recognizing it explicitly. Mozart accomplished a similar effect using repetitions.
Knowledge of these group type devices that he uses is very useful for playing his music, because it tells you exactly what you should and should not do. Another example of this can be found in the
3rd movement of his Waldstein sonata, where the entire movement is based on a 3-note theme represented by 155 (the first CGG at the beginning). He does the same thing with the initial arpeggio of the
1st movement of the Appassionata, with a theme represented by 531 (the first CAbF). In both cases, unless you maintain the beat on the last note, the music loses its structure, depth and excitement.
This is particularly interesting in the Appassionata, because in an arpeggio, you normally place the beat on the first note, and many students actually make that mistake. As in the Waldstein, this
initial theme is repeated throughout the movement and is made increasingly obvious as the movement progresses. But by then, the audience is addicted to it and does not even notice that it is
dominating the music. For those interested, you might look near the end of the 1st movement of the Appassionata where he transforms the theme to 315 and raises it to an extreme and almost ridiculous
level at bar 240. Yet most in the audience will have no idea what device Beethoven was using, except to enjoy the wild climax, which is obviously ridiculously extreme, but by now carries a mysterious
familiarity because the construct is the same, and you have heard it hundreds of times. Note that this climax loses much of its effect if the pianist does not bring out the theme (introduced in the
first bar!) and emphasize the beat note.
Beethoven tells us the reason for the inexplicable 531 arpeggio in the beginning of the Appassionata when the arpeggio morphs into the main theme of the movement at bar 35. That is when we discover
that the arpeggio at the beginning is an inverted and schematized form of his main theme, and why the beat is where it is. Thus the beginning of this piece, up to bar 35, is a psychological
preparation for one of the most beautiful themes he composed. He wanted to implant the idea of the theme in our brain before we heard it! That may be one explanation for why this strange arpeggio is
repeated twice at the beginning using an illogical chord progression. With analysis of this type, the structure of the entire 1st movement becomes apparent, which helps us to memorize, interpret, and
play the piece correctly.
The use of group theoretical type concepts might be just an extra dimension that Beethoven wove into his music, perhaps to let us know how smart he was, in case we still didn't get the message. It
may or may not be the mechanism with which he generated the music. Therefore, the above analysis gives us only a small glimpse into the mental processes that inspire music. Simply using these devices
does not result in music. Or, are we coming close to something that Beethoven knew but didn't tell anyone?
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188 helpers are online right now
75% of questions are answered within 5 minutes.
is replying to Can someone tell me what button the professor is hitting...
• Teamwork 19 Teammate
• Problem Solving 19 Hero
• Engagement 19 Mad Hatter
• You have blocked this person.
• ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.
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System of Equations with Sinc
November 12th 2012, 06:53 AM
System of Equations with Sinc
I have the following system:
$A\frac{\sin (B(f_1 - f_0))}{B(f_1 - f_0)} = X_1$
$A\frac{\sin (B(f_2 - f_0))}{B(f_2 - f_0)} = X_2$
where the unknowns are $A$ and $f_0$.
Can you solve such a system analytically? I can reduce it to k*sinc(...) = sinc(...) but what do I do from there?
November 12th 2012, 08:06 PM
Re: System of Equations with Sinc
Hey fobos3.
Can you show us what you have tried?
November 13th 2012, 01:33 PM
Re: System of Equations with Sinc
I tried solving for A in the two equations and then equating the result. This gives you something like:
$k\times sinc(B(f_1-f_0)) = sinc(B(f_2-f_0))$
I also tried expressing the two equations in exponential form but couldn't figure out how to solve it. I am not sure if it is possible to solve the system analytically.
This is not a homework problem and it is related to calculating the frequency and amplitude of a sine wave using two recursive DFTs.
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Some niceties in C.
Many large scale computation problems are sparse matrix inversions of one kind or another. And the way to parallelize, or Beowulfize them, is to take a matrix shaped like this,
What I would like to do is to be able to write the set of procedures that do this in a way that will enable others not to have to rewrite the wheel. Hence my libstripe effort. Libstripe wants to be
independent of the structure of the sparse matrix. It works on the assumption that the matrix rows have some sort of defined coherent organization and proceeds from there. Now, in most cases, we can
rationally expect each process in this lineup to be dependent only on its neighbors for continued work on its dataset.
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Using a Protractor
------ Note: The Information above this point will not be sent to your printer --------
------ Note: The Information below this point will not be sent to your printer --------
Related Resources
The various resources listed below are aligned to the same standard, (4MD06) taken from the CCSM (Common Core Standards For Mathematics) as the Geometry Worksheet shown above.
Measure angles in whole-number degrees using a protractor. Sketch angles of specified ...
Similar to the above listing, the resources below are aligned to related standards in the Common Core For Mathematics that together support the following learning outcome:
Geometric measurement: understand concepts of angle and measure angles
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RE: st: Re: Loop syntax
[Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index]
RE: st: Re: Loop syntax
From "Nick Cox" <n.j.cox@durham.ac.uk>
To <statalist@hsphsun2.harvard.edu>
Subject RE: st: Re: Loop syntax
Date Wed, 26 Mar 2008 16:15:05 -0000
You can count occurrences of t > 0 separately by patient.
egen tcount = sum(t > 0), by(patient)
Then counting patients, not recordings, is possible e.g. by tagging, say
egen tag = tag(patient)
count if tag & tcount
For much more detail, see the thread started on 18 March by Daniel
Bill Gould gave the first principles answer, while I gave an -egen-
They are complementary.
Johannes Geyer
I have some results on 236 patients and each patient has 57 recordings
stored in a single variable t. I am trying to count how many of these
patients have a score of t>0. I tried the following code:
gen tcount=0
forvalues i=1/236{
forvalues j=1/57{
if patient==`i' & t>0 {
then tcount==tcount+1
I think a loop is unnecessary here - you just want to count, right?
are many solutions, e.g.
count if t >0 & t<.
(saved in r(N))
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
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2002 Keystone RV Prices, Values and Specs
A Goshen, Indiana company created in 1996, Keystone RV produces an extensive range of travel and fifth wheel trailers. Designing trailers between 17 and 42 feet in length, lightweight construction
and an aerodynamic profile is a leading quality of Keystone RV products. Keystone RV has quickly gained popularity in the recreational vehicle industry becoming a top-selling trailer name in the
United States.
Read more Read less
Notes: Manufacturer note(s): TRAVEL TRAILERS/5TH WHEELS - Prices include air conditioner, awning, stabilizer jacks, microwave, AM/FM cassette stereo and water heater with DSI. Challenger and Hornet
previously listed under Damon Corporation.
Year note(s): TRAVEL TRAILERS/5TH WHEELS - Challenger models for 2002 have less standard equipment than in previous years.
Read more Read less
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4-4 Calculating CFU from dilution plating results
(177665 Reads)
Table of Contents| Chapter Article List| Printable Version | Printable Chapter
How des a count on a plates get converted to CFUs per gram or ml of sample? Let's illustrate the procedure with an example. Imagine that we perform the following experiment:
Five ml of milk are added to 45 ml of sterile diluent. From this suspension, two serial, 1/100 dilutions are made, and 0.1 ml is plated onto Plate Count Agar from the last dilution. After incubation,
137 colonies are counted on the plate.
This problem may be illustrated as follows:
Figure 4-2 A drawing of the dilution problem
It is normally a good idea to draw out dilution problems until you are comfortable doing them.
Note that it is often a good idea to draw out dilution problems until you are comfortable doing them. It will help you to develop a clear picture of what is being done.
The first step in solving this problem is to work out the total dilution of the sample. First 5 ml is added to 45 ml; This is a 1/10 dilution.
Figure 4-3 Initial dilution
The initial dilution is a 1 to 10 dilution.
Remember, there are many ways to make 1/10 and 1/100 dilutions. A 0.1 ml to 0.9 ml dilution is the same as a 1 ml to 9 ml dilution and a 13 ml to 117 ml dilution. Next, 1 ml of the first dilution is
added to 99 ml to make the second dilution, that is a 1/100 dilution. This is repeated with third dilution giving another 1/100 dilution. Then 0.1 ml of the third dilution is plated out on a plate of
PCA. The total dilution of the sample is cumulative and can be represented mathematically as....
Figure 4-4 Calulating total dilution
The total dilution for the problem
Notice that the amount put on the plate is also a dilution. Normally CFUs are reported per ml or per gram. In some cases less than 1 ml is put on the plate and this must be taken into account. One
way to solve this, is to factor it into the total dilution. In this problem 0.1 ml was added to the plate, or 1/10th of a ml. So multiply the total dilution by 1/10 for the amount added to the plate.
This leaves the total dilution as one-one millionth. The next step is to work out the dilution factor. The dilution factor is the reciprocal of the total dilution. In this case it would be......
Figure 4-5 Dilution factor
A mathematical representation of the diluction factor.
Finally, multiply the total dilution by the average number of colonies in the plate(s) and report your answer in CFUs/ml or CFUs/gram depending upon where the sample came from; in this case ml
because we used milk as a sample.
Figure 4-6 Total colony forming units
A calculation of the total number of CFUs in the original milk sample.
With enough practice, dilution problems can be worked out quite easily and rapidly. The method described above is just a suggested approach, if you find another way to do these problems which is more
intuitive for you, use it. When doing dilution problems, remember the following:
• Note that using this method, the answer in CFUs per one milliliter or per one gram is derived. Answers may need to be adjusted if the number of CFUs per sample (other than a milliliter or gram)
is requested. Assume 1 gram = 1 ml. (1 ml of water does indeed weigh 1 gram. That is actually how the ml is defined.)
• Use only those plates with colony counts between 30 and 300. With duplicate or triplicate plating from the same dilution, take the average of the plate counts and then proceed.
• Note that all individual dilutions and the amount plated are multiplied together.
• The initial dilution is often different from the subsequent dilutions. This is generally due to the nature of the sample available for analysis.
• Decimal (1:10) dilutions can be made by adding 1 ml to 9 ml. Proportional amounts can be utilized such as by adding 0.1 ml to 0.9 ml or 11 ml to 99 ml.
• Centimal (1:100) dilutions can be made by adding 1 ml to 99 ml. This can also be done by adding 0.1 ml to 9.9 ml.
• Note that plating 0.1 ml of a 10^-4 dilution results in the same dilution factor (10^5) as plating 1 ml of a 10^-5 dilution.
Here is another sample problem. Using any method you choose, solve the problem.
One ml of a bacterial culture is pipetted into a 9 ml dilution blank. One-tenth ml of this dilution is pipetted into a 9.9 ml dilution blank. From this dilution one-tenth ml is plated using 25 ml of
Plate Count Agar. 219 colonies arise after incubation. How many colony-forming units were present per ml of the original culture?
The correct answer: 2.19 X 10^6 CFUs / ml. (Note, the 25 ml of Plate Count Agar plated is irrelevant. Why?)
Table of Contents| Chapter Article List| Printable Version Printable Chapter
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Comparison of Waiting Times
On the Comparison of Waiting Times in Tandem Queues
Using a definition of partial ordering of distribution functions, it is proven that for a tandem queueing system with many stations in series, where each station can have either one server with an
arbitrary service distribution or a number of constant servers in parallel, the expected total waiting in system of every customer decreases as the interarrival and service distributions become
smaller with respect to that ordering. Some stronger conclusions are also given under stronger order relations. Using these results, bounds for the expected total waiting time in system are then
readily obtained for wide classes of tandem queues.
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Physics Forums - View Single Post - Billiard Balls
My problem involves rolling a billiard ball with an initial x velocity of 2.4ms and an initial y velocity of 0.7ms. The rolling friction is 0.1 and I have to work out the final resting position of
the ball. There is no sliding friction involved. I have to break this into 0.1seconds and return the results, i think that I have to work out the initial decelleration then the speed and new
position, and then the next decelleration and so on?
the mass of the ball is 0.17kg
I'm pretty new to this, so sorry if its a bit basic.
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Differential Geometry of Curves and Surfaces 1st Edition Chapter 4.4 Solutions | Chegg.com
b) If the geodesic is a plane curve, then its torsion becomes zero.
Then reverse the statement of part (a)
It is a line of curvature
Therefore if geodesic is a plane curve then it is a line of curvature.
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NA Digest Sunday, February 4, 1990 Volume 90 : Issue 05
NA Digest Sunday, February 4, 1990 Volume 90 : Issue 05
Today's Editor: Cleve Moler
Today's Topics:
From: T. J. Garratt <tjg%maths.bath.ac.uk@nsfnet-relay.ac.uk>
Date: Mon, 29 Jan 90 15:30:32 GMT
Subject: Roommate Needed at Copper Mountain Conference
WANTED: Person to share room for conference:
"ITERATIVE METHODS", Copper Mountain, Colorado,
1st - 5th April, 1990.
I am a male postgraduate studying for my PhD in Numerical Analysis
at Bath University, and will be attending the above conference.
To help with the costs of accommodation, I am looking for someone to
share a lodge room or deluxe studio.
Perhaps a student in a similar situation might be interested.
If you are interested or know someone who may be, then please contact:
Tony Garratt,
School of Mathematical Sciences,
Univeristy of Bath,
Claverton Down, Bath.
AVON. BA2 7AY.
United Kindgom.
E-mail: tjg@uk.ac.bath.baths
(OR na.spence@edu.stanford.na-net)
From: Bob Ward <ward@rcwsun.EPM.ORNL.GOV>
Date: Tue, 30 Jan 90 10:42:07 EST
Subject: Liz Jessup Wins Householder Fellowship at Oak Ridge
Elizabeth R. Jessup has been selected as the winner of the first
Householder Fellowship at the Oak Ridge National Laboratory (ORNL).
Dr. Jessup, who received her doctorate degree in Computer Science in
1989 from Yale University, is currently an Assistant Professor of
Computer Science at the University of Colorado at Boulder. Her
research interests are in parallel computing and numerical linear
Dr. Jessup will be collaborating with the researchers in ORNL's
Mathematical Sciences Section and with applied computational scientists
in various divisions at ORNL on scientific problems involving high
performance computing. Her primary interest will be on parallel
algorithms for solving large-scale eigenproblems on a
distributed-memory MIMD multiprocessor. Her fellowship appointment
will begin this summer.
Alston S. Householder was the organizer and founding Director of the
Mathematics Division (precursor of the current Mathematical Sciences
Section) at ORNL. In recognition of the seminal research contributions
of Dr. Householder to the fields of numerical analysis and scientific
computing, a distinguished postdoctoral fellowship program was
established and named in his honor. Householder Fellows will be
appointed annually for a term of one year, renewable for a second
The Householder Fellowship Program is supported by the Applied
Mathematical Sciences Subprogram of the U.S. Department of Energy.
From: Jorge More <more@antares.mcs.anl.gov>
Date: Wed, 31 Jan 90 09:21:30 CST
Subject: Barry Smith Wins Wilkinson Fellowship at Argonne
We are pleased to announce that Barry Smith from the Courant
Institute of Mathematical Sciences is the 1990 Wilkinson fellow.
Barry is a student of Olof Widlund working on domain decomposition
algorithms for the partial differential equations of linear elasticity.
In addition to Courant, he has worked at the IBM T. J. Watson Research
Center, Los Alamos National Laboratory, and at the University of Bergen.
He will join the Mathematics and Computer Science Division of Argonne
National Laboratory in the summer.
From: Henry Wolkowicz <hwolkocz@orion.waterloo.edu>
Date: Mon, 29 Jan 90 15:48:23 EST
Subject: Distance of a Matrix to a Subspace
How would one find (numerically) the distance between a given real
n by n matrix A and the given subspace S, where S is the subspace
of upper triangular matrices which are themselves made up of
k by k upper triangular blocks ?
The distance is the inf of spectral norms (largest singular value).
Henry Wolkowicz; Department of Combinatorics and Optimization;
Faculty of Mathematics; University of Waterloo;
Waterloo, Ontario, Canada N2L 3G1 (519-888-4597 office; 746-6592 FAX)
{hwolkowicz@water.bitnet; na.wolkowicz@na-net.stanford.edu}
{hwolkowicz@water.uwaterloo.ca; usersunn@ualtamts.bitnet }
From: Ben Lotto <ben@cps3xx.egr.msu.edu>
Date: 1 Feb 90 20:10:23 GMT
Subject: Numerical Integration Program Wanted
I would like a numerical integration program that will handle a Cauchy
principal value integral of the following form:
\lim_{\epsilon\to 0}
\int_{\epsilon}^{\pi} (f(\theta - t) - f(\theta + t)) / tan(t/2) dt
(this computes the conjugate function of f) where f is a function that
has a a couple of jump discontinuities (I could probably fudge things
and get rid of this) and a log x-type singularity. In particular, I
would like the algorithm to work for the function
f(x) = log |x|, if |x| < \pi / 2
0, if |x| >= \pi / 2
Reply by e-mail, please, as I don't read this newsgroup regularly.
Thanks in advance.
-B. A. Lotto (ben@nsf1.mth.msu.edu)
Department of Mathematics/Michigan State University/East Lansing, MI 48824
From: Bill Anderson <XB.N64@Forsythe.Stanford.EDU>
Date: Thu, 1 Feb 90 20:41:31 PST
Subject: Summer Programs for Undergraduates
Last week's NA Digest included an announcement of a Summer program
for undergraduates at CNSF at Cornell. Are there additional Summer
programs to which I could encourage two highly qualified
undergraduates to apply? One is a math major, the other CS.
Thanks in advance!
Bill Anderson
email: xa.e71@forsythe.stanford.edu
From: G. W. Stewart <stewart@cs.UMD.EDU>
Date: Fri, 2 Feb 90 07:47:33 -0500
Subject: Nominations Sought for Fifth Householder Prize
Alston S. Householder Award V (1990)
(Second Posting)
In recognition of the outstanding services of Alston Householder,
former Director of the Mathematics Division of the Oak Ridge National
Laboratory and Professor at the University of Tennessee, to numerical
analysis and linear algebra, it was decided at the Fourth Gatlinburg
Symposium (now renamed the Householder Symposium) in 1969 to
establish the Householder Award. This award is in the area in which
Professor Householder has worked and its natural developments, as
exemplified by the international Gatlinburg Symposia [see A. S.
Householder, The Gatlinburgs, SIAM Review 16:340-343 (1974)]. Recent
recipients of the award include James Demmel (Berkeley), Ralph Byers
(Cornell), and Nicholas Higham (Manchester).
The Householder Prize V (1990) will be awarded to the author of the
best thesis in Numerical Algebra. The term Numerical Algebra is
intended to describe those parts of mathematical research which have
both algebraic aspects and numerical content or implications. Thus
the term covers, for example, linear algebra that has numerical
applications or the algebraic aspects of ordinary differential,
partial differential, integral, and nonlinear equations.
The thesis will be assessed by an international committee consisting
of Chandler Davis (Toronto), Beresford Parlett (Berkeley), Axel Ruhe
(Gothenburg), Pete Stewart (Maryland), and Paul Van Dooren (Phillips,
To qualify, the thesis must be for a degree at the level of an
American Ph.D. awarded between 1 January 1987 and 31 December 1989.
An equivalent piece of work will be acceptable from those countries
where no formal thesis is normally written at that level. The
candidate's sponsor (e.g., supervisor of his research) should submit
five copies of the thesis (or equivalent) together with an appraisal
Professor G. W. Stewart
Department of Computer Science
University of Maryland
College Park, MD 20742
by 28 February 1990. The award will be announced at the
Householder XI meeting and the candidates on the short list will
receive invitations to that meeting.
From: Michael Mascagni <mascagni@ncifcrf.gov>
Date: Fri, 2 Feb 90 13:04:41 EST
Subject: Washington, DC Area E-mailing List
I am happy to announce a newly formed mailing list. The list's purpose is
to distribute information on scholarly talks, meetings, and other events of
interest to the "greater" Washington, DC area community involved in applied
mathematics, computer science, numerical analysis, high performance computing,
and scientific computing. We have identified people at several sites in the
area who have agreed to serve as site contributors. We are quite biased, and
have no doubt left out several sites, group, etc. Our purpose was not to
offend, but to get things going ASAP. So if you wish to be a site contributor,
please send in a request. If you wish to be placed on the mailing list also
send us e-mail. DO NOT E-MAIL TO MY NA-NET ADDRESS. Instead, send mail to
mascagni@jvncf.csc.org with your request. As soon as we have a reasonable
number of announcements, the first mailing will go out. Until then, spread
the word, and please communicate with mascagni@jvncf.csc.org!!
Thanks for your help in this.--Michael Mascagni
(na.mascagni, but mascagni@jvncf.csc.org for this)
From: Jerzy Wasniewski <mfci!wasniews@uunet.UU.NET>
Date: Mon, 29 Jan 90 07:38:27 EST
Subject: Dr. Zahari Zlatev Visiting Multiflow Computer, Inc.
Dr. Zahari Zlatev
National Environmental Research Institute,
Division for Emissions and Air Pollution,
Frederiksborgvej 399,
4000 Roskilde, Denmark
visiting Multiflow Computer, Inc. Feb 14 - 16, 1990. Dr. Zlatev
will present two lectures.
1) Thursday, February 15th, 1990 - 12:00 a.m.
Multiflow Computer, Inc.
31 Business Park Drive
Branford, CT 06405
Tel: (203) 488-6090
A b s t r a c t
The long-range transport of air pollutants ( LRTAP )
over Europe is studied, at the Air Pollution Laboratory of
the Danish Agency of Environmental Protection, by a
mathematical model based on a system of partial
differential equations ( PDE's ) . Four different
physical processes, advection, diffusion, deposition
and chemical reactions (together with emission sources),
are the main components of the LRTAP . These four
processes are described by different terms in the model
(the system of PDE's). Since the space domain is very
large (including the whole of Europe together with parts of
the Atlantic Ocean, Asia and Africa), the discretization of
the system of PDE's leads to huge systems of linear
algebraic equations ( LAE's ) . In the three
dimensional case on a 32 x 32 x 9 grid the number of
LAE's that are to be solved at each time-step is more
than 10**6 when 29 chemical species are involved
in the model. Even if the model is considered as a
two-dimensional model, the number of LAE's is still
very large; more than 10**5 . This explains why one
should make some simplifications in the model description
(which are not always very well justified physically, but
lead to a model that can be handled on the computer used)
and/or one should use high-speed computers. In the latter
case, high performance can be achieved by efficiently
implementing certain kernels which perform the bulk of the
computational work. Fortunately, regular grids are to be
used during the discretization of the LRTAP model. This
leads to the solution of LAE's whose coefficient
matrices are banded and whose solution dominates the
computational load. Several such kernels for solving banded
systems of LAE's will be described. Experimental results
obtained on AMDAHL VP1100, CRAY X-MP and ALLIANT will
be presented and discussed.
2) Friday, February 16, 1990 - 11:00 a.m.
Yale University - Numerical Analysis
A. K. Watson Hall - 51 Prospect Street - room 200
New Haven, CT 06520
A b s t r a c t
Consider the system ! Ax = b !. Assume that !A! is a
large and sparse, but neither any special property of this
matrix (such as symmetry and/or positive definiteness)
nor any structure of its non-zero elements (such as
bandedness) can be exploited. For such systems direct
methods may be both time and storage consuming, while
iterative methods may not converge. A hybrid method, which
attempts to avoid the drawbacks of both direct methods and
iterative methods, is proposed. We start with some factors
!L! and !U! obtained by removing "small" non-zero
elements during Gaussian elimination and use them to
precondition the system. Then one of three conjugate
gradients-type methods (ORTHOMIN, GMRES and CGS) can be
used. If the iterative process does not converge, then the
criterion used in the decision whether a non-zero element
is small or not is made more stringent and new factors are
calculated and used to precondition the system. This
process can, if necessary, be repeated several times. If
after a prescribed number of trials the iterative method
is still not convergent, then a switch is made to Gaussian
elimination. Thus, with regard to the accuracy
requirements the hybrid method is not worse than Gaussian
elimination. However, even more important is the fact that
the method is often less time and storage consuming than
Gaussian elimination. This is demonstrated by many
numerical examples (including the well-known
Boeing-Harwell test-matrices).
From: Mikko Tarkiainen <mcsun!sunic!tut!tukki!tarkiain@uunet.uu.net>
Date: 29 Jan 90 16:41:45 GMT
Subject: Conference on Numerical Methods for Free Boundary Problems
Second Announcement of the
July 23-27, 1990 in Jyvaskyla, Finland
TOPICS OF THE MEETING. The topics covered at the conference will be:
Free boundary problems in fluid mechanics, in hydrodynamics, in
mechanics, in ground freezing and in optimal shape design, capillary
free boundaries, shape memory problems, inverse and identification
problems, control of phase transition, solidification process, etc.
PARTICIPANTS. So far, among others, the following persons are
intending to attend:
Barbu, V. (Romania), Bossavit, A. (France), Chizikalov, V.A. (USSR),
Cuvelier, C. (The Netherlands), Fage, D. (USSR), Fasano, A. (Italy),
Gets, I. (USSR), Grossman, Ch. (DDR), Haslinger, J. (Czechoslovakia),
Hoffmann, K-H (BRD), Kaliev, I. (USSR), Kenmochi, N. (Japan),
Khludnev, A.M. (USSR), Knabner, P. (BRD), Kurtze, D.A. (USA),
Magenes, E. (Italy), Maximov, A. (USSR), Meirmanov, A. (USSR),
Mittelmann, H. (USA), Myslinski, A. (Poland), Niezgodka, M. (Poland),
O'Carrol, M.J. (USA), Paolini, M. (Italy), Primicerio, M. (Italy),
Rivkind, V. (USSR), Rogers, J.C.W. (USA), Sahm, P.R. (BRD),
Schulkes, R.M.S.M.(The Netherlands), Shemetov, N. (USSR),
Shopov, P.J. (Bulgaria), Verdi, C. (Italy).
REGISTRATION. Registration forms can be ordered from the address
below. Notice that the registration must be done before March 31, 1990.
A detailed program and abstracts of the lectures will be issued to
those attending. Registration forms should be sent to Professor Pekka
Neittaanm{ki. You may contact us also by email.
CONFERENCE FEE. The conference fee, which includes attendance at the
conference, conference material, refreshments during breaks, ship
cruise on Lake P{ij{nne and conference dinner, will be $ 100.
Participants especially from East and Southeast Europe may be given
some support for the conference fee and local expenses (travel in
Finland, living costs in Finland). Please inform us about required
financial support in the registration form.
ACCOMMODATION. Accommodation for the conference is available at the
Hotel Alba on the University campus. Also, student hotels are
available (2 km from the University). Please make the reservation for
the accommodation, including the dates, on the accommodation
registration form. If you want another hotel please inform us. If you
want to stay longer in Finland before or after the conference we can
help you to make reservations (hotels, summer houses, camping places,
THIRD ANNOUNCEMENT including a preliminary conference program,
information on preparing the paper for the conference proceedings,
travel connections in Finland, etc., will be sent at the end of April
Prof. Pekka Neittaanmaki
University of Jyvaskyla
Department of Mathematics
Seminaarinkatu 15
SF-40100 Jyvaskyla, Finland
email: Neittaanmaki@finjyu.bitnet
tel.: (+358 41)602733
telefax: 358-41602701
telex: 28219 JYK SF
Mikko Tarkiainen e-mail: mtt@jylk.jyu.fi
Department of Mathematics tarkiain@tukki.jyu.fi
University of Jyvaskyla, Finland phone: +358 41 292715
From: Germund Dahlquist <dahlquis@nada.kth.se>
Date: Fri, 2 Feb 90 12:56:51 +0100
Subject: SIAM Nordic Section meeting, June 1990
Third Annual Meeting of
June 26-27 1990
Stockholm, Sweden
SIAM Nordic Section was founded in 1987. The objectives of the section
are within the Nordic countries
- to further the application of mathematics to industry and science
- to promote basic research in mathematics leading to new methods and
techniques useful to industry and science
- to unite the community of researchers and graduate students in applied
- to provide media for the exchange of information and ideas between
mathematicians and other technical and scientific personnel.
The first annual meeting was held in 1988 in Bergen, Norway, the second
one in Espoo, Finland.
All kinds of contributions of 25 minutes duration (including
discussion) are welcome, but presentations from doctoral students and
nonacademic organisations are especially invited.
Please send a title of your talk and an abstract (at most one page
long) before April 18, 1990.
At the SIAM Nordic Section Meeting The GOLUB PRIZE will be awarded for
the best contributed paper presented at the Section Meeting by a
student who is from a Nordic country and has not yet finished PhD. The
second Golub Prize was given to Rune Karlsson from Linkoping at the
1989 meeting in Helsinki.
In addition to the contributed talks, there will be a number of talks
by leading researchers from the Nordic countries.
There will be a registration fee of 200 Sw.Cr. For members of the SIAM
Nordic Section, 150 Sw.Cr. only.
Membership can be arranged at the meeting.
There will be no registration fee for graduate students from the
Nordic countries.
There will be a "Wine & Carrots" -party on Tuesday, June 26, at 5 p.m.
The local organizer of the meeting is the Department of Numerical
Analysis and Computing Science (NADA) at the Royal Institute of
Housing has been arranged at a tourist class hotel, Hostel Frescati,
located at the University campus, about 5 km north of Stockholm
centre, while the meeting takes place at the Royal Institute of
You can either have a nice (?) walk (less than 3 kms) or go by bus and
subway. The same bus can also bring you downtown in about 10 minutes.
Rates per night are 170 Sw Crs (about US$ 27) for a single room, 130
Sw Crs per person in a double room. The reception of the hotel is open
all the time There is an extra cost (30 Sw Crs) for linen unless you
bring linen yourself. Breakfast is not included but is served in a
Campus restaurant. If you want us to book a room for you on Hostel
Frescati, please send in the enclosed registration form as soon as
possible. Hotel prices in Stockholm are high, about 1000 Sw Crs for a
single room.
For more information and questions, please contact:
Berit Gudmundson Germund Dahlquist
K T H K T H
S-100 44 Stockholm S-100 44 Stockholm
Sweden Sweden
Tel. +46 (8) 790 8077 +46 (8) 790 7142
Email: dahlquis@nada.kth.se
We like to mention that during the week June 18-22 there are two
Applied Mathematics meetings in the Nordic countries:
1) The 1990 Conference on Solution of Ordinary Differential Equations,
Helsinki, Finland (Register before April 30,1990)
Information from Prof Olavi Nevanlinna, Institute of Mathematics,
Helsinki University of Technology, 02150 Espoo 15, Finland
Email: mat-on@finhut.bitnet
2) The Householder Symposium XI Meeting in Numerical Algebra,
Tylosand,Sweden. (Deadline was Novenber 1, 1989)
Information from Prof Ake Bjorck, Dept of Mathematics,
Linkoping Univ, S-581 83 Linkoping, Sweden
So, if you decide to participate in one of the above meetings, you are
encouraged to extend your visit to the Nordic countries by attending
to the SIAM Nordic Section meeting. In between there is the famous
Nordic Midsummer Weekend, with midnight sun and all that + a Monday
for recovery.
By the way, there is also a great meeting in the week June 11-15:
3) 3rd International Conference on Hyperbolic Problems, Uppsala, Sweden
Information from Lena Jutestal, Dept of Scientific Computing,
Uppsala Univ, Stureg 4B, S-752 23 Uppsala, Sweden,
Email: lena@tdb.uu.se
From: Sven Hammarling <NAGSVEN%vax.oxford.ac.uk@nsfnet-relay.ac.uk>
Date: Mon, 29 Jan 90 18:04 GMT
Subject: NAG Floating-point Test Package
FPV is a program which attempts to test the floating-point operations + - * /
sqrt, and comparisons .LT. .GT. etc., on a systematically chosen set of
operands. The code is written with all floating-point operations in loops that
will vectorise easily. It can test that the arithmetic is rounded according to
a number of rounding rules, including all the IEEE rules. There are currently
Fortran-77 and ISO standard Pascal versions of FPV. Unlike Paranoia though, FPV
is a commercial product. Anyone interested in receiving more information should
contact The Numerical Algorithms Group.
Sven Hammarling.
From: Andy Sherman <cs.yale.edu!topcat!sherman-andy@CS.YALE.EDU>
Date: 30 Jan 90 20:56:06 GMT
Subject: PCGPAK2 for Solving Sparse Linear Equations
SCIENTIFIC Computing Associates, Inc. is pleased to announce the
availability of PCGPAK2, its new package of subroutines for the
iterative solution of large, sparse systems of linear equations.
PCGPAK2 offers a choice of solution methods based on a collection
of preprocessing, preconditioning, and iterative techniques
that includes some of the most robust and efficient methods known.
The entire package is written in portable Fortran 77, so it can be
easily merged with the large amount of existing scientific and
engineering software that depends on solving sparse linear systems.
Four basic iterative methods are available in PCGPAK2:
--- the conjugate gradient method (CG);
--- the generalized minimal residual method (GMRES(k));
--- ORTHOMIN(k);
--- the restarted generalized conjugate residual method (GCR(k)).
All of these are Krylov subspace methods that minimize a norm of the
residual error at each step. CG is applicable only to symmetric,
positive definite systems; the others are general methods designed
mainly for systems having nonsymmetric or non-positive-definite
symmetric coefficient matrices.
PCGPAK2 includes several options that can enhance the performance of the
basic iterative methods. Among these are:
1. Incomplete factorization preconditioning --
The system is preconditioned with an approximate factorization of the
coefficient matrix generated with sparse Gaussian elimination, ignoring
some or all of the fill-in. A levelparameter is used to control the
amount of fill-in that is neglected, and a relaxation parameter is
available to fully or partially preserve the matrix row sums.
2. Reduced system preprocessing --
A preprocessing step generates a smaller, denser system that is solved
using one of the preconditioned basic iterative methods.The solution to
the full system is recovered by postprocessing the solution to the
smaller reduced system.
3. Block iteration --
All of the methods in PCGPAK2 can exploit general block structure in the
coefficient matrix. This leads to iterative methods that are extremely
robust and natural for problems with underlying block structure arising
from geometric or modeling considerations. Both constant and variable
blocksizes are supported.
PCGPAK2 is applicable to a wide range of engineering and scientific
problems that depend on the solution of large sparse systems of linear
equations. Examples of application areas include structural engineering
analysis, aerodynamic and hydrodynamic modeling, oil reservoir
simulation, ocean acoustics, simulation of VLSI circuit designs and
combustion physics. For many problems, PCGPAK2 is substantially faster
and uses far less storage than alternative banded or sparse Gaussian
elimination methods. For example, on one relatively-small nonsymmetric
system of order 3969 arising from a nine-point discretization of an
elliptic partial differential equation on the unit square,
PCGPAK2 required less than one-fourth of the time and less than
one-fifth of the storage required by the band Gaussian elimination
routines from LINPACK. For larger two-dimensional and three-dimensional
partial differential equations, the savings are far greater.
The standard Fortran version of PCGPAK2 will run on essentially any
computer. Optimized versions of PCGPAK2 are available for a number of
vector machines, including the Cray 1, Cray XMP, Cray YMP, Cray 2, IBM
3090, Convex C-1, Convex C-2, and DEC VAX 9000.
For further information, contact SCIENTIFIC at
SCIENTIFIC Computing Associates, Inc.
246 Church Street, Suite 307
New Haven, CT 06510
Tel.: (203) 777-7442
FAX: (203) 776-4074
Email: sca@yale.edu or yale!sca
PCGPAK2 is a registered trademark of SCIENTIFIC Computing Associates, Inc.
Computers mentioned may be trademarks of their respective manufacturers.
From: Iain Duff <duff@antares.mcs.anl.gov>
Date: Sun, 28 Jan 90 16:32:16 CST
Subject: IMA Journal of Numerical Analysis Contents
The contents of the current issue of the IMA Journal of Numerical
Analysis are given below.
IMA Journal of Numerical Analysis - Volume 10, Number 1
A Iserles Stability and dynamics of numerical methods
for non-linear ordinary differential
M Z Liu and M N Spijker The stability of the i-methods in the
numerical solution of delay differential
J Gilbert and W A Light Envelope solutions for implicit ordinary
differential equations
D Funaro Convergence analysis for pseudospectral
multidomain approximations of linear
advection equations
J Solar Vortex filament method
A Bellen, A Jackiewicz, Stability analysis of Runge-Kutta methods
R Vermiglio and for Volterra integral equations of the
M Zennaro second kind
R Coquereaux, A Grossmann Iterative method for calculation of the
and B E Lautrup Weierstrass elliptic function
H Brass Optimal estimation rules for functions of
high smoothness
N Dyn, D Levin and Data dependent triangulations for piecewise
S Rippa linear interpolation
The annual subscription rate for IMAJNA is $216 (120 pounds
outside North America and 92 pounds in UK), with a reduced rate
for members of the IMA of 38.50 pounds. There are four issues
(each of approximately 150 pages) each year. Note that it is now
possible to pay for IMA journals and IMA membership using major
credit cards.
From: Bob Plemmons <plemmons%matple@ncsuvx.ncsu.edu>
Date: Wed, 31 Jan 90 14:35:45 EST
Subject: SIMAX April Contents
Table of Contents
SIAM J. on Matrix Analysis and Applications
April 1990, Vol. 11 no. 2.
1. On Perhermitian Matrices
Richard D. Hill, Ronald G. Bates, and Steven R. Waters
2. A Matrix Approach to the Design of Low-Order Regulators
L.H. Keel and S.P. Bhattacharyya
3. Some 0-1 Solutions to the Matrix Equation A(m) - A(n) = I
Chi Fai Ho
4. Sets of Positive Operators with Suprema
W.N. Anderson, Jr., T.D. Morley, and G.E. Trapp
5. Algebraic Polar Decomposition
Irving Kaplansky
6. The Laplacian Spectrum of a Graph
Robert Grone, Russell Merris, and V.S. Sunder
7. Robust Stability and Performance Analysis for State Space Systems
via Quadratic Lyapunov Bounds
Dennis S. Bernstein and Wassim M. Haddad
8. On the Singular Values of a Product of Operators
Rajendra Bhatia and Fuad Kittaneh
9. Points of Continuity of the Kronecker Canonical Form
Immaculada de Hoyas
10. On Rutishauser's Approach to Self-Similar Flows
D.S. Watkins and L. Elsner
11. Incremental Condition Estimation
Christian Bischof
12. A New Algorithm for Finding a Pseudoperipheral Node in a Graph
Roger G. Grimes, Daniel J. Pierce, and Horst D. Simon
***Looking ahead -
The July and October issues will contain, in part, invited papers
from the Salishan, Oregon, Sparse Matrix Symposium held in 1989.
From: K. McKinnon <EFTM11%emas-a.edinburgh.ac.uk@nsfnet-relay.ac.uk>
Date: 01 Feb 90 10:07:38 gmt
Subject: Lectureship in Mathematics at Edinburgh University
Lectureship in Mathematics
Particulars of Appointment
Applications are invited for a LECTURESHIP IN MATHEMATICS tenable in the
above Department. The appointment will commence on 1 October 1989 or at a date
to be decided between the department and the successful candidate.
The Department wishes to appoint an applied mathematician with strong
research interests. The ideal candidate will work in optimization theory or
numerical analysis, but strong candidates in other areas of applied
mathematics will be considered seriously. The successful candidate will have
the opportunity to interact fruitfully with the research groups in the
Department, and with other departments in the University.
There are three established chairs. The chair in Applied Mathematics is held
by D.F.Parker, whose interests include nonlinear wave propagation in solids
and optics. The other two are held by T.J.Lyons (currently Head of
Department) whose interests relate to probability theory, particularly in
analysis and geometry; and E.G.Rees whose interests are in topology and
geometry. There are five Readers, twenty four other teaching staff, two
computing officers and a number of other research workers. The interests of
the other teaching staff include optimization, numerical analysis, dynamical
systems, differential equations, analysis, probability, algebra, topology and
The Department is responsible for teaching and research in Pure and
Applied Mathematics, and also runs (jointly with Heriot-Watt University) an
MSc course in Nonlinear Mathematics, supported by the SERC. There are separate
departments of Chemical, Electrical and Mechanical Engineering, Computer
Science, Statistics, Artificial Intelligence, Geology and Geophysics, as well
as a large Theoretical Physics group within the Department of Physics. The
Mathematics Department has strong links with the new Edinburgh-based
SERC-funded programme for the development of new techniques for design,
optimisation and control in the process engineering industries. The
Department is housed in the James Clerk Maxwell Building on the King's
Buildings site of the University, together with the combined mathematics
libraries of the University and of the Edinburgh Mathematical Society. There
are excellent computing facilities, including a 400-transputer parallel
processing facility and two Distributed Array Processors (DAPs), in the same
building. Edinburgh is an internationally recognised centre for parallel
In addition to research, duties would involve lecturing in Mathematics to
Honours and Ordinary Degree students and to postgraduate students, preparing
and attending tutorials, supervising undergraduates, examining, supervising
postgraduate students and assisting generally in the work of the Department.
The appointee is expected to join the Universities Superannuation Scheme
(USS), and to contribute 6.35% of annual salary, in which case the University
will contribute an additional sum equal to 18.55% of annual salary. The
current salary scales for lecturers A and B are 10,458 to 20,469 pounds.
The University is prepared to contribute towards removal expenses of staff
coming from other parts of the United Kingdom to Edinburgh on a first
appointment to an established post within the University, the full cost of any
reasonable vouched expenditure on removal of furniture and effects, including
insurance thereon, and the cost of fares of bringing the family to Edinburgh.
Claims in respect of travel etc from overseas will be considered on their
Applications (7 copies), including curriculum vitae and the names and
addresses of three referees, should be sent to Professor T.J.Lyons, Department
of Mathematics, Room 5320, JCMB, The King's Buildings, Edinburgh EH9 3JZ, not
later than 2nd March 1990. In the case of overseas candidates, later
applications may be considered. Such candidates need supply only one copy of
their application.
PLEASE QUOTE REFERENCE NUMBER 1486
From: Bo Kagstrom <BOKG%SEUMDC51.BITNET@Forsythe.Stanford.EDU>
Date: Thu, 1 Feb 90 13:09 EDT
Subject: Chair in Scientific Computing at Umea
Announcement of SWEDEN's first chair as Professor in
Computer Graphics and Visualization in Scientific Computing
at the University of Umea, Sweden (Reference number: Dnr 321-189-90)
Umea university is a young university that lies at the mouth of the
river Ume, equidistant from both the capital, Stockholm, and Sweden+s
most northerly town Kiruna. Today the campus has some 3 000 employees
and 11 000 students. The university has achieved prominence in many
fields, of which bio-technology, environmental ecology and information
technology are some of those in which now intensive activity is taking
Expertise in the field of information technology in its broadest sense
is rapidily growing and in certain areas such as Scientific computing
great progress has already been made, and international collaboration
established, primarily with European and American researchers.
A couple of years ago a special action program for Information technology
- Scientific Computing was established at the faculty of Mathematical
and Natural Sciences. The program aims towards development of advanced
methods, algorithms and software in Scientific computing for different
parallel computer architectures.
The university is together with the Technical University of Lulea,
the Institute of Space Physics i Kiruna and the Industrial Development
Center in Skelleftea, founder of Supercomputer Center North (SDCN).
SDCN is one of two national centers for supercomputing in Sweden and
is connected to all swedish universities through the Swedish University
Network (SUNET). Thereby scientists have access to an IBM 3090-600 E/VF,
placed in Skelleftea, soon to be upgraded to a 600 J-model.
At the university we have a distributed-memory multiprocessor-system
Intel iPSC/2 hypercube with 64 nodes of which 16 nodes have a vector
facility and are about to aquire a shared-memory multiprocessor-system
with both high-performance computing power and advanced graphic
facilities for visualization.
Due to the partnership in SDCN Sweden+s first chair as professor in
Computer Graphics and Visualization in Scientific Computing is now
established at the university.
The field is very wide and interdisciplinary to its nature and candidates
for the chair can have different scientific profiles ranging from
research in tools and methods for Computer Graphics and Visualization
in Scientific Computing to graphics computing and visualization in
Scientific Computing with an emphasis on applications from biology,
biotechnology, chemistry, physics and medicine.
At the university we have applications/possible applications in for
instance biotechnology - molecular biology, chemometry, environmental
chemistry, geographical information systems, industrial design,
medicine, physical chemistry, psychology, theoretical physics and
space physics.
The professorship is placed at the department of Computing Science.
At the department there are professor chairs in numerical analysis,
computer science, and numerical analysis and parallel computing.
Since a couple of years there has been an intense development of knowledge
in the fields of parallel computing and environments and tools for
parallel computer architectures.
The university now announces a professorship in Computer Graphics and
Visualization in Scientific Computing as vacant, reference number
Dnr 321-189-90. Notice that the reference number must be mentioned on
the application!
To get started in this field as soon as possible the position can also
be a visiting professorship.
Send the application to Rektorsambetet, University of Umea, S-901 87 UMEA,
Sweden before the 30th of March 1990. Enclosed to the application should
be curriculum vitae, short summary of scientific and educational work,
and publications and ev. interest of a visiting professorship.
Questions will be answered by Professor Bo Kagstrom, Dept of Computing
Science, Umea University, S-901 87 Umea, phone +46-90165419,
email: bokg@biovax.umdc.umu.se (or na.kagstrom@na-net.stanford.edu)
or by Project coordinator Torbjorn Johansson, Supercomputer Center North,
Umea University, S-901 87 Umea, phone +46-90166585, email:
From: David Womble <dewombl@sandia.gov>
Date: 2 Feb 90 13:32:00 MST
Subject: Fellowship at Sandia National Labs
(Please distribute this announcement to colleagues and
students who do not receive the NANET distributions.)
Mathematics and Computational Science Department
Sandia National Laboratories
Sandia National Laboratories is seeking outstanding
candidates in the areas of numerical analysis, scientific
computing, or symbolic computing to fill its 1990 Applied
Mathematical Sciences Research Fellowship. The Fellowship is
supported by a special grant from the Applied Mathematical
Sciences Research Program at the U.S. Department of Energy.
The Fellowship is intended to provide an exceptional
opportunity for young researchers. Sandia's Mathematics and
Computational Science Department maintains strong programs in
theoretical computer science, analytical and computational
mathematics, computational physics and engineering, advanced
computational approaches for parallel computers, graphics, and
architectures and languages. Sandia provides a unique parallel
computing environment, including a 1024-processor NCUBE 3200
hypercube, a 1024-processor NCUBE 6400 hypercube, a Connection
Machine-2, and several large Cray supercomputers. The successful
candidate must be a U.S. citizen, must have earned a Ph.D. degree
or the equivalent, and should have a strong interest in advanced
computing research.
The fellowship appointment is for a period of one year, and
may be renewed for a second year. It includes a highly
competitive salary, moving expenses, and a generous professional
travel allowance. Applications from qualified candidates, or
nominations for the Fellowship, should be addressed to Robert
H. Banks, Division 3531-24B, Sandia National Laboratories, P.O.
Box 5800, Albuquerque, NM 87185. Applications should include a
resume, a statement of research goals, and the names of three
references. The closing date for applications is April 30, 1990.
The position will commence during 1990. Further inquiries can be
made by calling (505) 844-2248 or by sending E-mail to
Equal Opportunity Employer M/F/V/H
U.S. Citizenship is Required
End of NA Digest
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Quarter Wit, Quarter Wisdom: Evading Calculations Part II
we discussed how to solve equations with the variable in the denominator. We also said that the technique generally works for PS questions but you need to be careful while working on DS questions.
Today, let’s look at the reason behind the caveat.
Say, the question stem of a DS question asks you to find the value of n, the number of people in the room. Statement 1 of the question gives you the following equation:
60/(n – 5) – 60/n = 2
We can easily figure out that a value of n that satisfies this equation is 15. Now, is that enough to say that statement 1 is sufficient alone? No! It could be a trap! The equation, when manipulated,
gives us a quadratic. It is important to find out whether the second solution of the quadratic works for us. When n is the number of people, it must be positive. So one extra step that we should take
is re-arrange the equation to get the quadratic. If the constant term i.e. the product of the roots is negative, it means one root is positive and one is negative. Since we have already found the
positive root, it is the only answer and hence we can say that the statement 1 is sufficient alone.
60/(n – 5) – 60/n = 2
60*n – 60*(n – 5) = 2*n*(n – 5)
n^2 – 5n – 150 = 0
The constant term, -150, is negative so the product of the roots must be negative. This means one root must be negative and the other must be positive. Since we have already found the positive root
i.e. the number of people in the room, we can say that statement 1 is sufficient alone.
Let’s look at an example where we could fall in the trap.
Say statement 1 gives us an equation which looks like this:
60/(n +5) – 10/(n – 5) = 2
As discussed last week, we will easily see that n = 10 satisfies this equation. So should we move on now and say that statement 1 is sufficient alone? No, not so fast! Let’s try to manipulate the
equation to get the quadratic.
60/(n +5) – 10/(n – 5) = 2
60*(n – 5) – 10*(n + 5) = 2*(n – 5)(n + 5)
n^2 – 25n + 150 = 0
n = 10 or 15
So actually, there are two values of n that satisfy this equation. In PS questions, since we have a single answer, there would be only one solution so once you get one, you are done. In DS questions,
you need to be certain that only one value satisfies. There is a possibility that both values satisfy your constraints in which case your answer would change.
Therefore, it may not be necessary to solve the equation for the PS question, but it is certainly necessary to solve it for DS. That’s counter intuitive, isn’t it? We hope you understand the reason.
Another related trap in DS questions: Statement 1 gives you a quadratic and asks you for the value of x (no constraints that x must be an integer or positive number etc). You know that it is a
quadratic and it will give you two values of x so you say that statement 1 is not sufficient alone and move on. But hold it! What if both the roots of the equation are same? It may not apparent to
you when you look at the equation. When you solve it, you realize that the roots are the same. Hence, ensure that you solve the equation in DS questions before you decide on the sufficiency.
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas
Prep in Detroit, Michigan, and regularly participates in content development projects such as this blog!
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Relativity of Simultaneity
Relativity of simultaneity is a particular feature of the Lorentz transform (in units where c=1):
[itex]t'=\gamma (t-vx)[/itex]
Here is a transform which has length contraction and time dilation, but not the relativity of simultaneity:
[itex]t'=\gamma (t)[/itex]
Here is a transform which has the relativity of simultaneity, but not length contraction or time dilation:
thanks Dalespam; I think you mentioned that before. I don't fully understand it from that, but all information is helpful
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Math Forum Discussions
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Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.
Topic: Geometrical Puzzle
Replies: 4 Last Post: Oct 29, 2010 12:09 PM
Messages: [ Previous | Next ]
Geometrical Puzzle
Posted: Oct 27, 2010 11:07 AM
Here is a nice geometrical puzzle-
In a triangle ABC such that angle ABC=40 degree and
angle ACB=30 degree.Point P is taken inside the triangle
ABC such that BP=PC.Find angle BAP.
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"Abaci" and "abacuses" redirect here. For the Turkish surname, see
. For the medieval book, see
Liber Abaci
The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use centuries before the adoption of the written modern numeral system and is still widely used
by merchants, traders and clerks in Asia, Africa, and elsewhere. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in
grooves in sand or on tablets of wood, stone, or metal. The user of an abacus is called an abacist.^[2]
The use of the word abacus dates before 1387 AD, when a Middle English work borrowed the word from Latin to describe a sandboard abacus. The Latin word came from Greek ἄβαξ abax "board strewn with
sand or dust used for drawing geometric figures or calculating" (the exact shape of the Latin perhaps reflects the genitive form of the Greek word, ἄβακoς abakos). Greek ἄβαξ itself is probably a
borrowing of a Northwest Semitic, perhaps Phoenician, word akin to Hebrew ʾābāq (אבק), "dust" (since dust strewn on wooden boards to draw figures in).^[3] The preferred plural of abacus is a subject
of disagreement, with both abacuses^[4] and abaci^[4] in use.
The period 2700–2300 BC saw the first appearance of the Sumerian abacus, a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system.^[5]
Some scholars point to a character from the Babylonian cuneiform which may have been derived from a representation of the abacus.^[6] It is the belief of Old Babylonian^[7] scholars such as Carruccio
that Old Babylonians "may have used the abacus for the operations of addition and subtraction; however, this primitive device proved difficult to use for more complex calculations".^[8]
The use of the abacus in Ancient Egypt is mentioned by the Greek historian Herodotus, who writes that the Egyptians manipulated the pebbles from right to left, opposite in direction to the Greek
left-to-right method. Archaeologists have found ancient disks of various sizes that are thought to have been used as counters. However, wall depictions of this instrument have not been discovered,^[9
] casting some doubt over the extent to which this instrument was used.
During the Achaemenid Persian Empire, around 600 BC the Persians first began to use the abacus.^[10] Under Parthian and Sassanian Iranian empires, scholars concentrated on exchanging knowledge and
inventions by the countries around them – India, China, and the Roman Empire, when it is thought to be expanded over the other countries.
The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC.^[11] The Greek abacus was a table of wood or marble, pre-set with small counters in wood or metal for
mathematical calculations. This Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, Ancient Rome and, until the French Revolution, the Western Christian world.
A tablet found on the Greek island Salamis in 1846 AD (the Salamis Tablet), dates back to 300 BC, making it the oldest counting board discovered so far. It is a slab of white marble 149 cm (59 in)
long, 75 cm (30 in) wide, and 4.5 cm (2 in) thick, on which are 5 groups of markings. In the center of the tablet is a set of 5 parallel lines equally divided by a vertical line, capped with a
semicircle at the intersection of the bottom-most horizontal line and the single vertical line. Below these lines is a wide space with a horizontal crack dividing it. Below this crack is another
group of eleven parallel lines, again divided into two sections by a line perpendicular to them, but with the semicircle at the top of the intersection; the third, sixth and ninth of these lines are
marked with a cross where they intersect with the vertical line.
Main article:
Roman abacus
The normal method of calculation in ancient Rome, as in Greece, was by moving counters on a smooth table. Originally pebbles (calculi) were used. Later, and in medieval Europe, jetons were
manufactured. Marked lines indicated units, fives, tens etc. as in the Roman numeral system. This system of 'counter casting' continued into the late Roman empire and in medieval Europe, and
persisted in limited use into the nineteenth century.^[12] Due to Pope Sylvester II's reintroduction of the abacus with very useful modifications, it became widely used in Europe once again during
the 11th century^[13]^[14] This abacus used beads on wires; unlike the traditional roman counting boards; which meant the abacus could be used that much faster.^[15]
Writing in the 1st century BC, Horace refers to the wax abacus, a board covered with a thin layer of black wax on which columns and figures were inscribed using a stylus.^[16]
One example of archaeological evidence of the Roman abacus, shown here in reconstruction, dates to the 1st century AD. It has eight long grooves containing up to five beads in each and eight shorter
grooves having either one or no beads in each. The groove marked I indicates units, X tens, and so on up to millions. The beads in the shorter grooves denote fives –five units, five tens etc.,
essentially in a bi-quinary coded decimal system, obviously related to the Roman numerals. The short grooves on the right may have been used for marking Roman "ounces" (i.e. fractions).
The earliest known written documentation of the Chinese abacus dates to the 2nd century BC.^[17]
The Chinese abacus, known as the suànpán (算盤, lit. "Counting tray"), is typically 20 cm (8 in) tall and comes in various widths depending on the operator. It usually has more than seven rods. There
are two beads on each rod in the upper deck and five beads each in the bottom for both decimal and hexadecimal computation. The beads are usually rounded and made of a hardwood. The beads are counted
by moving them up or down towards the beam. If you move them toward the beam, you count their value. If you move away, you don't count their value.^[18] The suanpan can be reset to the starting
position instantly by a quick jerk along the horizontal axis to spin all the beads away from the horizontal beam at the center.^[19]
Suanpans can be used for functions other than counting. Unlike the simple counting board used in elementary schools, very efficient suanpan techniques have been developed to do multiplication,
division, addition, subtraction, square root and cube root operations at high speed. There are currently schools teaching students how to use it.
In the famous long scroll Along the River During the Qingming Festival painted by Zhang Zeduan (1085–1145 AD) during the Song Dynasty (960–1297 AD), a suanpan is clearly seen lying beside an account
book and doctor's prescriptions on the counter of an apothecary's (Feibao).
The similarity of the Roman abacus to the Chinese one suggests that one could have inspired the other, as there is some evidence of a trade relationship between the Roman Empire and China. However,
no direct connection can be demonstrated, and the similarity of the abaci may be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model (like most
modern Japanese) has 4 plus 1 bead per decimal place, the standard suanpan has 5 plus 2. (Incidentally, this allows use with a hexadecimal numeral system.) Instead of running on wires as in the
Chinese and Japanese models, the beads of Roman model run in grooves, presumably making arithmetic calculations much slower.
Another possible source of the suanpan is Chinese counting rods, which operated with a decimal system but lacked the concept of zero as a place holder. The zero was probably introduced to the Chinese
in the Tang Dynasty (618-907 AD) when travel in the Indian Ocean and the Middle East would have provided direct contact with India, allowing them to acquire the concept of zero and the decimal point
from Indian merchants and mathematicians.
First century sources, such as the Abhidharmakosa describe the knowledge and use of abacus in India.^[20] Around the 5th century, Indian clerks were already finding new ways of recording the contents
of the Abacus.^[21] Hindu texts used the term shunya (zero) to indicate the empty column on the abacus.^[22]
In Japanese, the abacus is called soroban (算盤, そろばん, lit. "Counting tray"), imported from China around 1600.^[23] The 1/4 abacus, which is suited to decimal calculation, appeared circa 1930,
and became widespread as the Japanese abandoned hexadecimal weight calculation which was still common in China. The abacus is still manufactured in Japan today even with the proliferation,
practicality, and affordability of pocket electronic calculators. The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental
calculation. Using visual imagery of a soroban, one can arrive at the answer in the same time as, or even faster than, is possible with a physical instrument.^[24]
The Chinese abacus migrated from China to Korea around 1400 AD.^[25] Koreans call it jupan (주판), supan (수판) or jusan (주산).^[26]
Native American
Some sources mention the use of an abacus called a nepohualtzintzin in ancient Mayan culture. This Mesoamerican abacus used a 5-digit base-20 system.^[27] The word Nepōhualtzintzin
[nepoːwaɬˈt͡sint͡sin] comes from the Nahuatl and it is formed by the roots; Ne - personal -; pōhual or pōhualli [ˈpoːwalːi] - the account -; and tzintzin [ˈt͡sint͡sin] - small similar elements. And its
complete meaning was taken as: counting with small similar elements by somebody. Its use was taught in the Calmecac [kalˈmekak] to the temalpouhqueh [temaɬˈpoʍkeʔ], who were students dedicated to
take the accounts of skies, from childhood. Unfortunately the Nepōhualtzintzin and its teaching were among the victims of the conquering destruction, when a diabolic origin was attributed to them
after observing the tremendous properties of representation, precision and speed of calculations.^[citation needed]
This arithmetic tool was based on the vigesimal system (base 20).^[28] For the Aztec the count by 20s was completely natural. The amount of 4, 5, 13, 20 and other cyclees meant cycles. The
Nepōhualtzintzin was divided in two main parts separated by a bar or intermediate cord. In the left part there were four beads, which in the first row have unitary values (1, 2, 3, and 4), and in the
right side there are three beads with values of 5, 10, and 15 respectively. In order to know the value of the respective beads of the upper rows, it is enough to multiply by 20 (by each row), the
value of the corresponding account in the first row.
Altogether, there were 13 rows with 7 beads in each one, which made up 91 beads in each Nepōhualtzintzin. This was a basic number to understand, 7 times 13, a close relation conceived between natural
phenomena, the underworld and the cycles of the heavens. One Nepōhualtzintzin (91) represented the number of days that a season of the year lasts, two Nepōhualtzitzin (182) is the number of days of
the corn's cycle, from its sowing to its harvest, three Nepōhualtzintzin (273) is the number of days of a baby's gestation, and four Nepōhualtzintzin (364) completed a cycle and approximate a year (1
¼ days short). The Nepōhualtzintzin amounted to the rank from 10 to the 18 in floating point, which calculated stellar as well as infinitesimal amounts with absolute precision, meant that no round
off was allowed, when translated into modern computer arithmetic.
The rediscovery of the Nepōhualtzintzin was due to the Mexican engineer David Esparza Hidalgo,^[29] who in his wanderings throughout Mexico found diverse engravings and paintings of this instrument
and reconstructed several of them made in gold, jade, encrustations of shell, etc.^[citation needed]. There have also been found very old Nepōhualtzintzin attributed to the Olmeca culture, and even
some bracelets of Mayan origin, as well as a diversity of forms and materials in other cultures.
George I. Sanchez, "Arithmetic in Maya", Austin-Texas, 1961 found another base 5, base 4 abacus in the Yucatán that also computed calendar data. This was a finger abacus, on one hand 0 1,2, 3, and 4
were used; and on the other hand used 0, 1, 2 and 3 were used. Note the use of zero at the beginning and end of the two cycles. Sanchez worked with Sylvanus Morley, a noted Mayanist.
The quipu of the Incas was a system of knotted cords used to record numerical data, like advanced tally sticks – but not used to perform calculations. Calculations were carried out using a yupana (
Quechua for "counting tool"; see figure) which was still in use after the conquest of Peru. The working principle of a yupana is unknown, but in 2001 an explanation of the mathematical basis of these
instruments was proposed by Italian mathematician Nicolino De Pasquale. By comparing the form of several yupanas, researchers found that calculations were based using the Fibonacci sequence 1, 1, 2,
3, 5 and powers of 10, 20 and 40 as place values for the different fields in the instrument. Using the Fibonacci sequence would keep the number of grains within any one field at minimum.^[30]
The Russian abacus, the schoty (счёты), usually has a single slanted deck, with ten beads on each wire (except one wire, usually positioned near the user, with four beads for quarter-ruble
fractions). Older models have another 4-bead wire for quarter-kopeks, which were minted until 1916. The Russian abacus is often used vertically, with wires from left to right in the manner of a book.
The wires are usually bowed to bulge upward in the center, to keep the beads pinned to either of the two sides. It is cleared when all the beads are moved to the right. During manipulation, beads are
moved to the left. For easy viewing, the middle 2 beads on each wire (the 5th and 6th bead) usually are of a different colour from the other eight beads. Likewise, the left bead of the thousands wire
(and the million wire, if present) may have a different color.
As a simple, cheap and reliable device, the Russian abacus was in use in all shops and markets throughout the former Soviet Union, and the usage of it was taught in most schools until the 1990s.^[31]
^[32] Even the 1874 invention of mechanical calculator, Odhner arithmometer, had not replaced them in Russia and likewise the mass production of Felix arithmometers since 1924 did not significantly
reduce their use in the Soviet Union.^[33] Russian abacus began to lose popularity only after the mass production of microcalculators had started in the Soviet Union in 1974. Today it is regarded as
an archaism and replaced by the handheld calculator.
The Russian abacus was brought to France around 1820 by the mathematician Jean-Victor Poncelet, who served in Napoleon's army and had been a prisoner of war in Russia.^[34] The abacus had fallen out
of use in western Europe in the 16th century with the rise of decimal notation and algorismic methods. To Poncelet's French contemporaries, it was something new. Poncelet used it, not for any applied
purpose, but as a teaching and demonstration aid.^[35]
School abacus
Around the world, abaci have been used in pre-schools and elementary schools as an aid in teaching the numeral system and arithmetic.
In Western countries, a bead frame similar to the Russian abacus but with straight wires and a vertical frame has been common (see image). It is still often seen as a plastic or wooden toy.
The type of abacus shown here is often used to represent numbers without the use of place value. Each bead and each wire has the same value and used in this way it can represent numbers up to 100.^[
citation needed]
Renaissance abaci gallery
Uses by the blind
An adapted abacus, invented by Tim Cranmer, called a Cranmer abacus is still commonly used by individuals who are blind. A piece of soft fabric or rubber is placed behind the beads so that they do
not move inadvertently. This keeps the beads in place while the users feel or manipulate them. They use an abacus to perform the mathematical functions multiplication, division, addition, subtraction
, square root and cubic root.^[36]
Although blind students have benefited from talking calculators, the abacus is still very often taught to these students in early grades, both in public schools and state schools for the blind. The
abacus teaches mathematical skills that can never be replaced with talking calculators and is an important learning tool for blind students.^[citation needed] Blind students also complete
mathematical assignments using a braille-writer and Nemeth code (a type of braille code for mathematics) but large multiplication and long division problems can be long and difficult. The abacus
gives blind and visually impaired students a tool to compute mathematical problems that equals the speed and mathematical knowledge required by their sighted peers using pencil and paper. Many blind
people find this number machine a very useful tool throughout life.^[36]
Binary abacus
The binary abacus is used to explain how computers manipulate numbers.^[37] The abacus shows how numbers, letters, and signs can be stored in a binary system on a computer, or via ASCII. The device
consists of a series of beads on parallel wires arranged in three separate rows. The beads represent a switch on the computer in either an 'on' or 'off' position.
See also
• Aimi, Antonio; De Pasquale, Nicolino (2005). "Andean Calculators" (PDF). translated by Del Bianco, Franca.
• Boyer, Carl B.; Merzbach, Uta C. (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0471543978.
• Brown, Lesley (ed.). "abacus". Shorter Oxford English Dictionary on Historical Principles. 2: A-K (5th ed.). Oxford, UK: Oxford University Press. p. 2. ISBN 978-0-19-860575-1.
• Warner, Deborah Jean, ed. (1998). Instruments of Science: An Historical Encyclopedia. Garland Encyclopedias in the History of Science. Garland Publishing. p. 7. ISBN 978-0815315612.
• Carruccio, Ettore (2006). Mathematics And Logic in History And in Contemporary Thought. Aldine Transaction. ISBN 978-0202308500.
• Crump, Thomas (1992). The Japanese Numbers Game: The Use and Understanding of Numbers in Modern Japan. The Nissan Institute/Routledge Japanese Studies Series. Routledge. ISBN 978-0415056090.
• Flegg, Graham (1983). Numbers: Their History and Meaning. Dover Books on Mathematics. Mineola, NY: Courier Dover Publications. ISBN 978-0233975160.
• Githens, Perry, ed. (August 1948). "Chinese Abacus". Popular Science 153 (2): 87–89.
• Good Jr., Robert C. (fall 1985). "The Binary Abacus: A Useful Tool for Explaining Computer Operations". Journal of Computers in Mathematics and Science Teaching 5 (1): 34–27.
• Gove, Philip Babcock, ed. (1976). "abacist". Websters Third New International Dictionary (17th ed.). Springfield, MA: G. & C. Merriam Company. p. 1. ISBN 0-87779-101-5.
• Hidalgo, David Esparza (1977). Nepohualtzintzin: Computador Prehispanico en Vigencia [The Nepohualtzintzin: a pre-Hispanic computer in use] (in Spanish). Mexico City, Mexico: Editorial Diana.
• Hudgins, Sharon (2004). The Other Side of Russia: A Slice of Life in Siberia and the Russian Far East. Eugenia & Hugh M. Stewart '26 Series on Eastern Europe. Texas A&M University Press. ISBN
• Huehnergard, John, ed. (2011). "Appendix of Semitic Roots, under the root ʾbq.". American Heritage Dictionary of the English Language (5th ed.). Houghton Mifflin Harcourt Trade. ISBN
• Ifrah, Georges (2001). Written at New York, NY. The Universal History of Computing: From the Abacus to the Quantum Computer. New York: John Wiley & Sons, Inc. ISBN 978-0471396710.
• Körner, Thomas William (1996). The Pleasures of Counting. Cambridge, UK: Cambridge University Press. ISBN 978-0521568234.
• Leushina, A. M. (1991). The development of elementary mathematical concepts in preschool children. National Council of Teachers of Mathematics. p. 427. ISBN 978-0873532990.
• Mish, Frederick C., ed. (2003). "abacus". Merriam-Webster's Collegiate Dictionary (11th ed.). Merriam-Webster, Inc. ISBN 0-87779-809-5.
• Mollin, Richard Anthony (September 1998). Fundamental Number Theory with Applications. Discrete Mathematics and its Applications. Boca Raton, FL: CRC Press. ISBN 978-0849339875.
• Pullan, J. M. (1968). The History of the Abacus. London: Books That Matter. ISBN 978-0090894109.
• Reilly, Edwin D., ed. (2004). Concise Encyclopedia of Computer Science. New York, NY: John Wiley and Sons, Inc. ISBN 978-0470090954.
• Smith, David Eugene (1958). History of Mathematics. Dover Books on Mathematics. 2: Special Topics of Elementary Mathematics. Courier Dover Publications. ISBN 978-0486204307.
• Stearns, Peter N.; Langer, William Leonard, eds. (2001). The Encyclopedia of World History (6th ed.). New York, NY: Houghton Mifflin Harcourt. ISBN 978-0395652374.
• Trogeman, Georg; Ernst, Wolfgang (2001). Trogeman, Georg; Nitussov, Alexander Y.; Ernst, Wolfgang, eds. Computing in Russia: The History of Computer Devices and Information Technology Revealed.
Braunschweig/Wiesbaden: Vieweg+Teubner Verlag. p. 24. ISBN 978-3528057572.
• Yoke, Ho Peng (2000). Li, Qi and Shu: An Introduction to Science and Civilization in China. Dover Science Books. Courier Dover Publications. ISBN 978-0486414454.
Further reading
External links
Abacus at Wikimedia Commons
Abacus curiosities
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Pumpkin Whoopie Pies With Chocolate Cream Cheese Filling
Yay! It is finally here! A week filled with nothing but my favorite bloggers sharing their pumpkin recipes AND some great giveaways! To kick things off my friend Aimee from Shugary Sweets is here
sharing her Pumpkin Whoopie pies!
Aimee's blog is filled with tons of sinful desserts and treats. If you want a fudge recipe or a killer cookie or cupcake. Be sure to check her out. I can tell you first hand... this
Root Beer Float Fudge
is amazing. I made it a month or so ago, my husbands co-workers gobbled it up. It will be making another appearance in my Christmas tins.
Not only do I have Aimee's Whoopie Pies to share with you.... KitchenAid has teamed up with me and has sponsored a giveaway you don't want to miss! Be sure to enter at the bottom of this post!
Hey Bakeaholic fans! My name's Aimee and blog over at Shugary Sweets. I'm so excited to be here...sharing one of my favorite things.
Yeah. Addicted.
Anyways, when Carrie asked me if I'd be interested in guest posting, for PUMPKIN WEEK, ummm yeah. No hesitation. I knew I wanted to share my Pumpkin Whoopie Pie recipe too, one that I haven't yet
shared on Shugary Sweets.
I know this time of year everyone is making whoopies.
WHOOPIE PIES. Get your mind out of the gutter! Whoopie Pies.
But I filled mine with a chocolate cream cheese filling. Cause I think it goes amazingly well with the pumpkin. Try it.
This recipe makes 32-36 sandwiches, so feel free to do what I do.
No. Not eat them all while watching Grey's Anatomy.
Freeze them! I stick them in a ziploc freezer bag so that my kids can grab one for their school lunches. By lunch time they are thawed and ready to eat.
And in case you're wondering...yes, they are delicious frozen too.
Pumpkin Whoopie Pie Recipe:
for the pumpkin pie:
1 cup brown sugar
1 cup granulated sugar
1 cup canola oil
1 can (15oz) pure pumpkin
2 eggs
1 tsp vanilla extract
3 1/2 cup all purpose flour
1 tsp baking soda
1 tsp baking powder
1 Tbsp cinnamon
1/2 tsp ground ginger
1/2 tsp ground nutmeg
1/4 tsp ground cloves
1/2 tsp kosher salt
for the filling:
1 pkg (8oz) cream cheese, softened
1/3 cup butter, softened
4 cup powdered sugar
1/3 cup unsweetened cocoa powder
3-4 Tbsp heavy cream
In large mixing bowl, mix all ingredients for whoopie pies until blended. Fill a large ziploc bag. Snip of corner of bag. Pipe circles of filling, using a spiral rotation, onto a parchment paper
lined baking sheet (my circles were about 2 1/2 inch). Bake in a 350 degree oven for 12-15 minutes. Remove and cool.
When completely cooled, make filling. Beat cream cheese and butter for 3 minutes in mixer. Add powdered sugar, cocoa and heavy cream. Beat an additional 3 minutes until fluffy. Add more cream if
necessary. Scoop tablespoons of filling onto one whoopie pie and top it with a matching sized cookie. Add sprinkles if desired. ENJOY. *recipe adapted from Martha Stewart*
**Aimee is the author and baker behind the blog, Shugary Sweets. You can also find her on Pinterest, Facebook and Twitter! Aimee... Thank you so much for sharing one of your pumpkin creations with us
all today! NOW for the KitchenAid giveaway. The other day I posted THIS recipe for Pumpkin Applesauce as well as a review of my new favorite kitchen tool. The KitchenAid Hand Blender. Great for
soups, sauces, smoothies, a quick chop of veggies and so many other uses! My friends at KitchenAid not only gave me one to try, they want one of you to try it as well! How to enter: Leave a comment
below telling me what your favorite pumpkin treat is. Additional entries: (leave an additional comment for each entry) 1) Follow Bakeaholic Mama on Facebook click HERE. 2) Follow Shugary Sweets on
Facebook click HERE. 3) Follow KitchenAid on Facebook click HERE. 4) Follow Bakeaholic Mama on Twitter click HERE. 5) Follow Shuary Sweets on Twitter click HERE. 6) Follow KitchenAid on Twitter click
HERE. 7) Tweet the following " I just entered to win @bakeaholicmama 's favorite kitchen tool from @KitchenAidUSA . " 8 entries per person, giveaway open to US residents only. Giveaway closes, 10/20/
2012 at 9pm. Winner will be announced at 9am 10/21/2012. (Disclaimer: I have received no monetary compensation to write this post. KitchenAid provided one Hand Blender for me and one for a giveaway.
All opinions are my own)
250 comments:
1. My fav pumpkin treat is pumpkin cream cheese swirl bread!
2. Our favorite treat is Pumpkin Waffles. We make them all year long.
3. My favorite pumpkin treat is pumpkin bagels... but I'm partial to anything pumpkin, such as these glorious whoopie pies :)
4. Not what some would consider a treat but I love pumpkin pancakes!
5. I Follow Bakeaholic Mama on Facebook.
6. I Follow Shugary Sweets on Facebook.
7. I Follow KitchenAid on Facebook.
8. I Follow Bakeaholic Mama on Twitter.
9. I Follow Shuary Sweets on Twitter.
10. I Follow KitchenAid on Twitter.
11. I tweeted: https://twitter.com/FireRunner2379/status/257475032463335425
12. Pumpkin and chocolate?! Great combo and fun recipe! My favorite pumpkin treat is my pumpkin apple snickerdoodle muffins - they're just plain heavenly.
13. I like you on fb
14. I like shugary sweets on fb
15. I like KA on fb
16. I follow you on twitter
17. I follow shugary sweets on twitter
18. I follow KA on twitter
19. tweeted ;)
20. My favorite is probably pumpkin pie bars
1. so how do I add my email, in case I win? its mperkins@gwtc.net
21. I like Bakeaholic Mama on FB
22. I like Shugary Sweets on FB
23. Like Kitchenaid on FB
24. I love pumpkin bread with loads of chocolate chips
25. My favorite pumpkin recipe is for pumpkin custard. It's an heirloom recipe that gets you right to the pumpkin goodness of pie without having to take the time to make a crust!
26. Andrea DowlingOctober 14, 2012 at 12:34 PM
I love all things pumpkin! But my all time favorite has to be warm pumpkin bread with cream cheese swirl....so heavenly
27. I love anything with pumpkin, but my favorite is pecan pumpkin rolls with cream cheese frosting!
28. I love my mother's pumpkin pie. Classics never get old
29. I Follow Bakeaholic Mama on Facebook
30. I Follow KitchenAid on Facebook
31. I Follow Shugary Sweets on Facebook
32. I Follow Bakeaholic Mama on Twitter @immortalb4
33. I Follow Shuary Sweets on Twitter @immortalb4
34. I Follow KitchenAid on Twitter @immortalb4
35. https://twitter.com/immortalb4/status/257532189279911936
36. Pumpkin bread!
37. I follow you on twitter @icywit
38. I follow you on Facebook.
39. Pumpkin pancakes with pumpkin butter cream cheese
40. I follow Kitchen Aid on FB
41. I follow Kitchen Aid on Twitter
42. I liked Shugary Sweets on FB
43. Pumpkin pie, always.
44. Saladgoddess following you on Twitter.
45. Saladgoddess following ShugarySweets on Twitter
46. Saladgoddess following KitchenAid on Twitter.
47. Tweeted: https://twitter.com/saladgoddess/status/257550402902437888
48. my favorite is pumpkin cheesecake!
49. i like bakeaholic mama on facebook
50. i like shugary sweets on facebook
51. i like kitchenaid on facebook
52. Aimee ColbyOctober 14, 2012 at 4:04 PM
How timely! I was seeking a pumpkin whoopie pie recipe just hours before this popped up on Facebook! So happy to see Aimee's recipe doesn't start with cake mix : D
This is going to be my new favorite pumpkin recipe!
53. This is a great contest, thanks for the chance to win!
It's super hard to come up with my favorite pumpkin treat ... I love pumpkin with anything! Pumpkin bread or pumpkin bread pudding would have to be my pick right now.
54. Following Bakeaholic Mama on FB.
55. Following Shugary Sweets on Facebook (Tammy Silverberg Gross).
56. Following KitchenAid on Facebook (Tammy Silverberg Gross).
57. Following @BakeaholicMama on Twitter (tamdoll).
58. Following @shugarysweets on Twitter (tamdoll).
59. Following @KitchenAidUSA on Twitter (tamdoll).
60. Tweeted - https://twitter.com/tamdoll/status/257583901575356416.
thanks again!
61. I love mini pumpkin pie bites
62. I Follow KitchenAid on Facebook
63. I Follow KitchenAid on Twitter
64. https://twitter.com/tweetyscute/status/257592553904435200
65. I love Pumpkin Pecan Pie and my kids like me to make Great Pumpkin Cookies and Harvest Loaf. Thanks for the giveaway!
66. Natalie V likes you on Facebook
67. Natalie V already likes Shugary Sweets on Facebook
68. @Lexiquin follows you on Twitter
69. @Lexiquin follows @shugarysweets on Twitter
70. @Lexiquin follows @KitchenAidUSA on Twitter
71. Tweet! https://twitter.com/lexiquin/status/257630436283535360
72. My favorite pumpkin treat is pumpkin pie!
73. I follow KitchenAid on FB
74. My favorite pumpkin treat is pumpkin muffins with butterscotch chips.
75. Pick one favorite pumpkin treat??? I'm really not sure how to narrow it to only one, since i love pumpkin in just about everything:-)
76. I follow Bakeaholic Mama on FB.
77. I follow you on twitter too!
78. I follow Shugary Sweets on FB
79. I follow Aimee on twitter too!
80. I added KitchenAid to my FB list!!!
81. And I follow KitchenAid on twitter too!!!
82. My favorite pumpkin treat is pumpkin cheesecake!
83. I follow Bakeaholic Mama on FB.
84. And I follow Shugary Sweets on FB.
85. I follow KitchenAid on FB.
86. I follow Bakeaholic Mama on Twitter.
87. And I follow Shugary Sweets on Twitter.
88. And I follow KitchenAid on Twitter.
89. And I just tweeted: https://twitter.com/ChipChipHooray/status/257840066041376769
90. These whoopie pies look incredible, Carrie! My absolute favorite is pumpkin cake, but these could wind up a quick second!
91. My favorite pumpkin treat is sticky pumpkin monkey bread! Love it!
92. I follow Bakeaholoic Mama on FB :)
93. I follow Shugary Sweets on Facebook
94. I follow KitchenAid on FB
95. I follow Bakaholic Mama on Twitter
96. I follow Shugary Sweets on Twitter
97. I follow KitchenAid on Twitter
98. Favorite pumpkin treat is pumpkin pasties.
99. I tweeted the message :) https://twitter.com/javogabo/status/257877448190984192
100. I tweeted about the giveaway via @MooshuJenne
101. I liked Bakeaholic Mama on Facebook
102. I liked Shugary Sweets on Facebook
103. I liked KitchenAid on Facebook
104. I follow Bakeaholic Mama on Twitter via @MooshuJenne
105. I followed Shugary Sweets on Twitter via @MooshuJenne
106. I follow KitchenAid on Twitter via @MooshuJenne
107. These look beautiful! And I need an immersion blender, mine just broke!
108. I love this post and all these delicious treats with pumpkin!!!
109. Follow you of course!
110. Following Shugary Sweets now on FB!
111. Tweeted! https://twitter.com/barefootbysea
112. Follow you on Twitter!
113. Following Shugary Sweets on Twitter!
114. I want a spoon, and a bowl of that filling. Holy crap.
115. Pumpkin roll with cream cheese filling!
116. I follow Bakeaholic Mama on Facebook
117. I follow Shugary Sweets on Facebook
118. I follow KitchenAid on Facebook
119. I follow Bakeaholic Mama on Twitter
120. I follow Shugary Sweets on Twitter
121. I follow KitchenAid on Twitter
122. I tweeted "I just entered to win @bakeaholicmama's favorite kitchen tool from @KitchenAidUSA"
123. My favorite pumpkin treat is pumpkin bread. I am still expanding my pumpkin experiences. lol
124. I follow kitchenaid on twitter
125. I follow bakeaholicmama on twitter
126. I just tweeted my entry. Thanks.
127. I like bakeaholicmama on Facebook
128. I like kitchaid on Facebook.
129. I am a pumpkin pie kinda girl. My favorite recipe is from the Cake Boss.
130. This is always an easy answer for me! I make a pumpkin roll with the cream cheese filling. My friends and family always start asking for it at the end of September. Can't wait to start baking.
FYI: Tweeted and following on FB KitchenAid and of course you! Thanks for the chance to win. xx
131. Pumpkin bars.
132. 1) Follow Bakeaholic Mama on Facebook click HERE.
ALREADY DID, THAT's How I saw this.
133. 2) Follow Shugary Sweets on Facebook click HERE.
134. 3) Follow KitchenAid on Facebook click HERE.
135. 4) Follow Bakeaholic Mama on Twitter click HERE.
136. 5) Follow Shuary Sweets on Twitter click HERE.
137. 6) Follow KitchenAid on Twitter click HERE.
138. 7) Tweet the following " I just entered to win @bakeaholicmama 's favorite kitchen tool from @KitchenAidUSA . "
DONE, DONE and DONE!
139. I love Bakeaholicmama! It's one of my favorite blogs. Just wanted to say that, first and foremost. And yes, it would be nice to win some goodies. My favorite dessert is a pumpkin roulade with a
mascarpone filling. It's so incredible!
140. turtle pumpkin pie : )
email kime00@yahoo.com
141. i also follow you on facebook :)
142. Anything with pumpkin in it. Today I made Pumpkin Cinnamon Rolls. Going to make crustless Pumpkin Pie with the leftover pumpkin puree.
143. My favorite pumpkin treat is this fabulous cake that I just made yesterday, Pumpkin Crumb Cake.....ooooooh so good! Nettie
144. I am following you on Facebook!
145. I am following Shugary Sweets on Facebook. Nettie
146. pumpkin roll of course
147. I am following Kitchen Aid on Facebook.
148. I am following Bakeaholic mama on Twitter. Nettie
149. I am following Shugary Sweets on Twitter. Nettie
150. I am following Kitchen Aid On Twitter, Nettie
151. I just tweeted the following " I just entered to win @bakeaholicmama 's favorite kitchen tool from @KitchenAidUSA . " Nettie
152. I love pumpkin rolls!
153. Following bakeaholic mama on facebook.
154. Following sugary sweets on facebook
155. Following kitchenaid on facebook
156. Followed bakeaholic mama on twitter
157. Followed sugary sweets on twitter
158. Followed kitchenaid on twitter
159. Tweeted I just entered to win @bakeaholicmama 's favorite kitchen tool from @KitchenAidUSA . "
160. My favorite pumpkin treat is either pumpkin bread, or something I made the other day for the first time: a pumpkin Dutch Baby pancake :)
Immersion blenders are awesome!
161. I've followed you on Facebook :)
162. Now following Shugary Sweets on FB
163. Following Kitchen Aid on Twitter :)
164. My favorite pumpkin treat is pumpkin cheesecake!
- Samantha M.
165. I follow Bakeaholic Mama on Facebook
- Samantha M.
166. I love Pumpkin cheesecake and pumpkin pancakes :)
167. I follow Shugary Sweets on Facebook
- Samantha M.
168. I follow KitchenAid on Facebook
- Samantha M.
169. Following you on Twitter!
170. My favorite treat is pumpkin bread!
171. Michael J. McCoyOctober 17, 2012 at 8:45 PM
I Follow Bakeaholic Mama on Facebook
172. I tweeted about the giveaway!
173. Michael J. McCoyOctober 17, 2012 at 8:47 PM
I Follow Shugary Sweets on Facebook
174. Michael J. McCoyOctober 17, 2012 at 8:48 PM
I Follow KitchenAid on Facebook
175. Michael J. McCoyOctober 17, 2012 at 8:49 PM
Pumpkin Pie is my favorite pumpkin sweet!!!!!!!!!!
176. GiGi WilsonOctober 17, 2012 at 9:24 PM
Classic always delish pumpkin pie!
177. pumpkin donuts
178. I like Bakeaholic Mama on Facebook
179. I like Shugary Sweets on Facebook
180. I like KitchenAid on Facebook
181. My favorite pumpkin treat is pumpkin cookies with raisins and frosting.
182. My favorite pumpkin treat is homemade pumpkin lattes. Yum!
183. I follow Bakeaholic Mama on Facebook
184. I follow Shugary Sweets on Facebook
185. I follow KitchenAid on Facebook
186. Love that pumpkin pie.
187. Pumpkin pancakes!
188. I follow bakeaholic mama on FB.
189. I follow shugary sweets on FB.
190. I follow kitchen aid on FB.
191. My fave is pumpkin pie - classic but so good!
192. My favorite is pumpkin bread. I follow both you and Aimee on Facebook.
193. I love everything pumpkin. Right now it is pumpkin whoppie pies
194. I follow you on FB
195. Dianna BradyOctober 20, 2012 at 9:03 AM
Pumpkin Bars are my favorite:)
196. Dianna BradyOctober 20, 2012 at 9:03 AM
I am a fan of yours on facebook.
197. Dianna BradyOctober 20, 2012 at 9:04 AM
I am a fan of Kitchenaid on Facebook.
198. I gotta go with a good ole fashioned pumpkin cheesecake! yummy!!
199. I follow you on Facebook!
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MathGroup Archive: September 1999 [00350]
[Date Index] [Thread Index] [Author Index]
Re: Coordinate Transformations
• To: mathgroup at smc.vnet.net
• Subject: [mg19910] Re: Coordinate Transformations
• From: Dave Grosvenor <dag at hplb.hpl.hp.com>
• Date: Tue, 21 Sep 1999 02:22:46 -0400
• Organization: Hewlett Packard Laboratories, Bristol, UK
• References: <7rsitq$3r3@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com
Organization: Hewlett Packard Laboratories, Bristol, UK
Jason Rupert wrote:
> Please Reply to: rupertj at email.uah.edu
Hi Jason
This is my solution for the problem I think you have.
I will post some mathematica code to help you but you still need to do
some work.
Calculate the rotation transformation between two right-handed
coordinate systems.
The appraoch is in two stages
i) Pick on one pair of corresponding axes, determine a transform which
will align the z-axes and the x-y planes. This is done by rotating about
the cross product of the corresponding z-axes, by the angle between the
two vectors defining the z-axes. Call this the intermediate coordinate
system where the z-axis are aligned but the x-y axes may be an arbitrary
rotation different.
ii) Now align the x-y planes of the two cordinate systems. This is done
by rotating about the z-axis (in the target and intermediate coordinate
system), by the angle between the intermediate x -axis and the target
x-axis. This is where the assumption that both coordinate systems are
right handed gets used.
There is loads of scope for going wrong (signs of angles,cross products
handedness of coordinate system,etc..) but it should be obvious when you
get it wrong. So you have plenty to do!
So making a general routine is a pain.
Rotation about an axis and point
Approach is to define a 3D coordinate transformation by specifying axis
of rotation (a point {0,0,0} and a vector), plus an angle of rotation.
This is standard (see Graphic Gems I "Rotation Tools", Michael E.
see yuz
Dave Grosvenor
Mathematica code
The two stages for the approach are something like below:-
i) Mat =
where a and b are the vector of the corresponding axes
ii) let {icx,icy,icz} = deHomogenise[Mat . homogenise[{cx,cy,cz]]
where homogenise[{x_,y_,z_}]:= {x,y,z,1}
deHomogenise[{x_,y_,z_,w_}]:= {x/w,y/w,z/w}
so now we rotate about
Mat1 =
{dx,dy,dz} is the corresponding axis in the target
Angle between Vector
These expressions are derived by dot and cross product for vectors.
magnitude[v_]:= Sqrt[v.v]
magnitude[Cross[u,v]]/(magnitude[u] magnitude[v])
cosineBetweenVectors[u_,v_]:= Dot[u,v]/(magnitude[u] magnitude[v])
Rotation 4x4 matrix routine
rotate routine:- from graphics gems, this routine is more general than
required (it allows a translate to an arbitrary point)--but is slow as
it does not pre-multiply the translation.
{{t x^2 + c,t x y + s z, t x z - s y},
{t x y - s z, t y^2 + c,t y z + s x},
{t x z + s y, t y z - s x, t z^2 + c}}]
Now a function to uplift the rotate matrix to a 4x4 transformation
RowBox[{"(", GridBox[{
{\(x\^2\ \((1 - Cos[theta])\) + Cos[theta]\),
\(x\ y\ \((1 - Cos[theta])\) + z\ Sin[theta]\),
\(x\ z\ \((1 - Cos[theta])\) - y\ Sin[theta]\), "0"},
{\(x\ y\ \((1 - Cos[theta])\) - z\ Sin[theta]\),
\(y\^2\ \((1 - Cos[theta])\) + Cos[theta]\),
\(y\ z\ \((1 - Cos[theta])\) + x\ Sin[theta]\), "0"},
{\(x\ z\ \((1 - Cos[theta])\) + y\ Sin[theta]\),
\(y\ z\ \((1 - Cos[theta])\) - x\ Sin[theta]\),
\(z\^2\ \((1 - Cos[theta])\) + Cos[theta]\), "0"},
{"0", "0", "0", "1"}
}], ")"}],
(MatrixForm[ #]&)]\)
Now we define a function to define a translate transformation as a 4x4
translate[{tx_,ty_,tz_}] := {{1,0,0,tx},{0,1,0,ty},{0,0,1,tz},{0,0,0,1}}
Conceptually we apply a transformation to move the origin to the desired
point and then perform the rotate and then we change the coordinate
back to undo the translate. We calculate and simplify this 4x4
FullSimplify[translate[{tx,ty,tz}] . uplift[Rotate3x3[{x,y,z},theta]] .
\!\({{x\^2 + Cos[theta] - x\^2\ Cos[theta],
x\ y - x\ y\ Cos[theta] + z\ Sin[theta],
x\ z - x\ z\ Cos[theta] - y\ Sin[theta],
\((tx\ \((\(-1\) + x\^2)\) + x\ \((ty\ y + tz\ z)\))\)\
\((\(-1\) + Cos[theta])\) + \((tz\ y - ty\ z)\)\ Sin[theta]},
x\ y - x\ y\ Cos[theta] - z\ Sin[theta],
y\^2 + Cos[theta] - y\^2\ Cos[theta],
y\ z - y\ z\ Cos[theta] + x\ Sin[theta],
\((tx\ x\ y + ty\ \((\(-1\) + y\^2)\) + tz\ y\ z)\)\
\((\(-1\) + Cos[theta])\) + \((\(-tz\)\ x + tx\ z)\)\
Sin[theta]}, {
x\ z - x\ z\ Cos[theta] + y\ Sin[theta],
y\ z - y\ z\ Cos[theta] - x\ Sin[theta],
z\^2 + Cos[theta] - z\^2\ Cos[theta],
\((\((tx\ x + ty\ y)\)\ z + tz\ \((\(-1\) + z\^2)\))\)\
\((\(-1\) + Cos[theta])\) + \((ty\ x - tx\ y)\)\ Sin[theta]},
0, 0, 1}}\)
RowBox[{"(", GridBox[{
{"1", "0", "0", "ax"},
{"0", "1", "0", "ay"},
{"0", "0", "1", "az"},
{"0", "0", "0", "1"}
}], ")"}],
(MatrixForm[ #]&)]\)
Then (lazilly) just define the desired routine as the un-simplified
matrix multiplications -- the assumption is that the vector defining the
axis of rotation is a unit vector. .
With[{unit = unitVector[{x0,y0,z0}]},
translate[{tx,ty,tz}] . uplift[Rotate3x3[{x,y,z},theta]] .
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Beta-Reduction As Unification
We define a unification problem ^UP with the property that, given a pure lambda-term M, we can derive an instance Gamma(M) of ^UP from M such that Gamma(M) has a solution if and only if M is
beta-strongly normalizable. There is a type discipline for pure lambda-terms that characterizes beta-strong normalization; this is the system of intersection types (without a "top" type that can be
assigned to every lambda-term). In this report, we use a lean version LAMBDA of the usual system of intersection types. Hence, ^UP is also an appropriate unification problem to characterize
typability of lambda-terms in LAMBDA. It also follows that ^UP is an undecidable problem, which can in turn be related to semi-unification and second-order unification (both known to be undecidable).
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The Joy of Math - review - Gifted Education
The Joy of Mathematics is a delightful overview of mathematics brought to us by the Teaching Company as part of The Great Courses series. Professor Arthur Benjamin is a dynamic and engaging speaker,
who clearly knows his subject matter backwards, forwards, and upside down. Benjamin is a math professor from Harvey Mudd College. His clear communication style is refreshing and his lectures truly
entertaining. This is no surprise, when one considers that he is also a trained magician, who has studied the art of entertainment. He knows how to speak to his audience, and his passion for
mathematics is very apparent.
The Joy of Math contains a total of 24 individual 30 minute lectures. I received a set of four dvds accompanied by a transcript and course guidebook. The first disc contains the easiest material, and
the last, the most difficult. The presenter occasionally mentions a fact learned in a previous session, but we skipped around a bit and did not suffer from too much lack of continuity. I watched
these lectures with my 12 year old son, who was already a fan of Dr. Benjamin's after attending one of his fantastic “mathemagic” presentations and obtaining a copy of Benjamin's book, “ Secrets of
Mental Math:The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks.” My son enjoyed these lectures tremendously, and often paused the lectures to share his own comments and
observations, and also to predict what might be coming up next.
Disc one contains the following segments:
1.The Joy of Math-The Big Picture
2.The Joy of Numbers
3.The Joy of Primes
4.The Joy of Counting
5.The Joy of Fibonacci Numbers
6.The Joy of Algebra
The first six lectures may be interesting to younger or less competent math students, but lectures 7-12 are better appreciated by math students who have mastered algebra and had some exposure to
geometry. Similarly, it will be helpful for students to have some knowledge of of trigonometry and calculus before watching the rest of the sessions, with the exception of those at the very end.
Discs two to four have these lectures:
7. The Joy of Higher Algebra
8.The Joy of Algebra Made Visual
9.The Joy of 9
10.The Joy of Proofs
11.The Joy of Geometry
12.The Joy of Pi
13.The Joy of Trigonometry
14.The Joy of the Imaginary Number i
15.The Joy of the Number e
16.The Joy of Infinity
17.The Joy of Infinite Series
18.The Joy of Differential Calculus
19.The Joy of Approximating with Calculus
20.The Joy of Integral Calculus
21.The Joy of Pascal's Triangle
22.The Joy of Probability
23.The Joy of Mathematical Games
24.the Joy of Mathematical Magic
This is a great refresher course for adults who are returning to school, or a fun supplemental course for “mathy” kids. Gifted and high ability math students who memorize digits of pi for fun or
think square roots are cool will find this series exceptionally fantabulous. Dr. Benjamin's playful manner and penchant for poems, puns, and wordplay will be sure to amuse as well as educate. His
enthusiasm is contagious. He explains Fibonacci numbers using a story about multiplying rabbits and probability with a horse (Harvey the Mudder!) who likes to run in the mud. Here's a math poem that
he wrote to honor the number e:
I think that I shall never see,
A number lovelier than e.
Whose digits are too great to state,
They're 2.71828.
And e has such amazing features,
It's loved by all, but mostly teachers.
With all of e's great properties,
Most integrals are done with e's.
Theorems are proved by fools like me,
But only Euler could make an e.
It's obvious that this guy really loves his work!
Throughout the course, sample math problems are solved on screen using a virtual chalkboard, so each step is shown in sequence. Viewers are advised to repeat any sections that are not immediately
clear. The companion guide has a helpful glossary, suggested reading lists, and internet resources to augment the course.
I highly recommend The Joy of Math for math lovers of any age, and also for those who might learn to love it with just a little encouragement.
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QUBIT READOUT VIA RESONANT SCATTERING OF JOSEPHSON SOLITONSAANM NAAMAN; OFERAACI Ellicot CityAAST MDAACO USAAGP NAAMAN; OFER Ellicot City MD USAANM Park; Jae I.AACI BoulderAAST COAACO USAAGP Park; Jae I. Boulder CO USAANM Pesetski; Aaron A.AACI GambrillsAAST MDAACO USAAGP Pesetski; Aaron A. Gambrills MD US
Patent application title: QUBIT READOUT VIA RESONANT SCATTERING OF JOSEPHSON SOLITONSAANM NAAMAN; OFERAACI Ellicot CityAAST MDAACO USAAGP NAAMAN; OFER Ellicot City MD USAANM Park; Jae I.AACI
BoulderAAST COAACO USAAGP Park; Jae I. Boulder CO USAANM Pesetski; Aaron A.AACI GambrillsAAST MDAACO USAAGP Pesetski; Aaron A. Gambrills MD US
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Systems and methods are provided for reading an associated state of a qubit. A first soliton is injected along a first Josephson transmission line coupled to the qubit. A velocity of the first
soliton is selected according to a physical length of the qubit and a characteristic frequency of the qubit. A second soliton is injected at the selected velocity along a second Josephson
transmission line that is not coupled to the qubit. A delay associated with the first soliton is determined relative to the second soliton.
An apparatus comprising: a Josephson transmission line; a soliton driver configured to provide a soliton to propagate along the Josephson transmission line; a soliton detector configured to determine
a time-of-flight of the soliton; and a phase qubit coupled to the Josephson transmission line, the phase qubit having a first characteristic frequency in a first state and a second characteristic
frequency in a second state; wherein at least one of the soliton detector, the Josephson transmission line, and the phase qubit are configured such that the phase qubit applies a first delay to the
propagation of the soliton along the Josephson transmission line when the phase qubit is in the first state and applies a second delay to the propagation of the soliton along the Josephson
transmission line when the phase qubit is in the second state.
The apparatus of claim 1, further comprising a coupling element configured to couple the qubit to the Josephson transmission line at multiple locations, such that the phase qubit applies the first
delay to the propagation of the soliton along the Josephson transmission line multiple times when the phase qubit is in the first state and applies the second delay to the propagation of the soliton
along the Josephson transmission line multiple times when the phase qubit is in the second state.
The apparatus of claim 1, wherein the Josephson transmission line is terminated with impedance mismatched ends to allow for internal reflection of the soliton within the Josephson transmission line,
such that the phase qubit applies the first delay to the propagation of the soliton along the Josephson transmission line multiple times when the phase qubit is in the first state and applies the
second delay to the propagation of the soliton along the Josephson transmission line multiple times when the phase qubit is in the second state.
The apparatus of claim 1, the soliton being a first soliton and the Josephson transmission line being a first Josephson transmission line, the soliton driver being configured to provide a second
soliton to propagate along the second Josephson transmission line.
The apparatus of claim 4, wherein the second Josephson transmission line is not coupled to the phase qubit, and the soliton detector is configured to determine a difference in the time-of-flight
between the first soliton and the second soliton.
The apparatus of claim 5, wherein the soliton driver is configured to provide the second soliton with a polarity opposite of a polarity of the first soliton, the soliton detector comprising a
combiner to determine coincident arrival of the solitons, such that if the solitons are coincident they will annihilate within the combiner without producing an output pulse.
The apparatus of claim 1, the phase qubit having a characteristic frequency, and the pulse emitter being configured to provide the soliton with a velocity based on the characteristic frequency of the
phase qubit.
The apparatus of claim 7, the characteristic frequency of the phase qubit being a first characteristic frequency of the phase qubit and the phase qubit having a second characteristic frequency, the
velocity of the soliton being determined such that a difference between a first delay experienced by the soliton when the phase qubit is in a first state associated with the first characteristic
frequency and a second delay experienced by the soliton when the phase qubit is in a second state associated with the second characteristic frequency is maximized.
The apparatus of claim 1, wherein the phase qubit is inductively coupled to the Josephson transmission line.
The apparatus of claim 1, wherein the phase qubit is capacitively coupled to the Josephson transmission line.
An apparatus, comprising: a first Josephson transmission line; a qubit coupled to the first Josephson transmission line; a second Josephson transmission line that is not coupled to the qubit; a
soliton driver configured to inject a first soliton along the first Josephson transmission line and a second soliton along the second Josephson transmission line, the first and second solitons having
a velocity selected according to a physical length of the qubit and a characteristic frequency of the qubit; and a soliton detector configured to detect an arrival of each of the first soliton and
the second soliton and determine a delay associated with the first soliton relative to the second soliton.
The apparatus of claim 11, wherein the qubit is a phase qubit.
The apparatus of claim 11, wherein the qubit is a transmon qubit.
The apparatus of claim 11, wherein the qubit is a quantronium qubit.
The apparatus of claim 11, wherein the velocity of the first and second solitons is selected such that the product of the velocity, the physical length of the qubit, and the characteristic frequency
of the qubit is substantially equal to one.
The apparatus of claim 11, the characteristic frequency of the qubit being a first characteristic frequency of the qubit, the determined delay being a first delay, and the qubit having a second
characteristic frequency, the velocity of the first and second solitons being determined such that a difference between the first delay, experienced by the first soliton when the qubit is in a first
state associated with the first characteristic frequency, and a second delay, experienced by the first soliton when the qubit is in a second state associated with the second characteristic frequency,
is maximized.
The apparatus of claim 11, wherein the qubit is dispersively coupled to a resonator, and the resonator is coupled to the first Josephson transmission line, such that the qubit is coupled to the first
Josephson transmission line through the resonator.
A method for reading an associated state of a qubit, comprising: producing a first soliton along a first Josephson transmission line coupled to the qubit, a velocity of the first soliton being
selected according to a physical length of the qubit and a characteristic frequency of the qubit; producing a second soliton at the selected velocity along a second Josephson transmission line that
is not coupled to the qubit; and determining a delay associated with the first soliton relative to the second soliton.
The method of claim 18, further comprising determining that the qubit is in a ground state if the determined delay is within a first range and determining that the qubit is in an excited state if the
determined delay is within a second range.
The method of claim 18, further comprising selecting the velocity such that the product of the velocity, the physical length of the qubit, and the characteristic frequency of the qubit is
substantially equal to one.
TECHNICAL FIELD [0001]
The present invention relates generally to quantum computing systems, and more particularly to a reading of a state of a qubit via resonant scattering of Josephson solitons.
BACKGROUND [0002]
A classical computer operates by processing binary bits of information that change state according to the laws of classical physics. These information bits can be modified by using simple logic gates
such as AND and OR gates. The binary bits are physically created by a high or a low energy level occurring at the output of the logic gate to represent either a logical one (e.g., high voltage) or a
logical zero (e.g., low voltage). A classical algorithm, such as one that multiplies two integers, can be decomposed into a long string of these simple logic gates. Like a classical computer, a
quantum computer also has bits and gates. Instead of using logical ones and zeroes, a quantum bit ("qubit") uses quantum mechanics to occupy both possibilities simultaneously. This ability means that
a quantum computer can solve certain problems with exponentially greater efficiency than that of a classical computer.
SUMMARY [0003]
In accordance with one aspect of the invention, an apparatus includes a Josephson transmission line and a soliton driver configured to provide a soliton to propagate along the Josephson transmission
line. A soliton detector is configured to determine a time-of-flight of the soliton, and a phase qubit is coupled to the Josephson transmission line. The phase qubit has a first characteristic
frequency and a second characteristic frequency corresponding to a first state and a second state of the qubit. At least one of the soliton detector, the Josephson transmission line, and the phase
qubit are configured such that the phase qubit applies a first delay to the propagation of the soliton along the Josephson transmission line when the phase qubit is in a first state associated with
the first characteristic frequency and applies a second delay to the propagation of the soliton along the Josephson transmission line when the phase qubit is in a second state associated with the
second characteristic frequency.
In accordance with another aspect of the invention, an apparatus includes a first Josephson transmission line, a qubit coupled to the first Josephson transmission line, and a second Josephson
transmission line that is not coupled to the qubit. A soliton driver is configured to inject a first soliton along the first Josephson transmission line and a second soliton along the second
Josephson transmission line. The first and second solitons have a velocity selected according to a physical length of the qubit and a characteristic frequency of the qubit. A soliton detector is
configured to detect an arrival of each of the first soliton and the second soliton and determine a delay associated with the first soliton relative to the second soliton.
In accordance with a further aspect of the invention, a method is provided for reading an associated state of a qubit. A first soliton is produced along a first Josephson transmission line coupled to
the qubit. A velocity of the first soliton is selected according to a physical length of the qubit and a characteristic frequency of the qubit. A second soliton is produced at the selected velocity
along a second Josephson transmission line that is not coupled to the qubit. A delay associated with the first soliton is determined relative to the second soliton.
BRIEF DESCRIPTION OF THE DRAWINGS [0006]
FIG. 1 illustrates an assembly for reading out a state of a qubit in accordance with an aspect of the present invention;
FIG. 2 illustrates a chart of the time evolution of the soliton velocity;
FIG. 3 illustrates one implementation of a Josephson transmission line;
FIG. 4 illustrates a chart representing the frequency dependence of a delay induced by a qubit for solitons of two different velocities;
FIG. 5 is a chart illustrating a sensitivity of solitons propagating on a JTL at varying velocities to a delay induced by a qubit as a function of a resonant frequency of the qubit;
FIG. 6 illustrates a first implementation of a system in accordance with an aspect of the present invention;
FIG. 7 illustrates a second implementation of a system in accordance with an aspect of the present invention; and
FIG. 8 illustrates a method for reading an associated state of a qubit in accordance with an aspect of the present invention.
DETAILED DESCRIPTION [0014]
Even under ideal circumstances, reading the state of a quantum bit projects the measured state on the qubit, effectively collapsing any superposition of states within the qubit (i.e., reducing the
amplitude of any non-measured states to zero). Current methods of measuring a state of some varieties of qubits, such as the tunneling readout of phase qubits, actually destroy the Hamiltonian
eigenstates of the qubit by substantially and irreversibly modifying the qubit spectrum, such that the qubit cannot be used again without a time-consuming reinitialization of the qubit. It will be
appreciated that this reinitialization will take considerable more time than a simple reset of a qubit to an initial state.
In accordance with an aspect of the present invention, a device can utilize sine-Gordon solitons (fluxons) propagating on a Josephson transmission line that is coupled to a qubit in a manner that
enables the use of such device for nondestructive readout. It is known that phase qubits and qubits of similar design lack a direct mapping of their states to a magnetic flux or reactance, making
them unsuitable for the so-called ballistic soliton readout approach. In accordance with an aspect of the present invention, however, the sensitivity of a soliton time-delay to an associated state of
a qubit can be enhanced for the purpose of qubit readout, allowing a ballistic readout methodology to be used for phase qubits and similar qubit types by exploiting resonant interactions between the
qubit and the solitons. The soliton time-of-flight can be made sensitive to small changes in the qubit resonance frequency and thereby to the qubit state. Accordingly, by measuring changes in the
time of flight of the soliton, the state of the qubit can be determined non-destructively.
To this end, the qubit state is mapped onto soliton propagation delay via the state-dependent resonance frequency of the qubit. The qubit is probed by one soliton, a train of solitons, or a single
soliton exposed to the qubit multiple times, and the propagation delay can be determined. By tuning the soliton velocity, which can be controlled by a DC (direct current) bias current through the JTL
or by an initial shape of the soliton pulse, the qubit-dependent response of the ballistic soliton can be maximized and measured as a function of the detuning of the soliton velocity and the qubit
transition frequency. By tuning the velocity of the incident fluxon, it is possible to create a scattering resonance between the fluxon and the qubit, where the interaction is strongest. This
resonant regime provides a number of advantages in reading out certain types of qubits, such as lower power dissipation, preservation of the qubit spectrum during readout, simplification of
integration, increased measurement speed, and faster acquisition rate of information about the state of the qubit per scattering event. Furthermore the measurement sensitivity may be increased with
an appropriate geometry that allows for multiple scattering of the same fluxon with the qubit.
FIG. 1 illustrates an assembly 10 for reading out a state of a qubit 12 in accordance with an aspect of the present invention. For example, the qubit 12 can be a phase qubit, a transmon-type qubit, a
quantronium-type qubit, or similar design. The assembly 10 includes a Josephson transmission line (JTL) 14 and a soliton driver 16 configured to provide solitons along the JTL. For example, the
soliton driver 16 can produce quanta of magnetic flux, referred to as fluxons, which propagate along the JTL 14. The qubit 12 is coupled, for example, capacitively or inductively, to the JTL 14, and,
depending on an associated velocity of the propagating soliton, can have a state-dependent effect on the time-of-flight. For example, the qubit 12 can produce a first delay in the propagation of the
soliton when the qubit is in a first state and produce a second delay in the propagation of the soliton when the qubit is in a second state, and the difference in the time-of-flight can be determined
at a soliton detector 18 coupled to the JTL 14. For example, the qubit 12 can be determined to be in the first state if the measured delay falls within a first range and in a second state if the
measured delay falls within a second range. It will be appreciated, however, that other methods can be utilized for exploiting the state-dependence of the delay introduced by the qubit 12 to readout
the qubit state.
In accordance with an aspect of the present invention, the delay induced by the soliton is caused by inelastic scattering of the soliton by the qubit, causing the soliton to lose kinetic energy and
slowing its propagation though the JTL 14. This inelastic scattering is state-dependent, with the kinetic energy lost from the soliton, and therefore its velocity after scattering, being dependent on
a transition frequency of the qubit 12 to higher energy levels. Accordingly, by exploiting the difference between the energy of the transition between the ground state and the first excited state and
the energy of the transition between the first excited state and the second excited state, it is possible to determine an associated state of the qubit 12 from the magnitude of the delay induced in
the soliton.
In one implementation, the soliton driver 16 can be configured to provide the solitons at a particular velocity, as to maximize a difference between the delays produced by the qubit in its first and
second states. Specifically, the velocity of the soliton can be selected according to a physical length of the qubit 12 and a characteristic frequency of the qubit. For example, the velocity can be
selected such that a product of the velocity, a physical length of the qubit 12, and the characteristic frequency of the qubit, such as a frequency associated with a state transition within the
qubit, is substantially equal to one. The velocity of the soliton can be controlled by altering an associated shape of the pulse produced at the soliton driver 16 or by adjusting a direct current
(DC) bias applied to the JTL 14.
For example, a propagating solution for the soliton can be written as:
φ ( x , t ) = 4 tan - 1 [ exp ( x - x 0 - ut 1 - u 2 ) ] Eq . 1 ##EQU00001##
wherein φ is the magnetic flux, x is a position along the JTL, x
is a reference position, t is time, and u is the soliton velocity.
Assuming the soliton generator is at the reference position, to provide a desired velocity, u, the voltage necessary at the generator can be described as:
V gen
( t ) = φ ( x , t ) t | x = x 0 = 2 u 1 - u 2 sech ( - ut 1 - u 2 ) Eq . 2 ##EQU00002##
Generalizing from the above, high velocity solitons can be generated from short, relatively high voltage pulses, while low velocity solitons can be generated from relatively long, low voltage pulses.
The velocity of the soliton can be further tuned by applying a DC current to the JTL 14. This bias current applies a Lorentz force on the traveling soliton, and can either increase or decrease the
velocity, depending on the polarity of the bias current.
To demonstrate the operation of the readout, the qubit 12 can be modeled as a parallel LCR oscillator, whose inductance is L
, capacitance is C
and losses are modeled by a resistor R. In this model, it is assumed that the qubit 12 and the JTL are coupled inductively, although, as mentioned previously, other configurations can be used. The
dynamical variable of the oscillator is the branch flux, φ
, which is related to the resonator inductance via φ
with I
representing the current through the resonator inductance. The coupling of the resonator to the JTL at a site N is represented by the transformer equation:
( φ N - φ N + 1 φ r ) = ( L - M M - L r ) ( I N I r ) Eq . 3 ##EQU00003##
where φ
are node fluxes of the JTL 14, M is a mutual inductance between the qubit 12 and the JTL, L is a series inductance of the JTL (e.g., 53 or 54 in FIG. 3), and I
is a current associated with an n
cell of the JTL (e.g., 60 or 70 in FIG. 3).
An equation of motion for the resonator can be written as:
φ r + 1 RC r φ . r + ω r 2 1 - k 2 φ r = ω r 2 M L ( 1 - k 2 ) ( φ N - φ N + 1 ) Eq . 4 ##EQU00004##
where the second term on the left accounts for resonator losses, ω
is the resonator frequency, and k=M/ {square root over (L
L)} is the qubit-JTL coupling strength.
The equation of motion for the JTL 14 in the continuum limit is:
∂ 2 φ ∂ t 2 = 1 L ∂ 2 φ ∂ x 2 - 1 L J sin φ + δ ' ( x ) { k 2 L ∂ φ ∂ x - M L r L ( 1 - k 2 ) φ r } Eq . 5 ##EQU00005##
The JTL 14 is described by a sine-Gordon equation of motion, having a soliton solution propagating along the line with a velocity u. The last term on the right results from current induced in the JTL
14 by its coupling to the qubit 12 and is treated as a perturbation. The perturbed sign-Gordon equation above can be transformed into two equations of motion for the position (X) and velocity (u) of
a soliton solution to give:
u t
= - tanh Θ 0 sech Θ 0 { 1 4 ( 1 - u 2 ) k 1 - k 2 L L r φ r - k 2 L L J sech Θ 0 } Eq . 6 ##EQU00006##
X t
= u + 1 4 u sech θ 0 { 2 k 2 1 - u 2 L L J sech θ 0 ( 1 - 2 θ 0 tanh θ 0 ) - k 1 - k 2 L L r ( 1 - θ 0 tanh θ 0 ) φ r } where Θ 0 = X 1 - u 2 . Eq . 7 ##EQU00007##
FIG. 2 illustrates a chart 30 of the time evolution of the soliton velocity, obtained by numerically integrating the above equations of motion. In this example, the resonator frequency is 10 GHz, the
JTL plasma frequency is 50 GHz, and the soliton initial velocity is 0.2 c, where c is the Swihart velocity. The Swihart velocity represents a characteristic velocity of a given Josephson transmission
line and represents a maximum velocity at which a soliton will propagate on a Josephson transmission line. The Swihart velocity of a JTL is determined as the product of a plasma frequency, ω
, of the JTL and an associated Josephson penetration length, λ
of the JTL. The Josephson penetration length can, in turn, be determined as the product of a cell length, a, of the JTL multiplied by a square root of a ratio of the series inductance of the JTL to a
shunt inductance of the JTL.
In the illustrated chart 30, the vertical axis 32 represents a normalized velocity of the soliton (e.g., the soliton velocity divided by the Swihart velocity for the transmission line) and the
horizontal axis 34 represents time in units of nanoseconds. The center of mass of the soliton passes the JTL site that is coupled to the resonator at the point where the time is zero. As can be seen
from the graph 36 of the soliton velocity, the interaction between the soliton and the resonator results in the soliton losing kinetic energy to the resonator (qubit). Accordingly, an overall delay
is caused in the propagation of the soliton down the JTL.
FIG. 3 illustrates one implementation of a Josephson transmission line 50. The Josephson transmission line is a transmission line that does not support propagating small-amplitude modes below an
associated junction plasma frequency, ω
. At frequencies higher than the plasma frequency, the Josephson transmission time supports propagating modes, referred to as plasmons. The Josephson transmission line includes a plurality of cells
60, 70, and 80 connected by series inductors 52-55. Each cell 60, 70, and 80 contains a Josephson junction 62, 72, and 82 shunted to ground through an associated capacitor 64, 74, and 84. The plasma
frequency of the Josephson transmission line is defined by a critical current, i
, for the Josephson junctions 62, 72, and 82 associated with each cell 60, 70, and 80 and a capacitance associated the shunt capacitors 64, 74, and 84. The series inductors 52-55 coupling the
individual cells 60, 70, and 80 each have an associated inductance referred to herein as a series inductance, L, of the transmission line, and a shunt inductance, L
, defined as an inductance of the Josephson junctions 62, 72, and 82, where L
, where h is the reduced Planck constant (h/2π) and e is the elementary charge. It will be appreciated that the Josephson transmission line provides a compact structure that is both compatible with
the low temperature operation desirable for quantum computing and capable of co-fabrication with a qubit using the same processing technology.
FIG. 4 illustrates a chart 90 representing the frequency dependence of a delay induced by a qubit for solitons of two different velocities. Specifically, the horizontal axis 92 represents the
frequency of the qubit, in gigahertz, and the vertical axis 94 represents a delay, measured in unit cells of the Josephson transmission line. In one implementation, the JTL can have a plasma
frequency of ω
/2π=58 GHz, with the Josephson junctions having associated inductances of 32 pH and shunt capacitors having associated capacitances of 235 f F. The series inductors of the JTL are selected to have an
inductance of 8 pH. In one example, the Josephson transmission line can be configured with two hundred cells and have a total length on the order of several millimeters.
A solid line 96 represents the frequency-dependent delay induced in a soliton by the qubit when the soliton is propagating at a first velocity, specifically 0.15 c, where c is the Swihart velocity. A
dashed line 98 represents the frequency-dependent delay induced in a soliton by the qubit when the soliton is propagating at a second velocity, specifically 0.2 c. It will be appreciated that the
frequency dependence of the soliton delay at the first velocity 96 is substantially more pronounced that the frequency dependence at the second velocity 98, and accordingly, the delay induced in the
soliton with be significantly more sensitive to frequency within the five to ten gigahertz range. The readout is sensitive to the change in the resonance frequency of the resonator (or qubit);
therefore the best sensitivity is expected for cases where the resonator is slightly detuned from the soliton-qubit resonance condition (maxima of respective lines 96 and 98 in FIG. 4), as this is
where the change in the soliton delay has a greater response to small changes in the resonator frequency. In FIG. 4, a soliton injected with initial velocity of 0.15 c will be most sensitive to small
changes in the resonator frequency about 7 GHz, whereas a soliton injected with velocity of 0.2 c will be most sensitive to resonators around 10 GHz.
FIG. 5 is a chart 100 illustrating a sensitivity of solitons propagating on a JTL at varying velocities to a delay induced by a qubit as a function of a resonant frequency of the qubit. The vertical
axis 102 represents the delay sensitivity, in unit cells per GHz, using the same parameters as FIG. 4 above. The horizontal axis 104 represents a resonant frequency of the qubit. A delay sensitivity
of a soliton having an initial velocity of 0.15 c is shown as a solid line 106, and a delay sensitivity of a soliton having an initial velocity of 0.2 c is shown as a dashed line 108.
As can be seen from the chart, solitons injected with initial velocity of 0.15 c could resolve a 200 MHz change in the frequency of a qubit that was initially at 7 GHz, by a change in their delay
amounting to 0.3 unit cells, or, given such a soliton travels approximately one hundred unit cells per nanosecond, three picoseconds, after traversing a 200-cell long JTL. For a different set of
parameters, where k=0.15, L
=20 pH, L=10 pH, L
=300 pH, ω
/2π=92 GHz (C=150 fF), and an injection velocity of 0.14 c, the same variation in the qubit frequency yields a delay of sixteen picoseconds, easily resolved with current technology. It will be
appreciated that the soliton is also sensitive to changes in resonators of lower frequencies. For the illustrated example, maximum sensitivity can be obtained at 1.7 GHz and 2.9 GHz with solitons
injected at 0.15 c and 0.2 c, respectively.
FIG. 6 illustrates a first implementation of a system 200 in accordance with an aspect of the present invention. In the illustrated system, two solitons with opposite polarities are injected by a
soliton driver 202 into two Josephson transmission lines 204 and 206. A first JTL 204 is coupled to a phase qubit 208 and a second JTL 206 is used as a timing reference. A soliton detector 210 is
coupled to each of the two JTLs 204 and 206 to detect the arrival of the injected solitons. In one implementation, the soliton detector 210 can include a combiner to determine coincident arrival of
the solitons. Specifically, if the solitons are coincident they will annihilate within the combiner, and no output pulse will be generated. If the transit times of the two paths are different, the
solitons will arrive at the combiner at the different times and thus generate an output pulse. It will be appreciated, however, that the soliton detector 210 can include any appropriate circuitry for
detecting a difference in arrival time between the two solitons.
It will be appreciated from the description above that, depending on the specific implementation, the phase qubit 208 can be coupled, either inductively or capacitively, to the first JTL 204 without
any intermediating structure. In the illustrated implementation, however, the first JTL 204 is not coupled directly to the phase qubit 208. Instead, since the soliton is sensitive to changes of the
resonant frequency of an oscillator, the phase qubit 208 can be coupled dispersively to a resonator 212, which, in turn, couples to the first JTL 204. While the qubit 208 is illustrated herein as a
phase qubit, this dispersive coupling arrangement can be particularly helpful to perform readout on qubits of the transmon or quantronium type.
In one implementation, the soliton-qubit interaction can be resonantly enhanced by restricting the volume of interaction between phase qubit and soliton, such that each incident soliton is made to
interact with the phase qubit multiple times, as quantitatively characterized by a finesse F=u/ex, where u is the velocity of the soliton, is the length of the line, and κ is the energy decay rate of
the cavity. To this end, the JTLs 204 and 206 are suitably terminated with impedance mismatched ends, allowing scattering of solitons off of each JTL 204 and 206 at intervals comparable to the
resonant frequency of the cavity formed by the JTL. Accordingly, incident solitons may be scattered with a state-dependent phase (transmission/reflection or delay) that is resolvable, and the
cumulative effect of the multiple exposures can be utilized to determine an associated state of the qubit.
FIG. 7 illustrates a second implementation of a system 250 in accordance with an aspect of the present invention. In this implementation, the qubit 252 can be coupled to multiple nodes along the JTL
254, which are separated by a distance l, such that the soliton interacts with the qubit multiple times and effectively presents the qubit with a periodic train of impulses. To this end, a coupling
element 256 is configured to couple the qubit 252 to the JTL 254 at multiple locations, such that when a pulse is generated at a soliton driver 258, the qubit applies a first delay to the propagation
of the soliton along the JTL multiple times when the qubit is in a first state and applies a second delay to the propagation of the soliton along the JTL multiple times when the qubit is in a second
state. The difference in the time-of-flight caused by these delays can be determined at a soliton detector 260 coupled to the JTL 254 to determine an associated state of the qubit.
FIG. 8 illustrates a method 300 for reading an associated state of a qubit in accordance with an aspect of the present invention. At 302, a velocity of a soliton is selected according to a physical
length of the qubit and a characteristic frequency of the qubit. For example, the velocity can be selected such that the product of the velocity, the physical length of the qubit, and the
characteristic frequency of the qubit is substantially equal to one. At 304, a first soliton is produced along a first Josephson transmission line coupled to the qubit. At 306, a second soliton is
produced at the selected velocity along a second Josephson transmission line that is not coupled to the qubit. At 308, a delay associated with the first soliton is determined relative to the second
soliton to determine a state of the qubit. For example, it can be determined that the qubit is a ground state if the determined delay is within a first range and the qubit is in a first excited state
if the determined delay is within a second range.
To summarize, systems and method in accordance with an aspect of the present invention provide means to read out a qubit, including a phase qubit, using coincidence measurements of sine-Gordon
solitons propagating on a Josephson transmission line. These methods rely on resonantly enhanced scattering of the soliton; in one implementation when the soliton transit time near the qubit is
commensurate with the qubit oscillation period and in another by ensuring repeated periodic interaction between the soliton and the qubit. An optimal readout sensitivity is obtained for frequencies
slightly detuned from the resonance condition. This readout scheme is advantageous as it avoids tunneling of the qubit, an improvement that avoids dissipation in the vicinity of the qubit, which may
activate unintended tunneling of neighboring qubits and emission of disruptive microwave radiation into the circuit, and relaxes qubit design constraints. The readout scheme also extracts the qubit's
information in a manner compatible with single-flux-quantum (SFQ) based qubit control.
What have been described above are examples of the invention. It is, of course, not possible to describe every conceivable combination of components or methodologies for purposes of describing the
invention, but one of ordinary skill in the art will recognize that many further combinations and permutations of the invention are possible. Accordingly, the invention is intended to embrace all
such alterations, modifications, and variations that fall within the scope of this application, including the appended claims.
Patent applications by Aaron A. Pesetski, Gambrills, MD US
Patent applications in class SPECIFIC SIGNAL DISCRIMINATING (E.G., COMPARING, SELECTING, ETC.) WITHOUT SUBSEQUENT CONTROL
Patent applications in all subclasses SPECIFIC SIGNAL DISCRIMINATING (E.G., COMPARING, SELECTING, ETC.) WITHOUT SUBSEQUENT CONTROL
User Contributions:
Comment about this patent or add new information about this topic:
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FOM: geometry
Robert Black Robert.Black at nottingham.ac.uk
Thu Oct 8 10:59:01 EDT 1998
The historical picture given in Mic Detlefsen's long and interesting
posting of 2 October seems to me to be pretty accurate, though of course
one could quibble over certain details. I'm a bit puzzled though by the
questions he thinks it would be profitable to discuss, in that it seems to
me he's posed these questions in a way which might have seemed natural to
Frege/Hilbert/Poincare but doesn't seem very natural today. Let me just
take the first one:
1: Is or should the asymmetry between arithmetic and geometry that Gauss
and nearly all other 19th century foundational thinkers believed in still
be treated as a fundamental 'datum' of the foundations of mathematics today?
Kant's view that we know a priori that euclidean geometry is true of
physical space is now dead as a dodo, so we can leave applied geometry,
'geometry as a branch of physics' as Steve puts it, to one side. So far as
pure geometry is concerned, ever since Bolyai and Lobachevsky we have had a
plurality of geometries, and now we have euclidean, non-euclidean, affine,
projective, riemannian, pseudoriemannian and God knows what else, each in
as many dimensions as you might happen to want. Particularly in the case
of projective and affine geometries we have synthetic axiomatizations in
terms of incidence etc. and analytic coordinatizations, i.e. models of
these axiomatizations in algebraic structures, together with theorems
relating the geometric to the algebraic point of view (e.g. Pappus' theorem
holds in a projective geometry iff the field underlying the
coordinatization is commutative).
As a result of all this, I'd have thought that *everybody* would agree that
the modern approach to (pure) geometry is structuralist, the subject matter
of geometry being a *plurality* of abstract structures. Hilbert's
'axiomatic method' is paradigmatic for this way of viewing things, though
of course it goes back to Dedekind and Riemann, perhaps even to Gauss. It
would have been totally foreign to Kant, however, and Frege had difficulty
with it, as is clear from his exchange of letters with Hilbert.
If one also takes a structuralist attitude to arithmetic, as many of us do,
then it would seem that there is no asymmetry between arithmetic and
geometry left. So it seems to me that Mic's question boils down to:
should we be structuralist about arithmetic? And the arguments on both
sides of that question are pretty familiar.
I'd like to ask another question about geometry though, roughly, just how
does it fit into the overall structure of modern mathematics? Geometrical
thinking is all-pervasive - e.g. every time one uses linear algebra one is
in effect thinking geometrically. Further: at least differential geometry
and algebraic geometry are major research areas. But Bourbaki, for
example, identifies the major structure-types of modern mathematics as
algebraic or topological: there's no volume called 'Geometrie Generale'.
Indeed Bourbaki clearly regards synthetic geometry as dead, or at best as
no more than an occasionally useful language for expressing pieces of
algebra - see in particular in his 'Elements d'histoire des mathematiques'
the chapter 'Formes quadratiques: geometrie elementaire'.
The most general definition of geometry that I'm aware of is: a geometry is
a set with a symmetric and reflexive 'incidence' relation. I suppose some
idea like that would come at the beginning of 'general geometry' the way
the definition of a magma comes at the beginning of Bourbaki's algebra.
So my (twofold) question is:
1. Should we identify and give separate treatment to 'geometrical
structures' as basic to modern mathematics, and
2. Why is the geometrical mode of thought - a mode abstracted from our
thought about the very special example of 3-dimensional physical space - so
pervasive in abstract mathematics?
Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD
tel. 0115-951 5845
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Math Goes Pop!
Last year marked the dawn of a new era in mathematical holidays. Spearheaded by Dr. Michael Hartl, Tau Day (celebrated today, June 28th) is an attempt to draw awareness to what he sees as a
fundamental error in the definition of the beloved circle constant . In particular, he (and others) argue that the more natural choice of the circle constant should be , which he affectionately dubs
. I outlined the reasons for this in a post last year, though if you have the time, I highly encourage you to read Hartl’s Tau Manifesto.
This year, I thought it would be nice to talk with Dr. Hartl in more detail about his inspirations for Tau Day, and where he envisions it in the future. He was gracious enough to agree to a brief
interview, which I humbly submit to you here.
Q: When did you first . . . → Read More: Second Annual Tau Day: Interview and Ideas!
For many of us, summer is thought of as the time between Memorial Day and Labor Day. For folks of a younger generation, though, trendier bookends are provided by two MTV Award shows: The Movie Awards
at the beginning of the summer, and the Video Music Awards at the end. Continuing this noble tradition, the 20th iteration of the MTV Movie Awards was broadcast this weekend. If you missed it, don’t
worry; I’m sure it will be shown another 300,000 or so times before the summer is out.
As a shining beacon of what is hip, MTV has a responsibility during its movie awards to highlight the most popular films of the year. This is in stark contrast to the priorities of higher brow award
shows such as the Oscars, for which artistic achievement is placed on the highest pedestal. This is not to say that these two goals need . . . → Read More: MTV/Oscar Showdown
MTV/Oscar Showdown
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Geometric Measurement and Dimension
Standards in this domain:
Explain volume formulas and use them to solve problems
• HSG-GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s
principle, and informal limit arguments.
• HSG-GMD.A.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
Visualize relationships between two-dimensional and three-dimensional objects
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How to find the order of SL(2,4)?
February 22nd 2009, 07:07 AM
How to find the order of SL(2,4)?
How do you find the order of a special linear group? For example, how do you find order of SL(2,4)? I think the answer's 60 but really would like a proof. How do you find the order of SL(2,5)?
Finally, here's a problem set on one of my problem sheets - "Show that the only scalar matrices in SL(2,5) are I and -I. This seems fairly obvious but I'm not sure how to begin a proof. And why
do you think this question was asked? (No proof required, I just want to know the reason behind asking this).
February 22nd 2009, 07:30 AM
You should know that $|\text{GL}(n,q)| = (q^n-1)(q^n - q)...(q^n - q^{n-1})$.
You should also know $\text{GL}(n,q)/\text{SL}(n,q) \simeq K^{\times}$ where $|K^{\times}| = q-1$, now you can solve for $|\text{SL}(n,q)|$.
February 22nd 2009, 10:45 AM
OK thanks, this is really helpful. I have not been given the results you quoted but was able to derive them. I've got two quick questions:
1) If A is a group and B is a subgroup then is the order of the quotient group A / B equal to the order of A divided by the order of B? ie is |A/B| = |A| / |B|? I appreciate the answer is
probably blindingly obvious, but I've thought about it a bit and just ended up confusing myself, so an answer and quick justification would be good here.
2) Why should the only scalar matrices in SL(2,5) be I or -I? I originally thought this looked obvious but actually now I'm not so sure...
February 22nd 2009, 10:51 AM
1) If A is a group and B is a subgroup then is the order of the quotient group A / B equal to the order of A divided by the order of B? ie is |A/B| = |A| / |B|? I appreciate the answer is
probably blindingly obvious, but I've thought about it a bit and just ended up confusing myself, so an answer and quick justification would be good here.
Do you remember the proof behind Lagrange's theorem? In the proof it is shown that if $r$ is the number of left cosets of $B$ in $A$ then $r|B| = |A| \implies r = |A|/|B|$. However, if $A$ is a
normal subgroup then $r = |A/B|$ and so $|A/B| = |A|/|B|$.
2) Why should the only scalar matrices in SL(2,5) be I or -I? I originally thought this looked obvious but actually now I'm not so sure...
I am not sure what you mean by "scalar matrices", do you mean $kI$ where $I$ is the identity matrix? If so then $\det (kI) = k^2$ and we need this determinant to be $1$, by definition of special
linear groups. Thus, $k = \pm 1$.
February 22nd 2009, 12:19 PM
Great response again. Surely $|A/B| = |A|/|B|$ [COLOR=Black]holds irrespective of whether B is a normal subgroup of A or just any subgroup of A? (I assume you mean B is a normal subgroup and not
A is a normal subgroup).
This is indeed what I meant by a scalar matrix - I was just slightly worried that in a general field it may not be obvious that $k^2 = 1$ implies k = 1 or -1, but actually this is of course
clear. Thanks.
EDIT: Actually I'm back to thinking that this last point may not be so obvious - what is clear is that 1 and -1 satisfy $k^2 = 1$, but how do we know there are no other such elements in a general
field satisfying this? (Above I was thinking about the Fundamental Theorem of Algebra and concluded that there were only two roots but this kind of argument won't work in general fields)
EDIT 2: This can be done by using the fact that all finite fields of the same order are isomorphic.
February 22nd 2009, 01:03 PM
If $B$ is not a normal subgroup of $A$ then you cannot form $A/B$!
However, you can still talk about $(A:B)$, the number of left cosets of $B$ in $A$.
Yes, $(A:B) = |A|/|B|$.
This is indeed what I meant by a scalar matrix - I was just slightly worried that in a general field it may not be obvious that $k^2 = 1$ implies k = 1 or -1, but actually this is of course
clear. Thanks.
If $k^2 = 1 \implies k^2 - 1 = 0\implies (k-1)(k+1) = 0 \implies k=1,-1$.
Of course if field charachteristic is two then there is only one solution since $1=-1$.
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When is a scheme a zero-set of a section of a vector bundle?
up vote 10 down vote favorite
Are there any general results on when a closed subscheme X of a quasi-projective smooth scheme M can be written as the zero-set of a section of a vector bundle E on M? To put it in a diagram: When is
X the fiber product of M -> E <- M , where one arrow is the zero section and the other arrow is the section I'm looking for.
If this is not possible, can X be written as a degeneracy locus?
add comment
4 Answers
active oldest votes
As for the first question, the class of X has to be the product of the Chern roots of the bundle, so in the Chow ring, it is the class of a complete intersection.
up vote 11 down As for the second question, you would have to find classes that will solve the class of X in the Thom-Porteus formula, see Fulton's intersection theory 14.4
vote accepted
Thanks for the answer! Is it possible to recover the bundle from the Chern roots, or even better, to actually get your hands on the section? And how restrictive is the condition
that it has to be the class of a complete intersection? A zero-set of a section can easily be a non-complete intersection, right? – Timo Schürg Oct 22 '09 at 8:37
@Timo, I only have very partial knowledge on how to continue: You can build bundles as sub-bundles or quotient bundles of other bundles until you get the desired chern roots. A
zero set can easily be non-complete intersection and be the same class of a complete intersection, but being the class of a complete intersection is a strong requirement. – David
Lehavi Oct 22 '09 at 8:54
add comment
Are you assuming that the rank of $E$ equals the codimension of the subscheme? You don't say so explicitly. If not, the answer is that every closed subscheme is a zero section, since it is
up vote the intersection of finitely many hypersurfaces.
6 down
Thanks for the answer! The way I posed the question was really for arbitrary rank of $E$.I also figured out another way of proving your statement in the meantime: Just take the first step
of a locally free resolution of the ideal sheaf of $X$ in $M$. That gives a surjective morphism $E \to I$, and thus a section. The case I really cared about was when $X$ has a perfect
obstruction theory $E^{-1} \to E^{0}$. I wanted to fix $dim(M)=rk(E^{0})$, and the rank of the vector bundle to be $rk(E^{0})$. If $X$ is affine, that actually works! It's Appendix A of
front.math.ucdavis.edu/1001.2719. – Timo Schürg Oct 1 '10 at 6:59
Sorry, the rank of the vector bundle in the comment should be $rk(E^{-1})$. – Timo Schürg Oct 1 '10 at 7:00
add comment
A necessary condition is that it be a locally complete intersection, since locally this is the same as asking that your scheme be the zero set of codimension many equations.
up vote 4 down vote
add comment
At least when the subvariety has codimension 2, this is known as "the Serre construction". There's a nice description of the case of points in a surface given in "Lectures on linear
up vote 2 series" by Lazarsfeld. I'm sure there are many other excellent references too, but that's the first that comes to mind.
down vote
add comment
Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry or ask your own question.
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boolean algebra sample multiple-choice questions
Author Message
ShatowRavem Posted: Saturday 30th of Dec 10:20
hi Gals I really hope some math master reads this. I am stuck on this test that I have to take in the next couple of days and I can’t seem to find a way to solve it. You see, my teacher
has given us this test covering "boolean algebra" sample "multiple-choice questions", subtracting fractions and logarithms and I just can’t understand it. I am thinking of paying
someone to help me solve it. If someone can give me some suggestions, I will very appreciative.
From: Kansas
Jahm Xjardx Posted: Monday 01st of Jan 08:04
The attitude you’ve adopted towards the "boolean algebra" sample "multiple-choice questions" is not the a good one. I do understand that one can’t really think of anything else in such
a situation. Its nice that you still want to try. My key to successful equation solving is Algebrator I would advise you to give it a try at least once.
From: Odense,
Denmark, EU
Momepi Posted: Tuesday 02nd of Jan 21:30
Algebrator is rightly a good software program that helps to deal with algebra problems. I remember facing troubles with least common measure, trigonometric functions and difference of
squares. Algebrator gave step by step solution to my algebra homework problem on typing it and simply clicking on Solve. It has helped me through several math classes. I greatly
recommend the program.
From: Ireland
keslixam Posted: Thursday 04th of Jan 07:08
Wow, that's amazing news ! I was so stressed but now I am quite happy that I will be able to improve upon my grades! Thank you for the info guys! So then I just have to get the software
and do my homework for tomorrow. Where can I find out more about it and buy it?
From: Leeds,
Noddzj99 Posted: Saturday 06th of Jan 07:28
I would advise using Algebrator. It not only helps you with your math problems, but also gives all the required steps in detail so that you can enhance the understanding of the subject.
From: the
MichMoxon Posted: Monday 08th of Jan 08:56
There you go http://www.linear-equation.com/systems-of-differential-equations.html.
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Implementation via Code of Rights
Implementation via Code of Rights∗
Semih Koray †and Kemal Yıldız ‡
February 4, 2008
Implementation of a social choice rule can be thought of as a design
of power (re)distribution in the society whose ”equilibrium outcomes”
coincide with the alternatives chosen by the social choice rule at any
preference profile of the society. In this paper, we introduce a new
societal framework for implementation which takes the power distri-
bution in the society, represented by a code of rights, as its point of
departure. We examine and identify how implementation via code of
rights (referred to as gamma implementation) is related to classical
Nash implementation via mechanism. We characterize gamma imple-
mentability when the state space on which the rights structure is to
be specified consists of the alternatives from which a social choice is
to be made. We show that any social choice rule is gamma imple-
mentable if it satisfies pivotal oligarchic monotonicity condition that
we introduce. Moreover, pivotal oligarchic monotonicity condition
combined with Pareto optimality is sufficient for a non-empty valued
social choice rule to be gamma implementable. Finally we revisit lib-
eral’s paradox of A.K. Sen, which turns out to fit very well into the
gamma implementation framework.
Keywords: Implementation, code of rights, Nash equilibrium,
pivotal oligarchic monotonicity, social choice rule.
1 Introduction
In classical implementation a rights structure among the members of the so-
ciety can be induced from the mechanism, designed to implement a social
Preliminary version.
Department of Economics, Bilkent University, Ankara.
Department of Economics, Bilkent University, Ankara.
choice rule under the given solution concept. In other words, in classical im-
plementation we have an implicit specification of a power distribution among
the members of the society. In this paper, we introduce a new institutional
design approach to implementation which depends directly on the alternative
set, and the rights structure in the society.
A constitution or a code of rights is used for the assignment of rights
to the members of the society. In Arrow [1] such a notion of constitution
is defined, where a ”well-behaved” social welfare function is considered as a
constitution. This notion leads us to the conclusion of well known Arrow’s
Impossibility Theorem. We define a code of rights as a set valued function,
which associates each ordered pair of alternatives with a family of coalitions,
indicating that each coalition in the specified family is given the right to
lead a switch from the first alternative to the second one. In our framework
code of rights is common knowledge, and is specified as being invariant of
The definition for code of rights that we use in this paper was introduced
in Sertel[7], where it is used as a design notion in the specification of a
Rechstaat. Parelelling the first and second welfare theorems of economics,
Sertel imparted to code of rights an invisible hand property and a property
of the preservation of the best public interest.
In a similar framework used in Sertel, Peleg[4] proposed a new definition
of constitution which specifies a rights structure among the members of the
society and investigated game forms that represent the distribution of power
which is dictated by the prevailing rights structure in the society.
In classical implementation there are various examples indicating the con-
nection between monotonicity and implementability. Maskin [3] showed that
any Nash implementable social choice rule is monotonic, and monotonicity
combined with some further assumptions as no veto power condition is suf-
ficient for Nash implementability. Danilov [2], proposed an essential mono-
tonicity condition which turned out to be both necessary and sufficient for
Nash implementability in case of having at least three agents.
Kaya and Koray [5] introduced the notion of oligarchy and oligarchic
monotonicity, where it is shown that; any oligarchic social choice rule satis-
fies oligarchic monotonicity and oligarchic monotonicity combined with una-
nimity condition is sufficient for characterization of oligarchic social choice
In section 2 we introduce the basic definitions and notation. The rela-
tion between Nash implementation and (A, γ)-implementation is examined
in section 3. In section 4, we introduce the pivotal oligarchic monotonicity
condition and related definitions. In sections 5 & 6, (A, γ)-implementation
is characterized in terms of pivotal oligarchic monotonicity, and Pareto op-
timality. In section 5, we show that any (A, γ)-implementable social choice
rule satisfies pivotal oligarchic monotonicity. The implementation theorem
is set in section 6, indicating that any non-empty valued, Pareto optimal
social choice rule, endowed with pivotal oligarchic monotonicity is (A, γ)-
implementable. In section 7, liberal’s paradox of Amartya K. Sen [6]is revis-
ited, and investigated from (A, γ)-implementation perspective.
2 Preliminaries
We use A to denote a non-empty, finite alternative set, while N ,as usual,
denotes the set of agents which is also assumed to be non-empty and finite.
We will use N to denote the collection of all subsets of N and any member of
N is said to be a coalition in N , denoted by generic element K; i.e K ∈ 2N =
N . A linear order on A is denoted by L(A), which is a complete, transitive,
and antisymmetric binary relation on A. The set of all linear order profiles
on A is denoted by L(A)N . For any i ∈ N and any a, b ∈ A, we represent,
agent i prefers b to a under R, by bRi a. Let R ∈ L(A)N and a ∈ A, the lower
contour set of R, for agent i with respect to alternative a ∈ A, is the set
consisting of alternatives to which a is preferred by agent i under preference
profile R, which is denoted by L(Ri , a). A social choice rule F maps every
linear order profile on A into a subset of A; i.e. F : L(A)N → 2A
A mechanism (or a game form) is a function g which maps every joint
strategy to an outcome in the alternative set; i.e. g : S → A, where S =
×i∈N Si , Si stands for agent i’s strategy set. A mechanism g, combined
with a linear order profile R ∈ L(A)N forms a normal form game and the
pure strategy Nash equilibria of the game is denoted by NE(g, R). We say
a social choice rule F is Nash implementable via a mechanism g if at each
preference profile R, alternatives chosen by F coincide with the alternatives
in the Nash equilibrium of the game for given R; i.e for any R ∈ L(A)N , we
have {g(s) | s ∈ N E(g, R)} = F (R).
Any social choice rule F is said to be monotonic if and only if for any
R, R ∈ L(A)N , and any a ∈ F (A) such that for any i ∈ N , we have
L(Ri , a) ⊂ L(Ri , a) implies a ∈ F (R ). We say F is Pareto optimal if
and only if there is no alternative in A which Pareto dominates a with re-
spect to given R; i.e for any R ∈ L(A)N and a ∈ F (R), there is no b ∈ A
such that for any i ∈ N , bRi a.
For any given preference profile R ∈ L(A)N , the benefit function βR :
A × A → 2N , maps any pair of alternatives (a, b) ∈ A × A, to a member of
2N ; i.e. the class of all coalition families. For any (a, b) ∈ A × A, any K ∈ N ,
K ∈ βR (a, b) implies that; all the members of the coalition K prefers b to a;
i.e. for any i ∈ K, bRi a.
We define a code of rights, as a function γ which maps any pair of alter-
natives (a, b) ∈ A × A, to a coalition family; i.e γ : A × A → 2N , where for
any (a, b) ∈ A × A, and any K ∈ N , K ∈ γ(a, b) implies that coalition K
is given the right to lead a switch from a to b, by the code of rights γ. We
assume that if any coalition is given the right to lead a switch from a to b,
then any coalition which contains this coalition preserves the same right;i.e
for any (a, b) ∈ A × A and for any K ∈ N , K ∈ γ(a, b) implies for any
K ∈ N where K ⊂ K , we have K ∈ γ(a, b). The collection of all code of
rights defined on A × A for given N is denoted by Γ(A, N ).
We assume that every coalition is able to make any switch, so we do
not specify an ability function α : A × A → 2N , which specifies the able
coalitions for leading a switch from an alternative to another one.
3 (A, γ)-implementation
Before introducing (A, γ)-implementability notion, we need to specify an
equilibrium condition which plays the role of solution concepts in classical
Definition 1 For any R ∈ L(A)N , and any a ∈ A, we say a is an (A, γ)-
equilibrium and denote it by a ∈ (A, γ, βR ) if and only if for any b ∈ A \
{a}, γ(a, b) ∩ βR (a, b) = ∅.
If for any alternative a, there is no willing coalition which is given the
right to lead a switch from a to any other alternative, then alternative a is
referred as an (A, γ )-equilibrium.1
Definition 2 Any social choice rule F is said to be (A, γ)-implementable if
there is a γ ∈ Γ(A, N ) such that for any R ∈ L(A)N , F (R) = (A, γ, βR ).
For any social choice rule F , if we can find a code of rights γ : A×A → 2N
such that; at each preference profile R, alternatives chosen by F coincide with
the alternatives in the (A, γ)-equilibria for given R, then F is said to be (A, γ
Example 1 Let N = {1, 2}, A = {a, b, c}, R and R be as specified below,
and the social choice rule F be such that; F (R) = {a), F (R ) = {b}
Notion of (A, γ )-equilibria as well as (A, γ) implementation can be extended to (S, γ
) implementation, where S stands for any arbitrary strategy set.
R R
a c c b
c b a c
b a b a
Firstly it is easy to check that F is not Nash implementable. Secondly,
let us construct a code of rights γ which would implement the given social
choice rule F . Let γ be such that;
∀x ∈ {b, c} γ(a, x) = {{1}, {1, 2}}
∀x ∈ {a, c} γ(b, x) = {{2}, {1, 2}}
∀x ∈ {a, b} γ(c, x) = {{1}, {2}, {1, 2}}
Now, for any x ∈ {b, c}, βR (a, x) = {{2}} but γ(a, x) = {{1}, {1, 2}}
implies βR (a, x) ∩ γ(a, x) = ∅ implies a ∈ (A, γ, βR ).
{2} ∈ βR (b, c) ∩ γ(b, c) implies b ∈ (A, γ, βR ).
{1} ∈ βR (c, a) ∩ γ(c, a) implies c ∈ (A, γ, βR ) implies a = (A, γ, βR ) =
F (R) and for any x ∈ {a, c}, βR (b, x) = {{1}} but γ(b, x) = {{2}, {1, 2}}
implies βR (b, x) ∩ γ(b, x) = ∅ implies b ∈ (A, γ, βR ).
{1} ∈ βR (a, c) ∩ γ(a, c) implies a ∈ (A, γ, βR ).
{2} ∈ βR (c, b) ∩ γ(c, b) implies c ∈ (A, γ, βR ) implies b = (A, γ, βR ) =
F (R ). Hence we can conclude that F defined on R and R , 2 is (A, γ)-
From Example 1, we can conclude that there are social choice rules which
are not Nash implementable, but (A, γ )-implementable. However, converse
of this holds as well; i.e there are social choice rules which are Nash imple-
mentable but not (A, γ )-implementable 3 . Following example establishes
this fact.
Example 2 Let N = {1, 2}, A = {a, b, c}, R, R and R be as specified
below, and the social choice rule F be such that; F (R) = {b}, F (R ) =
F (R ) = {a}.
2 ˜
We can extend F to the full domain by inducing F (R) from the (A, γ )-equilibria for
˜ ˜ ˜
any given R; i.e for any R ∈ L(A)N , F (R) = (A, γ, βR ).
In the (S, γ )-implementation framework one can show that any Nash implementable
social choice rule F is (S, γ)-implementable.
R R R
a c c b b c
b b a a a a
c a b c c b
First let us show that F is Nash implementable. Consider the following
mechanism; let S1 = S2 = {{a, b}, {a, c}, {b, c}}, g : S → A, where for any
s ∈ S = S1 × S2 , g(s) = s1 ∩ s2 , if there is only one x ∈ A such that
x ∈ s1 ∩ s2 , otherwise ties are broken with respect to the first component
of first agent’s strategy. Note that, for any s ∈ S, there is only one x ∈ A
such that x ∈ g(s). Now for given R, let s = ({a, b}, {b, c}), g(¯) = {b}. For
¯ s
given s1 = {a, b}, player 2 should choose either a or b, where bR2 a implies
∀s2 ∈ S2 , g(¯)R2 g(¯1 , s2 ) implies s ∈ N E(g, R). Moreover it is easy to check
s s ¯
s is the unique Nash equilibrium of the defined game under R. If one of R
or R is given, then we can similarly conclude that {a} is the unique Nash
equilibrium outcome. Moreover, one can extend F to the full domain by
inducing F from the Nash equilibria outcomes of the defined mechanism.
Now let us show that F is not (A, γ)-implementable. Suppose not; i.e.
there exists a γ ∈ Γ(A, N ) such that for any R ∈ L(A)N , F (R) = (A, γ, βR )
implies F (R ) = (A, γ, βR ) = {a} and {2} ∈ βR (a, b) implies {2} ∈ γ(a, b),
similarly from F (R ) = {a}, we get {2} ∈ γ(a, c), with {{2}} = βR (a, b) =
βR (a, c) implies for any x ∈ A \ {a}, γ(a, x) ∩ βR (a, x) = ∅ implies {a} ∈
(A, γ, βR ) = F (R), contradicting F (R) = {b}. Hence we can conclude that
F is not (A, γ)-implementable.
4 Pivotal oligarchic monotonicity
In order to state our monotonicity condition, first we need to introduce some
auxiliary notions.
Definition 3 For any R ∈ L(A)N , and any (a, b) ∈ A × A, MR (a, b) stands
for the maximal coalition in the coalition family βR (a, b); i.e MR (a, b) ∈
βR (a, b) and for any K ∈ βR (a, b), K ⊂ MR (a, b).
Since N is finite we know that; there always exists a unique maximal
coalition, possibly empty set, in the coalition family βR (a, b).
Definition 4 A social choice rule F is said to be monotonic if and only if
for any R, R ∈ L(A)N , any a ∈ F (R) satisfying condition
∀b ∈ A, MR (a, b) ⊂ MR (a, b) (1)
implies a ∈ F (R ).
Maskin introduced the monotonicity condition in terms of sets consisting
alternatives, specified for each agent; here we restate the monotonicity con-
dition by specifying coalitions for each alternative associated with the ones
chosen by F .
Definition 5 For any (a, b) ∈ A × A, any K ∈ 2N , K is said to be an
(a, b)-oligarchy if and only if for any R ∈ L(A)N , bRK a implies a ∈ F (R).
If there is a coalition K such that; b is preferred to a by all the members
of K implies a is not chosen by F , then we call K; an a-oligarchy via b or
simply an (a, b)-oligarchy.
Definition 6 For any R ∈ L(A)N , any a ∈ F (R), any b ∈ A, and any K ∈
2N , K is said to be a pivotal (a, b, R) oligarchy if and only if MR (a, b) ∪ K
is an (a, b)-oligarchy.
Any coalition K is considered as a pivotal coalition for having an (a, b)-
oligarchy, if the coalition formed by unification of the largest coalition which
prefers b to a under R, and K forms an (a, b)-oligarchy.
Definition 7 For any R ∈ L(A)N , any a ∈ F (R), any b ∈ A , and
any K ∈ 2N , K is said to be a non-pivotal (a, b, R)-oligarchy denoted by
K ∈ C N P O (a, b, R) [C N P O (a, b, R) stands for family of non-pivotal (a, b, R)-
oligarchies] if and only if K is not a pivotal (a, b, R) oligarchy. More-
over, K is said to be a maximal non-pivotal (a, b, R)-oligarchy denoted by
K ∈ C M N P O (a, b, R) if and only if K ∈ C N P O (a, b, R) and there is no
K ∈ C N P O (a, b, R) such that K ⊂ K .
Remark 1 Any alternative a, being chosen by F under R indicates that;
MR (a, b) is not an (a, b) oligarchy, if not clearly a should not be chosen by
F , hence we know that MR (a, b) is in the family of non-pivotal (a, b, R)-
oligarchies, C N P O (a, b, R), and clearly any member of CM N P O (a, b, R) con-
tains MR (a, b).
Definition 8 (Pivotal oligarchic monotonicity, POM) Any social choice rule
F satisfies POM if and only if for any R, R ∈ L(A)N and any a ∈ F (R)
satisfying condition
∀b ∈ A, ∃K ∈ C M N P O (a, b, R) : MR (a, b) ⊂ MR (a, b) ∪ K (2)
implies a ∈ F (R ).
Intuitively, POM means that alternative a continues to be chosen by F ,
unless there is an (a, b)-oligarchy which prefers b to a under R .
Lemma 1 Any social choice rule F endowed with POM is monotone.
Proof. Take any R, R ∈ L(A)N , and a ∈ F (R), where condition (1) is
satisfied. Now for any b ∈ A, MR (a, b) ⊂ MR (a, b) implies (2) holds, hence
a ∈ F (R ).
5 Necessity of POM for (A, γ) implementabil-
Lemma 2 For any (A, γ)-implementable social choice rule F , let γ be a code
of rights which implements F , for any(a, b) ∈ A × A, and any K ∈ 2N such
that K = ∅, we have K ∈ γ(a, b) if and only if K is an (a, b)-oligarchy.
Proof. (⇒) For any(a, b) ∈ A × A, assume that ∅ = K ∈ γ(a, b). Now
K ∈ γ(a, b) implies for any R ∈ L(A)N such that K ∈ βR (a, b), K ∈ γ(a, b) ∩
βR (a, b), and K = ∅ implies γ(a, b)∩βR (a, b) = ∅ hence we get a ∈ (A, γ, βR ),
now sinceF is (A, γ)-implementable we get a ∈ F (R).
(⇐) Assume not; i.e. K is an (a, b)-oligarchy but K ∈ γ(a, b). Take any
R such that for any i ∈ N \ K, aRi b, and bRK a; [ i.e. K = MR (a, b)] . Now
K is an (a, b)-oligarchy implies a ∈ F (R), and F is(A, γ)-implementable
indicates that a ∈ (A, γ, βR ) thus, we can conclude that ∃K ⊂ K such that
K ∈ γ(a, b) implies K ∈ γ(a, b) contradicting K ∈ γ(a, b).
Theorem 2 Any (A, γ)-implementable social choice rule F satisfies POM.
Proof. Take any (A, γ)-implementable social choice rule F , any a ∈ F (R),
and any R, R ∈ L(A)N such that condition (2) holds.
Now condition (2) implies for any b ∈ A, there exists K ∈ C M N P O (a, b, R)
such that MR (a, b) ⊂ MR (a, b) ∪ K where MR (a, b) ∪ K is not an (a, b)-
oligarchy, hence MR (a, b) is not an (a, b)-oligarchy, by the lemma above
we get; MR (a, b) ∈ γ(a, b) combined with MR (a, b) being maximal implies
γ(a, b) ∩ βR (a, b) = ∅, so a ∈ (A, γ, βR ) now, F being (A, γ)-implementable
implies a ∈ F (R ) hence F satisfies POM.
6 The implementation theorem
In this section we state a converse result to Theorem 1. We construct a code
of rights to implement a social choice rule F , which is non-empty valued
Pareto optimal, and which satisfies pivotal oligarchic monotonicity.
Theorem 3 Any non-empty valued, Pareto optimal social choice rule F,
endowed with POM, is (A, γ)-implementable.
Proof. First let us construct the code of rights, γ such that; for any (a, b) ∈
A × A, and any K ∈ 2N , we have K ∈ γ(a, b) if and only if K is an (a, b)-
oligarchy. Now, for any R ∈ L(A)N , a ∈ F (R), and b ∈ A; a ∈ F (R)
implies MR (a, b) is not an (a, b)-oligarchy indicating that MR (a, b) ∈ γ(a, b),
MR (a, b) being maximal implies γ(a, b) ∩ βR (a, b) = ∅, so a ∈ (A, γ, βR ).
This implies F (R) ⊂ (A, γ, βR ).
Conversely to show that; (A, γ, βR ) ⊂ F (R), for any R ∈ L(A)N , take
any a ∈ (A, γ, βR ), and assume that a ∈ F (R). Now F is non-empty valued
implies there exists b ∈ A \ {a} such that b ∈ F (R). Since F is Pareto
optimal, there exists K ∈ 2N such that K = ∅, and K ∈ βR (a, b). Assume
without loss of generality that K = MR (a, b).
Now construct a new preference profile R such that for any j ∈ N \ K,
L(Rj , a) = A, and for any c = a, L(Rj , c) \ {a} = L(Rj , c) \ {a}, moreover
let for any i ∈ K, Ri = Ri . We claim that a ∈ F (R ), suppose not; i.e.
a ∈ F (R ). Take any c ∈ A, and consider MR (a, c), clearly we have MR (a, c)
⊂ K, andMR (a, c) = MR (a, c) ∩ K, as RK = RK . Let K ∈ 2N such that
¯ ¯
K = MR (a, c) ∩ (N \ K); i.e. K is the maximal subcoalition in N \ K which
prefers c to a under R, it is clear that K ∪ MR (a, c) ∈ βR (a, c). Now, a ∈
(A, γ, βR ) implies γ(a, c)∩βR (a, c) = ∅ hence K ∪MR (a, c) ∈ γ(a, c) implies
K ¯
¯ ∪ MR (a, c) is not an (a, c)-oligarchy, thus we get K is an non-pivotal
(a, c, R )-oligarchy. This implies that, there exists K ∈ C M N P O (a, c, R )
¯ ˜
such that K ⊂ K. Now we have shown that; for any c ∈ A, there exists
K ˜
˜ ∈ C M N P O (a, c, R ) such that MR (a, c) ⊂ MR (a, c) ∪ K. Thus by POM
we can say that a ∈ F (R), contradicting that a ∈ F (R). Hence we can
conclude that a ∈ F (R ).
Let preference profile, R be such that for any j ∈ N \ K, Rj = Rj , and
for any i ∈ K, L(Ri , a) = A\{b}, and for any c ∈ A\{a, b}, L(Ri , c)\{a, b} =
L(Ri , c) \ {a, b}. We claim that; a ∈ F (R ), assume contrary; i.e. a ∈ F (R ).
Now, take any c ∈ A \ {a, b}, we have MR (a, c) = ∅. Let K be such that
K = MR (a, c) ∩ K, note that by construction of R we have; MR (a, c) =
MR (a, c)∩K, and clearly K ∈ βR (a, c). Now a ∈ (A, γ, βR ) implies γ(a, c)∩
βR (a, c) = ∅ implies K ∪MR (a, c) = K ∪∅ = K ∈ γ(a, c) indicating K is not
an (a, c)-oligarchy, so K is an non-pivotal (a, c, R )−oligarchy. This implies
˜ ˜
there exists K ∈ C M N P O (a, c, R ) such that K ⊂ K. Moreover if c = b, we
have MR (a, b) = K = MR (a, b) implies there exists K ∈ C M N P O (a, b, R )
such that ∅ ⊂ K. ˜
˜ Thus for any c ∈ A, there exists K ∈ C M N P O (a, c, R ) such
that MR (a, c) ⊂ MR (a, c) ∪ K by POM, implies a ∈ F (R ), contradicting
that a ∈ F (R ). Hence we can conclude a ∈ F (R ).
/ /
Now we know that; a ∈ F (R ) where K = MR (a, b) ; i.e. K is the
largest coalition which prefers b to a under R , moreover for any R ∈ L(A)N
such that bR ˜
˜ K a, we clearly have; for any i ∈ N , L(Ri , a) ⊂ L(Ri , a)
combined with monotonicity which is known to be implied by POM from
Lemma 1 shows that a ∈ F (R) indicating that K is an (a, b)-oligarchy, thus
K ∈ γ(a, b) implies K ∈ γ(a, b) ∩ βR (a, b), with K = ∅ we can say that
γ(a, b) ∩ βR (a, b) = ∅, contradicting a ∈ (A, γ, βR ). Hence we can conclude
that; a ∈ F (R), indicating; (A, γ, βR ) ⊂ F (R).
7 (A, γ )-implementation and Sen’s liberal
In this section, we consider Sen’s paradox of the Paretian liberal from the
(A, γ)-implementation perspective that we have introduced in section 3. We
show that; we can design codes of rights that are consistent with Sen’s min-
imal liberalism, and Pareto optimality. Finally we revisit Sen’s conclusion
of impossibility of a Paretian liberal in terms of (A, γ)-implementability. To
establish the desired result we first introduce the familiar definitions used by
Sen, under the general framework that is described in section 2.
Definition 9 Any social choice rule F satisfies minimal liberalism if there
exist {i, j} ⊂ N such that i = j, and for any l ∈ {i, j}, there exist xl , y l ∈ A
such that for any R ∈ L(A)N , xl Rl y l implies y l ∈ F (R), and respectively
y l Rl xl implies xl ∈ F (R).
Minimal liberalism implies that there are at least two individuals such
that for each of them there are at least a pair of alternatives (x,y) over
which he is decisive, that is whenever he prefers x to y, y is not chosen, and
respectively whenever he prefers y to x, x is not chosen. In other words
any social choice rule F satisfies minimal liberalism if there are at least two
individuals {i, j} ⊂ N such that i = j, where for each of them there are at
least a pair of alternatives (xi ,yi ), (xj ,yj ) such that i is an (xi ,yi )-oligarchy,
and j is an (xj ,yj )-oligarchy. Moreover, let us characterize minimal liberalism
in terms of codes of rights.
Definition 10 Any code of rights γ is said to satisfy minimal liberalism,
and denoted by γ L , if
(3) There exist {i, j} ⊂ N such that i = j, and for any l ∈ {i, j} there
exists xl , y l ∈ A such that for any K ∈ 2N , K ∈ γ(xl , y l ) or K ∈ γ(y l , xl ) if
and only if l ∈ K holds.
Now, let us show that for any social choice rule F , being (A, γ L )-
implementable that is; having code of rights which satisfies minimal lib-
eralism and which implements F , implies F satisfies minimal liberalism.
Lemma 3 Any (A, γ L )-implementable social choice rule F satisfies minimal
Proof. Let F be an (A, γ L )-implementable social choice rule then there is
a code of rights,γ, which implements F and satisfies (3) implies there exist
{i, j} ⊂ N such that i = j ,and for any l ∈ {i, j} there exist xl , y l ∈ A
such that for any R ∈ L(A)N such that xl Rl y l , [{l} ∈ γ L (xl , y l ) ∩ βR (xl , y l )]
implies y l ∈ (A, γ L , βR ), thus y l ∈ F (R) as F is (A, γ L )-implementable.
Similarly for any R ∈ L(A)N such that y l Rk xl , {l} ∈ γ L (y l , xl ) ∩ βR (y l , xl )
implies xl ∈ (A, γ, βR ), so xl ∈ F (R) indicating that F satisfies minimal
Moreover, via Lemma 2 it can easily be shown that; any social choice rule
F which is (A, γ )-implementable, and which satisfies minimal liberalism is
indeed (A, γ L )-implementable.
Definition 11 Any code of rights γ is said to satisfy Pareto optimality, and
denoted by γ P , if for any a, b ∈ A such that a = b, N ∈ γ(a, b).
Lemma 4 Any (A, γ P )-implementable social choice rule F satisfies Pareto
Proof. Assume not; i.e. F is (A, γ P )-implementable, but F is not Pareto
optimal implies there exists R ∈ L(A)N , and there exist a, b ∈ A such that
a ∈ F (R), for any i ∈ N bRi a implies N ∈ βR (a, b) thus N ∈ βR (a, b) ∩
γ(a, b) indicating a ∈ (A, γ, βR ) this implies that a ∈ F (R) as F is (A, γ
)-implementable, contradicting a ∈ F (R).
Now we can state the theorem indicating impossibility of a Paretian lib-
eral, in terms of (A, γ)-implementability.
Theorem 4 There is no non-empty valued social choice rule F which is
(A, γ P L )-implementable [i.e implementable by a γ, which satisfies minimal
liberalism, and Pareto optimality].
Proof. Assume not; i.e. there is a non-empty valued social choice rule F such
that for any R ∈ L(A)N , and F is (A, γ P L )-implementable for N = {1, 2}
implies (3) that is; there exist x, y, z, w ∈ A such that for any K ∈ 2N , K ∈
γ(x, y) or K ∈ γ(y, x) if and only if 1 ∈ K and K ∈ γ(z, w) or K ∈ γ(w, z)
if and only if 2 ∈ K holds. Now, if (x, y) = (z, w), then let A = {x, y},
and consider R such that xR1 y, yR2 x, implies {1} ∈ βR (x, y) ∩ γ(x, y), and
{2} ∈ βR (y, x) ∩ γ(y, x) implies (A, γ, βR ) = ∅, hence we getF (R) = ∅,
contradicting F being non-empty valued. Assume without loss of generality,
x = z, and y = w. Now for A = {x, y, w} consider R given below, note that
only Pareto optimal outcomes are x, y, this implies (A, γ, βR ) ⊂ {x, y}.
x y
y w
w x
However, {2} ∈ βR (x, w) ∩ γ(x, w) implies x ∈ (A, γ, βR ); {1} ∈ βR (y, x) ∩
γ(y, x) implies y ∈ (A, γ, βR ), so (A, γ, βR ) = ∅, but F is non-empty
valued, contradicting F is (A, γ P L )-implementable.
Now if x,y,z,w are all distinct then consider R given below, again note
that only Pareto optimal outcomes are w, y implies (A, γ, βR ) ⊂ {w, y}
w y
x z
y w
z x
However, {2} ∈ βR (w, z) ∩ γ(w, z) implies w ∈ (A, γ, βR ); {1} ∈ βR (y, x) ∩
γ(y, x) implies y ∈ (A, γ, βR ), thus (A, γ, βR ) = ∅, but F is non-empty
valued, contradicting F is (A, γ P L )-implementable.
8 Conclusion
In this paper we introduced the notion of (A, γ)-implementation, and pro-
vided a characterization in terms of Pareto optimality, and pivotal oligarchic
monotonicity. (A, γ)-implementation differs from classical implementation
mainly in two respects: (i) In (A, γ)-implementation, we explicitly specify a
rights structure among the members of the society, which is independent of
their preferences, where outcomes are determined as a result of this rights
structure and preferences. (ii) In classical implementation we deal with gen-
eral strategy sets whereas in (A, γ)-implementation we choose the strategy
set being equivalent to the alternative set, which leads to a rather simple
Our work in this paper also paves the way for the analysis of (S, γ)-
implementation, and its characterization. Moreover, identifying the rela-
tion between implementation under other solution concepts, and (A, γ)-
implementation are other subjects for further research.
[1] Arrow, K.J., Values and collective decision-making. In: Laslett p, Runci-
man WG (eds) Philosophy, politics, and society, Third Series. Basil Black-
well, Oxford, pp 215-232.
[2] Danilov, V., Implementation via Nash Equilibrium. Econometrica, 60
(1992), 43-56.
[3] Maskin, E., Nash Equilibrium and Welfare Optimality. Review of Eco-
nomic Studies, 66 (1998), 23-38.
[4] Peleg, B., Effectivity functions, game forms, games, and rights. Social
Choice and Welfare, 15 (1998) 67-80.
[5] Kaya,A., Two Essays on Social Choice Theory. Master’s Thesis. Bilkent
University, Ankara, 2000.
[6] Sen, A., The Impossibility of a Paretian Liberal. Journal of Political
Economy, 78 (1970) 152-157.
[7] Sertel, R.M., Designing Rights: Invisible Hand Theorems, Covering and
Membership. Mimeo: Bogazici University
|
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Topic: How can we teach kids when we can't teach teachers?
Replies: 2 Last Post: Sep 18, 2002 11:47 PM
Messages: [ Previous | Next ]
How can we teach kids when we can't teach teachers?
Posted: Apr 14, 1999 2:35 AM
I come at this from a little different perspective than I've seen in
the many messages I have read on this forum. I am making a career
change into teaching after 20 years in the private sector working as
an actuary. I am in California so I have to take a year's worth of
education courses plus do student teaching to get my credential.
Anyone care to guess what percent of my classwork will involve
learning anything about teaching math? Anyone care to guess what
percentage of my coursework will be done with a professor who has ever
taught math? If you guessed 0 you would be right on.
Instead what I am faced with is 5 courses; an introductory class of
mixed elementary and secondary candidates, two secondary methods
courses, a reading course and a multicultural course. To my mind this
is just crazy.
The intro course was moderately worthwhile. We watched a bunch of
Harry Wong's video series and learned about things like reporting
requirements in cases of suspected abuse. The actual content of the
course could have easily fit into an afternoon.
The first methods course was disappointing, particularly when I
realize it was 50% of the methods education I was going to receive.
Absolutely nothing specific to math, of course. All of the math
specific content was supposed to come from studying the CA Math
framework, whcih really has little about *how* to teach math. Then
the crowning finish was teaching a 30-minute mini-class (and having to
watch the lessons of the 15 other students). And the only feedback we
got on the lessons was from each other, the prof being a big believer
in peer review even when it was not at all establshed that the peers
in question (me included) would recognize the difference between a
good lesson and a KFC commercial.
My current course is the reading course. The biggest part of this
course is doing 25 hours of one-on-one reading tutoring of a secondary
student and submitting audio tales and writeups of the sessions. Can
any of you working math teachers out there explain to me the relevance
of this to teaching a class of 30-35 kids math? I mean, come on. If
the kid hasn't learned to read after 6 to 8 years in school, 4-5 of
which were heavily focused on reading, how am I supposed to teach him
to read en passant of teaching him math in the 45 minutes a day I have
him? It sure sounds good, though, "every teacher is a reading
teacher". And 20% of my teacher training is spent on this.
I don't know what will come in the second methods class. I understand
that we will put together a unit, but unless by some (unlikely)
miracle the prof is a math person I suspect it will be just like the
first methods class, i.e., no real training on teaching *math*. So
what's the deal? Is teaching supposed to be like sales? You know
they say that a good salesman can sell anything. Am I supposed to
believe that a good teacher can teach anything? Are the techniques of
teaching phys ed. the same as the techniques of teaching math? *Are
there* techniques of teaching math? Is so they are kept in secret in
my program.
And then there is the multicultural class. This smacks a little too
much of a bow to the forces of political correctness for me to warm up
to the course. But even if it is very worthwhile, should it assume a
higher priority than giving me a course in which they will give me
some actual training in how to teach math?
Maybe my years in the private sector have made me too practical. When
I had to train people I trained them in the things they were going to
be doing. Education school seems to make great effort to *avoid*
teaching me about the thing I will be doing.
Maybe it is all for the best. I'll probably end up teaching my
students math the same way I was taught, since I'm not being trained
to do anything different. But the real irony is that all of the
things with which we are supposed to infuse our lessons, e.g.,
relevance, motivation, real world applications, guided practice, etc,
are completely lacking in the training I am getting. Maybe the old
witticism needs to be revised to "those who can't, train teachers".
Rich Bednarski
Date Subject Author
4/14/99 How can we teach kids when we can't teach teachers? Rich Bednarski
4/23/99 Teachers teaching teachers to teach what? James R. Frysinger
9/18/02 Re: How can we teach kids when we can't teach teachers? Nizam
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November 6th 2012, 09:35 AM #1
Junior Member
Dec 2009
Suppose p is a polynomial of degree 3 with three zeros at x=1, x=2, and x=3. Find p if p(x) evaluated at 5 is 48.
Re: Polynomials
Since we know that 1,2,3 are roots, we can write that:
$P(5)=a(5-1)(5-2)(5-3)=24a=48$ -> $a=2$.
Now distribute 2 in $P(x)=a(x-1)(x-2)(x-3)$:
So $P(x)=2x^3-12x^2+22x-12$.
I hope you understand now.
Re: Polynomials
Thank you so much! Huge help!!
November 6th 2012, 09:50 AM #2
November 6th 2012, 12:17 PM #3
Junior Member
Dec 2009
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Power Balancing of Inline Multicylinder Diesel Engine
Advances in Mechanical Engineering
Volume 2012 (2012), Article ID 937917, 9 pages
Research Article
Power Balancing of Inline Multicylinder Diesel Engine
^1Department of Mechanical Engineering, M. E. Society's College of Engineering, Pune 411001, India
^2Department of Mechanical Engineering, Government College of Engineering, Pune 411005, India
Received 26 July 2012; Accepted 3 October 2012
Academic Editor: Mehdi Ahmadian
Copyright © 2012 S. H. Gawande et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
In this work, a simplified methodology is presented for power balancing by reducing the amplitude of engine speed variation, which result in excessive torsional vibrations of the crankshaft of inline
six-cylinder diesel engine. In modern fuel injection systems for reciprocating engines, nonuniform cylinder-wise torque contribution is a common problem due to nonuniform fuel supply due to a defect
in fuel injection system, causing increased torsional vibration levels of the crankshaft and stress of mechanical parts. In this paper, a mathematical model for the required fuel adjustment by using
amplitude of engine speed variation applied on the flywheel based on engine dynamics is suggested. From the found empirical relations and FFT analysis, the amplitude of engine speed variation (i.e.,
torsional vibration levels) of the crankshaft of inline six-cylinder diesel engine genset can be reduced up to 55%. This proposed methodology is simulated by developing MATALB code for uniform and
nonuniform working of direct injection diesel engine of SL90 type manufactured by Kirloskar Oil Engine Ltd., Pune, India.
1. Introduction
The internal combustion engine plays an important role in our society as means for transforming liquid and gaseous fuels to other more useful energy forms. Internal combustion engines are used in
applications ranging from automotive to power generation. One of the drawbacks with combustion of fossil fuels is the emissions of carbon oxides (), nitrogen oxides (), and sulphur oxides (). The
emissions of these chemical compounds from mainly power generation industry and automotive vehicles have been in focus during the last 40 years and have been one of the main drivers of the
development of internal combustion engines. With the advances in electronics and digital technologies in the 1970s, it became feasible to electronically control the fuel injections to increase the
fuel combustion efficiency and at the same time reduce emissions. The problem with cost efficient electronic fuel-injection systems is the need of periodic calibration of the cylinder-wise fuel
injections. Without calibration, the amounts of fuel injected into the cylinders deviate significantly. In diesel engines, fuel can be delivered in several distinct injection pulses. The injection
timing and quantity of each injection is important to provide better control of the combustion process. Depending on the operating conditions, different injection strategies are used; therefore, the
ability to distinguish between fueling imbalances is important. The three most prominent approaches to estimate cylinder imbalance are based on three different feedback measurement variables: exhaust
oxygen concentration, in-cylinder pressure, and crankshaft speed. The exhaust oxygen concentration-based approaches focus on estimation of air-fuel ratio imbalances, whereas in-cylinder pressure and
the crankshaft speed-based approaches focus on estimated torque (fuel) imbalances. Due to these reasons in modern fuel injection systems for reciprocating engines, nonuniform cylinder-wise torque
contribution is a common problem, causing increased torsional vibration levels of the crankshaft and stress of mechanical components. Engine balancing is the process of tuning an engine so that all
of its cylinders produce the same amount of power for a given load. Therefore in this work the focus is on correcting power imbalance due to misfuel/misfire in medium-speed power plant engine.
The control objective of automotive cylinder-balancing methods is usually to equalize the cylinder-wise torque contributions [1–3]. Given a rigid crankshaft, this control objective is equal to
minimizing the torsional vibrations of the crankshaft. In practice, the crankshaft can only be assumed rigid for lower vibratory frequencies, which consequently limits the types of engines that can
be balanced. In addition, considering pump and valve torques, and misalignments and unbalances in the rotating system, it follows that a minimization of torsional vibration level implies more uniform
cylinder torque contributions [4].
Compared to automotive applications, there is a set of specific problems which needs to be addressed for medium-speed engines. First of all, to reconstruct the superposed oscillating gas torque from
angular speed measurements of medium-speed engines, the dynamic influence of the flexible coupling and load needs to be considered [5]. Secondly, the dynamics of flexible couplings include
uncertainties, which for some coupling types may significantly deteriorate the accuracy of the reconstructed gas torque. To ensure the performance, these uncertainties should be taken into account.
Thirdly, for engines with many cylinders, frequencies need to be analyzed for which the crankshaft can no longer be assumed rigid. In order to accurately balance the engine, means for considering the
dynamics of the crankshaft have therefore to be developed. Given that the engine is decoupled from the load for the set of torque frequencies generated by the fuel combustions, the superposed
oscillating gas torque can be calculated from measurements of the angular speed of the flywheel, that is, it can be assumed that the load torque affecting the engine is more or less constant. Due to
the lower rotational speed of medium-speed engines, the lower torque-order excitations from the fuel combustions are usually in the vicinity of the lowest resonant frequencies of the crankshaft
system. As a consequence, the dynamic influence of the flexible coupling and load on the engine needs to be considered [6] in order to reconstruct the superposed oscillating gas torque correctly. The
dynamics of flexible couplings include uncertainties which for some coupling types need to be considered [5]. In addition, the dynamics also depend on many different factors such as age, temperature,
and vibratory frequencies. These uncertainties may induce significant errors in the estimated oscillating gas torque which affects the overall performance of the cylinder balancing. The problem of
relating the calculated superposed oscillating torque to the consecutive cylinder firings is simplified if the crankshaft can be assumed rigid for the considered set of frequencies. For medium-speed
engines, this assumption is generally valid only for lower vibratory orders. As a consequence, cylinder balancing using the assumption of an inflexible crankshaft can only be used for engines with
few cylinders. Some methods have been suggested which take into account the dynamics of the crankshaft [7–9]. The usual approach is to increase the order of the lumped mass-spring model in order to
capture the fundamental dynamics of the crankshaft. For engines where not all parameters are sufficiently known, off-line parameter identification methods have been proposed [10]. For manufacturers
with a large engine portfolio and many various types of installations, the parameter management becomes a problem, making the study of methods for online parameter estimation well motivated.
According to Kiencke and Nielsen [11], the demanded and the actual amounts of fuel injected in the cylinders differ up to 25%, due to different characteristics of the fuel injectors [12], varying
pressure differences between the rail and the cylinders, clogging of injector nozzles, [13], and so forth. This deteriorates the control of fuel injections and results in variations from average
torque contributions of the various cylinders. The varying torque applied on the crankshaft causes increased torsional vibrations, imposing increased stresses and inevitable wear of mechanical
components. The problem is therefore to adjust the cylinder-wise fuel injections so that the torque contributions can be balanced.
Balancing the cylinder-wise torque contributions of automotive and other high-speed engines was addressed by reconstructing the cylinder-wise net indicated torques [14–16] by direct use of the
angular acceleration [17, 18] or by reconstructing only the relative torque contributions of the cylinders [19]. Taraza et al. [8] suggested a method where the measured angular speed was directly
related to the nonuniform torque contribution. Hence, in order to determine the cylinder-wise net indicated torque, the oscillating torque applied on the flywheel is reconstructed from measurements
of the angular speeds of the crankshaft. Hence it is required to assume that the crankshaft is rigid and that the engine is sufficiently decoupled from the transmission and load and it is observed
that the oscillating torque can be reconstructed by using a single mass engine model and one angular speed measurement, [11, 14, 19, 20]. To obtain an estimate of the prevailing load torque, the
authors of [11] use the fact that the instantaneous torque from the engine is zero at top-dead center (TDC) and bottom-dead-center (BDC), whereas Rizzoni [14] use a linear relationship between the
root mean-square (RMS) value of the oscillating torque amplitude and net indicated torque applied on the flywheel. Instead of estimating the engine load, the authors of [7] proposed an observer-based
method, which uses the measured engine load torque directly to reconstruct the cylinder-wise net indicated torques of a 6-cylinder engine. Kim et al. [7] used an engine model for a genset, in which
the load was modeled with an additional mass. However, the dynamic link between the generator and grid was not included, although it has a significant impact on the dynamics. Moreover, engine models
which include flexible couplings generally assume that the stiffness and damping are constant [7, 11]. However, as the stiffness and damping of flexible couplings used in these kinds of generator
sets depend significantly on the vibration frequency, the nonlinearities of the couplings should be taken into account to accurately reconstruct the oscillating torque.
During this work, it is seen that when applying any cylinder fault/misfire/imbalance detection method on medium speed diesel engine, it is required to focus on three aspects: (a) the modeling of the
engine, (b) the modeling of the flexible coupling, and (c) the modeling of the load. As medium-speed engines have a nominal speed which is normally below 1500rpm, the excitation orders of the
cylinder-wise torque contributions are very close to the first natural frequency of the rotating system. This makes it necessary to include the dynamics of the load in the engine model. Therefore in
order to overcome the above stated deficiencies and to suggest alternative solution to the existing problems in current fuel injection system of diesel engine, the problem of cylinder balancing of
medium-speed internal combustion engines is investigated in this research with the objective of minimizing torsional vibrations due to engine speed variation by harmonic analysis. By analyzing the
gas torque of cylinder and angular speed of the crankshaft, fuel-injection adjustments are determined to minimize the variation in engine speed which results in reduction of torsional vibrations of
the engine crankshaft.
2. Problem Formulation and Objective
As per the past literature and industrial survey carried in engine manufacturing industry located in MIDC, Pune, namely, Greaves Cotton Ltd., and Kirloskar Oil Engine Ltd., in an internal combustion
reciprocating diesel engine, the quantity of fuel actually injected into each cylinder and at each injection may be different from the nominal fuel quantity requested by the electronic control unit
(ECU) which is used to determine the energizing time of the injector [3, 11, 12]. The energizing time of the injector depends on the dispersion and the time-drift variations of the injector’s
characteristics, due to the production process spread and aging of the injection system. In fact, the current injector production processes are not accurate [12] enough to produce injectors with
tight tolerances; moreover, these tolerances become worse with aging during the injector life-time. As a result, for a given energization time and a given rail pressure, the quantity of fuel actually
injected may be different from one injector to another. This difference in fuel injected quantity results in a cylinder-by-cylinder torque imbalance, causing some problems such as differences in
pressure peak, differences in heat release, and dynamic effects on a crankshaft which ultimately results in excessive torsional vibrations.
Hence in order to overcome the above stated disadvantages in current fuel injection system of diesel engine, following objectives were set to satisfy the purpose of engine power balancing:(i)tune the
engine to some nominal state specified by the vendor for performance and fuel consumption,(ii)balance the average power within each cylinder so as to minimize the engine vibration and stresses on the
engine components.
3. Experimental Setup and Measurements
In order to study the effect of detection of imbalance and balancing, the required experimental setup was developed as shown in Figure 1.
Figure 1 shows schematic layout of test setup developed for the measurement of engine speed to analyze and measure the variations in time and engine speed. Figure 1 shows position of a six-cylinder
engine, flywheel with alternator, gear wheel, sensor, and FFT spectrum analyzer. Crankshaft angular speed of internal combustion engines is usually measured by means of a gear or measurement disk and
a speed-pickup. As the gear wheel rotates the tooth on gear or mark passes the sensor, a step formed voltage is generated, called pulse train, which is used for calculating the angular speed. The
power balancing method proposed in this work uses the determination of the angular velocity at every edge of the gear wheel signal as it rotates. The crankshaft angle was measured at every ten
degrees, that is, when a new edge on the gear is sensed by sensor as shown in Figure 2. The measured speed responses in time domain as shown in Figure 4 and Figure 5 for uniform and non-uniform
engine operation are obtained to calculate angular velocity to decide the position of the crank shaft. The angular velocity is calculated, when a positive edge appears, using the differential
Equation (1) as follows: where is the known sector angle described by the set of pulses for which the engine speed is measured and is the measured time, and is the number of teeth on gear. The time
is measured by a digital timer set in FFT spectrum analyzer which is controlled by the zero crossings of pulse signal. Figure 2 shows the gear wheel mounted on crankshaft next to flywheel with
position of hall effect sensor when engine is in rotating position with speed of 1500rpm. Here speed was measured by digital display mounted on engine housing as well as digital tachometer. Figure 3
shows the measurement of engine speed in terms of the speed step response in time domain. Here FFT spectrum analyzer is used to plot the harmonic spectrum of speed step response corresponding to
engine harmonic order.
Figures 7 and 8 show the comparison of the speed signal for the normal operation and non-uniform operation when cylinder 5 is cut off, in Cartesian and polar coordinates, respectively. This shows
that average engine speed is 1501.359rpm for normal working and 1503.437rpm for misfuel in cylinder no. 5.
Figure 6 shows measured time for six-cylinder diesel engine for normal working and misfuel in cylinder no. 5.
Figure 7 illustrates a graph of engine speed (in rpm) versus crankshaft position (in degrees) for a considered six-cylinder engine over one complete engine cycle. It is seen that the actual
instantaneous speed of the engine varies significantly from its average speed (1500rpm) as each of the engine cylinders fires in turn (the peaks in the figure represent successive firings of the
engine cylinders). It is also seen that the peaks do not all lie at the same value, indicating different power contributions from each cylinder as it fires. A cylinder having a greater power
contribution will increase the engine speed to a higher level than the firing of cylinders having a lower power contribution. This work comprehends the use of fast Fourier transform (FFT), in order
to relate the data to engine order, using the engine crank angle as the independent variable. It is theoretically possible to calculate the cylinder power contribution of a six-cylinder engine from
the first three engine orders. When all FFT components are zero, the power contribution of the engine cylinders is equally balanced. Once the FFT calculation has been completed, then the focus is to
calculate the fuel adjustments to be applied to the fuel injection system for each cylinder in order to drive the FFT components to zero.
4. Development of Algorithm for Cylinder Power Balancing
In this section an algorithm for engine cylinder power balancing is explained which comprise the following steps:(1)measurement of instantaneous speed of the engine crankshaft during a working cycle
by speed pick up,(2)perform a fast Fourier transform (FFT) upon the sensed engine speed, thereby producing at least one Fourier transform component corresponding to the harmonic orders 0.5, 1, and
1.5 of the engine,(3)determine a cylinder power imbalance condition from a phase of the Fourier transform component,(4)balancing is carried by using predetermined adjustment selected based on the
observations in step (3). Figure 9 illustrates the relationship between each of the first three engine harmonic orders and common cylinder imbalance conditions which increase the magnitude of these
orders in the FFT results. As shown in Figure 9(b), contribution to power imbalance at the 0.5 order is primarily due to one cylinder imbalance (cylinder 1). This indicates that the relative fueling
is different for cylinders, causing the under or over fueled cylinder to produce substantially different power. Figure 9(c) illustrates cylinder bank-to-cylinder bank symmetric imbalances which
contribute primarily to the 1.0 order. Finally, Figure 9(d) illustrates cylinder bank-to-cylinder bank offsets, in which one bank of cylinders has substantially different fueling from the other
cylinder bank, which contributes primarily to the magnitude of the 1.5 order.
Figures 9(b), 9(c), and 9(d) illustrate the general shape of the FFT data for each of the first three engine orders. As seen in Figure 9(b), the 0.5 engine order results in a cosine wave having a
period of 720° crank degrees. The 1.0 engine order illustrated in Figure 9(c) is also a cosine wave, having a period of 360° degrees. Finally, the 1.5 engine order illustrated in Figure 9(d) is a
cosine wave having a period of 240° crank degrees.
For the Kirloskar six-cylinder engine, the cylinder firing order is 1-5-3-6-2-4. The cylinders firing instantaneous positions are shown in Figures 9(b), 9(c), and 9(d) to indicate which cylinder is
firing at the time the data was produced.
Figure 9(b) illustrates the 0.5 order component of the FFT when cylinder 1 is high (higher than average output power developed by the cylinders) or when cylinder 6 is low (lower than average output
power developed by the cylinders). The phase of the waveform in Figure 9(b) would be translated to the left or to the right if another engine cylinder was high or low. For example, the peak in the
waveform of Figure 9(b) would occur at 240° of crank angle if cylinder 4 was high or cylinder 3 was low. A substantially flat waveform for the 0.5 order component of the FFT indicates that no
substantial single cylinder imbalances are occurring within the engine. Hence in the present work, an attempt is made to apply the fuelling correction to the engine in order to iteratively drive the
0.5 order component of the FFT to zero to achieve a balanced condition.
Similarly, the presence of a 1.0 order component in the FFT, as illustrated in Figure 9(c), indicates that pair of cylinders on opposite banks of the engine is either high or low. For example, the
waveform of Figure 9(c) indicates that cylinder 1 and cylinder 6 of the engine are both high with respect to the average power developed by the cylinders. Similar to 0.5 order waveform, the phase of
the 1.0 order waveform shown in Figure 9(c) will be translated to the left or to the right when other pairs of cylinders are either high or low. The present work is therefore operative to make
changes in the fueling correction to the engine in order to iteratively drive the waveform of Figure 9(c) to zero which results in balanced state of the engine.
Figure 9(d) illustrates the 1.5 order FFT component, indicating bank-to-bank offsets in the engine. The waveform illustrated in Figure 9(d) indicates that cylinders 1, 2, and 3 are high, while
cylinders 4, 5, and 6 are low. The opposite condition in the engine (cylinders 1, 2, and 3 low and cylinders 4, 5, and 6 high) will produce a 120° phase shift in the waveform of Figure 9(d). The
presence of a 1.5 order component in the FFT data needs to adjust the fueling correction to the engine to iteratively drive the 1.5 order component of the FFT to zero to achieve a balanced condition.
5. Determination of Fueling Corrections
Once the first three engine order components of the FFT are determined, the present work then utilizes this information in order to determine the fueling corrections to be applied to the fuel system
for each engine cylinder. Therefore a matrix is defined for the fueling corrections for a 6-cylinder engine as follows: where is fueling correction for cylinder 1, is fueling correction for cylinder
2, is fueling correction for cylinder 3, is fueling correction for cylinder 4, is fueling correction for cylinder 5, and is fueling correction for cylinder 6.
From Figure 9(b), it is seen that in order to correct the 0.5 order component when cylinder 1 is high, it will be necessary to reduce the fueling to cylinder 1 by a relatively great amount, increase
the fueling for cylinder 6 by relatively great amount, reduce the fueling to cylinders 4 and 5 by a relatively smaller amount, and increase the fueling to cylinders 2 and 3 by a relatively smaller
amount. These changes to the fueling of the cylinders will have a tendency to flatten out the waveform of Figure 9(b). Likewise, Figure 9(b) indicates that the fueling to cylinders 1 and 6 should be
reduced by a relatively greater amount, while the fueling to cylinders 2, 3, 4, and 5 should be increased by a relatively smaller amount. Again, this will have the tendency to flatten out the
waveform of Figure 9(c). Finally, Figure 9(d) indicates that the fueling to cylinders 1, 2, and 3 should be reduced by the same amount that the fueling to cylinders 4, 5, and 6 are increased by the
same amount. These changes to the fueling for the situations indicated in Figures 9(b), 9(c), and 9(d) may be expressed in matrix form as shown in equation (3) as follows: where “” is an iteratively
determined constant based on trial error method. Value of “” is iteratively determined for the engine and fuel system. Using the FFT waveform magnitude of fueling correction is determined. The
magnitude of fueling corrections for each of the cylinder is represented as shown in equation (3) by the factors 1 and 2 corresponding to minimum and maximum values, respectively.
For 6-cylinder medium speed engine the value of “” is selected and used for determinations of fueling correction as per equation (4) as follows: Further, by taking the summation of equation (3),
changes to the fueling may be expressed in matrix form as follows: As per the above discussion, it is seen that the following matrices are used to correct any combination of engine cylinder power
imbalance in a 6-cylinder engine.
Adjustments for 0.5 order components are as follows:Cyl.1 high or cyl.6 low:.
Similarly;cyl.2 high or cyl.5 low: ,cyl.3 high or cyl.4 low: ,cyl.4 high or cyl.3 low: ,cyl.5 high or cyl.2 low: ,cyl.6 high or cyl.1 low: .Adjustments for 1.0 Order Components: cyl.1, 6 high:
Similarly;cyl.2, 5 high: ,cyl.3, 4 high: ,cyl.3, 4 low: ,cyl.2, 5 low: ,cyl.1, 6 low: ,Adjustments for 1.5 Order Components: Cyl.1, 2, 3, high: Similarly;Cyl.4, 5, 6, high: Based on these
investigations the following condition is proposed.
Condition 1. For a well power-balanced six-cylinder 4-strokes (4-S) reciprocating diesel engine, the resultant of magnitude of the amplitudes of first three harmonic orders (0.5, 1, and 1.5) is
always zero.
Proof. A matrix is defined for the experimentally measured fueling corrections for a 6-cylinder engine (as per Figure 9) as follows: From the above investigation, it is seen that of sum of the
corrections of the FFT components of 0.5, 1.0, and 1.5 engine orders is found to be zero, zero, and zero which implies that for well power balanced engine the sum of magnitude of the amplitudes of
first three harmonic orders (0.5, 1, and 1.5) of four stroke reciprocating diesel engine is always zero.
6. Simulation Code for Uniform and Nonuniform Engine Operation
In order to validate and execute the calculated and simulated fuel adjustments in an operating six-cylinder diesel engine of SL90 type, MATLAB code was developed for uniform and nonuniform engine
operation. Here the focus is to observe the dynamic behavior of engine by simulating different parameters such as piston velocity, gas torque, mass torque, mass gravity torque due to piston weight
and reciprocating parts, total torque, and engine speed. The combined effect of gas torque, mass torque, and mass gravity torque on engine speed for uniform and non-uniform engine operation is
investigated and effect on lower engine order is noted and explained in details.
6.1. Comparison of FFT Waveform of Simulated Engine Speed for Six-Cylinder Engine Order
In this subsection, the comparison of FFT waveform of lower engine harmonic order (0.5, 1, 1.5) for uniform and nonuniform engine operation with experimental and simulated results is explained. A
closed match between experimental and simulated fueling correction is observed. From simulated results with engine nonuniform operation as shown in Figures 10, 11, and 12, it is observed that after
applying fueling correction, the level of torsional vibration reduces from 0.8 (1/s) to 0.5 (1/s) for 0.5 engine order, from 0.42 5(1/s) to 0.32 (1/s) for 1 engine order, and from 0.4 (1/s) to 0.18
(1/s) for 1.5 engine order.
7. Conclusion
The primary objective of this work was to get insight into how torsional vibrations due to engine speed variations play an important role in basic design calculations, performance diagnosis of
reciprocating internal combustion engine by detecting and correcting power imbalance in operating six-cylinder diesel engines. The objective was achieved with the help of extensive analytical work,
computer aided simulation tools, commercially available softwares, and experimental investigations. The new approach presented for power balancing to reduce the torsional vibrations due speed
variation can be effectively used for considered six-cylinder diesel engine. The use of Fourier transform of engine speed signal for one complete cycle (0–720°) can be effectively used to detect
power imbalance and balancing, as an imbalanced engine results in nonzero magnitudes and balanced engine results in zero (0) magnitudes of the FFT components that correspond to the 0.5, 1.0, and 1.5
engine orders. The compensation scheme suggested in this work uses the iteration to reduce the magnitudes of these components to zero (0) if the engine is detected to be imbalanced. From further
investigation, it is found that when all FFT components (of 0.5, 1.0, and 1.5 engine orders) are flatten nearer to zero (0), and the power contribution of the engine cylinders is said to be equally
balanced. To execute and validate the findings, simulation code is developed in MATLAB. From experimental work and simulated results, it is seen that the proposed fueling corrections for power
balancing are in good agreement. From simulated results with engine nonuniform operation, it is observed that after applying fueling corrections, the level of torsional vibration reduces from 0.8 (1/
s) to 0.5 (1/s) for 0.5 engine order, from 0.425 (1/s) to 0.32 (1/s) for 1 engine order, and from 0.4 (1/s) to 0.18 (1/s) for 1.5 engine order.
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Vibration, vol. 295, no. 1-2, pp. 141–164, 2006. View at Publisher · View at Google Scholar · View at Scopus
14. G. Rizzoni, “Estimate of indicated torque from crankshaft speed fluctuations: a model for the dynamics of the IC engine,” IEEE Transactions on Vehicular Technology, vol. 38, no. 3, pp. 168–179,
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675–686, 1994. View at Scopus
16. G. Rizzoni, S. Drakunov, and Y. Y. Wang, “On-line estimation of indicated torque in IC engines via sliding mode observers,” in Proceedings of the American Control Conference, pp. 2123–2127,
Seattle, Wash, USA, June 1995. View at Scopus
17. F. Kimmich, A. Schwarte, and R. Isermann, “Fault detection for modern diesel engines using signal- and process model-based methods,” Control Engineering Practice, vol. 13, no. 2, pp. 189–203,
2005. View at Publisher · View at Google Scholar · View at Scopus
18. J. Yang, L. Pu, Z. Wang, Y. Zhou, and X. Yan, “Fault detection in a diesel engine by analysing the instantaneous angular speed,” Mechanical Systems and Signal Processing, vol. 15, no. 3, pp.
549–564, 2001. View at Publisher · View at Google Scholar · View at Scopus
19. H. Straky, Fault detection and fault correction for cylinder balancing in a common-rail diesel engine [M.S. thesis], University of Karlsruhe, 1997.
20. S. H. Gawande, L. G. Navale, M. R. Nandgaonkar, D. S. Butala, and S. Kunamalla, “Fault detection of inline reciprocating diesel engine: a mass and gas torque approach,” Advances in Acoustics and
Vibration, vol. 2012, Article ID 314706, 6 pages, 2012. View at Publisher · View at Google Scholar
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Affine transformation
Affine transformation
From Wiki.GIS.com
This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more
precise citations where appropriate. (March 2009)
In geometry, an affine transformation or affine map or an affinity (from the Latin, affinis, "connected with") between two vector spaces (strictly speaking, two affine spaces) consists of a linear
transformation followed by a translation:
$x \mapsto A x+ b.$
In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, satisfying certain properties described below.
Geometrically, an affine transformation in Euclidean space is one that preserves
1. The collinearity relation between points; i.e., three points which lie on a line continue to be collinear after the transformation
2. Ratios of distances along a line; i.e., for distinct collinear points p[1], p[2], p[3], the ratio | p[2] − p[1] | / | p[3] − p[2] | is preserved
In general, an affine transformation is composed of linear transformations (rotation, scaling or shear) and a translation (or "shift"). Several linear transformations can be combined into a single
one, so that the general formula given above is still applicable.
[edit] Representation of affine transformations
Ordinary vector algebra uses matrix multiplication to represent linear transformations, and vector addition to represent translations. Using an augmented matrix, it is possible to represent both
using matrix multiplication. The technique requires that all vectors are augmented with a "1" at the end, and all matrices are augmented with an extra row of zeros at the bottom, an extra column —
the translation vector — to the right, and a "1" in the lower right corner. If A is a matrix,
$\begin{bmatrix} \vec{y} \\ 1 \end{bmatrix} = \begin{bmatrix} A & \vec{b} \ \\ 0, \ldots, 0 & 1 \end{bmatrix} \begin{bmatrix} \vec{x} \\ 1 \end{bmatrix}$
is equivalent to the following
$\vec{y} = A \vec{x} + \vec{b}.$
This representation exhibits the set of all invertible affine transformations as the semidirect product of K^n and GL(n, k). This is a group under the operation of composition of functions, called
the affine group.
Ordinary matrix-vector multiplication always maps the origin to the origin, and could therefore never represent a translation, in which the origin must necessarily be mapped to some other point. By
appending a "1" to every vector, one essentially considers the space to be mapped as a subset of a space with an additional dimension. In that space, the original space occupies the subset in which
the final index is 1. Thus the origin of the original space can be found at (0,0, ... 0, 1). A translation within the original space by means of a linear transformation of the higher-dimensional
space is then possible (specifically, a shear transformation). This is an example of homogeneous coordinates.
The advantage of using homogeneous coordinates is that one can combine any number of affine transformations into one by multiplying the matrices. This device is used extensively by graphics software.
[edit] Properties of affine transformations
An affine transformation is invertible if and only if A is invertible. In the matrix representation, the inverse is:
$\begin{bmatrix} A^{-1} & -A^{-1}\vec{b} \ \\ 0,\ldots,0 & 1 \end{bmatrix}$
The invertible affine transformations form the affine group, which has the general linear group of degree n as subgroup and is itself a subgroup of the general linear group of degree n + 1.
The similarity transformations form the subgroup where A is a scalar times an orthogonal matrix. If and only if the determinant of A is 1 or –1 then the transformation preserves area; these also form
a subgroup. Combining both conditions we have the isometries, the subgroup of both where A is an orthogonal matrix.
Each of these groups has a subgroup of transformations which preserve orientation: those where the determinant of A is positive. In the last case this is in 3D the group of rigid body motions (proper
rotations and pure translations).
For any matrix A the following propositions are equivalent:
• A – I is invertible
• A does not have an eigenvalue equal to 1
• for all b the transformation has exactly one fixed point
• there is a b for which the transformation has exactly one fixed point
• affine transformations with matrix A can be written as a linear transformation with some point as origin
If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation. This may make it easier to classify and understand the transformation. For
example, describing a transformation as a rotation by a certain angle with respect to a certain axis is easier to get an idea of the overall behavior of the transformation than describing it as a
combination of a translation and a rotation. However, this depends on application and context. Describing such a transformation for an object tends to make more sense in terms of rotation about an
axis through the center of that object, combined with a translation, rather than by just a rotation with respect to some distant point. As an example: "move 200 m north and rotate 90°
anti-clockwise", rather than the equivalent "with respect to the point 141 m to the northwest, rotate 90° anti-clockwise".
Affine transformations in 2D without fixed point (so where A has eigenvalue 1) are:
• pure translations
• scaling in a given direction, with respect to a line in another direction (not necessarily perpendicular), combined with translation that is not purely in the direction of scaling; the scale
factor is the other eigenvalue; taking "scaling" in a generalized sense it includes the cases that the scale factor is zero (projection) and negative; the latter includes reflection, and combined
with translation it includes glide reflection.
• shear combined with translation that is not purely in the direction of the shear (there is no other eigenvalue than 1; it has algebraic multiplicity 2, but geometric multiplicity 1)
[edit] Affine transformation of the plane
To visualise the general affine transformation of the Euclidean plane, take labelled parallelograms ABCD and A′B′C′D′. Whatever the choices of points, there is an affine transformation T of the plane
taking A to A′, and each vertex similarly. Supposing we exclude the degenerate case where ABCD has zero area, there is a unique such affine transformation T. Drawing out a whole grid of
parallelograms based on ABCD, the image T(P) of any point P is determined by noting that T(A) = A′, T applied to the line segment AB is A′B′, T applied to the line segment AC is A′C′, and T respects
scalar multiples of vectors based at A. [If A, E, F are colinear then the ratio length(AF)/length(AE) is equal to length(A′F′)/length(A′E′).] Geometrically T transforms the grid based on ABCD to that
based in A′B′C′D′.
Affine transformations don't respect lengths or angles; they multiply area by a constant factor
area of A′ B′ C′ D′ / area of ABCD.
A given T may either be direct (respect orientation), or indirect (reverse orientation), and this may be determined by its effect on signed areas (as defined, for example, by the cross product of
[edit] Example of an affine transformation
The following equation expresses an affine transformation in GF(2) (with "+" representing XOR):
$\{\,a'\,\} = M\{\,a\,\} + \{\,v\,\},$
where [M] is the matrix
$\begin{bmatrix} 1&0&0&0&1&1&1&1 \\ 1&1&0&0&0&1&1&1 \\ 1&1&1&0&0&0&1&1 \\ 1&1&1&1&0&0&0&1 \\ 1&1&1&1&1&0&0&0 \\ 0&1&1&1&1&1&0&0 \\ 0&0&1&1&1&1&1&0 \\ 0&0&0&1&1&1&1&1 \end{bmatrix}$
and {v} is the vector
$\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}.$
For instance, the affine transformation of the element {a} = x^7 + x^6 + x^3 + x = {11001010} in big-endian binary notation = {CA} in big-endian hexadecimal notation, is calculated as follows:
$a_0' = a_0 \oplus a_4 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 1 = 0 \oplus 0 \oplus 0 \oplus 1 \oplus 1 \oplus 1 = 1$
$a_1' = a_0 \oplus a_1 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 1 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 1 = 0$
$a_2' = a_0 \oplus a_1 \oplus a_2 \oplus a_6 \oplus a_7 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1$
$a_3' = a_0 \oplus a_1 \oplus a_2 \oplus a_3 \oplus a_7 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1$
$a_4' = a_0 \oplus a_1 \oplus a_2 \oplus a_3 \oplus a_4 \oplus 0 = 0 \oplus 1 \oplus 0 \oplus 1 \oplus 0 \oplus 0 = 0$
$a_5' = a_1 \oplus a_2 \oplus a_3 \oplus a_4 \oplus a_5 \oplus 1 = 1 \oplus 0 \oplus 1 \oplus 0 \oplus 0 \oplus 1 = 1$
$a_6' = a_2 \oplus a_3 \oplus a_4 \oplus a_5 \oplus a_6 \oplus 1 = 0 \oplus 1 \oplus 0 \oplus 0 \oplus 1 \oplus 1 = 1$
$a_7' = a_3 \oplus a_4 \oplus a_5 \oplus a_6 \oplus a_7 \oplus 0 = 1 \oplus 0 \oplus 0 \oplus 1 \oplus 1 \oplus 0 = 1.$
Thus, {a′} = x^7 + x^6 + x^5 + x^3 + x^2 + 1 = {11101101} = {ED}.
[edit] See also
• The transformation matrix for an affine transformation
• Affine geometry
• Homothetic transformation
• Linear transformation (the second meaning is affine transformation in 1D)
• Flat (geometry)
[edit] External links
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Anna University Chennai Me 2301 — thermal engineering - november/december 2010 model question papers
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Posted Date: 13 Dec 2010 Posted By:: VEERAIYAN BOSE Member Level: Gold Points: 5 (Rs. 1)
2010 Anna University Chennai B.E Mechanical Engineering Me 2301 — thermal engineering - november/december 2010 Question paper
Course: B.E Mechanical Engineering University/board: Anna University Chennai
B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2010
Fifth Semester
Mechanical Engineering
ME 2301 — THERMAL ENGINEERING
(Regulation 2008)
Time : Three hours Maximum : 100 Marks
Answer ALL questions
PART A — (10 × 2 = 20 Marks)
1. Why is Carnot cycle not used in real applications?
2. Draw the P-V diagram for a dual cycle.
3. Draw the valve timing diagram for a CI engine.
4. What is the indicated power of four cylinder engine if BP with 4-cylinder
working is 18.75 kW and BP with 3-cylinder working is 13.06 kW.
5. What are the factors reducing the final velocity of steam in nozzle flow?
6. What is the difference between impulse and reaction turbine?
7. How is the inter cooler used to reduce the power consumption of compressor?
8. List the advantages of multistage compressor over single stage compressor.
9. What is dew point temperature?
10. Define the COP of refrigerators.
PART B — (5 × 16 = 80 Marks)
11. (a) Derive an expression for the air standard efficiency of Diesel cycle and
then deduce it for mean effective pressure. [Marks 16]
(b) A six cylinder four stroke petrol engine has a swept volume of 300 cubic
cm per cylinder, a compression ratio of 10 and operates at a speed of
35000 rpm. If the engine is required to develop an output of 73.5 kW at
this speed, calculate the cycle efficiency, the necessary rate of heat
addition, the mean effective pressure, maximum temperature of the cycle
and efficiency ratio. The pressure and temperature before is entropic
compression are 1.0 bar and 15°C respectively, take Cv = 0.72 and
? = 1.4. [Marks 16]
12. (a) (i) Explain the working principle of 4-stroke engine. (Marks 8)
(ii) With a neat diagram explain the working of battery ignition
system. (Marks 8)
(b) (i) Describe the working of Diesel fuel pump. (Marks 8)
(ii) Explain the pressure feed lubrication system with a neat diagram.
(Marks 8)
13. (a) In a steam nozzle, the steam expands from 4 bar to 1 bar. The initial
velocity is 60 m/s and initial temperature is 200°C. Determine the exit
velocity if the nozzle efficiency is 92% and the dryness fraction at exit. [Marks 16]
(b) A single row impulse turbine develops 132.4 kW at a blade speed of
175 m/s using 2 kg of steam per sec. Steam leaves the nozzle at 400 m/s.
Velocity coefficient of the blade is 0.9. Steam leaves the turbine blades
axially. Assuming no shock determine the nozzle angle, blade angles at
entry and exit. [Marks 16]
14. (a) A single acting 14 cm × 10 cm reciprocating compressor is operating at
P1 = 1 bar, T1 = 20°C, P2 = 6 bar and T2 = 180°C. The speed of compressor
is 1200 rpm and shaft power is 6.25 kW. If the mass of air delivered is
1.7 kg/min, calculate the actual volumetric efficiency, the indicated
power, the isothermal efficiency, the mechanical efficiency and the
overall efficiency. [Marks 16]
(b) A single stage reciprocating air compressor has clearance volume 5% of
stroke volume of 0.05 m3/sec. The intake conditions are 95 kN/m2, 300 K.
The delivery pressure is 720 kN/m2. Determine the volumetric efficiency
referred to (i) intake conditions (ii) atmospheric conditions of 100 kN/m2
and 290 K (iii) FAD and (iv) power required to drive the compressor, if
the ratio of actual to indicated power is 1.5. Take index of compression
and expansion as 1.3. [Marks 16]
15. (a) One kg of air at 35°C DBT and 60% RH is mixed with 2 kg of air at 20°C
DBT and 13°C dew point temperature. Calculate the vapour pressure and
dew point temperature of stream one, enthalpy of both the streams and
specific humidity of the mixture. [Marks 16]
(b) The temperature range in a Freon-12 plant is –6°C to 27°C. The
compression is is entropic and there is no cooling of the liquid. Find the
COP assuming that the refrigerant (i) after compression is dry and
saturated (ii) leaving the evaporator is dry and saturated. The properties
of F-12 are given in the table :
Sl.No. t°C hf hg sf sg Cp
1 –6 413 571 4.17 4.76 0.641
2 27 445 585 4.28 4.75 0.714 [Marks 16]
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Here's the question you clicked on:
Find mBAC in circle O. (The figure is not drawn to scale.)
• one year ago
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A. 170 B. 95 C. 47.5 D. 42.5
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do u go to connexus
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no i go to James Madison
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i was on the unit 6 test and was wondering if someone could help me with some problems
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im on exam 6 test to
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Oh, hey. Real simple. If you can figure out the measure of the arc that the inscribed angle subtends, then the angle will simply be: \[\Large \text{inscribed angle} = \frac{\text{measure of arc
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i dont know how to figure that out the arc -_-
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so it would be 85 .. then what do you do? can you show me
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naw gurl. it aint even.
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im confused ...
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I can tell.
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Helppp lol
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would it be 90? 95? I dont know lol
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I'm sorry. The whole guessing business just kills me. Stop guessing, stop begging for answers, and show some effort.
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i am i dont know how to do this, Im trying ..
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All of the facts you need: Two angles that form a straight line must add up to 180 degrees. An arc has the same measure as a central angle that includes it. An inscribed angle is half of the
measure of the arc it includes.
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|dw:1357910861134:dw| @ErinWeeks We should never guess in math, or else you will always have 25% or thereabouts for your grades! :( The above diagram shows the relation between the angles
subtended by the same chord AC, at the circumference B or at the centre O. If you join OB, then OA,OB,OC are all equal to the radius. Thus triangl OAB is isosceles, therefore mBAO=mABO. Similarly
mBCO=mCBO. But mABO+mCBO=mAOC (exterior angles), which means finally \( mAOC=2 * mABC.\) Hope this helps. Note: this is similar to the question I answered previously. Hope that this more detailed
explanation helps you work on other problems. Please, do NOT guess your answers if you want a good grade. Understanding is faster than guessing.
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Ralated rates
November 17th 2007, 07:45 PM
Ralated rates
A stone is dropped into a deep, dark mine. A clunk is heard 7 seconds later. Estimate the depth of the shaft in feet. Ignore air resistance. Take the speed of sound as 1000 feet/second.
Velocity of sound= 1000 ft/sec
d = 16t^2
can anyone help me to solve this question?
Thank you.
November 17th 2007, 08:01 PM
A stone is dropped into a deep, dark mine. A clunk is heard 7 seconds later. Estimate the depth of the shaft in feet. Ignore air resistance. Take the speed of sound as 1000 feet/second.
Velocity of sound= 1000 ft/sec
d = 16t^2
can anyone help me to solve this question?
Thank you.
see here
it is a very similar problem, do you understand it?
November 17th 2007, 08:41 PM
Thank you
I get it now, i just need to solve for s.
$<br /> \boxed{ \frac{\sqrt{s}}{4}+ \frac{s}{1000} = 7 }<br />$
November 17th 2007, 08:44 PM
i hope you really understood what happened and didn't just plug in the relevant numbers into TPH's equation.
could you, for instance, tell me how he came up with the $\frac {\sqrt{s}}4$?
November 17th 2007, 08:59 PM
s = 16 t^2
t= (s/16)^1/2
$t=\frac {\sqrt{s}}4$
November 17th 2007, 09:06 PM
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Intuition behind Pincus' "injectively bounded statements"
up vote 6 down vote favorite
David Pincus, Zermelo-Fraenkel Consistency Results by Fraenkel-Mostowski Methods, The Journal of Symbolic Logic, Vol. 37, No. 4 (Dec., 1972), pp. 721-743
Pincus introduces the notion of injectively bounded statements, which he proves are sentences which can be transferred from a (permutation) model of ZFA to a (symmetric) model of ZF. I have no
intuition for what these statements are, and he only gives a couple of examples (allow me to ignore the special case of projectively bounded statements, I am after generality here).
I would like, if possible, a more structural explanation of what it means for a statement to be injectively bounded, rather that something that looks like a mess of codings via ordinals.
lo.logic set-theory independence-results
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1 Answer
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(Note: this isn't something I really know, so this might be wildly off base.)
To start with, let's look at a weaker transfer principle: the Jech-Sochor Embedding Theorem.
Jech-Sochor says that sentences depending only on a bounded amount of the cumulative hierarchy above the set $A$ of atoms can be "passed over" to a model of genuine ZF. More precisely,
Jech-Sochor states that
Let $\gamma$ be a fixed ordinal, and $V$ a model of $ZFA$ with a set $A$ of atoms. Then there is a model $W$ of $ZF$ and an embedding $i: V\rightarrow W: x\mapsto \tilde{x}$ such
that $(P_\alpha(A)^V, \in^V)\cong (P_\alpha(\tilde{A})^W, \in^W)$.
So statements of "fixed depth" can be transferred. For example, "non-well-orderability" can be preserved by setting $\gamma=\omega+2$ (really, we just care about the powerset, but we
also want to talk about maps from $\omega$ so we need to go up $\omega+1$ many levels to get $\omega$ into $P_\gamma(A)$, and then one more level to get the desired maps).
In Pincus, this property of the truth of $\phi(X)$ only depending on some fixed level of the cumulative hierarchy over $X$ is called boundability; so, for example, on page 722 Pincus
phrases the Jech-Sochor theorem as:
"A boundable statement is transferable."
The question is whether we can improve this result to transfer statements that don't necessarily depend just on $P_\gamma(X)$ for some fixed $\gamma$, but are still "locally determined"
in some sense. This "locally determined" is his condition that
$\vert x\vert\le\sigma(y)$
in a (sur/in)jectively boundable statement. So now we've switched from caring about the number of powersets required to reach a set, to caring about its "cardinality" being small. Note,
though, that the bound on the size of $x$ itself must be boundable, so the idea of Jech-Sochor isn't really going away.
The requirement that each element of $x$ have no intersection with the transitive closure of $y$ seems more technical, and I'm not sure if there's a clean intuition behind it. His
example 2B1 shows why I feel okay not caring about this part of the definition too much - in the end, we take some class of potential counterexamples (field expansions that might be
algebraic closures) but which don't satisfy this disjointness condition, and just slide them over in an appropriately definable manner (in this case, $x\mapsto \lbrace (w, y): w\in x\
rbrace$). I suspect that in general something like this will be possible without much difficulty (although I am not sure on this point).
up vote 4 So what we're left with is that a (sur/in)jectively bounded sentence is essentially a $\Sigma_2$-sentence where
down vote
accepted • the universal quantification is taken over sets of boundable size, and
• the matrix of the sentence is boundable in the original Jech-Sochor sense.
(Of course, this isn't really $\Sigma_2$, since this "matrix" might well have quantifiers, but oh well.) What's really new here is this universal quantifier - note that
At this point it would be nice to see a injectively boundable statement which is not boundable. I think the clearest example is Pincus' 2B6 on page 724 (actually, he uses this as an
example of an injectively boundable statement which is not surjectively boundable, but that distinction seems less intuitively crucial to me). The statement here is
"Every infinite partially ordered set has either an infinite chain or an infinite antichain but there is an infinite, Dedekind-finite set."
This sentence is a conjunction $\Phi\wedge\Psi$, where
• $\Phi\equiv$ "Every infinite poset has a chain or antichain," and
• $\Psi\equiv$"There is a strictly Dedekind-finite set."
Now $\Psi$ already transferable by Jech-Sochor (since "is a Dedekind-finite set" depends just on the powerset). $\Phi$, however, is a bit trickier, since it involves quantifying over the
class of all posets! And this certainly can't be done with Jech-Sochor.
Instead, we use a trick. First, we can rewrite $\Phi$: $$\Phi\equiv "\forall x(\vert x\vert_-\le\omega\implies (\text{ if $x$ is an infinite poset, then $x$ has an infinite chain or
antichain})) "$$ since if $\vert x\vert_->\omega$ then we can already build a chain or antichain without choice. Now the conclusion of this implication is a boundable formula of $x$,
since it really only talks about the powerset! So even though the whole sentence $\Phi$ wasn't boundable, by massaging it a bit we got it to the point where the universal quantifier
causing all the trouble was just over sets of small "size," and this was enough for it to be injectively boundable.
Note that here, the specific notion of size we use is crucial: $\Phi$ isn't surjectively boundable, since $\vert x\vert^-$ can behave more weirdly on Dedekind-finite sets. So this is
what should motivate injective boundability: it's the broadest obvious way to push up the strength of Jech-Sochor to allow some non-powerset-bounded universal quantification.
Hopefully, this helps. Tl;dr: injectively-bounded properties will generally look like "This simple property $(\neg\Psi)$ does not always $(y)$ have small witnesses $(\forall x[\vert x\
vert_-< . . .])$."
I should point out that $|X|_-=\aleph(X)$ (its Hartogs number) and $|X|^-=\aleph^*(X)$ (its Lindenbaum number), in modern notation. – Asaf Karagila Jul 24 '13 at 12:05
Good point, thanks! (I actually didn't know the modern notation - I forget where, but I originally learned the $\vert X\vert_-$/$\vert X\vert^-$ way.) – Noah S Jul 24 '13 at 12:53
Noah, your profile says that you're a grad student in Berkeley. So I'm assuming that you're probably not over 50 years old. This makes me somewhat surprised about your previous
remark! – Asaf Karagila Jul 24 '13 at 13:46
I don't recall seeing the notation $\aleph^*$ or the name "Lindenbaum number" before. But I'm considerably over 50 years old, so anything I say that begins "I don't recall" can be
attributed to senility. In any case, I wouldn't use any of these notations (not even $\aleph(X)$, which seems rather standard nowadays) in a paper without saying what I meant by it. –
Andreas Blass Jul 24 '13 at 15:17
@Andreas: Truss used $\aleph^*$ already in the 70's (see his paper about models with perfect sets), and I am remembering other people using that notation as well. The term Lindenbaum
1 number is completely new, and I am trying to make it stick. Much like Hartogs number is named after Hartogs who proved that the totality of $\leq$ implies the axiom of choice,
Lindenbaum proved that the totality of $\leq^*$ implies the axiom of choice. But then he was killed by the Nazis and the proof was only published by Sierpinski in 1948. – Asaf
Karagila Jul 24 '13 at 15:20
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Breaking News: Two Great New Measurements
Two new ground-breaking measurements reported results in the last 24 hours! Here are very quick summaries.
A group of atomic physicists, called the ACME collaboration, has performed the best search so far for the electric dipole moment (EDM) of the electron. Unfortunately they didn’t find the EDM, but
the limit
is 12 times stronger than the previous one. While this is still a billion times larger than what is expected in the Standard Model of particle physics (the equations used for the known elementary
particles and forces), there are various types of as-yet unknown particles and forces that could easily produce a much larger electron EDM, through new violations of T symmetry (or, almost
equivalently, CP symmetry). These effects could have been large enough to have been discovered by this experiment, so those types of possible phenomena are now more constrained than before.
Fortunately, there’s more to look forward to; the method these folks are using can eventually be improved by another factor of 10 or so, meaning that a discovery using this technique is still
This morning the LUX dark matter experiment reported new results, and knocked everyone’s socks off. They have understood their backgrounds from radioactivity much better and more quickly than most
of us expected, using new calibration methods and a much better characterization of their backgrounds than has previously been possible. Although they have a detector only a bit larger than XENON100
and have only run the detector underground for three months, compared to the year or so that XENON100 ran previously, their limits on the rate for a dark matter particle to hit a Xenon nucleus beats
XENON100′s results by a factor of 2 for a dark matter particle of mass 1000 GeV/c², increasing to about a factor of 3 for a dark matter particle in the 100 GeV/c² mass range, and soaring to a factor
of 20 for a dark matter particle in the 10 GeV/c² mass range. Consequently, LUX pretty definitively rules out the possibility, hinted at by several dark matter experiments (as discussed in the
second half of the article I wrote about this in April), of a dark matter particle in the 5 – 20 GeV/c² mass range. (See the figure below.) While XENON100 seemed to contradict this possibility
already, it didn’t do so by a huge factor, so there were questions raised as to whether their result was convincing. But the sort of ~10 GeV/c² dark matter that people were talking about is ruled out
by LUX by such a large factor that finding ways around their result seems nigh impossible. And again, there’s more to look forward to; by 2015 their results should improve by another factor of 5 or
so… so they get another shot at a discovery, as will XENON1T, the successor to XENON100.
Congratulations to both groups for their spectacular achievements!
17 responses to “Breaking News: Two Great New Measurements”
1. “But the sort of ~10 GeV/c² dark matter that people were talking about …”
Matt, regarding what “people were talking about”, are we speaking here of sheer speculation on their part? Or mathematical/statistical predictions based on something more concrete (though still
having a speculative foundation).
□ See the second half of http://profmattstrassler.com/articles-and-posts/relativity-space-astronomy-and-cosmology/dark-matter/current-hints-of-dark-matter-413/
2. Does the EDM result confirms the existence of unknown particles and forces or not yet ?
What is the impact of excluding 5-20GeV WIMPS on Supersymmetry ?
□ 1) Since the EDM is not observed, but only bounded from above, the result simply excludes some types of unknown particles and forces; it confirms nothing.
2) Not much.
3. Matt,interesting report. What is the range of believable theoretical models prediction of WIMP-nucleon cross section? Or one can get any answer by fudging couplings?!! This will be a serious
problem for dark matter theories. If I understand MOND has been ruled out by terrestrial experiments.
□ There are plenty of dark-matter theories that can evade the XENON100 and LUX results, which are putting increasing pressure specifically on WIMP-type models, where the particle in question
interacts with matter via the Higgs or via the Z particle. If dark matter is of a very different sort, these measurements may not be relevant.
□ I have seen figures of 10^-39 cited for a neutrino cross-section (which should be similar to any Z mediated interactions with dark matter) and 10^-42 to 10^-46 for a Higgs mediated
cross-section in an old post by Jester that I have lost the link to since I accidentally linked to the entire blog rather than the pertinent post.
☆ Link found: http://resonaances.blogspot.com/2011/04/xenon100-nothing.html
○ “There exists another natural possibility for WIMP dark matter: a particle interacting via Higgs boson exchange. This would lead to the cross section in the 10^-42-10^-46 cm2 ballpark
(depending on the Higgs mass and on the coupling of dark matter to the Higgs). This generic possibility is now getting disfavored thanks to Xenon100′s efforts, unless the Higgs is
heavier than we expect. Therefore, even though models predicting the cross section below 10^-44 cm2 certainly do exist, it may be a good moment to start thinking more seriously about
alternatives to WIMP.”
So this is pre-Higgs discovery and could be made far more specific now?
4. to seek supersymmetry might occur strongest violation of operator T or CP to levels greater than given by standard model.i believe will appear chanes in the spacetime structures to 4-dimension
manifolds as observed by s. donaldson-that implies differents smooth topological 4-dimension manifolds with spin tensor-with torsion
5. Is there not a case that the EDM is in fact what we now believe is the Higgs Particle. Is there any chance of ambiguities here?
6. It was suggested at the time of the Higgs measurement that it is only the same thing ECM? Please indicate
7. But heavy WIMPS does not solve the hierarchy problem, in addition we are still waiting for your explanation of naturalness problem .
□ Yep, still waiting… and holding my breath :)
8. What limits does the LUX result impose on LSP production at LHC energies?
This entry was posted in Astronomy, Particle Physics and tagged astronomy, atoms, DarkMatter, particle physics. Bookmark the permalink.
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Surface analog of clothoid: curvatures covering $\mathbb{R}$
up vote 1 down vote favorite
The clothoid $C$, a.k.a. the Euler spiral, is one among many curves with the property that its curvatures cover $\mathbb{R}$ in the sense that, for every $x \in \mathbb{R}$, there is a point $p \in
C$ such that the curvature at $p$ is $x$:
I am seeking surface analogs:
Are there examples of surfaces $S$ embedded in $\mathbb{R}^3$ with the property that for every $x \in \mathbb{R}$, there is a point $p \in S$ such that the Gaussian curvature at $p$ is $x$?
Although my main interest is in surfaces in 3D, one could ask the same question for Riemannian manifolds whose sectional curvatures cover $\mathbb{R}$.
dg.differential-geometry curves-and-surfaces curvature
en.wikipedia.org/wiki/Moirai#The_three_Moirai – Will Jagy Jul 11 '13 at 18:41
It's got to be named after Clotho... – Will Jagy Jul 11 '13 at 18:59
@Will: Right! Or rather, Clotho is named after the root "to spin." clothoid: "From Gk. kloth, from klothein "to spin" + epenthetic vowel -o- + eides "form," → -oid; because the curve is reminiscent
of the thread that winds around a weaving loom." – Joseph O'Rourke Jul 11 '13 at 19:11
Alright, now we need pictures of thread winding around a weaving loom. Or paintings or.. what did that Albrecht Durer do? – Will Jagy Jul 11 '13 at 19:29
Dürer---etchings. – Joseph O'Rourke Jul 11 '13 at 20:16
add comment
3 Answers
active oldest votes
Constructing such a surface is not too hard.
Note that for a smooth surface, Gaussian curvature is a continuous function. Hence if you have a connected surface it suffices to have points of arbitrarily large (in absolute value)
Gaussian curvatures of either sign.
So consider the graph of the function
$$ z = f(x,y) = \cos(2\pi x) + x y^2 $$
We have that
$$ \mathrm{d}f = (-2\pi \sin (2\pi x) + y^2)\mathrm{d}x + 2xy \mathrm{d}y $$
and in particular
$$ \mathrm{d}f(n,0) = 0 $$
for $n \in \mathbb{Z}$. At those points the Gaussian curvature is simply the determinant of the Hessian
up vote 3
down vote $$ K(n,0) = \det\begin{pmatrix} - 4\pi^2\cos(2\pi n) & 2\cdot 0 \\ 2\cdot 0 & 2n \end{pmatrix} = - 8 \pi^2 n$$
Do you perhaps intend to add other criteria to your surface? The clothoid has the property that every curvature value is realised by exactly one point. This is of course not possible for
a surface, but maybe you want a surface where every the sets $K^{-1}(k)$ are all homeomorphic or something like that?
If you want a surface that is contained in a compact set in $\mathbb{R}^3$, consider the following map:
Let $D = \{ (\theta,s)\in \mathbb{R}^2 : \theta\in (-\pi,\pi), |s| < 1, |s^2 \tan \theta| < \frac12\}$
Let $\phi:D\to \mathbb{R}\times\mathbb{R}_+ \times\mathbb{S}^1$, the cylindrical coordinate representation of $\mathbb{R}^3$, be given by
$$\phi(\theta,s) = (s,\frac12 \tan\theta s^2,\theta) $$
The principle curvatures at point $(0,s)$ are $\{ 1, -\tan\theta\}$ and so the Gauss curvataure is $-\tan\theta$.
Our choice of domain guarantees that $\phi$ is an embedding and that $\phi(D)$ is contained in a ball of sufficiently large radius.
As a side remark: the graph of the function $y = x\sin x$ also attains, as curvature, all real numbers. – Willie Wong Jul 11 '13 at 10:34
3 Considering that the Gauss curvature is a scalar function and that a surface is parametrised by two, perhaps a more interesting question to ask is for an example of a surface where
the ordered pair (mean curvature, Gauss curvature) (or similarly, the pair of principle curvatures) cover $\mathbb{R}^2$. – Willie Wong Jul 11 '13 at 10:51
I like these questions, Willie! The pair of principle curvatures is especially appealing. – Joseph O'Rourke Jul 11 '13 at 11:27
@Willie Wong: For a surface in $\mathbb{R}^3$, one always has $H^2\ge K$, so the image of the map $(H,K)$ will never be onto $\mathbb{R}^2$. Maybe you want the image to be onto the
set allowed by this inequality. – Robert Bryant Jul 11 '13 at 19:28
@RobertBryant: right. AM-GM of course. There's also the small complication that the pair of principle curvatures are not uniquely ordered for the other version. Perhaps for pair of
principle curvatures version a better target will be $\{(x,y)\in\mathbb{R}^2: x \geq y\}$. – Willie Wong Jul 12 '13 at 7:37
add comment
Here is Willie Wong's function $f(x,y) = \cos(2\pi x) + x y^2 $ at two different scales:
up vote 1 down vote
add comment
when reading about the problem, I almost immediately had the idea to define a surface via the combination of two clothoids in a similar fashion as two circles are combined to define a
The first clothoid is defined via the $u$ parameter in the $xy$-plane as usual and will be traced out by a second clothoid that is defined via the $v$ parameter and, as the origin of the
up vote 1 second clothoid moves along the first clothoid, $x(u)$ corresponds to $z(v)$ and, $y(u)$ corresponds to the orthogonal distance to the $xy$-plane's clothoid after a point's projection
down vote into the $xy$-plane.
I apologize for this rather coarse description; hopefully the example helps despite.
I just noticed that I have misinterpreted the problem; what I tried to do, is to find a surface such that for each pair of principal curvatures there is a point of S with exactly that
pair of principal curvatures. – Manfred Weis Jul 12 '13 at 18:50
You are answering the followup question, "Surface in 3D that realizes all pairs of principal curvatures"! – Joseph O'Rourke Jul 12 '13 at 19:01
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Not the answer you're looking for? Browse other questions tagged dg.differential-geometry curves-and-surfaces curvature or ask your own question.
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Order of Operations
Sometimes order matters! In this BrainPOP movie, Tim and Moby introduce you to the order of operations that you should use when solving a math problem. You’ll learn which items in a problem come
first; when you deal with exponents; and which operations are worked from left to right. Find out what happens when you ignore the order of operations, and why we need a specific order of operations.
You’ll also discover what to do when operations occur within parenthesis — and you’ll get tips on a few great ways to remember the correct order! This movie is definitely made to order.
Watch the Math movie about Order of Operations »
What is the correct order in which to solve equations?Why do we use order of operations? What if you need to do something first inside the parentheses that isn’t in order?
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Bloomfield, NJ SAT Math Tutor
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search results
Expand all Collapse all Results 1 - 5 of 5
1. CMB Online first
New Characterizations of the Weighted Composition Operators Between Bloch Type Spaces in the Polydisk
We give some new characterizations for compactness of weighted composition operators $uC_\varphi$ acting on Bloch-type spaces in terms of the power of the components of $\varphi,$ where $\varphi$
is a holomorphic self-map of the polydisk $\mathbb{D}^n,$ thus generalizing the results obtained by Hyvärinen and Lindström in 2012.
Keywords:weighted composition operator, compactness, Bloch type spaces, polydisk, several complex variables
Categories:47B38, 47B33, 32A37, 45P05, 47G10
2. CMB 2011 (vol 56 pp. 55)
Cliquishness and Quasicontinuity of Two-Variable Maps
We study the existence of continuity points for mappings $f\colon X\times Y\to Z$ whose $x$-sections $Y\ni y\to f(x,y)\in Z$ are fragmentable and $y$-sections $X\ni x\to f(x,y)\in Z$ are
quasicontinuous, where $X$ is a Baire space and $Z$ is a metric space. For the factor $Y$, we consider two infinite ``point-picking'' games $G_1(y)$ and $G_2(y)$ defined respectively for each $y\in
Y$ as follows: in the $n$-th inning, Player I gives a dense set $D_n\subset Y$, respectively, a dense open set $D_n\subset Y$. Then Player II picks a point $y_n\in D_n$; II wins if $y$ is in the
closure of ${\{y_n:n\in\mathbb N\}}$, otherwise I wins. It is shown that (i) $f$ is cliquish if II has a winning strategy in $G_1(y)$ for every $y\in Y$, and (ii) $ f$ is quasicontinuous if the
$x$-sections of $f$ are continuous and the set of $y\in Y$ such that II has a winning strategy in $G_2(y)$ is dense in $Y$. Item (i) extends substantially a result of Debs and item (ii) indicates
that the problem of Talagrand on separately continuous maps has a positive answer for a wide class of ``small'' compact spaces.
Keywords:cliquishness, fragmentability, joint continuity, point-picking game, quasicontinuity, separate continuity, two variable maps
Categories:54C05, 54C08, 54B10, 91A05
3. CMB 2009 (vol 53 pp. 11)
Approximation and Interpolation by Entire Functions of Several Variables
Let $f\colon \mathbb R^n\to \mathbb R$ be $C^\infty$ and let $h\colon \mathbb R^n\to\mathbb R$ be positive and continuous. For any unbounded nondecreasing sequence $\{c_k\}$ of nonnegative real
numbers and for any sequence without accumulation points $\{x_m\}$ in $\mathbb R^n$, there exists an entire function $g\colon\mathbb C^n\to\mathbb C$ taking real values on $\mathbb R^n$ such that \
begin{align*} &|g^{(\alpha)}(x)-f^{(\alpha)}(x)|\lt h(x), \quad |x|\ge c_k, |\alpha|\le k, k=0,1,2,\dots, \\ &g^{(\alpha)}(x_m)=f^{(\alpha)}(x_m), \quad |x_m|\ge c_k, |\alpha|\le k, m,k=0,1,2,\
dots. \end{align*} This is a version for functions of several variables of the case $n=1$ due to L. Hoischen.
Keywords:entire function, complex approximation, interpolation, several complex variables
4. CMB 2009 (vol 52 pp. 535)
A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$
We strengthen certain results concerning actions of $(\Comp,+)$ on $\Comp^{3}$ and embeddings of $\Comp^{2}$ in $\Comp^{3}$, and show that these results are in fact valid over any field of
characteristic zero.
Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine space
Categories:14R10, 14R20, 14R25, 13N15
5. CMB 2005 (vol 48 pp. 622)
Hyperplanes of the Form ${f_1(x,y)z_1+\dots+f_k(x,y)z_k+g(x,y)}$ Are Variables
The Abhyankar--Sathaye Embedded Hyperplane Problem asks whe\-ther any hypersurface of $\C^n$ isomorphic to $\C^{n-1}$ is rectifiable, {\em i.e.,} equivalent to a linear hyperplane up to an
automorphism of $\C^n$. Generalizing the approach adopted by Kaliman, V\'en\'ereau, and Zaidenberg which consists in using almost nothing but the acyclicity of $\C^{n-1}$, we solve this problem for
hypersurfaces given by polynomials of $\C[x,y,z_1,\dots, z_k]$ as in the title.
Keywords:variables, Abhyankar--Sathaye Embedding Problem
Categories:14R10, 14R25
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F=ma Confusion about answer given for F
I'm reading a book and one of the questions asks: how much force is required to give an object weighing 3000lb an acceleration of 12ft/sec^2?
I calculated this as F=3000x12=36,000pdl. The answer given is "1125 force" (36,000/32). Why is this? Why isn't it just 36,000pdl?
The "confusing" part is this: If you push a one-pound mass (1 lbm) with a one-pound force (1 lbf) the object will accelerate at 32 ft/sec2, NOT 1 ft/sec^2. This is because the 1 lbm object weighs 1
lbf here on the surface of the earth, yet when you drop it, it falls with an acceleration of 32 ft/sec/sec. Slugs and poundals were invented to disguise this fact and avoid confusion. For me, I think
they cause more problems then they solve.
This doesn't come up in SI where, if you push 1 kg with a force of 1 N, it accelerates at 1 m/sec2.
Out of curiosity, in Canada, when a kid's height and weight are measured in school, his/her height is given in _____ and weight in ____. Please fill in the blanks.
I'm also curious about this, does anyone anywhere say "I weigh 690 newtons" or does everyone say "...70 kilos" ?
... Conversion is a very slow process.
Apparently the reasons for converting aren't strong enough to overcome the obstacles. I learned physics with "the metric system" but I work in engineering in the US. At first I thought the units were
funny (pounds per hour, gallons per minute, etc.) but soon they became familiar. As long as you understand what is going on, any system of units "works" and you may see the advantages of one or the
other, depending on the given situation.
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Data Mining: A Mathematical Perspective
CS 391D/CAM 395T
CS Unique No. 54950 / CAM Unique No. 66117
Fall 2009
TTh 9:30-11am
WEL 2.312
Instructor: Prof. Inderjit Dhillon (send email)
Office: ACES 2.332
Office Hours: Tue 11am-noon and by appointment
TA: Wei Tang (send email)
Office: TAY 137
Office Hours: MW 3:30-5:30pm
Course Description
Data mining is the automated discovery of interesting patterns and relationships in massive data sets. This graduate course will focus on various mathematical and statistical aspects of data mining.
Topics covered include supervised methods (regression, classification) and unsupervised methods (clustering, principal components analysis, dimensionality reduction). The technical tools used in the
course will draw from linear algebra, multivariate statistics and optimization. The main tools from these areas will be covered in class, but undergraduate level linear algebra is a pre-requisite
(see below). A substantial portion of the course will focus on research projects, where students will choose a well defined research problem. Projects can vary in their theoretical/mathematical
content, and in the implementation/programming involved. Projects will be conducted by teams of 2-3 students.
Pre-requisites: Basics (undergraduate level) of linear algebra (M341 or equivalent) and some mathematical sophistication.
Reading Material
Class Presentations
Class Projects
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Wolfram Demonstrations Project
Parity Recurrence in Thue-Morse Sequence
The Thue-Morse sequence gives the parity for the sum of ones in binary numbers. It can be obtained by steps which append the binary complement of the previous step. The recurrence plot shows the mod
2 differences between the and terms of the sequence. The heads for rows and columns are obtained by this procedure which can start from intial values 0 or 1.
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several calculus problems from homework assignment that i couldn't solve II
April 28th 2011, 07:33 PM #1
Apr 2011
several calculus problems from homework assignment that i couldn't solve II
3. Let C be the arc of y=8-(x^2)/2 which lies above the x-axis. Find the longest line segment both of whose endpoints lie on C.
(A) 7.7
(B) 8
(C) 8.9
(D) 9.1
(E) 12.5
if you could answer this question, help me please
Last edited by Ackbeet; April 29th 2011 at 02:21 AM. Reason: Splitting off problem from other thread.
$L=\displaystyle\int_{-4}^{\;4}\sqrt {1+(y')^2}\;dx=2\displaystyle\int_{0}^4\sqrt {1+x^2}\;dx=\ldots$
April 29th 2011, 03:35 AM #2
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[aprssig] Position Ambituity in APRS!
Robert Bruninga bruninga at usna.edu Tue Jan 8 15:58:10 UTC 2008
Welcome to the 15 year debate!
>> Is, or is not, position ambiguity at the
>> transmit end simply the
>> truncation of lat/lon digits?
It is not. It is NOT truncation. It is simply the transmission
of the AVAILABLE DIGITS to the degree of precision desired or
known. If you have only degrees, you transmit only degrees,
which give a position to the nearest degree (60 miles). If you
have only degrees and minutes, you transmit only degrees and
minutes which is the position to the nearest minute (1 mile).
If you have a position known only to the nearest tenth of a
minute, then you only transmit the degrees, minutes and the
tenth of a minute that you have (position known to the nearest
tenth of a mile). This is not truncation. It is transmitting
what you know, and NOT implying additional digits of precision
that you do not know.
That is all that APRS position ambiguity means. You transmit
the position only with the number of digits that match the
precision that you have. And you do NOT add precise decimal
digits beyond your knowledge. The position field in APRS is a
CHARACTER STRING that happens to have room for digits of
precision down to hundredths of a minute. It is NOT a numeric
field which many programmers incorrectly implemented.
> You have to understand the way Bob's mind works.
Yes, it is simple. If the position is known to be 38 degrees 58
minutes, then you transmit ONLY "3858. N" It is absolutely
WRONG to send "3858.00N". Any middle school science teacher can
tell us that.
> This tops my most-hated of Bob's excursions.
> The engineering way to handle uncertainty with
> position is as has been suggested, a precise
> position representing the best guess, and an
> altitude-like extension representing the
> approximate radius of uncertainty.
Absolutely wrong. That precise estimate implies a PRECISION
that does not exist. Such simplifications by APRS clones
unwilling to properly implement this simplest of concepts
undermines the integrity of information from sender to receiver.
If the sender does not know the precise position, then he should
not under any circumstances send it as a precise position. He
should transmit only what he knows so that the recepient cleary
sees the same level of ambiguity.
> Google does this, Garmin does this, Trimble does
> this, but Bob? To save a few bytes in the protocol,
> Bob reused bytes in the lat/lon.
Not true. I did not reuse "bytes". What I refused to do was to
put in higher digits of precision when those digits ARE NOT
KNOWN. To do so would violate every principle of "precison" as
taught in middle school.
> His intention was not that this be interpreted as a
> question mark in the lat/lon, a literal uncertainty
> interpreted as a polygon, as an engineer would, but
> rather simply as a magnitude of uncertainty.
Partly right. Because the uncertainty is not a precise polygon;
it is a lack of additional precision. It is an uncertanty of
the number of digits of precision by the sender, and an EXACT
transfer of that same uncertany to the recepient. In that sense
it conveys the "magnitude of uncertainty" from the sender to the
recepient in an exact format that cannot be missinterpreted.
> The problem is that this representation does
> not fit the reality.
It may not fit with the reality of some APRS implementations
that took the simplistic approach of truncating digits, but it
does transfer exactly from the sender to the receiver the
knowledge of the ambiguity if displayed properly. If one
doesn't have a digit of precision, then he should NOT stick in a
ZERO. Stick in a SPACE character, just like he would write it
on a piece of paper.
> So, think of ambiguity as representing a circle,
> taking the center of the polygon described by the
> lowest and highest values of the missing digits,
> and with the radius of the magnitude of the missing
> digits.
Yes, now we are talking about how to display it. This now is
why it is so important to do it consistently across all APRS
clients so that everyone gets the same visualization that the
sender intended...
What you describe above is what I intended for display but with
one additional tweak as implemented in APRSdos. And that is to
provide a SLIGHT random offset within that area of uncertanty so
that if multiple APRS positions are reported in that same area,
that they are not all stacked on top of each other so that only
the top one appears.
If they all use the same precise center of the area, then only
the latest ICON shows on the map and only one CIRCLE of
ambiguity shows. This can be very missleading to the casual
viewer of the map. But in APRSdos, if there are 6 such stations
reporting ambiguity in the same polygon, then each of their
circles of ambiguity will each show, but slightly offset so that
they all individually appear and so at a glance, one can see
that there are 6 stations there.
It is very simple. This is the definition of APRS ambiguity:
1) The APRS position field is a CHARACTER string
2) The sender only includes the digits he knows
3) On receipt a circle of ambiguity is displayed that represents
the possible ambiguity due to the lack of precision
4) For display purposes, thse circles are offset slightly so
that multiple stations reporting the same ambiguous position do
not all appear as a single display.
In addition, the original APRSdos does the following:
A) The SYMBOL is only shown as long as the size of the symbol
overlaps the size of the circle. In this case the circle is
hidden or not displayed. Example, viewing a .1 mile ambiguous
station on a 100 mile map scale, you see the symbol and all
looks normal.
B) As one zooms in, and the circle becomes larger than the
symbol, then the SYMBOL disappears and only the circle is
displayed. This avoids the appearance of the symbol as an
"exact location inside a circle". It is not. At this point,
the circle is the best representation for that station, the
symbol is not.
C) Originally, APRSdos simply let the SIZE of the symbol expand
so that it always covered the area of ambiguiy as the map was
zoomed. But this can clutter the map. My favorite example, is
when I arrive in a city airport and I enter the estimated
position of the city into my HT with a 10 mile ambiguity just to
show where I am (without carrying a GPS). I do not want my
SYMBOL to cover the entire city!
So that is why I fell back to (B) above as the best way to
convey the ambiguity to the recepient at high map zooms.
Bob, WB4APR
More information about the aprssig mailing list
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Here's the question you clicked on:
Show that the area of a regular 8-gon is equal to the product of its longest diagonal and its shortest diagonal.
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|dw:1318922329389:dw| the octagon is regular giving it equal sides and angles thus all the triangles have the same area and both rectangles have the same area. now, we have 8 triangles with area=
1/2ab giving, total area of the triangles =8(1/2)ab =4ab also we have 2 rectangles with area =bc giving total area of the rectangles =2bc thus total area of the octagon =4ab+2bc ***** now, the
length of the longest diagonal call it X = a+c+a =2a +c the length of the shortest diagonal call it Y = 2b multiplying X and Y gives XY=(2a +c)(2b)=4ab+2bc =area of octagon end of proof
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is replying to Can someone tell me what button the professor is hitting...
• Teamwork 19 Teammate
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.
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Advanced Number Theory with Applications
• Applies tools, such as algebraic number theory, to Diophantine equations
• Presents the application of elliptic curve cryptography
• Discusses modular forms and functions, including applications to elliptic curves used to prove FLT—topics not found in similar books
• Describes sieve methods, including Bombieri’s asymptotic sieve and the number field sieve
• Offers an accessible overview of the proof of FLT
• Contains nearly 50 mini-bios of relevant mathematicians, more than 330 problems, and solutions to odd-numbered exercises
Solutions manual available for qualifying instructors
Exploring one of the most dynamic areas of mathematics, Advanced Number Theory with Applications covers a wide range of algebraic, analytic, combinatorial, cryptographic, and geometric aspects of
number theory. Written by a recognized leader in algebra and number theory, the book includes a page reference for every citing in the bibliography and more than 1,500 entries in the index so that
students can easily cross-reference and find the appropriate data.
With numerous examples throughout, the text begins with coverage of algebraic number theory, binary quadratic forms, Diophantine approximation, arithmetic functions, p-adic analysis, Dirichlet
characters, density, and primes in arithmetic progression. It then applies these tools to Diophantine equations, before developing elliptic curves and modular forms. The text also presents an
overview of Fermat’s Last Theorem (FLT) and numerous consequences of the ABC conjecture, including Thue–Siegel–Roth theorem, Hall’s conjecture, the Erdös–Mollin-–Walsh conjecture, and the
Granville–Langevin Conjecture. In the appendix, the author reviews sieve methods, such as Eratothesenes’, Selberg’s, Linnik’s, and Bombieri’s sieves. He also discusses recent results on gaps between
primes and the use of sieves in factoring.
By focusing on salient techniques in number theory, this textbook provides the most up-to-date and comprehensive material for a second course in this field. It prepares students for future study at
the graduate level.
Table of Contents
Algebraic Number Theory and Quadratic Fields
Algebraic Number Fields
The Gaussian Field
Euclidean Quadratic Fields
Applications of Unique Factorization
The Arithmetic of Ideals in Quadratic Fields
Dedekind Domains
Application to Factoring
Binary Quadratic Forms
Composition and the Form Class Group
Applications via Ambiguity
Equivalence Modulo p
Diophantine Approximation
Algebraic and Transcendental Numbers
Minkowski’s Convex Body Theorem
Arithmetic Functions
The Euler–Maclaurin Summation Formula
Average Orders
The Riemann zeta-function
Introduction to p-Adic Analysis
Solving Modulo p^n
Introduction to Valuations
Non-Archimedean vs. Archimedean Valuations
Representation of p-Adic Numbers
Dirichlet: Characters, Density, and Primes in Progression
Dirichlet Characters
Dirichlet’s L-Function and Theorem
Dirichlet Density
Applications to Diophantine Equations
Lucas–Lehmer Theory
Generalized Ramanujan–Nagell Equations
Bachet’s Equation
The Fermat Equation
Catalan and the ABC-Conjecture
Elliptic Curves
The Basics
Mazur, Siegel, and Reduction
Applications: Factoring and Primality Testing
Elliptic Curve Cryptography (ECC)
Modular Forms
The Modular Group
Modular Forms and Functions
Applications to Elliptic Curves
Shimura–Taniyama–Weil and FLT
Appendix: Sieve Methods
Solutions to Odd-Numbered Exercises
Index: List of Symbols
Index: Alphabetical Listing
Author Bio(s)
Richard A. Mollin is a professor in the Department of Mathematics and Statistics at the University of Calgary. In the past twenty-three years, Dr. Mollin has founded the Canadian Number Theory
Association and has been awarded six Killam Resident Fellowships. Over the past thirty-three years, he has written more than 190 publications.
Editorial Reviews
The reader following this book will obtain a thorough overview of some very deep mathematics which is still in active research today. … I readily recommend this book to advanced undergraduates and
beginning graduate students interested in advanced number theory. This book can also be read by the enthusiast who is well-acquainted with the author's previous book Fundamental Number Theory with
—IACR Book Reviews, May 2011
… each section comes with a large number of illustrating examples and accompanying exercises. … The rich bibliography contains 106 references, where maximum information is imparted by explicit page
reference for each citing of a given item within the text. The carefully compiled index has more than 1,500 entries presented for maximum cross-referencing. Overall, this excellent textbook bespeaks
the author’s outstanding expository mastery just as much as his mathematical erudition and elevated taste. Presenting a wide panorama of topics in advanced classical and contemporary number theory,
and that in an utmost lucid and comprehensible style of writing, the author takes the reader to the forefront of research in the field, and on a truly exciting journey over and above.
—Werner Kleinert, Zentralblatt MATH, 2010
When I was looking over books for my course, I was very pleased by yours, and look forward to teaching from it. … after much thought I found that I liked yours best for its completeness, its
problems, and for the way you weave current results and conjectures into the text. … Among other things that pleased me about your book, I’m so glad continued fractions come where they do. … a worthy
book …
—David Barth-Hart, Associate Head, School of Mathematical Sciences, Rochester Institute of Technology, New York, USA
This terrific book is testimony to Richard Mollin’s mathematical erudition, wonderful taste, and also his breadth of culture. … Mollin’s treatment of elliptic curves is a model of clear exposition …
[It] succeeds very well in its goal of providing a means of transition from more or less foundational material to papers and advanced monographs, i.e., research in the field. … a wondrous book,
successfully fulfilling the author’s purpose of effecting a bridge to modern number theory for the somewhat initiated. … it’s very nice to find in Mollin’s book a high quality and coherent treatment
of this beautiful material and pointers in abundance to where to go next.
—Michael Berg, Loyola Marymount University, MAA Review, 2009
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new kind of interactive problem book released
I found a newly released problem book on intermediate and college algebra, with medium and advanced level problems. What it brings new is its structure: it does not only contain the problems and
their solutions, but the solver passes through three intermediary stages to the final solution: hints, solving algorithms and proofs, which are separate in the book.
It is listed on Amazon and B&N, but it can be acquired cheaper on publisher's website at www.infarom.com/new_releases.html .
It is titled MATHEMATICS PROBLEMS WITH SEPARATE PROGRESSIVE SOLUTIONS: HINTS, ALGORITHMS, PROOFS. VOLUME 1 - INTERMEDIATE AND COLLEGE ALGEBRA.
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[plt-scheme] HTDP: 9.5.5
From: Todd O'Bryan (toddobryan at gmail.com)
Date: Sat Sep 13 22:34:03 EDT 2008
That's the absolutely essential part! Let's look at the template for a
list-of-number function:
;; lon-fun: list-of-number -> ?
;; ?
(define (lon-fun alon)
[(empty? alon) ]
[(cons? alon) (first alon)
(lon-fun (rest alon)]))
Now replace lon-fun with convert:
;; convert: list-of-number -> number
;; consumes a list of digits from least to most significant
;; produces the corresponding number
(define (convert alon)
[(empty? alon) ]
[(cons? alon) (first alon)
(convert (rest alon)]))
(check-expect (convert (cons 4 (cons 3 (cons 2 (cons 1 empty))))) 1234)
The template itself gives you two pieces when the list is non-empty,
the first element and the function called on the rest of the list. In
this example, you get 4 and 123. Your job is to figure out how to
combine those two pieces together to get the answer you desire.
That's the key point that I constantly push with my students. The
shape of the data encourages a particular shape for the functions that
manipulate the data. The shape of the function already does much of
the work for you. Your job is to look at what you get "for free" and
figure out how to massage it to get what you want.
On Sat, Sep 13, 2008 at 7:43 PM, Matthias Felleisen
<matthias at ccs.neu.edu> wrote:
> Now discover Jens's hint by following the design recipe.
> On Sep 13, 2008, at 7:33 PM, Grant Rettke wrote:
>> On Sat, Sep 13, 2008 at 5:57 PM, Jens Axel Soegaard
>> <jensaxel at soegaard.net> wrote:
>>> Think recursively:
>>> 1234 = 4 + 10*123
>> I see! Thanks Jens.
>> _________________________________________________
>> For list-related administrative tasks:
>> http://list.cs.brown.edu/mailman/listinfo/plt-scheme
> _________________________________________________
> For list-related administrative tasks:
> http://list.cs.brown.edu/mailman/listinfo/plt-scheme
Posted on the users mailing list.
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Don’t Store That in a Float
I promised in my last post to show an example of the importance of knowing how much precision a float has at a particular value. Here goes.
As a general rule this type of data that should never be stored in a float:
Elapsed game time should never be stored in a float. Use a double instead. I’ll explain why below.
As an extra bonus, because switching to double is not always the best solution, this post demonstrates the dangers of unstable algorithms, and how to use the guarantees of floating-point math to
improve them.
How long has this been going on?
A lot of games have some sort of GetTime() function that returns how long the game has been running. Often these return a floating-point number because it allows for convenient use of seconds as the
units, while allowing sub-second precision.
GetTime() is typically implemented with some sort of high frequency timer such as QueryPerformanceCounter. This allows time resolution of a microsecond or better. However it’s worth looking at what
happens to this resolution if the time is returned as a float, or stored in a float. We can do that using one of the TestFloatPrecision functions from the last post – just call them from the watch
window of the debugger. In the screen shot below I tested the precision available at one minute, one hour, one day, and one week:
It’s important to understand what this data means. The number ‘60’, like all integers up to 16777216, can be exactly represented in a float. The watch window shows that the next value after 60 that
can be represented by a float is about 60.0000038. Therefore, if we use a float to store “60 seconds” then the next time that we can represent is 3.8 microseconds past 60 seconds. If we try to store
a value in-between then it will be rounded up or down.
How long did it take?
One of the most common things to do with time values is to subtract them. For instance, we might have code like this:
double GetTime();
float TimeSomethingBadly()
float fStart = GetTime();
float elapsed = GetTime() – fStart;
return elapsed;
The implication of the precision calculations above is that if ‘fStart’ is around 60, then ‘elapsed’ will be a multiple of 3.8 microseconds (two to the negative eighteenth seconds). That is the most
precision you can get. If less than 3.8 microseconds has elapsed then ‘elapsed’ will either be rounded down to zero, or rounded up to 3.8 microseconds.
Therefore, if our game timer starts at zero and we store time in a float then after a minute the best precision we can get from our timer is 3.8 microseconds. After our game has been running for an
hour our best precision drops to 0.24 milliseconds. After our game has been running for a day our precision drops to 7.8 milliseconds, and after a week our precision drops to 62.5 milliseconds.
This is why storing time in a float is dangerous. If you use float-time to try calculating your frame rate after running for a day then the only answers above 30 fps that are possible are infinity,
128, 64, 42.6, or 32 (since the possible frame lengths are 0, 7.8, 15.6, 23.4, or 31.2 milliseconds). And it only gets worse if you run longer.
As another example consider this code:
double GetTime();
void ThinkBadly()
float startTime = (float)GetTime();
// Do AI stuff here
float elapsedTime = GetTime() – startTime;
assert(elapsedTime < 0.005); //
The purpose of this code is to warn the developers whenever the AI code takes inordinately long. However when the game has been running for a day (actually the problem reaches this level after 65,536
seconds) GetTime() will always be returning a multiple of 0.0078 s, and ‘elapsedTime’ will always be a multiple of that duration. In most cases ‘elapsedTime’ will be equal to zero, but every now and
then, no matter how fast the AI code executes, the time will tick over to the next representation during the AI calculations and ‘elapsedTime’ will be 0.0078 s instead of zero. The assert will then
trigger even though the AI code is actually still under budget.
It’s a catastrophe for base-ten also
The general term for what is happening with these time calculations is catastrophic cancellation. In all of these examples above there are two time values that are accurate to about seven digits.
However they are so close to each other that when they are subtracted the result has, in the worst case, zero significant digits.
We can see the same thing happening with decimal numbers. A float has roughly seven decimal digits of precision so the decimal equivalent would be getting a time value of 60.00000 and having the next
possible time value be 60.00001. Given a seven-digit decimal float we can’t get more than a tenth of a microsecond precision when dealing with time around 60 seconds. When we subtract 60.00000 from
60.00001 then six of the seven digits cancel out and we end up with just one accurate digit. For times less than a tenth of a microsecond we have a complete catastrophe – all seven digits cancel out
and we get zero digits of precision, just like with a binary float.
Double down
The solution to all of this is simple. GetTime() must return a double, and its result must always be stored in a double. The cancellation still occurs, but it is no longer catastrophic. A double has
enough bits in the mantissa that even if your game runs for several millennia your double-precision timers will still have sub-microsecond precision. You can verify this by using the double-precision
variation of TestFloatPrecisionAwayFromZero():
union Double_t
Double_t(double val) : f(val) {}
// Portable extraction of components.
bool Negative() const { return (i >> 63) != 0; }
int64_t RawMantissa() const { return i & ((1LL << 52) – 1); }
int64_t RawExponent() const { return (i >> 52) & 0x7FF; }
int64_t i;
double f;
#ifdef _DEBUG
{ // Bitfields for exploration. Do not use in production code.
uint64_t mantissa : 52;
uint64_t exponent : 11;
uint64_t sign : 1;
} parts;
double TestDoublePrecisionAwayFromZero(double input)
union Double_t num(input);
// Incrementing infinity or a NaN would be bad!
assert(num.RawExponent() < 2047);
// Increment the integer representation of our value
num.i += 1;
// Subtract the initial value find our precision
double delta = num.f – input;
return delta;
You can see in the screenshot below that if you store time in doubles then after your game has been running for a week you will have sub-nanosecond precision, and after three millennia you will still
have sub-millisecond precision.
Clearly a double is overkill for storing time, but since a float is underkill a double is the right choice.
Aside: my initial calculation of the precision remaining after three millennia was wrong because the calculation of the number of seconds was done with integer math, and it overflowed and gave a
completely worthless answer. Which proves that integer math can be just as tricky as floating-point math.
Changing your units doesn’t help
All along I am assuming that you are storing your time in seconds. However your choice of units doesn’t significantly affect the results. If you decide that your time units are milliseconds, or days,
then the precision available after your game has been running for a day will be about the same. It is the ratio between the elapsed time and the time being measured that matters. I like seconds
because they are intuitive and human friendly, and that does matter.
Or use integers
Tom Forsyth points out that the same issues happen with world coordinates and that switching to integer types can give you greater worst-case precision, as well as consistent precision. The Windows
GetTickCount() and GetTickCount64() functions use this technique, using milliseconds as the units. This alternative to using a double for time is quite reasonable, especially if you encapsulate it
well. A uint32_t with milliseconds as units will overflow every 50 days or so but you can avoid that by using a uint64_t. However despite Tom’s threats to invoke his OffendOMatic rule for all who use
doubles, I still prefer doubles for game time because of the combination of convenient units (seconds), more than sufficient precision, and easy calculations.
While Tom and I appear to disagree over whether you should use double in situations like this, we agree that ‘float’ won’t work.
Recently John Carmack said “Time should be a double of seconds” – that’s a good vote of confidence to have.
Note that while GetTickCount() and GetTickCount64() are millisecond precision they are often actually less accurate than you would expect. Unless you have changed the Windows timer frequency with
timeBeginPeriod() the GetTickCount functions will only return a new value every 10-20 milliseconds (insert pithy comment about precision versus accuracy here).
Four billion dollar question
Even if you use doubles for time, the precision available will still change as game time marches on from zero to the length of your game. These precision changes – while smaller with doubles than
with floats – can still be dangerous. Luckily there is a convenient way to get the consistent precision of an integer, with the convenient units of a double.
If you start your game clock at about 4 billion (more precisely 2^32, or any large power of two) then your exponent, and hence your precision, will remain constant for the next ~4 billion seconds, or
~136 years.
And, when using doubles, this precision is approximately one microsecond.
So there you have it. The one-true answer. Store elapsed game time in a double, starting at 2^32 seconds. You will get constant precision of better than a microsecond for over a century, and if you
accidentally store time in a float you will precision errors immediately instead of after hours of gameplay. You read it here first.
Time deltas fit in a float
It is important to understand that the limited precision of a float is only a problem if you do an unstable calculation, such as catastrophic cancellation cancelling out most of the digits. The code
below, on the other hand, is fine:
double GetTime();
float TimeSomethingWell()
double dStart = GetTime(); // Store time in a double
float elapsed = GetTime() – dStart; // Store *result* in a float
return elapsed;
In TimeSomethingWell() we store the result of the subtraction in a float – after the catastrophic cancellation. Therefore our elapsed time value will have tons of precision.
Similarly, if you are using floats in your animation system to represent short times, such as the location of key-frames in a 60 second animation, then floats are fine. However when you add these to
the current time you need to store the result of the addition in a double.
Forrest Smith made a pretty table showing how the precision of a float changes as the magnitude increases, and I mangled it to suit my needs. Here it is for time:
┃ Float Value │ Time Value │ Float Precision │ Time Precision ┃
┃ 1 │ 1 second │ 1.19E-07 │ 119 nanoseconds ┃
┃ 10 │ 10 seconds │ 9.54E-07 │ .954 microsecond ┃
┃ 100 │ ~1.5 minutes │ 7.63E-06 │ 7.63 microseconds ┃
┃ 1,000 │ ~16 minutes │ 6.10E-05 │ 61.0 microseconds ┃
┃ 10,000 │ ~3 hours │ 0.000977 │ .976 milliseconds ┃
┃ 100,000 │ ~1 day │ 0.00781 │ 7.81 milliseconds ┃
┃ 1,000,000 │ ~11 days │ 0.0625 │ 62.5 milliseconds ┃
┃ 10,000,000 │ ~4 months │ 1 │ 1 second ┃
┃ 100,000,000 │ ~3 years │ 8 │ 8 seconds ┃
┃ 1,000,000,000 │ ~32 years │ 64 │ 64 seconds ┃
And here is the table showing how the precision of a float diminishes when you use it to measure large distances, with meters being the units in this case:
┃ Float Value │ Length Value │ Float Precision │ Length Precision │ Precision Size ┃
┃ 1 │ 1 meter │ 1.19E-07 │ 119 nanometers │ virus ┃
┃ 10 │ 10 meters │ 9.54E-07 │ .954 micrometers │ e. coli bacteria ┃
┃ 100 │ 100 meters │ 7.63E-06 │ 7.63 micrometers │ red blood cell ┃
┃ 1,000 │ 1 kilometer │ 6.10E-05 │ 61.0 micrometers │ human hair width ┃
┃ 10,000 │ 10 kilometers │ 0.000977 │ .976 millimeters │ toenail thickness ┃
┃ 100,000 │ 100 kilometers │ 0.00781 │ 7.81 millimeters │ size of an ant ┃
┃ 1,000,000 │ .16x earth radius │ 0.0625 │ 62.5 millimeters │ credit card width ┃
┃ 10,000,000 │ 1.6x earth radius │ 1 │ 1 meter │ uh… a meter ┃
┃ 100,000,000 │ .14x sun radius │ 8 │ 8 meters │ 4 Chewbaccas ┃
┃ 1,000,000,000 │ 1.4x sun radius │ 64 │ 64 meters │ half a football field ┃
Stable algorithms also matter
Some time ago I investigated some asserts in a particle animation system. Values were going out of range after less than an hour of gameplay and I traced this back to an out-of-range ‘t’ value being
passed to the Lerp function, which expected it to always be from 0.0 to 1.0. Clamping was one obvious solution but I first investigated why ’t’ was going out of range.
One problem with the code was that the three parameters were all floats, so over long periods of time it would inevitably have insufficient precision. However we were getting instability much earlier
than expected and it felt like switching to double immediately might just mask an underlying problem.
The parameters to the function, all time values in seconds, corresponded to the end of an animation segment, the length of that segment, and the current time, which was always between the start of
the segment (segmentEnd-segmentLength) and ‘segmentEnd’. Because the start time of the segment was not passed in this code calculated it, and then did a straightforward calculation to get ‘t’:
float CalcTBad(float segmentEnd, float segmentLength, float time)
float segmentStart = segmentEnd – segmentLength;
float t = (time – segmentStart) / segmentLength;
return t;
Straightforward, but unstable. Because ‘segmentLength’ is presumed to be quite small compared to ‘segmentEnd’, there is some rounding during the first subtraction and the difference between
‘segmentStart’ and ‘segmentEnd’ will be a bit larger or smaller than ‘segmentLength’. The resulting difference will always be a multiple of the current precision, so it will degrade over time, but
even very early in the game the result will not be perfect. Because the value for ‘segmentStart’ is slightly wrong the value of “time – segmentStart” will be slightly wrong, and occasionally ‘t’ will
be outside of the 0.0 to 1.0 range.
This will happen even if you use doubles. The errors will be smaller, but ‘t’ can still go slightly outside the 0.0 to 1.0 range. As the game goes on ‘t’ will range farther outside of the correct
range, but from just a few minutes into the game the results will show signs of instability.
The natural tendency is to say “floating-point math is flaky, clamp the results and move on”, but we can do better, as shown here:
float CalcTGood(float segmentEnd, float segmentLength, float time)
float howLongAgo = segmentEnd – time;
float t = (segmentLength – howLongAgo) / segmentLength;
return t;
Mathematically this calculation is identical to CalcTBad, but from a stability point of view it is greatly improved.
If we assume that ‘time’ and ‘segmentEnd’ are large compared to ‘segmentLength’, then we can reasonably assume that ‘segmentEnd’ is less than twice as large as time. And, it turns out that if two
floats are that close then their difference will fit exactly into a float. Always. So the calculation of ‘howLongAgo’ is exact. Ponder that for a moment – given a few reasonable assumptions we have
exact results for one of our floating-point math operations.
With ‘howLongAgo’ being exact, if ‘time’ is within its prescribed range then ‘howLongAgo’ will be between zero and ‘segmentLength’, and so will ‘segmentLength’ minus ‘howLongAgo’. IEEE floating-point
math guarantees correct rounding so when we divide by ‘segmentLength’ we are guaranteed that ‘t’ will be from 0.0 to 1.0. No clamping needed, even with floats.
This real example demonstrates a few things:
• Any time you add or subtract floats of widely varying magnitudes you need to watch for loss of precision
• Sometimes using ‘double’ instead of ‘float’ is the correct solution, but often a more stable algorithm is more important
• CalcT should probably use double (to give sufficient precision after many hours of gameplay)
Your compiler is trying to tell you something…
With Visual C++ on the default warning level you will get warning C4244 when you assign a double to a float:
warning C4244: ‘initializing’ : conversion from ‘double’ to ‘float’, possible loss of data
Possible loss of data is not necessarily a problem, but it can be. Suppressing warnings, with #pragma warning or with a cast, is something that should be done thoughtfully, after understanding the
issue. Otherwise the compiler might say “I told you so” when your game fails after a twenty-four hour soak test.
Does it matter?
For some game types this problem may be irrelevant. Many games finish in less than an hour and a float that holds 3,600 (seconds) still has sub-millisecond accuracy, which is enough for most
purposes. This means that for those game types you should be fine storing time in a float, as long as you reset the zero-point of GetTime() at the beginning of each game, and as long as the clock
stops running when the game is paused.
For other game types – probably the majority of games – you need to do your time calculations using a double or uint64_t. I’ve seen problems on multiple games who failed to follow this rule. The
problems are particularly tedious to track down and fix because they may take many hours to show up.
Store your time values in a double, starting at 2^32 seconds, and then you don’t need to worry, at least not as much, as long as you avoid unstable algorithms.
A lot of people have commented on this article and said that the justification for using double instead of 64-bit integers is not very strong. I agree that either one will work, however I think that
double has a couple of advantages. One is developer convenience. A floating point number like 1.73 is far easier to comprehend than 1730 (fixed-point with ms accuracy) and it has more precision. The
more precision you give to a fixed-point integer the more unwieldy the numbers get, and there is a real cost to this.
The other reason is game industry specific. When a game does time calculations it typically uses the time values for physics, AI, and graphics, and these systems typically need floating-point
numbers. So, it turns out that you cannot avoid floating-point time. Therefore, you might as well do it in the first place, and do it right. Most games already use floating-point numbers for time – I
just want to encourage them to not use ‘float’.
It’s also interesting to note that Apple uses double for time – NSTimeInterval is a double. As they say: “NSTimeInterval is always specified in seconds; it yields sub-millisecond precision over a
range of 10,000 years.”
Next time…
On the next post I think it might finally be time to start jumping into the delicate subject of how to compare floating-point numbers, with the many subtleties involved. Previous articles in this
series, and other posts, can be found here.
5 Responses to Don’t Store That in a Float
1. Your writing is very clean and easy to understand. I don’t have any use for most of your topics covered (currently in C# land) but I still enjoy reading it. Thanks.
2. You are off by an order of magnitude on the diameter of the sun. It’s about 100x as wide as Earth, not 10x. So the last 2 entries in your distance table need to be updated.
□ Good catch. I’m not sure how I missed that. Fixed.
3. Dekker found some nice properties about summation and multiplication of floating point values, and how to make them accurate. Take a look at that (template versions of the dekker alorithms
published by Takeshi Ogita and S.M. Rump et al in SIAM Journal on Scientific Computing 26
(2005), Nr. 6, S. 1995–1988:
void two_sum ( T a , T b , T & x , T & y ) {
x = a + b;
T z = x – a;
y = (a – (x-z ) ) + (b-z ) ;
void split ( T & x , T & y , T const& a ) {
T c = T ( ( 1UL << ( ( float_traits : : mantissa_bits >> 1)
+ float_traits : : mantissa_bits%2)) + 1) * a ;
x = c – ( c-a );
y = a – x;
void two_product ( T a , T b , T & x , T & y ) {
x = a * b;
T a1 , a2 ;
split ( a1 , a2 , a ) ;
T b1 , b2 ;
split ( b1 , b2 , b ) ;
y = a2*b2 – ( ( ( x – a1 * b1 ) – a2 * b1 ) – a1 * b2 ) ;
I hope it gets reasonably formated. The first one makes x = float( a + b ) and x + y = a + b if float would have infinite precision. So y contains the error of the limited floating point
Something similar can be stated for two_product. x = float(a*b) and x+y =a*b.
All this only works if the compiler does optimize away the floating point operations.
The nice thing about this is that one can easily create summations and multiplications with higher accuracy, by keeping the error term (y) and reusing it in following operations. I once
implemented a matrix expression template library that execute 100% accurate scalar products, using the following algorithm (invented by S.M Rump, Takeshi Ogita et al “Accurate Floating Point
Summation” 2006):
template // faster than two_sum works only for a >= b
void fast_two_sum ( T a , T b , T & x , T & y ) {
x = a + b;
T q = x – a;
y = b-q;
typename T::value_type accurate_sum ( T & vec )
typedef typename T::value_type value_type;
size_t n = num_elements ( vec );
if ( n == 0 ) return 0;
value_type mu = std::abs ( vec ( 0 ) );
for ( size_t i = 1; i != n; ++i )
mu = std::max ( std::abs( vec(i) ), mu );
value_type Ms = next_power_two ( value_type ( n+2));
value_type sigma = Ms*next_power_two( mu );
value_type phi = std::numeric_limits::epsilon ( ) * Ms;
value_type factor = value_type(2)*phi*Ms;
if ( ! check_extraction_parameters ( phi, sigma, factor ) )
return simple_sum ( vec );
value_type t = 0;
T q;
while ( true ) {
q = elementwise_sub ( elementwise_add ( sigma, vec ), sigma );
value_type tau = simple_sum ( q );
vec = vec – q;
value_type tau1, tau2;
fast_two_sum ( t, tau, tau1, tau2 );
if( std::abs ( tau1 ) >= factor*sigma
|| sigma <= std::numeric_limits::denorm_min() )
return tau1 + ( tau2 + simple_sum ( vec ) );
t = tau1;
sigma = phi*sigma;
return 0;
The algorithm walks through the exponent until the operands no longer add relevant values. If the vector has all operands within a similar mantissa range the algorithm terminates sooner. The
result is accurate for the given floating point type.
□ Very cool.
A similar property is that if you have a compiler that generates fmadd instructions (fused multiply add where rounding doesn’t occur until after the add) then this calculation:
a * b + a * -b
typically gets compiled as fmad(a, b, a * -b). That is, the “a * -b” is done as a normal multiply, and the “a * b” is done as part of an fmadd. The net result is that the result is the error
in a * b. That’s a cool property I think.
This entry was posted in AltDevBlogADay, Floating Point, Programming and tagged double, float, floating point, game time, microseconds, precision, time resolution. Bookmark the permalink.
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What is the Taylor polynomial of degree 2 for g near 8?
May 18th 2010, 10:12 PM #1
Junior Member
Apr 2010
What is the Taylor polynomial of degree 2 for g near 8?
Suppose (a) What is the Taylor polynomial of degree 2 for
I got -4-1(x-8), but it's wrong.
(b) What is the Taylor polynomial of degree 3 for
I got -4-1(x-8) + (3/2)(x-8)^2, but it's wrong.
(c) Use the two polynomials that you found in parts (a) and (b) to approximate
I got -4-1(.1), but it's wrong
Do you understand what "degree" means? A polynomial is of degree "n" if and only if the highest power in the polynomial is n (I bet you learned that long ago!).
4- 1(x- 8) is wrong because it is of degree 1, not 2.
$-4-1(x-8) + (3/2)(x-8)^2$ is wrong because it is of degree 2, not 3.
You seem be thinking that "degree" is the number of terms in the polynomial- it isn't. A polynomial of degree n typically has n+1 terms.
the nth degree adds the term...
Thanks! do you know how to do part c?
c) Use the two polynomials that you found in parts (a) and (b) to approximate
May 19th 2010, 03:30 AM #2
MHF Contributor
Apr 2005
May 19th 2010, 03:48 PM #3
May 2010
May 25th 2010, 03:16 PM #4
Junior Member
Apr 2010
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NAG Library
NAG Library Routine Document
1 Purpose
D03PFF integrates a system of linear or nonlinear convection-diffusion equations in one space dimension, with optional source terms. The system must be posed in conservative form. Convection terms
are discretized using a sophisticated upwind scheme involving a user-supplied numerical flux function based on the solution of a Riemann problem at each mesh point. The method of lines is employed to
reduce the PDEs to a system of ordinary differential equations (ODEs), and the resulting system is solved using a backward differentiation formula (BDF) method.
2 Specification
SUBROUTINE D03PFF ( NPDE, TS, TOUT, PDEDEF, NUMFLX, BNDARY, U, NPTS, X, ACC, TSMAX, RSAVE, LRSAVE, ISAVE, LISAVE, ITASK, ITRACE, IND, IFAIL)
INTEGER NPDE, NPTS, LRSAVE, ISAVE(LISAVE), LISAVE, ITASK, ITRACE, IND, IFAIL
REAL (KIND=nag_wp) TS, TOUT, U(NPDE,NPTS), X(NPTS), ACC(2), TSMAX, RSAVE(LRSAVE)
EXTERNAL PDEDEF, NUMFLX, BNDARY
3 Description
D03PFF integrates the system of convection-diffusion equations in conservative form:
$∑j=1NPDEPi,j ∂Uj ∂t + ∂Fi ∂x =Ci ∂Di ∂x +Si,$ (1)
or the hyperbolic convection-only system:
$∂Ui ∂t + ∂Fi ∂x =0,$ (2)
$i=1,2,\dots ,{\mathbf{NPDE}}\text{, }a\le x\le b\text{, }t\ge {t}_{0}$
, where the vector
is the set of solution values
$U x,t = U 1 x,t ,…, U NPDE x,t T .$
The functions
depend on
; and
depends on
, where
is the spatial derivative of
. Note that
must not depend on any space derivatives; and none of the functions may depend on time derivatives. In terms of conservation laws,
$\frac{{C}_{i}\partial {D}_{i}}{\partial x}$
are the convective flux, diffusion and source terms respectively.
The integration in time is from ${t}_{0}$ to ${t}_{\mathrm{out}}$, over the space interval $a\le x\le b$, where $a={x}_{1}$ and $b={x}_{{\mathbf{NPTS}}}$ are the leftmost and rightmost points of a
user-defined mesh ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{NPTS}}}$. The initial values of the functions $U\left(x,t\right)$ must be given at $t={t}_{0}$.
The PDEs are approximated by a system of ODEs in time for the values of
at mesh points using a spatial discretization method similar to the central-difference scheme used in
, but with the flux
replaced by a
numerical flux
, which is a representation of the flux taking into account the direction of the flow of information at that point (i.e., the direction of the characteristics). Simple central differencing of the
numerical flux then becomes a sophisticated upwind scheme in which the correct direction of upwinding is automatically achieved.
The numerical flux vector,
say, must be calculated by you in terms of the
values of the solution vector
(denoted by
respectively), at each mid-point of the mesh
, for
$j=2,3,\dots ,{\mathbf{NPTS}}$
. The left and right values are calculated by D03PFF from two adjacent mesh points using a standard upwind technique combined with a Van Leer slope-limiter (see
LeVeque (1990)
). The physically correct value for
is derived from the solution of the Riemann problem given by
$∂Ui ∂t + ∂Fi ∂y =0,$ (3)
, i.e.,
corresponds to
, with discontinuous initial values
, using an
approximate Riemann solver
. This applies for either of the systems
; the numerical flux is independent of the functions
. A description of several approximate Riemann solvers can be found in
LeVeque (1990)
Berzins et al. (1989)
. Roe's scheme (see
Roe (1981)
) is perhaps the easiest to understand and use, and a brief summary follows. Consider the system of PDEs
or equivalently
. Provided the system is linear in
, i.e., the Jacobian matrix
does not depend on
, the numerical flux
is given by
$F^=12 FL+FR-12∑k=1NPDEαkλkek,$ (4)
) is the flux
calculated at the left (right) value of
, denoted by
); the
${\lambda }_{k}$
are the eigenvalues of
; the
are the right eigenvectors of
; and the
${\alpha }_{k}$
are defined by
$UR-UL=∑k=1NPDEαkek.$ (5)
An example is given in
Section 9
If the system is nonlinear, Roe's scheme requires that a linearized Jacobian is found (see
Roe (1981)
The functions
not ${F}_{i}$
) must be specified in a
. The numerical flux
must be supplied in a separate
. For problems in the form
, the actual argument D03PFP may be used for
. D03PFP is included in the NAG Library and sets the matrix with entries
to the identity matrix, and the functions
to zero.
The boundary condition specification has sufficient flexibility to allow for different types of problems. For second-order problems, i.e.,
depending on
, a boundary condition is required for each PDE at both boundaries for the problem to be well-posed. If there are no second-order terms present, then the continuous PDE problem generally requires
exactly one boundary condition for each PDE, that is
boundary conditions in total. However, in common with most discretization schemes for first-order problems, a
numerical boundary condition
is required at the other boundary for each PDE. In order to be consistent with the characteristic directions of the PDE system, the numerical boundary conditions must be derived from the solution
inside the domain in some manner (see below). You must supply both types of boundary conditions, i.e., a total of
conditions at each boundary point.
The position of each boundary condition should be chosen with care. In simple terms, if information is flowing into the domain then a physical boundary condition is required at that boundary, and a
numerical boundary condition is required at the other boundary. In many cases the boundary conditions are simple, e.g., for the linear advection equation. In general you should calculate the
characteristics of the PDE system and specify a physical boundary condition for each of the characteristic variables associated with incoming characteristics, and a numerical boundary condition for
each outgoing characteristic.
A common way of providing numerical boundary conditions is to extrapolate the characteristic variables from the inside of the domain. Note that only linear extrapolation is allowed in this routine
(for greater flexibility the routine
should be used). For problems in which the solution is known to be uniform (in space) towards a boundary during the period of integration then extrapolation is unnecessary; the numerical boundary
condition can be supplied as the known solution at the boundary. Examples can be found in
Section 9
The boundary conditions must be specified in
in the form
$GiLx,t,U=0 at x=a, i=1,2,…,NPDE,$ (6)
at the left-hand boundary, and
$GiRx,t,U=0 at x=b, i=1,2,…,NPDE,$ (7)
at the right-hand boundary.
Note that spatial derivatives at the boundary are not passed explicitly to
, but they can be calculated using values of
at and adjacent to the boundaries if required. However, it should be noted that instabilities may occur if such one-sided differencing opposes the characteristic direction at the boundary.
The problem is subject to the following restrictions:
(i) ${P}_{i,j}$, ${F}_{i}$, ${C}_{i}$ and ${S}_{i}$ must not depend on any space derivatives;
(ii) ${P}_{i,j}$, ${F}_{i}$, ${C}_{i}$, ${D}_{i}$ and ${S}_{i}$ must not depend on any time derivatives;
(iii) ${t}_{0}<{t}_{\mathrm{out}}$, so that integration is in the forward direction;
(iv) The evaluation of the terms ${P}_{i,j}$, ${C}_{i}$, ${D}_{i}$ and ${S}_{i}$ is done by calling the PDEDEF at a point approximately midway between each pair of mesh points in turn. Any
discontinuities in these functions must therefore be at one or more of the mesh points ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{NPTS}}}$;
(v) At least one of the functions ${P}_{i,j}$ must be nonzero so that there is a time derivative present in the PDE problem.
In total there are ${\mathbf{NPDE}}×{\mathbf{NPTS}}$ ODEs in the time direction. This system is then integrated forwards in time using a BDF method.
For further details of the algorithm, see
Pennington and Berzins (1994)
and the references therein.
4 References
Berzins M, Dew P M and Furzeland R M (1989) Developing software for time-dependent problems using the method of lines and differential-algebraic integrators Appl. Numer. Math. 5 375–397
Hirsch C (1990) Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows John Wiley
LeVeque R J (1990) Numerical Methods for Conservation Laws Birkhäuser Verlag
Pennington S V and Berzins M (1994) New NAG Library software for first-order partial differential equations ACM Trans. Math. Softw. 20 63–99
Roe P L (1981) Approximate Riemann solvers, parameter vectors, and difference schemes J. Comput. Phys. 43 357–372
5 Parameters
1: NPDE – INTEGERInput
On entry: the number of PDEs to be solved.
Constraint: ${\mathbf{NPDE}}\ge 1$.
2: TS – REAL (KIND=nag_wp)Input/Output
On entry: the initial value of the independent variable $t$.
On exit
: the value of
corresponding to the solution values in
. Normally
Constraint: ${\mathbf{TS}}<{\mathbf{TOUT}}$.
3: TOUT – REAL (KIND=nag_wp)Input
On entry: the final value of $t$ to which the integration is to be carried out.
4: PDEDEF – SUBROUTINE, supplied by the NAG Library or the user.External Procedure
must evaluate the functions
which partially define the system of PDEs.
may depend on
may depend on
is called approximately midway between each pair of mesh points in turn by D03PFF. The actual argument D03PFP may be used for
for problems in the form
. (D03PFP is included in the NAG Library.)
The specification of
SUBROUTINE PDEDEF ( NPDE, T, X, U, UX, P, C, D, S, IRES)
INTEGER NPDE, IRES
REAL (KIND=nag_wp) T, X, U(NPDE), UX(NPDE), P(NPDE,NPDE), C(NPDE), D(NPDE), S(NPDE)
1: NPDE – INTEGERInput
On entry: the number of PDEs in the system.
2: T – REAL (KIND=nag_wp)Input
On entry: the current value of the independent variable $t$.
3: X – REAL (KIND=nag_wp)Input
On entry: the current value of the space variable $x$.
4: U(NPDE) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{U}}\left(\mathit{i}\right)$ contains the value of the component ${U}_{\mathit{i}}\left(x,t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$.
5: UX(NPDE) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{UX}}\left(\mathit{i}\right)$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}\left(x,t\right)}{\partial x}$, for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}
6: P(NPDE,NPDE) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{P}}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of ${P}_{\mathit{i},\mathit{j}}\left(x,t,U\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$ and $\mathit
{j}=1,2,\dots ,{\mathbf{NPDE}}$.
7: C(NPDE) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{C}}\left(\mathit{i}\right)$ must be set to the value of ${C}_{\mathit{i}}\left(x,t,U\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$.
8: D(NPDE) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{D}}\left(\mathit{i}\right)$ must be set to the value of ${D}_{\mathit{i}}\left(x,t,U,{U}_{x}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$.
9: S(NPDE) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{S}}\left(\mathit{i}\right)$ must be set to the value of ${S}_{\mathit{i}}\left(x,t,U\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$.
10: IRES – INTEGERInput/Output
On entry: set to $-1\text{ or }1$.
On exit
: should usually remain unchanged. However, you may set
to force the integration routine to take certain actions as described below:
Indicates to the integrator that control should be passed back immediately to the calling subroutine with the error indicator set to ${\mathbf{IFAIL}}={\mathbf{6}}$.
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{IRES}}=3$ when a physically meaningless
input or output value has been generated. If you consecutively set ${\mathbf{IRES}}=3$, then D03PFF returns to the calling subroutine with the error indicator set to ${\mathbf{IFAIL}}={\
must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D03PFF is called. Parameters denoted as
be changed by this procedure.
5: NUMFLX – SUBROUTINE, supplied by the user.External Procedure
must supply the numerical flux for each PDE given the
values of the solution vector
is called approximately midway between each pair of mesh points in turn by D03PFF.
The specification of
SUBROUTINE NUMFLX ( NPDE, T, X, ULEFT, URIGHT, FLUX, IRES)
INTEGER NPDE, IRES
REAL (KIND=nag_wp) T, X, ULEFT(NPDE), URIGHT(NPDE), FLUX(NPDE)
1: NPDE – INTEGERInput
On entry: the number of PDEs in the system.
2: T – REAL (KIND=nag_wp)Input
On entry: the current value of the independent variable $t$.
3: X – REAL (KIND=nag_wp)Input
On entry: the current value of the space variable $x$.
4: ULEFT(NPDE) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{ULEFT}}\left(\mathit{i}\right)$ contains the left value of the component ${U}_{\mathit{i}}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$.
5: URIGHT(NPDE) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{URIGHT}}\left(\mathit{i}\right)$ contains the right value of the component ${U}_{\mathit{i}}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$.
6: FLUX(NPDE) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{FLUX}}\left(\mathit{i}\right)$ must be set to the numerical flux ${\stackrel{^}{F}}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$.
7: IRES – INTEGERInput/Output
On entry: set to $-1\text{ or }1$.
On exit
: should usually remain unchanged. However, you may set
to force the integration routine to take certain actions as described below:
Indicates to the integrator that control should be passed back immediately to the calling subroutine with the error indicator set to ${\mathbf{IFAIL}}={\mathbf{6}}$.
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{IRES}}=3$ when a physically meaningless
input or output value has been generated. If you consecutively set ${\mathbf{IRES}}=3$, then D03PFF returns to the calling subroutine with the error indicator set to ${\mathbf{IFAIL}}={\
must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D03PFF is called. Parameters denoted as
be changed by this procedure.
6: BNDARY – SUBROUTINE, supplied by the user.External Procedure
must evaluate the functions
which describe the physical and numerical boundary conditions, as given by
The specification of
SUBROUTINE BNDARY ( NPDE, NPTS, T, X, U, IBND, G, IRES)
INTEGER NPDE, NPTS, IBND, IRES
REAL (KIND=nag_wp) T, X(NPTS), U(NPDE,3), G(NPDE)
1: NPDE – INTEGERInput
On entry: the number of PDEs in the system.
2: NPTS – INTEGERInput
On entry: the number of mesh points in the interval $\left[a,b\right]$.
3: T – REAL (KIND=nag_wp)Input
On entry: the current value of the independent variable $t$.
4: X(NPTS) – REAL (KIND=nag_wp) arrayInput
On entry: the mesh points in the spatial direction. ${\mathbf{X}}\left(1\right)$ corresponds to the left-hand boundary, $a$, and ${\mathbf{X}}\left({\mathbf{NPTS}}\right)$ corresponds to the
right-hand boundary, $b$.
5: U(NPDE,$3$) – REAL (KIND=nag_wp) arrayInput
On entry
: contains the value of solution components in the boundary region.
If ${\mathbf{IBND}}=0$, ${\mathbf{U}}\left(\mathit{i},\mathit{j}\right)$ contains the value of the component ${U}_{\mathit{i}}\left(\mathrm{x},t\right)$ at $x={\mathbf{X}}\left(\mathit{j}\
right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$ and $\mathit{j}=1,2,3$.
If ${\mathbf{IBND}}e 0$, ${\mathbf{U}}\left(\mathit{i},\mathit{j}\right)$ contains the value of the component ${U}_{\mathit{i}}\left(x,t\right)$ at $x={\mathbf{X}}\left({\mathbf{NPTS}}-\
mathit{j}+1\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$ and $\mathit{j}=1,2,3$.
6: IBND – INTEGERInput
On entry
: specifies which boundary conditions are to be evaluated.
BNDARY must evaluate the left-hand boundary condition at $x=a$.
${\mathbf{IBND}}e 0$
BNDARY must evaluate the right-hand boundary condition at $x=b$.
7: G(NPDE) – REAL (KIND=nag_wp) arrayOutput
On exit
must contain the
th component of either
, depending on the value of
, for
$\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$
8: IRES – INTEGERInput/Output
On entry: set to $-1\text{ or }1$.
On exit
: should usually remain unchanged. However, you may set
to force the integration routine to take certain actions as described below:
Indicates to the integrator that control should be passed back immediately to the calling subroutine with the error indicator set to ${\mathbf{IFAIL}}={\mathbf{6}}$.
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{IRES}}=3$ when a physically meaningless
input or output value has been generated. If you consecutively set ${\mathbf{IRES}}=3$, then D03PFF returns to the calling subroutine with the error indicator set to ${\mathbf{IFAIL}}={\
must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D03PFF is called. Parameters denoted as
be changed by this procedure.
7: U(NPDE,NPTS) – REAL (KIND=nag_wp) arrayInput/Output
On entry: ${\mathbf{U}}\left(\mathit{i},\mathit{j}\right)$ must contain the initial value of ${U}_{\mathit{i}}\left(x,t\right)$ at $x={\mathbf{X}}\left(\mathit{j}\right)$ and $t={\mathbf{TS}}$,
for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{NPTS}}$.
On exit: ${\mathbf{U}}\left(\mathit{i},\mathit{j}\right)$ will contain the computed solution ${U}_{\mathit{i}}\left(x,t\right)$ at $x={\mathbf{X}}\left(\mathit{j}\right)$ and $t={\mathbf{TS}}$,
for $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{NPTS}}$.
8: NPTS – INTEGERInput
On entry: the number of mesh points in the interval $\left[a,b\right]$.
Constraint: ${\mathbf{NPTS}}\ge 3$.
9: X(NPTS) – REAL (KIND=nag_wp) arrayInput
On entry: the mesh points in the space direction. ${\mathbf{X}}\left(1\right)$ must specify the left-hand boundary, $a$, and ${\mathbf{X}}\left({\mathbf{NPTS}}\right)$ must specify the right-hand
boundary, $b$.
Constraint: ${\mathbf{X}}\left(1\right)<{\mathbf{X}}\left(2\right)<\cdots <{\mathbf{X}}\left({\mathbf{NPTS}}\right)$.
10: ACC($2$) – REAL (KIND=nag_wp) arrayInput
On entry
: the components of
contain the relative and absolute error tolerances used in the local error test in the time integration.
is the estimated error for
at the
th mesh point, the error test is
Constraint: ${\mathbf{ACC}}\left(1\right)$ and ${\mathbf{ACC}}\left(2\right)\ge 0.0$ (but not both zero).
11: TSMAX – REAL (KIND=nag_wp)Input
On entry: the maximum absolute step size to be allowed in the time integration. If ${\mathbf{TSMAX}}=0.0$ then no maximum is imposed.
Constraint: ${\mathbf{TSMAX}}\ge 0.0$.
12: RSAVE(LRSAVE) – REAL (KIND=nag_wp) arrayCommunication Array
need not be set on entry.
must be unchanged from the previous call to the routine because it contains required information about the iteration.
13: LRSAVE – INTEGERInput
On entry
: the dimension of the array
as declared in the (sub)program from which D03PFF is called.
Constraint: ${\mathbf{LRSAVE}}\ge \left(11+9×{\mathbf{NPDE}}\right)×{\mathbf{NPDE}}×{\mathbf{NPTS}}+\left(32+3×{\mathbf{NPDE}}\right)×{\mathbf{NPDE}}+7×\phantom{\rule{0ex}{0ex}}{\mathbf{NPTS}}
14: ISAVE(LISAVE) – INTEGER arrayCommunication Array
need not be set on entry.
must be unchanged from the previous call to the routine because it contains required information about the iteration. In particular:
Contains the number of steps taken in time.
Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the
functions in the boundary conditions.
Contains the number of Jacobian evaluations performed by the time integrator.
Contains the order of the last backward differentiation formula method used.
Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the $LU$ decomposition of
the Jacobian matrix.
15: LISAVE – INTEGERInput
On entry
: the dimension of the array
as declared in the (sub)program from which D03PFF is called.
Constraint: ${\mathbf{LISAVE}}\ge {\mathbf{NPDE}}×{\mathbf{NPTS}}+24$.
16: ITASK – INTEGERInput
On entry
: the task to be performed by the ODE integrator.
Normal computation of output values ${\mathbf{U}}$ at $t={\mathbf{TOUT}}$ (by overshooting and interpolating).
Take one step in the time direction and return.
Stop at first internal integration point at or beyond $t={\mathbf{TOUT}}$.
Constraint: ${\mathbf{ITASK}}=1$, $2$ or $3$.
17: ITRACE – INTEGERInput
On entry
: the level of trace information required from D03PFF and the underlying ODE solver.
may take the value
No output is generated.
Only warning messages from the PDE solver are printed on the current error message unit (see X04AAF).
Output from the underlying ODE solver is printed on the current advisory message unit (see X04ABF). This output contains details of Jacobian entries, the nonlinear iteration and the time
integration during the computation of the ODE system.
If ${\mathbf{ITRACE}}<-1$, then $-1$ is assumed and similarly if ${\mathbf{ITRACE}}>3$, then $3$ is assumed.
The advisory messages are given in greater detail as
increases. You are advised to set
, unless you are experienced with
sub-chapter D02M–N
18: IND – INTEGERInput/Output
On entry
: indicates whether this is a continuation call or a new integration.
Starts or restarts the integration in time.
Continues the integration after an earlier exit from the routine. In this case, only the parameters TOUT and IFAIL should be reset between calls to D03PFF.
Constraint: ${\mathbf{IND}}=0$ or $1$.
On exit: ${\mathbf{IND}}=1$.
19: IFAIL – INTEGERInput/Output
On entry
must be set to
$-1\text{ or }1$
. If you are unfamiliar with this parameter you should refer to
Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$-1\text{ or }1$
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit
unless the routine detects an error or a warning has been flagged (see
Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by
Errors or warnings detected by the routine:
On entry, ${\mathbf{TS}}\ge {\mathbf{TOUT}}$,
or ${\mathbf{TOUT}}-{\mathbf{TS}}$ is too small,
or ${\mathbf{ITASK}}=1$, $2$ or $3$,
or ${\mathbf{NPTS}}<3$,
or ${\mathbf{NPDE}}<1$,
or ${\mathbf{IND}}e 0$ or $1$,
or incorrect user-defined mesh, i.e., ${\mathbf{X}}\left(i\right)\ge {\mathbf{X}}\left(i+1\right)$ for some $i=1,2,\dots ,{\mathbf{NPTS}}-1$,
or LRSAVE or LISAVE are too small,
or ${\mathbf{IND}}=1$ on initial entry to D03PFF,
or ${\mathbf{ACC}}\left(1\right)$ or ${\mathbf{ACC}}\left(2\right)<0.0$,
or ${\mathbf{ACC}}\left(1\right)$ or ${\mathbf{ACC}}\left(2\right)$ are both zero,
or ${\mathbf{TSMAX}}<0.0$.
The underlying ODE solver cannot make any further progress, with the values of
, across the integration range from the current point
. The components of
contain the computed values at the current point
In the underlying ODE solver, there were repeated error test failures on an attempted step, before completing the requested task, but the integration was successful as far as $t={\mathbf{TS}}$.
The problem may have a singularity, or the error requirement may be inappropriate. Incorrect specification of boundary conditions may also result in this error.
In setting up the ODE system, the internal initialization routine was unable to initialize the derivative of the ODE system. This could be due to the fact that
was repeatedly set to
in one of
when the residual in the underlying ODE solver was being evaluated. Incorrect specification of boundary conditions may also result in this error.
In solving the ODE system, a singular Jacobian has been encountered. Check the problem formulation.
When evaluating the residual in solving the ODE system,
was set to
in at least one of
. Integration was successful as far as
The values of ${\mathbf{ACC}}\left(1\right)$ and ${\mathbf{ACC}}\left(2\right)$ are so small that the routine is unable to start the integration in time.
In either,
was set to an invalid value.
${\mathbf{IFAIL}}=9$ (D02NNF)
A serious error has occurred in an internal call to the specified routine. Check the problem specification and all parameters and array dimensions. Setting
may provide more information. If the problem persists, contact
The required task has been completed, but it is estimated that a small change in the values of
is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when
${\mathbf{ITASK}}e 2$
An error occurred during Jacobian formulation of the ODE system (a more detailed error description may be directed to the current advisory message unit when ${\mathbf{ITRACE}}\ge 1$).
Not applicable.
Not applicable.
One or more of the functions ${P}_{i,j}$, ${D}_{i}$ or ${C}_{i}$ was detected as depending on time derivatives, which is not permissible.
7 Accuracy
D03PFF controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on both the number of mesh points and on their
distribution in space. In the time integration only the local error over a single step is controlled and so the accuracy over a number of steps cannot be guaranteed. You should therefore test the
effect of varying the components of the accuracy parameter,
D03PFF is designed to solve systems of PDEs in conservative form, with optional source terms which are independent of space derivatives, and optional second-order diffusion terms. The use of the
routine to solve systems which are not naturally in this form is discouraged, and you are advised to use one of the central-difference schemes for such problems.
You should be aware of the stability limitations for hyperbolic PDEs. For most problems with small error tolerances the ODE integrator does not attempt unstable time steps, but in some cases a
maximum time step should be imposed using
. It is worth experimenting with this parameter, particularly if the integration appears to progress unrealistically fast (with large time steps). Setting the maximum time step to the minimum mesh
size is a safe measure, although in some cases this may be too restrictive.
Problems with source terms should be treated with caution, as it is known that for large source terms stable and reasonable looking solutions can be obtained which are in fact incorrect, exhibiting
non-physical speeds of propagation of discontinuities (typically one spatial mesh point per time step). It is essential to employ a very fine mesh for problems with source terms and discontinuities,
and to check for non-physical propagation speeds by comparing results for different mesh sizes. Further details and an example can be found in
Pennington and Berzins (1994)
The time taken depends on the complexity of the system and on the accuracy requested.
9 Example
For this routine two examples are presented. There is a single example program for D03PFF, with a main program and the code to solve the two example problems given in Example 1 (EX1) and Example 2
Example 1 (EX1)
This example is a simple first-order system which illustrates the calculation of the numerical flux using Roe's approximate Riemann solver, and the specification of numerical boundary conditions
using extrapolated characteristic variables. The PDEs are
$∂U1 ∂t + ∂U1 ∂x + ∂U2 ∂x = 0, ∂U2 ∂t +4 ∂U1 ∂x + ∂U2 ∂x = 0,$
$x\in \left[0,1\right]$
$t\ge 0$
. The PDEs have an exact solution given by
$U1 x,t = 12 exp x + t + expx-3t + 14 sin 2 π x-3t 2 - sin 2 π x+t 2 + 2 t2 - 2 x t , U2 x,t = expx-3t - expx+t + 12 sin 2 π x-3t 2 + sin 2 π x - 3 t 2 + x2 + 5 t2 - 2 x t .$
The initial conditions are given by the exact solution. The characteristic variables are
corresponding to the characteristics given by
respectively. Hence a physical boundary condition is required for
at the left-hand boundary, and for
at the right-hand boundary (corresponding to the incoming characteristics); and a numerical boundary condition is required for
at the left-hand boundary, and for
at the right-hand boundary (outgoing characteristics). The physical boundary conditions are obtained from the exact solution, and the numerical boundary conditions are calculated by linear
extrapolation of the appropriate characteristic variable. The numerical flux is calculated using Roe's approximate Riemann solver: Using the notation in
Section 3
, the flux vector
and the Jacobian matrix
$F= U1+U2 4U1+U2 and A= 1 1 4 1 ,$
and the eigenvalues of
with right eigenvectors
${\left[\begin{array}{cc}1& 2\end{array}\right]}^{\mathrm{T}}$
${\left[\begin{array}{cc}-1& 2\end{array}\right]}^{\mathrm{T}}$
respectively. Using equation
${\alpha }_{k}$
are given by
$U1R-U1L U2R-U2L =α1 1 2 +α2 -1 2 ,$
that is
$α1 = 14 2 U1R - 2 U1L + U2R - U2L and α2 = 14 -2 U1R + 2 U1L + U2R - U2L .$
is given by
$FL = U1L+U2L 4U1L+U2L ,$
and similarly for
. From equation
, the numerical flux vector is
$F^ = 12 U1L+U2L+0U1R+U2R 4U1L+U2L+4U1R+U2R - 12 α1 3 1 2 - 12 α2 -1 -1 2 ,$
that is
$F^ = 12 3U1L-0U1R+32U2L+12 U2R 6U1L+ 2U1R+ 3U2L-0U2R .$
Example 2 (EX2)
This example is an advection-diffusion equation in which the flux term depends explicitly on
$∂U ∂t +x ∂U ∂x =ε ∂2U ∂x2 ,$
$x\in \left[-1,1\right]$
$0\le t\le 10$
. The parameter
is taken to be
. The two physical boundary conditions are
and the initial condition is
. The integration is run to steady state at which the solution is known to be
across the domain with a narrow boundary layer at both boundaries. In order to write the PDE in conservative form, a source term must be introduced, i.e.,
$∂U ∂t + ∂xU ∂x =ε ∂2U ∂x2 +U.$
As in Example 1, the numerical flux is calculated using the Roe approximate Riemann solver. The Riemann problem to solve locally is
in the flux term is assumed to be constant at a local level, and so using the notation in
Section 3
. The eigenvalue is
and the eigenvector (a scalar in this case) is
. The numerical flux is therefore
$F^= xUL if x≥0, xUR if x<0.$
9.1 Program Text
9.2 Program Data
9.3 Program Results
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Speed of shadow.
July 5th 2012, 09:49 AM #1
Jul 2012
Speed of shadow.
Hi guys,
I'm stuck on this problem.
A ball is dropped from a height of 100 ft, at which time its shadow is 500 ft from the ball. How fast is the shadow moving when the ball hits the ground? The ball falls with velocity 32 ft/sec,
and the shadow is cast by the sun.
The only thing I have so far is that the derivative of the height of the ball is -32. I have no idea where to go from there.
Re: Speed of shadow.
calculate the time in which ball hits the ground, and from that information u can get the akceleration of the shadow rest is easy. sry dont have much time maybe later i repost.
Last edited by Imo; July 5th 2012 at 10:41 AM.
Re: Speed of shadow.
... not projected, so initial velocity = 0.
I suspect this was 500 ft from the point on the ground directly under the ball, since the question is about the velocity of the shadow along the ground. If not, use Pythagoras to find this
horizontal distance.
Acceleration, surely?
So you can take the angle of its rays as constant.
So vertical and horizontal distances covered, and therefore velocities, will remain in proportion.
And the proportion is 5:1, or else root 24 to one, on the alternative interpretation of 'shadow is 500 ft from ball'.
Use v^2 = u^2 + 2as.
Last edited by tom@ballooncalculus; July 5th 2012 at 10:52 AM.
Re: Speed of shadow.
The problem is exactly how it's phrased in my book.
The way I see it is, you can make a right triangle from the initial position of the ball, the initial position of the shadow and the point on the ground where the ball will land. Are you saying
the angles within that triangle are constant, cause to me they're not,though I could be visualizing it wrong.
Re: Speed of shadow.
Word for word? What about 'falls with velocity 32 m/s' ? Not acceleration? If initial velocity, why not say? ...
Not equal to each other, but equal to their own corresponding counterparts in any smaller triangle later on.
The sun being so hugely far and high in the sky, we must be expected to assume that the vertical height of the triangle remains one fifth of the horizontal base.
Re: Speed of shadow.
Yes, that would be my interpretation. Initially, you have a right triangle with the ball at one vertex, the right angle 100 ft below the ball, and the third vertex 500 ft to one side of the right
angle. At each point, as the ball falls, the line from the ball to its shadow is parallel to the hypotenuse of the original triangle. By "similar triangles", when the ball is x feet above the
ground, taking y as the distance to its shadow, x/y= 100/500.
Re: Speed of shadow.
Well, if the angles are constant then the answer is -64(6)^(1/2).
Re: Speed of shadow.
Drop the minus sign as direction doesn't matter.
Now, where's the 6 from?
Are you using v^2 = u^2 + 2as ...?
And when you've got the vertical speed, scale appropriately (i.e. times 5 or, just conceivably, root 24).
Re: Speed of shadow.
Square root of 24 is equal to 2 times the square root of 6.
I'm assuming that the ball is falling at a constant velocity of 32 ft/sec. 32*2(6)^(1/2) is equal to 64(6)^(1/2).
What formula are you referring to cause I don't recognize it.
Re: Speed of shadow.
Oh, fair enough. But do this:
v^2 = u^2 + 2as
first. (Or be careful.) Where v is final velocity (of the ball when it reaches ground level), u = initial velocity = 0, a = acceleration = 32 m/s^2, and s = distance = 100.
Equations of motion - Wikipedia, the free encyclopedia #SUVAT_equations
One is bound to ask, "on which planet?!"
Last edited by tom@ballooncalculus; July 5th 2012 at 12:54 PM.
Re: Speed of shadow.
Forgot all about that equation, I calculated the time first and then used it to calculate the speed. That wouldn't have been necessary with your equation.
In this case I am assuming that the ball is accelerating at 32 ft/s^2.
July 5th 2012, 10:34 AM #2
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July 5th 2012, 12:18 PM #7
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Items where Subject is "H Engineering > H143 Structural Mechanics"
Number of items at this level: 46.
Autoparametric structure
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Matthew, P. (2013) Linear modal analysis of L-shaped beam structures. Mechanical Systems and Signal Processing, 38 (2). pp. 312-332. ISSN
Cauchy-Born rule
Zhang, Yu and Sansour, Carlo and Bingham, Chris (2013) Single-walled carbon nanotube modelling based on Cosserat surface theory. In: 12th WSEAS International Conference on Applications of Electrical
Engineering (AEE '13), January 30th - February 1st 2013, Cambridge, Mass..
Cosserat surface
Zhang, Yu and Sansour, Carlo and Bingham, Chris (2013) Single-walled carbon nanotube modelling based on Cosserat surface theory. In: 12th WSEAS International Conference on Applications of Electrical
Engineering (AEE '13), January 30th - February 1st 2013, Cambridge, Mass..
Dynamic Stiffness
Georgiades, Fotios and Scholtz, Peter (2010) Sensitivity of experimental dynamic stiffness of the vibrator-earth system. In: Near Surface 2010 – 16th European Meeting of Environmental and Engineering
Geophysics, 6 - 8 September 2010, Zurich.
Empirical interatomic potential
Zhang, Yu and Sansour, Carlo and Bingham, Chris (2013) Single-walled carbon nanotube modelling based on Cosserat surface theory. In: 12th WSEAS International Conference on Applications of Electrical
Engineering (AEE '13), January 30th - February 1st 2013, Cambridge, Mass..
Essential nonlinearity
Panagopoulos, Panayotis and Georgiades, Fotios and Tsakirtzis, Stylianos and Vakakis, Alexander, F. and Bergman, Lawrence, A. (2007) Multi-scaled analysis of the damped dynamics of an elastic rod
with an essentially nonlinear end attachment. International Journal of Solids and Structures, 44 (18-19). pp. 6256-6278. ISSN 0020-7683
Euler-Bernoulli beams
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Matthew, P. (2013) Linear modal analysis of L-shaped beam structures. Mechanical Systems and Signal Processing, 38 (2). pp. 312-332. ISSN
Flexible systems
Georgiades, Fotios and Vakakis, Alexander, F. and Kerschen, Gaetan (2007) Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end
attachment. International Journal of Non-Linear Mechanics, 42 (5). pp. 773-788. ISSN 0020-7462
Georgiades, Fotios and Latalski, Jaroslaw and Warminski, Jerzy (2011) Mode shapes variation of a composite beam with piezoelectric patches. Transaction of Aviation Institute / Prace- Instytut
Lotnictwa , 218 . pp. 36-43. ISSN 0509-6669
Georgiades, Fotios and Peeters, Maxime and Kerschen, Gaetan and Golinval, Jean-Claude and Ruzzene, Massimo (2007) Localization of energy in a perfectly symmetric bladed disk assembly due to
nonlinearities. ASME Proceedings | Advances in Aerospace Technology, 1 . pp. 229-237. ISSN UNSPECIFIED
Georgiades, Fotios and Scholtz, Peter (2010) Sensitivity of experimental dynamic stiffness of the vibrator-earth system. In: Near Surface 2010 – 16th European Meeting of Environmental and Engineering
Geophysics, 6 - 8 September 2010, Zurich.
Georgiades, Fotios and Vakakis, Alexander, F. (2007) Dynamics of a linear beam with an attached local nonlinear energy sink. Communications in Nonlinear Science and Numerical Simulation, 12 (5). pp.
643-651. ISSN 1007-5704
Georgiades, Fotios and Vakakis, Alexander, F. (2009) Passive targeted energy transfers and strong modal interactions in the dynamics of a thin plate with strongly nonlinear attachments. International
Journal of Solids and Structures, 46 (11-12). pp. 2330-2353. ISSN 0020-7683
Georgiades, Fotios and Vakakis, Alexander, F. and Kerschen, Gaetan (2007) Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end
attachment. International Journal of Non-Linear Mechanics, 42 (5). pp. 773-788. ISSN 0020-7462
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Mathhew, P. (2013) Towards linear modal analysis for an L-shaped beam: equations of motion. Mechanics Research Communications, 47 . pp. 50-60.
ISSN 0093-6413
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Matthew, P. (2013) Linear modal analysis of L-shaped beam structures. Mechanical Systems and Signal Processing, 38 (2). pp. 312-332. ISSN
Panagopoulos, Panayotis and Georgiades, Fotios and Tsakirtzis, Stylianos and Vakakis, Alexander, F. and Bergman, Lawrence, A. (2007) Multi-scaled analysis of the damped dynamics of an elastic rod
with an essentially nonlinear end attachment. International Journal of Solids and Structures, 44 (18-19). pp. 6256-6278. ISSN 0020-7683
L-shaped beam
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Mathhew, P. (2013) Towards linear modal analysis for an L-shaped beam: equations of motion. Mechanics Research Communications, 47 . pp. 50-60.
ISSN 0093-6413
L-shaped beam structure
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Matthew, P. (2013) Linear modal analysis of L-shaped beam structures. Mechanical Systems and Signal Processing, 38 (2). pp. 312-332. ISSN
Georgiades, Fotios and Warminski, Jerzy and Cartmell , Matthew (2012) Linear modal analysis of L-shaped beam structures: parametric studies. Journal of Physics: Conference Series, 382 (1). p. 2006.
ISSN 1742-6588
Localized NNMs
Georgiades, Fotios and Peeters, Maxime and Kerschen, Gaetan and Golinval, Jean-Claude and Ruzzene, Massimo (2007) Localization of energy in a perfectly symmetric bladed disk assembly due to
nonlinearities. ASME Proceedings | Advances in Aerospace Technology, 1 . pp. 229-237. ISSN UNSPECIFIED
Modal analysis
Georgiades, Fotios and Latalski, Jaroslaw and Warminski, Jerzy (2011) Mode shapes variation of a composite beam with piezoelectric patches. Transaction of Aviation Institute / Prace- Instytut
Lotnictwa , 218 . pp. 36-43. ISSN 0509-6669
Multi-scaled analysis
Panagopoulos, Panayotis and Georgiades, Fotios and Tsakirtzis, Stylianos and Vakakis, Alexander, F. and Bergman, Lawrence, A. (2007) Multi-scaled analysis of the damped dynamics of an elastic rod
with an essentially nonlinear end attachment. International Journal of Solids and Structures, 44 (18-19). pp. 6256-6278. ISSN 0020-7683
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Matthew, P. (2012) Nonlinear modal analysis of an L-shape beam structure. In: 4th International Conference on Localization, Energy Transfer and
Nonlinear Normal Modes in Mechanics and Physics - NNM2012, July 1-5, 2012, Haifa. (Unpublished)
Non-linear resonance
Georgiades, Fotios and Vakakis, Alexander, F. and Kerschen, Gaetan (2007) Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end
attachment. International Journal of Non-Linear Mechanics, 42 (5). pp. 773-788. ISSN 0020-7462
Non-linear targeted energy pumping
Georgiades, Fotios and Vakakis, Alexander, F. and Kerschen, Gaetan (2007) Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end
attachment. International Journal of Non-Linear Mechanics, 42 (5). pp. 773-788. ISSN 0020-7462
Nonlinear damped transitions
Panagopoulos, Panayotis and Georgiades, Fotios and Tsakirtzis, Stylianos and Vakakis, Alexander, F. and Bergman, Lawrence, A. (2007) Multi-scaled analysis of the damped dynamics of an elastic rod
with an essentially nonlinear end attachment. International Journal of Solids and Structures, 44 (18-19). pp. 6256-6278. ISSN 0020-7683
Nonlinear equations of motion
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Matthew, P. (2012) Nonlinear modal analysis of an L-shape beam structure. In: 4th International Conference on Localization, Energy Transfer and
Nonlinear Normal Modes in Mechanics and Physics - NNM2012, July 1-5, 2012, Haifa. (Unpublished)
Nonlinear modal analysis
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Matthew, P. (2012) Nonlinear modal analysis of an L-shape beam structure. In: 4th International Conference on Localization, Energy Transfer and
Nonlinear Normal Modes in Mechanics and Physics - NNM2012, July 1-5, 2012, Haifa. (Unpublished)
Nonlinear targeted energy transfer
Georgiades, Fotios and Vakakis, Alexander, F. (2007) Dynamics of a linear beam with an attached local nonlinear energy sink. Communications in Nonlinear Science and Numerical Simulation, 12 (5). pp.
643-651. ISSN 1007-5704
Nonlinear targeted energy transfers
Georgiades, Fotios and Vakakis, Alexander, F. (2009) Passive targeted energy transfers and strong modal interactions in the dynamics of a thin plate with strongly nonlinear attachments. International
Journal of Solids and Structures, 46 (11-12). pp. 2330-2353. ISSN 0020-7683
Pareto optimality
Riley, Mike and Dunning, Peter D. and Brampton, Christopher J. and Kim, H. Alicia (2014) Multi-objective robust topology optimization with dynamic weighting. In: International Conference on
Engineering and Applied Sciences Optimization , 4-6 June 2014, Kos, Greece. (In Press)
Single-walled carbon nanotube
Zhang, Yu and Sansour, Carlo and Bingham, Chris (2013) Single-walled carbon nanotube modelling based on Cosserat surface theory. In: 12th WSEAS International Conference on Applications of Electrical
Engineering (AEE '13), January 30th - February 1st 2013, Cambridge, Mass..
Topology optimization
Riley, Mike and Dunning, Peter D. and Brampton, Christopher J. and Kim, H. Alicia (2014) Multi-objective robust topology optimization with dynamic weighting. In: International Conference on
Engineering and Applied Sciences Optimization , 4-6 June 2014, Kos, Greece. (In Press)
elastic continua dynamics
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Matthew, P. (2013) Linear modal analysis of L-shaped beam structures. Mechanical Systems and Signal Processing, 38 (2). pp. 312-332. ISSN
ensiferan ear
Lomas, Kathryn and Montealegre-Z, Fernando and Parsons, Stuart and Field, Larry H. and Robert, Daniel (2011) Mechanical filtering for narrow-band hearing in the weta. Journal of Experimental Biology,
214 (5). pp. 778-785. ISSN 0022-0949
laser vibrometry
Lomas, Kathryn and Montealegre-Z, Fernando and Parsons, Stuart and Field, Larry H. and Robert, Daniel (2011) Mechanical filtering for narrow-band hearing in the weta. Journal of Experimental Biology,
214 (5). pp. 778-785. ISSN 0022-0949
linear equations of motion
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Mathhew, P. (2013) Towards linear modal analysis for an L-shaped beam: equations of motion. Mechanics Research Communications, 47 . pp. 50-60.
ISSN 0093-6413
mechanical tuning
Lomas, Kathryn and Montealegre-Z, Fernando and Parsons, Stuart and Field, Larry H. and Robert, Daniel (2011) Mechanical filtering for narrow-band hearing in the weta. Journal of Experimental Biology,
214 (5). pp. 778-785. ISSN 0022-0949
modal analysis
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Mathhew, P. (2013) Towards linear modal analysis for an L-shaped beam: equations of motion. Mechanics Research Communications, 47 . pp. 50-60.
ISSN 0093-6413
Georgiades, Fotios and Warminski, Jerzy and Cartmell, Matthew, P. (2013) Linear modal analysis of L-shaped beam structures. Mechanical Systems and Signal Processing, 38 (2). pp. 312-332. ISSN
multi-objective optimisation
Riley, Mike and Dunning, Peter D. and Brampton, Christopher J. and Kim, H. Alicia (2014) Multi-objective robust topology optimization with dynamic weighting. In: International Conference on
Engineering and Applied Sciences Optimization , 4-6 June 2014, Kos, Greece. (In Press)
nonlinear modal interactions
Georgiades, Fotios and Vakakis, Alexander, F. (2009) Passive targeted energy transfers and strong modal interactions in the dynamics of a thin plate with strongly nonlinear attachments. International
Journal of Solids and Structures, 46 (11-12). pp. 2330-2353. ISSN 0020-7683
shock isolation
Georgiades, Fotios and Vakakis, Alexander, F. (2009) Passive targeted energy transfers and strong modal interactions in the dynamics of a thin plate with strongly nonlinear attachments. International
Journal of Solids and Structures, 46 (11-12). pp. 2330-2353. ISSN 0020-7683
tympanum vibration
Lomas, Kathryn and Montealegre-Z, Fernando and Parsons, Stuart and Field, Larry H. and Robert, Daniel (2011) Mechanical filtering for narrow-band hearing in the weta. Journal of Experimental Biology,
214 (5). pp. 778-785. ISSN 0022-0949
Lomas, Kathryn and Montealegre-Z, Fernando and Parsons, Stuart and Field, Larry H. and Robert, Daniel (2011) Mechanical filtering for narrow-band hearing in the weta. Journal of Experimental Biology,
214 (5). pp. 778-785. ISSN 0022-0949
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Application of Pythagoras:Shaded Region
A circle is a simple shape of Euclidean geometry consisting of the set of points in a plane that is a given distance from a given point, the centre. The distance between any of the points and the
centre is called the radius.
Circles are simple closed curves which divide the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary
of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.
A circle is a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to
the axis of the cone.
Area enclosed
Area of the circle = π x area of the shaded square
As proved by Archimedes, the area enclosed by a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which
comes to π multiplied by the radius squared:
(Our solved example in mathguru.com uses this concept).
Equivalently, denoting diameter by d,
that is, approximately 79 percent of the circumscribing square (whose side is of length d).
The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would
be denoted as ABCD.
A so-called crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. Its angles are not right angles. Other
geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.
If a rectangle has length l and width w
It has area A = lw (Our solved example in mathguru.com uses this concept).
The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.
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Wojciech Samotij
I am a post-doctoral researcher at the School of Mathematics at Tel Aviv University hosted by
Noga Alon
Michael Krivelevich
, and
Ron Peled
. I am also a junior research fellow at Trinity College, Cambridge. My areas of interest include various branches of extremal and probabilistic combinatorics, such as extremal (hyper)graph theory,
the theory of random graphs, and Ramsey theory, as well as some topics in additive number theory. I completed my PhD at the University of Illinois at Urbana-Champaign in 2010 under the supervision of
Jozsi Balogh
. Prior to coming to Urbana-Champaign, I received master's degrees in mathematics and computer science from the Unversity of Wroclaw.
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Matches for:
Contemporary Mathematics
1988; 312 pp; softcover
Volume: 74
ISBN-10: 0-8218-5082-2
ISBN-13: 978-0-8218-5082-4
List Price: US$46
Member Price: US$36.80
Order Code: CONM/74
The representations of a finitely generated group in a topological group \(G\) form a topological space which is an analytic variety if \(G\) is a Lie group, or an algebraic variety if \(G\) is an
algebraic group. The study of this area draws from and contributes to a wide range of mathematical subjects: algebra, analysis, topology, differential geometry, representation theory, and even
mathematical physics. In some cases, the space of representations is the object of the study, in others it is a tool in a program of investigation, and, in many cases, it is both.
Most of the papers in this volume are based on talks delivered at the AMS-IMS-SIAM Summer Research Conference on the Geometry of Group Representations, held at the University of Colorado in Boulder
in July 1987. The conference was designed to bring together researchers from the diverse areas of mathematics involving spaces of group representations. In keeping with the spirit of the conference,
the papers are directed at nonspecialists, but contain technical developments to bring the subject to the current research frontier. Some of the papers include entirely new results. Readers will gain
an understanding of the present state of research in the geometry of group representations and their applications.
• W. Abikoff -- Kleinian groups--geometrically finite and geometrically perverse
• G. W. Brumfiel -- The real spectrum compactification of Teichmuller space
• G. W. Brumfiel -- A semi-algebraic Brower fixed point theorem for real affine space
• G. W. Brumfiel -- The tree of a non-archimedean hyperbolic plane
• K. Corlette -- Gauge theory and representations of Kahler groups
• D. R. Farkas -- The Diophantine nature of some constructions at infinity
• B. Fine and G. Rosenberger -- Complex representations and one-relator products of cyclics
• M. Gerstenhaber and S. D. Schack -- Sometimes \(H^1\) is \(H^2\) and discrete groups deform
• W. M. Goldman -- Geometric structures on manifolds and varieties of representations
• W. M. Goldman and Y. Kamishima -- Topological rigidity of developing maps with applications to conformally flat structures
• W. J. Harvey -- Modular groups and representation spaces
• A. Lubotzky and A. R. Magid -- Local structures of representation varieties: examples
• J. J. Millson -- Deformations of representations of finitely generated groups
• K. Morrison -- Connected components of representation varieties
• J. O'Halloran -- A characterization of orbit closure
• R. C. Penner -- Calculus on moduli spaces
• D. M. Snow -- Affine homogeneous spaces
• C. W. Stark -- Deformations and discrete subgroups of loop groups
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exactly simulating a random walk from infinity
up vote 10 down vote favorite
In diffusion-limited aggregation on the square lattice, one lets a particle do "random walk from infinity" until it hits the current aggregate, at which point the site occupied by the particle is
added to the aggregate; then the particle is started from infinity again, and so on.
This begs the question: What does it mean to let a particle do random walk from infinity? Or, in a more practical vein, how does one simulate such a walk?
The answer to the first question is, Random walk from infinity is just the limit as $v$ goes to infinity of random walk started at $v$. More precisely, for each $v$, we can look at random walk that
starts from $v$ and stops when it hits (0,0) $T$ time-steps later (where the hitting-time $T$ is random), and we can re-index it so that it starts at $v$ at time $-T$ and hits (0,0) at time 0. This
gives us a probability measure on paths that end at (0,0). Now we take the limit as $v$ gets farther and farther from (0,0), and we do some work and show that the law of the random path approaches a
limit, and that this limit doesn't depend on how $v$ goes off to infinity. Actually I'm bluffing; I don't know how to prove this. I assume it's in the literature; can anyone point me to a relevant
book or article?
Now we come to the problem of simulation. What we want is a black box that prints out the path in reverse, chugging away until we turn it off. Note that this is not the same as having a black box
that, for any $n$, prints out the last $n$ steps of the path; for, after the box has printed out that path, if we decide we want to know "What happened before that?", we can't find out by increasing
$n$ and consulting the box again, because then the black box will give us the last $n+1$ steps of a DIFFERENT random path. On the other hand, if we had a box that, given the last $n$ steps of the
path, could sample from its (up to 4 different) continuations one step back into the past with the appropriate conditional probabilities, one could use this box iteratively to build the sort of black
box I want. It might be very hard to compute these conditional probabilities in the case of the square grid, but I suppose that would be one way to do it. A more robust (and, to my way of thinking,
prettier) solution would be something more combinatorial, using ideas like coupling and domination.
pr.probability random-walk simulation
It should just be the same thing as a random walk conditioned not to hit the origin at any positive time. – George Lowther Jun 7 '11 at 23:23
That's believable, but can you prove it to me? (Indeed, what does it mean to condition on this event, given that the event has measure zero? I guess it means, condition on the event
no-return-to-the-origin-up-to-time-T and then take the limit as T goes to infinity.) Also, assuming that your assertion is true, does it give a workable simulation scheme? I don't think that the
law of this conditioned walk is just "pick a random neighbor as long as it isn't the origin". – James Propp Jun 8 '11 at 12:40
I edited my answer to explain the equivalence. I may add more later. – Ori Gurel-Gurevich Jun 8 '11 at 17:34
add comment
1 Answer
active oldest votes
To answer your first question (regarding existence of the limit): I never remember references, but all you have to do here is show that for any two far enough starting points, there is a
coupling of the RWs started at them, such that with high probability the two paths hit the aggregate at the same point (for example, because they start walking together before hitting the
aggregate). In $\mathbb{Z}^2$ it's pretty straightforward to do: if you let one walker walk till it hits the aggregate, then with high probability its path will separate the aggregate from
the starting point of the second walker. Then you let the second walker walk till it hits the first path and follow this path thereafter.
As for the second question: as I said, I never remember references, but I believe that you can find how to calculate the harmonic measure exactly using the 2d potential kernel in Spitzer's
"Principles of Random Walks".
up vote 5 To elaborate on George's comment and your reply (again, I'm pretty sure all this appears in Spitzer): Let $A$ be a finite set of vertices and start a SRW at $X_0=x$ and for some $y\in A$
down vote we look at $\mathbb{P}(X_{\tau_A}=y)$, where $\tau_A$ is the hitting time of $A$. Then this probability is equal to $\sum_w \mathbb{P}(w)$ where the sum is over all paths starting at $x$
accepted and ending at $y$ and not going through $A$. Since the SRW on $\mathbb{Z}^2$ is reversible this is equal to the sum over paths strating at $y$ and ending at $x$ and not going through $A$.
This is exactly the expected number of visits to $x$ for a SRW started at $y$ and killed upon returning to $A$. This is proportional to the probability of hitting $x$ before returning to
$A$ (again, when starting at $y$). If we take $x$ to be far away we see that conditioning on hitting $x$ before returning to $A$ is the same (asymptotically) as conditioning on the walk
not returning to $A$ for a long time.
More can be said about the distribution of the conditioned RW, but right now I have to go.
This looks like the right way to think about backwards walk, but I got lost at "This is exactly the expected number of visits to x for a SRW started at x and killed upon returning to A."
Is there a typo here? What has happened to the dependence on y? – James Propp Jun 9 '11 at 2:36
That was a typo, I edited it now. – Ori Gurel-Gurevich Jun 9 '11 at 3:37
The inclusion of y helps, but I still don't get it. Try it on the 3-vertex path-graph {0,1,2} with transition probabilities p(0,1) = p(2,1) = 1, p(1,0) = p(1,2) = 1/2, taking y=0, x=1,
and A = {y}. Then the path-sum is just p(0,1) = 1. On the other hand, the expected number of visits to 1 for a SRW started at 0 and killed upon returning to 0 is 2 (and even without
knowing anything about geometric random variables one can see that it's strictly greater than 1, since the walk gets to visit 1 once for free, and gets to visit again with positive
probability). Am I missing something? – James Propp Jun 9 '11 at 12:26
For Uri's argument to work you need to be doing the walk on a regular graph, otherwise there is another factor which is the ratio of the degrees of $x$ and $y$. – Louigi Addario-Berry
Jun 9 '11 at 13:00
What Louigi said (also, Ori, not Uri). – Ori Gurel-Gurevich Jun 9 '11 at 15:04
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Uniform boundedness of feasible per capita output streams under convex technology and non-stationary labor
Maćkowiak, Piotr (2004): Uniform boundedness of feasible per capita output streams under convex technology and non-stationary labor.
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This paper shows that under classical assumptions on technological mapping and presence of an indispensable production factor there is a bound on long-term per capita production. The bound does not
depend on initial state of economy. It is shown that all feasible processes converge uniformly over every bounded set of initial inputs p.c. to some set (dependent on technology).
Item Type: MPRA Paper
Original Uniform boundedness of feasible per capita output streams under convex technology and non-stationary labor
Language: English
Keywords: boundedness of trajectories, output path, multi-sector growth model
O - Economic Development, Technological Change, and Growth > O4 - Economic Growth and Aggregate Productivity > O41 - One, Two, and Multisector Growth Models
Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C61 - Optimization Techniques; Programming Models;
Dynamic Analysis
Item ID: 41891
Depositing Piotr Maćkowiak
Date 13. Oct 2012 16:54
Last 11. Apr 2014 14:57
W. Brock, On existence of weakly-maximal programmes in a multi-sector economy. Review of Economic Studies, 37:275-280, 1970.
D.Gale. On optimal development in a multi-sector economy. Review of Economic Studies, 34:1-18, 1967.
S.Joshi. Existence in undiscounted non-stationary non-convex multisector environments. Journal of Mathematical Economics, 28:111-126, 1997.
V.Levin. Some applications of set-valued mappings in mathematical economics. Journal of Mathematical Economics, 20:69-87, 1991.
References: R.Lucas, N.Stokey. Recursive Methods in Economic Dynamics. Academic Press, 1989.
P.Ma\'ckowiak. Some remarks on lower hemicontinuity of convex multivalued mappings. To appear in: Economic Theory.
\bibitem{McKenzie1986} L.~McKenzie. \newblock Optimal economic growth, turnpike theorems and comparative dynamics. \newblock w: {\em Handbook of Mathematical Economics, Vol.~4} (red.
K.~Arrow, M.~Intriligator), strony 1281--1353. Elsevier Science Publishers, 1986. \bibitem{Nikaido1968} H.~Nikaido. \newblock {\em Convex Structures and Economic Theory}. \newblock
Academic Press, 1968.
\bibitem{PelegRyder1972} B.~Peleg, H.~Ryder. \newblock On optimal consumption plans in a~multi-sector economy. \newblock {\em Review of Economic Studies}, 39:159--169, 1972.
URI: http://mpra.ub.uni-muenchen.de/id/eprint/41891
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Wednesday, March 30, 2011 at 12:07am
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Wednesday, March 30, 2011 at 12:03am
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Sunday, March 27, 2011 at 10:40pm
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No. One way to solve this problem is like this: (1/2) * (33/1) = 33/2 = 16 1/2 16 * 33 = 528 528 + 16 1/2 = 544 1/2 You could also use 16.5 for 16 1/2 and multiply. The answer is the same: 544.5
Sunday, March 27, 2011 at 5:44pm
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Sunday, March 27, 2011 at 5:42pm
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Saturday, March 12, 2011 at 3:41am
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Monday, February 28, 2011 at 8:10pm
3rd grade math hw help!
1) 2/4,2/3,2/7,2/5 2/4 = 0.5 2/3 = 0.66666 2/7 = 0.28571 2/5 = 0.4 2) 1/2,6/7,5/7,1/10 1/10=0.1 1/2=0.5 5/7=0.71429 6/7=0.85714 3) 1/2,6/7,1/12,2/3,11/12,1/6,1/3, 1/4, 4/5 1/12=0.08333 1/6=0.16666 1/
4=0.25 1/3=0.33333 1/2=0.5 2/3=0.66666 4/5=0.8 6/7=0.85714 11/12=0.91666
Thursday, February 24, 2011 at 9:47pm
3rd grade math - arrays
Mary says that the greater number of counters, the greater the number of different arrays you can form. Give an example that shows that Mary is wrong.
Wednesday, February 9, 2011 at 1:45am
3rd grade math
Mrs Gainey needs to buy string for 5 students to do a science activity. Each student needs 2 feet of blue string and 1 foot of red string. How many yards of string does Mrs. Gainey need to buy?
Tuesday, February 8, 2011 at 6:37pm
3rd grade math
You could use 25 file cards (or old greeting cards?) to represent the birthday cards. Ask the child to make stacks with 5 cards in each stack. Number each stack with a post-it note. Each stack is
labeled 1 hour. The child can then count the hours.
Monday, February 7, 2011 at 5:48pm
3rd grade math
What patterns help you divide? DRaw a picture or make a chart to show how to divide 1600 by 4. Give the quotient. We know the answer but having trouble to make a chart to show the outcome. Please
help. Thanks in advance
Sunday, January 30, 2011 at 9:47pm
Math/3rd grade
Use a calculator. Or better yet -- remember that the digits of the product or 9 times any number will always add up to 9. 18, 27, 36, 45, 54, 63, 72, 81, 90 Another way is to multiply 9 * 10 = 90 and
subtract 9 from 90.
Thursday, January 13, 2011 at 5:39pm
3rd grade Math
The original data is given in terms of area, but the width is not in the same units. If area 1 = x, then area 2 = 2x and area 3 = x + 3. If the total area = 18 square inches, then x + 2x + x + 2 = 18
Then you can solve for x. 4x + 2 = 18 4x = 16 x = 4
Thursday, January 6, 2011 at 12:12pm
3rd grade social studies
http://www.google.com/search?sourceid=chrome&ie=UTF-8&q=acropolis Read carefully.
Tuesday, January 4, 2011 at 9:23pm
3rd grade Shurley English
invisible you-subject put-verb some- indefinite adjective cheese-adjective slices-noun (direct object) and-coordinating conjunction bread-noun(direct object) on-preposition your-possessive adjective
plate-noun(object of preposition)
Wednesday, December 15, 2010 at 8:25pm
math 3rd grade
5.10.15.20.25.30.... while... 10,20,30........... so Each 10 has two 5's.
Tuesday, December 14, 2010 at 9:29pm
If a grade was drawn at random from the data shown below, what is the probability that the grade is not a B? Grade Frequency A 2 B 8 C 11 D 2 F 1 I think it is 1 out of 5 or 4 but I am not sure?
Please help :) Thank you much!
Sunday, December 5, 2010 at 10:02am
Spanish 1
1. this one must be a question because of the word order = ¿A quién le gustan los animales? 2. ¿A quiénes les gusta la música? Gustar is nearly always used in the 3rd person singular if ONE thing is
pleasing or the 3rd person plural if 2 or ...
Monday, November 29, 2010 at 7:35pm
pre algebra
1st number = x, 2nd number = 4x - 5, 3rd numbe = 2x - 9, 4(2x - 9) - x 6, 8x - 36 - x = 6, 7x 42, x = 6 = 1st number. (4x - 5) = 4*6 - 5 = 19 = 2nd number. (2x - 9) = 2*6 - 9 = 3 = 3rd number. .
Monday, November 29, 2010 at 5:56pm
3rd grade math
Marlon has 4 cards, Jake has 4 cards, and Sam has 3 cards. Can you write a multiplication sentence to find how many cards they have in all? Explain in general 4+4+3=11 In multiplication sentence=
(3*3)+2=11 OR (3*4)-1 BUT I AM NOT 100% SURE
Sunday, November 28, 2010 at 4:52pm
3rd grade math
coach took 16 players to a game. he said they could have a drink, hot dog or both. 12 had a drink, 9 had a hot dog. how many had both?
Tuesday, November 23, 2010 at 4:23pm
4th grade
My answer to this question is: Circus Clowns were 1st with 12 members Marching Band 2nd with 24 members Girl Scouts 3rd with 30 members Jugglers 4th with 12 members Boy Scouts 5th with 15 members
Dancers 6th with 6 members and Football Team 7th with 28 members I am not sure ...
Thursday, November 18, 2010 at 10:25pm
3rd grade math
Emma has 36 feet of fence. She wants to make the largest rectangular area possible for her rabbit to play in. What length should she make each side of the rabbit pen? Show your work and explain how
you found the largest area.
Thursday, November 18, 2010 at 6:18pm
math grade 11
F(x) = Y = 4x^2 - 3x + 2kx + 1. This problem was solved by using EXCEL spread sheets and trial & error. First, I temporarily ignored the 3rd term (2kx); and I changed b (the coefficient of x) until I
found the required zeroes. Then I calculated the corresponding value of k: b...
Thursday, November 4, 2010 at 7:45pm
Pages: <<Prev | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | Next>>
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Cranbury Trigonometry Tutor
...I have been speaking Hebrew for over thirty years, in addition to teaching Religious School to various grade levels for the past twelve years. I have also taught Adult Bar/Bat Mitzvah classes
in order to prepare them for conversion or marriage. In addition, I currently tutor several students in preparation for their Bar/Bat Mitzvah.
53 Subjects: including trigonometry, reading, geometry, English
...As a tutor, I will strive to incorporate fun activities to help my students learn in a way that works for them. I am currently in the process of getting my certification to teach students K-12.
I have met all qualification and am only waiting for final approval.I taught Algebra 1 in my student teaching experience.
9 Subjects: including trigonometry, geometry, algebra 2, SAT math
...He has been a full-time teacher for 2 years, including 2 years as a substitute classroom teacher in Middlesex County for grades K-12. Uri also tutored Spanish to children and adults of all
levels, as well as other subjects at Middlesex County College. He loves to cook, watch documentaries, listen to all kinds of music, and traveling.
19 Subjects: including trigonometry, Spanish, statistics, geometry
...I'm an expert at providing writing help with schoolwork or application materials. SAT Math is the single subject that I've spent the most time tutoring. I work with students at all skill
levels, with extra time and without.
36 Subjects: including trigonometry, English, chemistry, calculus
...It can afford the student an opportunity to ask questions they might be too shy or embarrassed to ask in front of all their peers. Or it can afford them the opportunity to grasp concepts at a
pace more in line with their learning abilities. But sometimes tutoring can be an individualized supplement to a student's classroom education.
10 Subjects: including trigonometry, geometry, precalculus, algebra 1
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Penny & Dell Logic and Sudoku Value Pack
5/3/2012 6:15:29 Penny/Dell,
I have a few questions about this particular value pack:
Purple Pisces 1) I know that the Sudoku Plus Variety magazine is no longer published, but is there a chance that back issues of this publication are included in this value pack?
Posts: 599
2) Is there a guarantee that I'll get at least one issue of Logic Lover's Math and Logic Problems?
3) Can you give an estimate in regard to the Sudoku magazine to Logic magazine ratio?
Thank you!
edited by Purple Pisces on 5/3/2012
• permalink
5/3/2012 9:12:08 Purple Pisces wrote:
creamchz3@aol.com I have a few questions about this particular value pack:
Posts: 879
1) I know that the Sudoku Plus Variety magazine is no longer published, but is there a chance that back issues of this publication are included in this value pack?
2) Is there a guarantee that I'll get at least one issue of Logic Lover's Math and Logic Problems?
3) Can you give an estimate in regard to the Sudoku magazine to Logic magazine ratio?
Thank you!
edited by Purple Pisces on 5/3/2012
I'm guessing they will tell you to contact customer service. Anybody else? CC
• permalink
5/4/2012 3:45:21 Purple Pisces wrote:
admin I have a few questions about this particular value pack:
Posts: 81 1) I know that the Sudoku Plus Variety magazine is no longer published, but is there a chance that back issues of this publication are included in this value pack?
2) Is there a guarantee that I'll get at least one issue of Logic Lover's Math and Logic Problems?
3) Can you give an estimate in regard to the Sudoku magazine to Logic magazine ratio?
Thank you!
edited by Purple Pisces on 5/3/2012
Purple Pisces,
Thanks for your inquiry. The pack will contain six Logic, six Sudoku and four Logic Plus Sudoku magazines. Unfortunately, we do not have enough back issues of the Dell Crazy for
Sudoku Plus Variety Sudoku title to include in this pack.
We don't offer the Logic Lover's Math & Logic Problems magazine in this pack, but you can find six of them in our 36-magazine Value Pack (to order, please call us at
We appreciate your interest in the Logic and Sudoku Value Packs!
• permalink
5/4/2012 4:45:50 Thank you for taking the time and looking into what the value pack contains. Even though the value pack is an excellent value, I think I may just reintroduce myself to an issue
PM of the Logic Lover's Math & Logic Problems, which is one of the publications I am mainly interested in.
Just for future reference, in case I decide to get a subscription of the Logic Lover's Math & Logic, does it come in an envelope or plastic wrap?
Purple Pisces
Posts: 599
• permalink
5/4/2012 7:31:57 Purple Pisces wrote:
Thank you for taking the time and looking into what the value pack contains. Even though the value pack is an excellent value, I think I may just reintroduce myself to an
issue of the Logic Lover's Math & Logic Problems, which is one of the publications I am mainly interested in.
Posts: 879 Just for future reference, in case I decide to get a subscription of the Logic Lover's Math & Logic, does it come in an envelope or plastic wrap?
Hey Purp, With my experience when they mail one magazine it comes in plastic. When they mail two or three together they come in an envelope. When I order a large sum ( more than
10 or so) they are boxed. Hope this helps! CC
• permalink
5/4/2012 9:03:28 Thanks CC! I've read on the forum here before that not all subscriptions are mailed in a protective wrapper and was wondering if the Dell Logic Lover's Math & Logic magazine
PM was.
edited by Purple Pisces on 5/4/2012
Purple Pisces
Posts: 599 • permalink
5/5/2012 7:59:24 Yes, PP, my subscription issues arrive each month in a plastic wrapper. It's very nice!
• permalink
Posts: 604
5/5/2012 1:27:35 Bernadette you suscribe to Dell Logic & Math? I just did a search and found someone who asked about the shipping method for this very publication 3 years ago, and they received
PM an answer that the address labe is affixed directly to the cover and is not in a protective wrapper. I'm hoping that's changed in the past few years.
edited by Purple Pisces on 5/5/2012
Purple Pisces
Posts: 599 • permalink
5/5/2012 1:55:34 No, PP, I do not have a subscription to that particular title but I do have subscriptions to four other different PennyDell publications, all of which have arrived each month
PM for the past year wrapped in a plastic sheet.
I cannot imagine that PennyDell would not mail out all of their magazines in the same manner, but if I am steering you wrong, I apologize.
Posts: 604
• permalink
5/5/2012 4:07:53 Bernadette no need to apologize!!
• permalink
Purple Pisces
Posts: 599
5/7/2012 9:02:14 Yes, I guess that way you will know for sure!
• permalink
Posts: 604
5/7/2012 10:11:28 Purple Pisces wrote:
Thank you for taking the time and looking into what the value pack contains. Even though the value pack is an excellent value, I think I may just reintroduce myself to an
issue of the Logic Lover's Math & Logic Problems, which is one of the publications I am mainly interested in.
Administrator Just for future reference, in case I decide to get a subscription of the Logic Lover's Math & Logic, does it come in an envelope or plastic wrap?
Posts: 81
Purple Pisces,
Yes, the magazine is shipped in plastic wrap. Happy solving!
• permalink
5/7/2012 1:13:49 Thank you!! I see a Math & Logic Problems subscription in my future!!
• permalink
Purple Pisces
Posts: 599
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[plt-scheme] Re: HTDP Exercise 12.4.2 ... Help! (Solved!!)
From: S Brown (ontheheap at gmail.com)
Date: Sat May 2 22:34:42 EDT 2009
I just wanted to update this topic in order to say that I have finally
solved this exercise the *correct* way! Thank you very much to
everyone here who provided input.
For anyone who finds this topic and is looking for help with this
1. Learn the right way to think about recursion. Don't make the
mistake I made and try to put the entire recursive process into your
head all at once. It simply doesn't work. Even writing down all of the
results of the recursion step-by-step won't help. Instead, use the
definition of the function itself to understand how the recursion
works, and then create the rest of the function as if the recursive
call already does what it's supposed to do.
2. Follow the design recipes. This exercise is probably the first one
in the book where you pretty much have to use the design recipes in
order to find the right solution. The specific recipe you should be
looking at here is in section 9.4
3. Walk away from this exercise if/when you get frustrated. Take a
walk, watch a movie, etc. Basically, go do something that doesn't
require a lot of thinking.
4. Don't give up on it until you have the solution. It's worth it once
you solve it because you gain a better understanding of how to use and
think about recursion, and how to use the design recipes.
Posted on the users mailing list.
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Why can an infinite-order polynomial be written as the ratio of two finite-order polynomials?
up vote 2 down vote favorite
I am studying GARCH processes in Time Series Analysis by Hamilton.
Something that has regularly been used in the book is the assumption that an infinite-order polynomial can be written as the ratio of two finite-order polynomials.
$\pi(L) = \sum_{j=1}^{\infty} \pi_j L^j$
And then
$\pi(L) = \frac{\alpha(L)}{1-\delta(L)} = \frac{\alpha_1 L^1 + \alpha_1 L^2 + ... + \alpha_m L^m}{1 - \delta_1 L^1 - \delta_1 L^2 - ... - \delta_r L^r}$
Followed by "assuming the roots of $1-\delta(L) = 0$ are outside the unit circle".
What is the reasoning behind this transformation? And is the assumption about the roots outside the unit circle required for the infinite order polynomial to have this ratio representation?
@Lee Mosher: the question could certainly be more clear and I do not claim it is right for this site, but your objection strikes me as strange. There is a clear context mentioned and AFAIK the
coefficients in that context will be in linear recurrecne, so it will be a rational. – quid Apr 18 '13 at 13:59
@Nathan Wilson: Likely the best thing to do for you is ask this on another site (for example the mathematics or the statistics site on the stackexchange network, see FAQs for details). However,
here are two remarks: pi(L) is the fraction of two polynomials since (or let us rather say if) its coefficients fulfil a linear recurrence relation that is you can compute a coefficient by a fixed
linear function of a fixed number of preceeding ones. And this should be the case in your context. For the question regarding the roots: you do not need this to express it as a fraction... – quid
Apr 18 '13 at 14:01
Thank you @quid. – Nathan Wilson Apr 18 '13 at 14:06
...this would be possible in a completely "formal" way (if it is possible at all). However, if one wishes to treat these as expressions as "functions of L" it is (or might be) important that the
denominator is never 0 for the L one wants to plug into the expression, relatedly that the series converges. And there might be a condition there that confines the L to the unit circle, so that if
it only vanishes outside the unit circle "everything is fine." Yet again, you might get a better response on other sites on the stackexchange network. – quid Apr 18 '13 at 14:09
You are welcome! Yet treat the information I gave with some critical distance; this is a bit of guess work on my part. – quid Apr 18 '13 at 14:10
show 1 more comment
1 Answer
active oldest votes
I am not sure how much background you have in analysis but it may be helpful to keep in mind that "the assumption about the roots outside the unit circle" is required not "for the
up vote 2 infinite order polynomial to have this ratio representation", which seems to be a separate assumption, but rather for the series to converge for all points in the unit disk.
down vote
add comment
Not the answer you're looking for? Browse other questions tagged polynomials or ask your own question.
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Analysis - Series
April 20th 2010, 10:26 AM #1
Junior Member
Dec 2009
Analysis - Series
The question is i need to prove the following series converge or diverge,
for the first part i get (e^-3+5)/5 which is bigger than 1 so it diverges and for the second part i get 1 ( so is this wrong as a series diverges if it is less than 1 and converges if it is more
than 1,or is it a special case)
Any help would be much appreciated
The question is i need to prove the following series converge or diverge,
for the first part i get (e^-3+5)/5 which is bigger than 1 so it diverges and for the second part i get 1 ( so is this wrong as a series diverges if it is less than 1 and converges if it is more
than 1,or is it a special case)
Any help would be much appreciated
$\frac{e^{-3n}}{(2n)^n+5}\leq \frac{1}{2^n}$
$\frac{5n}{n^3+n^2}=\frac{5}{n^2+n}\leq \frac{5}{n^2}$
Now use the comparison test and we're through.
April 20th 2010, 11:16 AM #2
Oct 2009
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SCU Dept. of Mathematics and Computer Science -- Sample Math Major
[Return to Department of Mathematics and Computer Science homepage]
[Return to Mathematics Major page]
Sample Curriculum
Class of 2013+
This chart is based on the new university and college requirements ("core curriculum 2009") in effect in Fall 2009 for the class of 2013.
│Quarter│Freshman Year │Sophomore Year │Junior Year │Senior Year │
│ │ │ │ │Math 103 (Adv. Lin. Alg.)│
│ │Math 11 (Calc I) │Math 14 (Calc IV) │Math 102 (Adv. Calc.) │or Math 111 (Abs. Alg.) │
│Fall │CSCI 10 (Intro CS) │Diversity │Math Up. Div. │or Math 176 (Comb.) │
│ │Crit Think Writing (STS) I │Foreign Lang. I │Math 100 (or Adv. Writing)│Rel Theol Culture III │
│ │Culture & Ideas I │Math 51 (Discr. Math)*│(Elective) │Elective │
│ │ │ │ │(Elective) │
│ │Math 12 (Calc II) │Math 52 (Abst. Alg.) │Math Up. Div. │Math Up. Div. │
│Winter │Physics 31 │Foreign Lang. II │Civic Engagement │Elective │
│ │Culture & Ideas II │Math 22 (Diff. Eq.) │Elective │Elective │
│ │Rel Theol Culture I │Rel. Theol Culture II │(Elective) │(Elective) │
│ │Math 13 (Calc. III) │Math 53 (Lin. Alg) │Math Up. Div. │Math Up. Div. │
│Spring │Physics 32 & 32L │Soc. Science │Culture & Ideas III │Elective │
│ │Crit Think Writing (STS) II │Arts │Elective │Elective │
│ │Elective │Ethics │(Elective) │(Elective) │
Students may not take more than 19 units a quarter without permission. Upper Division courses taken in some "elective" slots in the Junior and Senior year may lead to an overload.
-- One course (which may be a core course) must be designated as an "experiential learning" course.
-- Students must declare a "pathway" by the end of their sophomore year and take 4 approved courses (which may fulfill other requirements) in that pathway.
A. General Comments
1. Students who show proficiency in a high-level programming language may substitute another course to fulfill the technology requirement in lieu of CSCI 10.
2. Students planning to go on to graduate school in pure mathematics should take Math 105 (Complex), 111 (Abst. Alg. I), 112 (Abst. Alg. II), 113 (Topology), 153 (Interm. Analysis I), 154 (Interm.
Analysis II).
3. Students planning to go on to graduate school in applied mathematics should complete the emphasis in applied mathematics (see Section C below) and take Math 105 (Complex), 111 (Abst. Alg. I), 144
(Partial Diff Eq), 153 (Interm. Analysis I), 154 (Interm. Analysis II), 155 (Ord Diff Eq).
4. Students are referred to the Bulletin for details concerning other recommendations and substitutions.
B. Pre-Secondary Teaching Options and Recommendations:
B-I: Special Recommendations for Those Preparing for Admission to California Teacher Training Credential Programs
The State of California requires that students seeking a credential to teach mathematics or computer science in California secondary schools must pass the California Subject Examination for Teachers
(CSET), a subject area competency examination. The secondary teaching credential additionally requires the completions of an approved credential program, which can be completed as a fifth year of
study and student teaching.
B-II: Students may consider completing the Emphasis in Mathematics Education. In addition to the general requirement for majoring in Mathematics,
1. Students must complete Math 101 (Geom.), 102, 111, 122 (Prob. & Stats. I), 123 (Prob. & Stats. II) or 8 (Stats), 170 (Devel. Math.), either 175 (Numb. Th.) or 178 (Cryptography).
2. Students must complete Educ 198B (Second. School Teach.).
3. Physics 11 and 12 (formerly 20 and 21) may be substituted for Physics 31 and 32 (formerly 4 and 5).
4. Students are strongly recommended to complete the Urban Education minor.
C. Special Recommendations for the Emphasis in Applied Mathematics
1. Students must complete Math 102, 122, 123 (Prob. & Stats. II), 166 (Num. Analysis), and either 103 or 176.
2. Students must also complete two courses from Math 125 (Financial Math), 144 (PDE), 155 (ODE), 164 (Comp. Simul.), 165 (Lin. Prog.), 178 (Cryptography), or an approved alternative (but not from
other upper division computer science courses).
D. Special Recommendations for the Emphasis in Financial Mathematics
In addition to the general requirement for majoring in Mathematics,
1. Students must complete Math 102, 122, 123, 125, 144, 166.
2. Students must also complete Business 70, Accounting 11, 12, and Finance 121, 124.
E. Special Recommendations for Students interested in Computer Applications
Students are referred to the requirements for C.S. minors and the upper division tracks recommended for C.S. majors.
At least 12 upper division courses (60 units) are required for graduation. Thus, if the English Composition course is not an upper division course, at least three of the free electives must be upper
divsion courses.
The information presented on this webpage is not intended as the official statement of graduation requirements. The student is referred to the current University Bulletin.
Last Updated: 14 April 2009
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This Article
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Bibliographic References
Add to:
September 1970 (vol. 19 no. 9)
pp. 859-860
ASCII Text x
K.N. Levitt, "R70-38 The Time Required for Group Multiplication," IEEE Transactions on Computers, vol. 19, no. 9, pp. 859-860, September, 1970.
BibTex x
@article{ 10.1109/T-C.1970.223066,
author = {K.N. Levitt},
title = {R70-38 The Time Required for Group Multiplication},
journal ={IEEE Transactions on Computers},
volume = {19},
number = {9},
issn = {0018-9340},
year = {1970},
pages = {859-860},
doi = {http://doi.ieeecomputersociety.org/10.1109/T-C.1970.223066},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - JOUR
JO - IEEE Transactions on Computers
TI - R70-38 The Time Required for Group Multiplication
IS - 9
SN - 0018-9340
EPD - 859-860
A1 - K.N. Levitt,
PY - 1970
KW - null
VL - 19
JA - IEEE Transactions on Computers
ER -
Applying some simple, easily understood principles, Spira, in extending some earlier work of Winograd, points the way to a powerful theory of computation complexity. Spira considers a (d, r)
combinational network which is an interconnection of r-input, single-output modules, with each input-output line carrying a value from the set {0, 1, ? , d -1}. A finite function f: X1 ? X2 ? ? Xn?Y
is to be computed, but it is assumed that before the inputs are inserted into the network, each input can be individually (and arbitrarily) transformed by a set of maps gj: Xj?Ij. It is also assumed
that there is a 1-1 output map h: Y?Oc, so in practice the (d, r) network will have as input [g1(x), ?, gn(xn)] and as output h(f(x1, ?, xn)). The problem is to bound the number of levels required of
the network. Given a f for a particular output mapping, it is not difficult to specify a lower bound on the number of levels required, by identifying for each output line the number of different
values of input variables which yield a different output value. The minimum number of levels required for each output line is then evaluated by noting that an output at level z can depend on at most
r' input lines whence the output line requiring the most levels provides the bound.
K.N. Levitt, "R70-38 The Time Required for Group Multiplication," IEEE Transactions on Computers, vol. 19, no. 9, pp. 859-860, Sept. 1970, doi:10.1109/T-C.1970.223066
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Department of Mathematics
As an art and a science, mathematics occupies a special place in the Saint Xavier University curriculum. It serves as an investigative tool for the natural sciences, the social sciences, business,
education and psychology. It also plays an important role in the development of human thought. Mathematics forms the bridge whereby the student enters the realm of abstract and precise scientific
The Department of Mathematics offers two major programs of study: mathematics and mathematics with secondary education certification. Decisions concerning the nature of each student's study of
mathematics are made with particular attention to previous preparation and individual educational plans. Students who major in mathematics are prepared for entry into graduate schools, and for entry
into business and industrial positions that require a bachelor's degree in mathematics. Students who major in mathematics with certification in secondary education fulfill the requirements for
teaching high school, middle school, or junior school level mathematics. Courses in computer science, natural science, physics and business may be taken to fulfill the application requirement of each
program. The department also offers minor programs of study in mathematics and mathematics education that will complement many majors at the University, especially majors in business, computer
science, education and science.
B. Becker, Chair; P. Army; A. Dagys; C. Gawlik; M. Hardy; H. Lyne; A. Mojiri; R. Narcaroti; P. Petkus; C. Riola; J. Smenos; A. Wazwaz.
Requirements for Admission to the Major in Mathematics
1. Application for admission to the mathematics major or the mathematics major with secondary education certification must be made on the appropriate application form or on the change of major form.
2. To be admitted as a major in the mathematics department, students with less than 15 hours of undergraduate credit must have four years of high school mathematics and an ACT mathematics sub-score
of at least 26.
3. To be admitted as a major in the mathematics department, students with 15 to 29 hours of existing University credits must have a GPA of 2.5 or higher, and students with 30 or more hours of
existing University credits must have a GPA of 2.3 or higher.
Major Programs of Study
Requirements for a Mathematics Major
Students majoring in mathematics have the choice of a bachelor of arts or a bachelor of science degree.
1. Required mathematics courses (31 credit hours)
The following courses are required for both B.A. and B.S. degree candidates: MATH 200, 201, 202, 203, 211, 301, 305, 306 and 399.
2. Application area (6-10 credit hours)
Two courses in an application area are required for both B.A. and B.S. degree candidates. Possible application areas include Computer Science, Physics (calculus based), Business, and others by
petition. Departmental approval is required.
3. Elective courses
B.A. candidates must select 2 courses (6 credit hours) and B.S. candidates must select 5 courses (15 credit hours) from the 300-level mathematics elective courses.
4. Students pursuing the B.A. in mathematics must complete 6 credit hours of foreign language, foreign culture, or global studies courses or a combination of these, as part of the university general
education requirements.
5. Students planning to attend graduate school are strongly encouraged to complete the requirements for a B.S. degree.
6. A grade of C or better is required in all courses counting toward the major.
7. Majors in the Department of Mathematics are expected to attend special events and lectures sponsored by the department.
Requirements for a Mathematics Major Preparing to Teach at the Secondary Level (6-12 Certification)
Students preparing to teach mathematics at the secondary level have the choice of a bachelor of arts degree or a bachelor of science degree.
1. Required mathematics courses (40 credit hours)
The following courses are required for both B.A. and B.S. degree candidates: MATH 200, 201, 202, 203, 211, 301, 305, 306, 307, 308, 309 and 399.
2. Application area (6-10 credit hours)
Two courses in an application area are required for both B.A. and B.S. degree candidates. Possible application areas include Computer Science, Physics (calculus based), Business and others by
petition. Departmental approval is required.
3. Elective courses
B.S. candidates must select 2 courses (6 credit hours) from the 300-level mathematics elective courses.
4. Students pursuing the B.A. in mathematics must satisfy the general education requirements by taking 6 credit hours of foreign language, or foreign culture, or global studies courses or a
combination of these.
5. Students must be admitted to the School of Education Secondary Education Program.
6. Students planning to attend graduate school are strongly encouraged to complete the requirements for a B.S. degree.
7. A grade of C or better is required in all courses for the major, all general education courses, and all professional education courses, and students must maintain a minimum cumulative GPA of
2.5. Consult the School of Education section of the catalog for specific requirements and procedures. It is the responsibility of each student to ascertain and fulfill the requirements for
the desired degree program. The major advisor will assist the student in this responsibility.
8. Majors in the Department of Mathematics are expected to attend special events and lectures sponsored by the department.
Minor Programs of Study
The department also offers minor programs of study in mathematics and mathematics education that complement many majors at the University, especially majors in business, computer science, education
and science.
Requirements for a Minor in Mathematics
1. Mathematics courses (a minimum of 18 credit hours is required for a minor in mathematics). Only the following courses may be counted toward a minor in mathematics: MATH 200, 201, 202, 203, 211,
301, 303, 305, 306, 307, 308, 309, 315, 321, 322, 331, 360.
2. A grade of C or better is required in each course counting toward the minor.
Requirements for a Minor in Mathematics Education
1. Required mathematics courses (16-17 credit hours) MATH 121, 122, 200, 112 or 201, 351*. Note: MATH 121 and MATH 122 apply toward the mathematics requirement for K-9 certification as well.
2. Required computer science course (3 credit hours) CMPSC 112.
3. Elective courses (6 credit hours): elect two courses from: MATH 221*, 222*, 223*, 224*. Note: MATH 222 applies toward the Mathematics requirement for K-9 certification as well.
4. A grade of C or better is required in all courses counting toward the minor.
*This course has MATH 121 and 122 as prerequisites.
Independent Study
Credit for courses in mathematics may be obtained on an independent study basis only if the following conditions are met:
1. The student has completed a minimum of 12 credit hours in the Department of Mathematics and Computer Science.
2. The student has earned a GPA of 3.0 or above in mathematics courses.
3. The student has obtained the consent of the department chairperson and the course instructor.
The faculty in the department is committed to the use of technology to enhance understanding of mathematical concepts and develop mathematical skills. Computers and hand-held calculators are
integrated into coursework in mathematics courses. Note: Calculators are required in all mathematics courses. Please see the course listing for the appropriate calculator.
The Compass Mathematics Placement Test is required to determine placement in mathematics courses. Students may be required to complete both MATH 090 and MATH 099 or just MATH 099 prior to enrollment
in a college-level mathematics course.
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The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes
This is a curious book. The introduction says "If you want a comprehensive, academic dictionary of mathematics, look elsewhere. If you want rigor and proof, try the next shelf. Herein you will find
only the unusual and the outrageous, the fanciful and the fantastic: a compendium of the mathematics they [those villains!] didn't teach you in school." The second sentence is spot on, and the author
is to be commended for putting in the first sentence, but since the word "only" in the third sentence is not accurate, I fear that some people will still buy this book (for $40) thinking that it is a
dictionary of mathematics. It is certainly a dictionary of something, with alphabetical entries from abacus to zonohedron. The subtitle, From Abracadabra to Zeno's Paradoxes, enhances the effect.
There are, in fact, entries for most common mathematical terms here, though the author is not very interested in many of them. The entry on ordinary differential equation has 16 words, 5 of which are
"Compare with partial differential equation", which gets 29 words. There is a slightly longer entry for differential equation, which says (among other things) that "if only nth powers of the
derivatives are involved, the equation is said to have degree n." The entry for abstract algebra reads as if written by someone who never took the course, though the entry for group is better. The
binomial theorem is never fully stated, though it could be pieced together from the brief entries for binomial theorem and binomial coefficient. In the latter, the author writes m choose n when he
means n choose m. There is no entry for Stirling's formula, though it is mentioned in the entries for π and e.
Quite a few of the entries have no obvious connection with mathematics, for example those on Jorge Luis Borges, John Cage, Ernst Florens Friedrich Chladni, John Dee and Lord Edward Plunkett Dunsany,
all of whom get more space than Cauchy. I got tired of looking for these pretty quickly, but catch-22 is here too, and so is swastika, apparently because it is a 20-sided polygon.
The author is clearly very fond of puzzles and recreational mathematics, especially recreational number theory; thus Frederick Schuh gets an entry, but Issai Schur does not. He also likes exotic
plane curves, games (backgammon gets half a page, with no attempt to justify its inclusion in what is ostensibly a mathematics book), optical illusions and extremely large numbers. (Ron Graham, who
could be the subject of a very good book, gets 1/3 the space of his eponymous number.) If you share these predilections, and if you are not too pedantic, then you would probably enjoy this book. I
didn't care for it much myself.
There are some embarrassing mistakes. Arthur Cayley's name is correct in his own entry, but he is called George in Cauchy's entry on the previous page. Erwin Schrodinger's name is correct at the top
of page 34, but he is called Wernher in the entry for William Rowan Hamilton. Jacobi's name is correct in his entry, but he is called Charles in the entry for God. Gödel gets another l in the heading
of page 135. Solomon Golomb's name is correct in his entry, but he is called Simon near the top of page 272. The author uses "loose" when he means lose in the entry for Jacobi. The entry on
triangular numbers says that every triangular number is a perfect number. Somewhere in the book — I can't find it now — e is said to be about 2.712, though a good value is given in the entry for e. I
suppose that Avogadro's constant is here because it's very large, but Avogadro is misspelled. The entry for Abel says that Galois died in a sword fight, but the duel was fought with pistols.
The author has a Ph.D in astronomy, so I find it a bit surprising that he seems to know so little about special functions. The entry on them has only 35 words, and one of the examples is Lagrange
polynomials, which presumably means either Legendre or Laguerre polynomials. (Neither Legendre nor Laguerre gets an entry; Legendre is perhaps the best mathematician without one, though Eisenstein
doesn't have one either, and Schur is another candidate.) The brief entry on Hermite says that he studied a class of differential equations now known as Hermite polynomials, thus confusing an
equation with its solution.
The book is for the most part well-written, but not uniformly so. The entry for calculus of variations has two sentences, the first being "Calculus problems, especially differentiation and
maximization, that involve functions on a set of functions of a real variable." The entry for partition number begins "A number that gives the number of ways of placing n indistinguishable balls into
n indistinguishable urns." Characteristically, the author is interested (just barely) in the number of partitions, but not at all interested in partitions as such — there is no entry for partition.
The book is very weak on enumeration. Stirling numbers are not mentioned at all, though Bell numbers get an entry. The entry for Catalan numbers is remarkably short for this sort of book — evidently
the author has never looked at Stanley's Enumerative Combinatorics or at his web site. Neither Euler numbers nor Eulerian numbers are mentioned. The entry on Bernoulli numbers would be improved by
inserting "nonzero" in the sentence "The first few Bernoulli numbers are...."
Ramanujan's name is rendered twice as "Ramanujan, Srinivasa Aaiyangar." The first three words of the obituary by P.V. Seshu Aiyar and R. Ramachandra Rao are Srinivasa Ramanujan Aiyangar, though the
title is just Srinivasa Ramanujan. In his outstanding biography of Ramanujan, Robert Kanigel (whose name is misspelled in the references) uses Srinivasa Ramanujan Iyengar once, explaining where each
name comes from, and then stops using Iyengar (and for that matter Srinivasa). I don't think G.H. Hardy uses the third name at all, and Berndt only very rarely, e.g., when talking about Ramanujan's
father. In any event, I don't know where the spelling Aaiyangar comes from. I also don't know what the source is for the assertion that Ramanujan's letters (to Baker and Hobson, though the author
does not say so) were returned unopened. Kanigel doesn't say this (although he doesn't rule it out), and neither do Berndt or Hardy. It seems the least likely of Kanigel's three possibilities (letter
ignored, discouraging reply) to me.
I am also uncomfortable with the sentence "But even in cases where [Ramanujan] arrived at conclusions already known, he'd often travel an original route, and, in many cases, almost purely by
intuition." For one thing, it is not one of the author's better contributions to English prose style. More important, I think that the last four words are liable to give a false impression. We still
do not know how Ramanujan found many of his theorems, but it should be emphasized that, however formidable his intuition may have been, it was the kind of intuition that comes from doing lots of very
difficult calculations. If you're bothered by this sort of thing, then this is definitely not the book for you.
Warren Johnson (
) teaches at Connecticut College.
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Compound Interest : Future Value and Present Value
Number of results: 45,497
compound interest math
calculate the compound interest on an investment of $45,000 at 6% interest compounded quarterly fro 3 years Formula for compound interest: A=P(1+i)^n A=amount at the end of n periods (future value) P
=amount invested (present value) i=interest per compounding period. 6% per ...
Monday, July 1, 2013 at 6:46am by MathMate
Find the future value of $10,000 invested now after five years if the annual interest rate is 8 percent. a. What would be the future value if the interest rate is a simple interest rate? b. What
would be the future value if the interest rate is a compound interest rate?
Wednesday, August 22, 2012 at 1:09am by tisha
Interest is calculated with the following formula I=Prt What is the future value ( present value + interrst ) of this account? Present value is $600; interest is 6% Term is 12 years ( Enter your
answer as dollars and cents. )
Thursday, November 10, 2011 at 10:59pm by brandon
Contempoary Mthematics
Question8: Using the Present Value Table on page 358 of your text to compute the present value (principal) for an investment with a compound amount of $20,000, a 30 moth term of investment, and a 14%
nominal interest rate compound semiannually. Question 9. What is the ...
Thursday, October 3, 2013 at 8:12am by Erica Walden
Future/Present Value Problems
present value = 40000 + 70000(1.08)^-10 = 40000 + 32423.54 = 72423.54 Are you not familiar with the basic formulas for compound interest?
Wednesday, February 25, 2009 at 11:40am by Reiny
it affects the real cost, either in present value, or in future value. Interest costs reflect in lost value, or lost purchase value in the future.
Thursday, September 30, 2010 at 9:08am by bobpursley
Using the present value formula you deposit $12,000 in an account that pays 6.5% interest compounded quarterly. A. find the future value after one year? B. Use the future value formula for simple
interest to determine the effective annual yield?
Sunday, August 14, 2011 at 5:47pm by marie
business math
We do not know if it is simple or compound interest. Since compounding frequency is not mentioned, we assume it is simple interest. The simple interest formula: Future value = present value(1+ni) n=
number of years, i=annual interest So 4,370.91 =PV(1+3*0.03) => PV=4,370.91...
Sunday, June 23, 2013 at 9:23am by MathMate
Find how much must be deposited now (present value) at % simple interest so that in 6 years an account will contain $8958 (future value). present value = $ WHAT FORMULA DO I NEED TO USE TO HELP ME
Friday, September 10, 2010 at 2:19pm by AMANDA
200- to 300-word description of the four time value of money concepts: present value, present value of an annuity, future value, and future value of annuity. Describe the characteristics of each
concept and provide an example of when each would be used.
Friday, February 12, 2010 at 8:24pm by Anonymous
Write a 200- to 300-word description of the four time value of money concepts: present value, present value of an annuity, future value, and future value of annuity. Describe the characteristics of
each concept and give an example of when each would be used
Thursday, August 13, 2009 at 9:30am by Joseph
• Write a 200- to 300-word description of the four time value of money concepts: present value, present value of an annuity, future value, and future value of annuity. Describe the characteristics of
each concept and provide an example of when each would be used.
Friday, December 18, 2009 at 9:38pm by Anonymous
Jean will receive $8,500 per year for the next 15 years from her trust. If a 7% interest rate is applied, what is the current value of the future payments? Describe how you solved this problem,
including which table (for example, present value and future value) was used and why.
Monday, November 23, 2009 at 10:28pm by Anonymous
Jean will receive $8,500 per year for the next 15 years from her trust. If a 7% interest rate is applied, what is the current value of the future payments? Describe how you solved this problem,
including which table (for example, present value and future value) was used and why.
Saturday, November 28, 2009 at 2:18am by Anonymous
Jean will receive $8,500 per year for the next 15 years from her trust. If a 7% interest rate is applied, what is the current value of the future payments? Describe how you solved this problem,
including which table (for example, present value and future value) was used and why.
Saturday, November 28, 2009 at 2:18am by Anonymous
Jean will receive $8,500 per year for the next 15 years from her trust. If a 7% interest rate is applied, what is the current value of the future payments? Describe how you solved this problem,
including which table (for example, present value and future value) was used and why.
Monday, November 23, 2009 at 10:28pm by Anonymous
As Reiny pointed out, we do not use tables for the past 40 years, so we can only guess what's shown in the tables. My guess would be that the factor is the ratio of future/present values, namely:
1.005^48=1.270489 Note: compound interest is calculated as: future value=present ...
Wednesday, July 3, 2013 at 2:16pm by MathMate
Simple interest is just: Interest = Principle x rate x time be sure to write % as a decimal to 5% = .05 and 3% = .03 You can add your interest to the original value to find the future value.
Compounded annually: Future value = P(1+ rate)^time tiffany's problem: Future value = ...
Thursday, February 21, 2013 at 9:33am by JJ
I assume 5% is annual interest compounded monthly. Use the compound interest formula: Future value =P((1+r)^n-1)/r where P=monthly payment, =$60 r=interest rate per period = .05/12 n=number of
periods = 30*12 then Future value =60*((1+0.05/12)^360-1)/(0.05/12) =49935.52 From ...
Monday, August 15, 2011 at 7:56pm by MathMate
financial 200
Write a 200-300 word description of the four time value of money concepts: present value, present value of an annuity, future value, and future value of annuity. Descrive the characteristics of each
concept and provide an example of when each of when each would be used.
Friday, July 23, 2010 at 11:09pm by Chris
Future Value/Present Value Problems
so you want the amount of 600 000 at the end of 15 years at 10% p.a. amount = 600000(1.1)^15 = $ 25 063 348.90 a simple application of the compound interest formula.
Wednesday, February 25, 2009 at 11:38am by Reiny
Which of the following actions wiling DECREASE the present value of an investment. A. Decrease the interest B. Decrease the future value C. Decrease the amount of time D. All of the above will
increase the present value
Sunday, January 19, 2014 at 9:00pm by Anthony
why is it necessary to an annuity present value. Why is the calculation of the present value of any future amount important? Why is the present value of any future amount greater when the discount
rate is lower?
Thursday, September 25, 2008 at 7:21pm by Anonymous
Use the following scenario to answer Discussion Question 2. Jean will receive $8,500 per year for the next 15 years from her trust. If a 7% interest rate is applied, what is the current value of the
future payments? Describe how you solved this problem, including which table (...
Thursday, September 17, 2009 at 8:16pm by Anonymous
DESCRIPTION OE THE FOUR TIME VALUE OF MONEY CONCEPTS Present value is the value of a cash flow today. Usage when a single cash flow is to be discounted to today’s value. Formula PV = FV / ((1+i) ^n))
Where, PV = Present value FV = Future Value i= interest rate per compounding...
Friday, December 18, 2009 at 9:38pm by Abacus
Find the future value one year from now of a $7,000 investment at a 3% annual compound interest rate. Also calculate the future value if the investment is made for 2 yeaars?
Monday, August 20, 2012 at 4:32pm by CANDICE
Calculate the present value of the investments using the compound interest formula over the past 10 years, or n=120 periods (t) at interest rate of i=0.0384/12=0.0032 per period. The monthly payment
P=$625 per period, and therefore PV = present value FV = (i.e. future value ...
Monday, December 24, 2012 at 11:04pm by MathMate
Classify the finacial problem. Assume a 7% interest rate compounded annually. Find the value of a $ 1,000 certificate in 4 years. a) sinking fund, b) ordinary annuity, c) future value, d) present
value e) amortization.
Saturday, March 2, 2013 at 1:17pm by Andrew
Classify the finacial problem. Assume a 7% interest rate compounded annually. Find the value of a $ 1,000 certificate in 4 years. a) sinking fund, b) ordinary annuity, c) future value, d) present
value e) amortization.
Saturday, March 2, 2013 at 1:17pm by Andrew
Classify the finacial problem. Assume a 7% interest rate compounded annually. Find the value of a $ 1,000 certificate in 4 years. a) sinking fund, b) ordinary annuity, c) future value, d) present
value e) amortization
Sunday, March 3, 2013 at 8:57am by Andrew
Math 209
1.9%=(1.9)/100 i am not sure that there is an annual interest formula but it might be compounded interest where interest is compounded once a year. the compound interest formula is A=p(1+r/n)^n*t
where p is the principle r is the rate n is the compounding period t is the time ...
Monday, August 15, 2011 at 1:22am by johnathon
Calculating Interest Rate. Find the interest rate implied by the following combinations of present and future values. PresentValue Years Future Value $400 11 $684 $183 4 $249 $300 7 $300 Since you do
not state otherwise, I am assuming that your interest rate is compounded ...
Sunday, March 18, 2007 at 11:45pm by Antoinette
Find the future value one year from now of a $7,000 investment at a 3 percent annual compound interest rate. Also calculate the future value if the investment is made for two years.
Friday, January 13, 2012 at 3:38pm by audrey
well, in all of them interest, present value, future value, and inflation costs are always being used.
Saturday, December 11, 2010 at 6:29am by bobpursley
The present value of the money in your savings acct is $420, and you're receiving 3% annual interest compounded monthly. What is the future value in 2 months?
Sunday, May 27, 2012 at 9:55pm by melissa
Compound math
(Future Value) Suppose you invest $8000 into an account that pays an annual interest rate of 6.2%. How much is in the account after 30 years if a. simple interest is compound monthly? b. interest is
compounded monthly? c. interest is compounded daily?
Tuesday, April 19, 2011 at 9:39am by Help
business maths
Compound interest =P[(1+r)^n-1] where P=present value r=rate of interest per period = 0.06/4 = 0.015 n=number of periods = 5*4 = 20 Compound interest =35000(1.015^20-1) =?
Thursday, May 26, 2011 at 6:20pm by MathMate
The present value of the money in your savings account is $420, and you're receiving 3% annual interest compounded monthly. What is the future value in two months?
Monday, November 15, 2010 at 4:59pm by Anonymous
The present value of the money in your savings account is $420, and you're receiving 3% annual interest compounded monthly. What is the future value in two months?
Friday, May 4, 2012 at 4:31pm by Paige
On June 1, 2012, Pitts Company sold some equipment to Gannon Company. The two companies entered into an installment sales contract at a rate of 8%. The contract required 8 equal annual payments with
the first payment due on June 1, 2012. What type of compound interest table is...
Wednesday, October 17, 2012 at 8:02pm by Jed
On June 1, 2012, Pitts Company sold some equipment to Gannon Company. The two companies entered into an installment sales contract at a rate of 8%. The contract required 8 equal annual payments with
the first payment due on June 1, 2012. What type of compound interest table is...
Wednesday, October 17, 2012 at 8:22pm by Jed
Compound Interest : Future Value and Present Value
Payments of $1800 and $2400 weere made on a $10,000 variable-rate loan 18 and 30 months after the date of the loan. The interest rate was 11.5% compounded semi-annually for the first two years and
10.74% compounded monthly thereafter. What amount was owed on the loan after ...
Friday, July 24, 2009 at 1:22am by Math
business math-73
Use the Present Value Table on page 358 of your text to compute the present value (principal) for an investment with a compound amount of $20,000, a 30-month term of investment, and a 14% nominal
interest rate compounded semiannually. (Points : 2.5)
Saturday, October 5, 2013 at 11:49pm by Minnie
The present value of an ordinary annuity is the sum of the present values of the future periodic payments at the point in time one period before the first payment. What is the amount that must be
paid (Present Value) for an annuity with a periodic payment of R dollars to be ...
Monday, November 28, 2011 at 8:20pm by tchrwill
Hint:Use the compound interest formula Future=present*(1+i)^n Here Future=40000 n=30 i=0.06 (6%) Solve for Present.
Saturday, June 1, 2013 at 8:21am by MathMate
a=16,00, r=11.5%, t=5 years determine the present value, p, you must invest to have the future value, a, at simple interest rate r after time t. round uo to nearest cent
Sunday, August 14, 2011 at 4:53pm by marie
future value = present value e^yr where y = 27 years r = .04 yearly interest rate e^1.08 = 2.945 so 2700 * 2.945 = 7950.63
Wednesday, December 19, 2012 at 3:46pm by Damon
Present value = 230,000 Inflation rate, r = 6% p.a. Period, n = 21 Future value = present value * (1 + r)n You should have no problem proceeding from here.
Sunday, July 12, 2009 at 2:07pm by MathMate
present value
The present value P that will amount to A dollars in n years with interest compounded annually at annual interest rate r, is given by P = A (1 + r) -^n. Find the present value that will amount to
$50,000 in 20 years at 8% compounded annually.
Monday, July 14, 2008 at 11:11am by Don
5. The present value of the money in your savings account is $420, and you're receiving 3% annual interest compounded monthly. What is the future value in two months? A. $424.11 B. $426 C. $422.10 D.
Friday, May 4, 2012 at 4:31pm by Ahlam
In the future value annuity table at any interest rate for one year, why is the future value interest factor of this annuity equal to 1.00?
Saturday, August 3, 2013 at 5:46am by April
Interest, i=9%=0.09 p.a. Future value, S Ordinary annuity for 6 years, n=6 yearly payment, R = $20,000 Find future value when child will be 24 years old: S = R((1+i)^n-1)/i = $20,000*(1.09^6-1)/0.09
= $20,000*(7.523335) = $150,466.69 Present value (when child is 17) P= S/(1+i...
Saturday, April 30, 2011 at 11:40pm by MathMate
Use the compound interest formula: Future = Present*(1+r)^n so 32=Present(1.062)^5 Solve for Present.
Sunday, February 26, 2012 at 11:44pm by MathMate
Personal Finance
Aaron wants to put $200.00 per month into an IRA account at 15% for four years. What is he solving for using his financial calculator? A. Present Value B. Future Value C. Interest Rate D. Payment
Friday, October 25, 2013 at 10:37pm by Sharon
If the principal P = $900, the rate r = 7 1/2 %, and time t = 1 year, find the following. What is the amount of interest? What is the future value? Ok found I think I found the interest 900x0.075=
67.50 is this correct.Now I need help in finding the future value.
Wednesday, November 23, 2011 at 1:34pm by MONICA
math- algebra
Interest for the first year is the principal, P=$900 multiplied by the rate of interest, r=0.045, multiplied by the number of periods, n=1. So interest I=$900*0.045*1=$40.5 The future value is the
sum of the interest and the principal. Note that in this case (n=1), the ...
Sunday, July 31, 2011 at 11:49pm by MathMate
Math-- Desperate for help!
Find the amount in an account if $2000 is invested at 6.125%,compounded semi-anually,for 2 years. A. $2,256.49 B. $2,252.50 C. $2,324.89 D. $544,757.84 One of these is the correct answer. I am coming
up with (D) 544,757.84 Can you check, because I am probably wrong. You are ...
Sunday, February 25, 2007 at 8:16pm by Hilda
an inheritance will be 20000. the interest rate for the the time value of money is 7%. How much is the inheritance worth now, if it will be received a) in 5 years? b)in 10 years? c)in 20 years I know
i'm supposed to use F=P(1+i)^n or P=F(1+i)^-n but I am not sure if the ...
Thursday, April 9, 2009 at 2:00am by jen
fin 200 question
Jean will receive $8,500 per year for the next 15 years from her trust. If a 7% interest rate is applied, what is the current value of the future payments? Describe how you solved this problem,
including which table (for example, present value and future value) was used and ...
Tuesday, July 14, 2009 at 7:29pm by meshelle
Financing and Accounting
The formula I'd use is: FV = PV(1+i)^n, where FV = future value PV = present value i = interest rate per period n = number of periods
Thursday, February 18, 2010 at 5:59pm by Destiney
find the present value of the following future amount. 600,000 at 6% compunded semiannually for 25 years what is the present value
Tuesday, April 23, 2013 at 6:57pm by stacy
I need an understanding of a few matters in my accounting class: 1) what are annuities and why is it necessary to calculate there present value? 2) How does the frequency of interest compounding,
regardless of the rate of interest or period of accumulation affect the future ...
Wednesday, March 31, 2010 at 5:25pm by Stumpped on Accounting
What you probably did was calculated simple interest for 15 years on $1000 and added to $2200 to give $2650. Compound interest formula are based on the number of periods, n, the interest was
compounded. The interest being compounded 4 times a year, so there are 15*4=60 periods...
Friday, September 2, 2011 at 4:48pm by MathMate
compound and interest Using the below values please calculate the amount accumulated (future value) *Initial principal=$2000 *Interest Rate=9% *Number of years 7 *Monthly compounding
Wednesday, April 13, 2011 at 1:59pm by John
compound and interest Using the below values please calculate the amount accumulated (future value) *Initial principal=$2000 *Interest Rate=9% *Number of years 7 *Monthly compounding
Wednesday, April 13, 2011 at 10:37pm by George
compound and interest Using the below values please calculate the amount accumulated (future value) *Initial principal=$2000 *Interest Rate=9% *Number of years 7 *Monthly compounding
Thursday, April 14, 2011 at 8:03am by John
Ms. Sue, Thank you for the websites. I will look into them. I do have another question for you: I do understand the formulas for future value and present value, but I want to make sure that I am on
the right track with this one. Future value: $5,000 compounded quarterly at 6% ...
Wednesday, March 31, 2010 at 5:25pm by Stumpped on Accounting
Math: Present Value
what is the present value of nine annual cash payments of 4 000 to be paid at the end of each year using an interest rate of 6%.
Sunday, January 11, 2009 at 12:11pm by Anonymous
Computing future value calculate the future value of a retirement account in which you deposit $2,000 a year for 30 years with an annual interest rate of 7 percent.
Tuesday, January 22, 2013 at 6:52pm by tammy
Comparing Future Value. Calculate the future value of a retirement account on which you deposit $2000.00 a year for 30 years with an annual interest rate of 7%.
Thursday, April 19, 2012 at 1:20pm by Valarie
How much money must be deposited now at 6% interest compounded semiannually to yield an annuity payment of $4,000 at the beginning of each six-month period for a total of five years answer needs to
be rounded to the nearest cent I got $29,440.36 choices are $38,120.80 or $35,...
Thursday, November 22, 2007 at 9:32pm by tchrwill
Math Finite
Use the compound interest formula for n=number of periods R=monthly interest rate, 5.75%/12=0.479167% P=present value of investment = $12000 F=future value of investment = $15000 Then F=ARn 15000=
12000(1.00479167)n 1.00479167n = 15000/12000=1.25 take log on both sides n*log(1....
Wednesday, December 8, 2010 at 12:41am by MathMate
bonds, present value concept
Many factors influence present value of bonds. Basically, it is an attempt to combine recent sales data, risk of default, probability of being called, and coupon interest rate compared to anticipated
inflation and the prevailing interest rate for similar maturity. If these ...
Wednesday, May 13, 2009 at 8:27pm by drwls
The compound interest formula is: 500000=P*(1.076)^15 Solve for P, the present value.
Saturday, July 30, 2011 at 7:44pm by MathMate
personal finances
computing future value. calucale the future value of a retirement account in which you deposit $2,000 a year for 30 years with an annual interest rate of 7 percent.
Saturday, January 28, 2012 at 12:51pm by shanty
financal problems
Computing future value. Calculate the future value of a retirement account in which you deposit 2,000 a year for 30 years with an annual interest rate of 7 percent.
Tuesday, April 24, 2012 at 5:48pm by gina
5. Compute the price of $3,461,181 received for the bonds by using the tables of present value in Appendix A. (Round to the nearest dollar.) Your total may vary slightly from the price given due to
rounding differences. Present value of the face amount $ Present value of the ...
Thursday, May 12, 2011 at 10:20am by johnetta
Compute the price of $7,936,343 received for the bonds by using the tables of present value in Appendix A. (Round to the nearest dollar.) Your total may vary slightly from the price given due to
rounding differences. Present value of the face amount $ Present value of the ...
Monday, April 25, 2011 at 8:40pm by Kieran McCamment
Compound interest
What is the future value of $800 invested for 14 years at 11 percent compounded annually
Saturday, May 12, 2012 at 3:28am by Anonymous
Selling price of a bond: Problem type 1 On December 31, 2008, $140,000 of 9% bonds were issued. The market interest rate at the time issuance was 11%. The bonds pay on June 30 and December 31 and
mature in 10 years. Compute the selling price of a single $1,000 bond on December...
Monday, June 13, 2011 at 7:46am by Nick
Rob has a balance of 1695$ in his bank account The account pays 2.9% interest per year, compounded annually. The compound interest formula is A=P(1+i)^n A=future value\P=principal/i+interest rate/n=
number of payments rods balance will reach 3000$ after how many years?
Thursday, October 3, 2013 at 1:50pm by sam
P=principal ($4000) or present value r=interest rate per period (5.2% p.a.= 1.3% per quarter) n=number of periods (3*4 quarters = 12) Amount after 3 years (future value) =P(1+r)^n =4000(1.013)^12 = ?
Sunday, May 15, 2011 at 7:02pm by MathMate
I am so confused on how to answer this question given from my instructor. I see many websites that pertain to present value. Is there an explanation somewhere on the internet that would help explain?
I googled the question and found"Net present value - Wikipedia, the free ...
Thursday, April 1, 2010 at 12:11pm by Highly confused
what formulas do i use for this: Investments Suppose $10,000 is invested at an annual rate of 5% for 10 years. Find the future value if interest is compounded as follows. A) Annually B) Quarterly C)
Monthly D)Daily (365 days) In each case, use the formula Future value = ...
Tuesday, June 19, 2007 at 11:46pm by student
Use financial calculator to solve for the interest rate involved in the following future value of an annuity due problem. The future value is $57,000, the annual payment is $7,500, and the time
period is six years
Saturday, April 28, 2012 at 2:00pm by Phoebe
It requires the solution for R of the following equation, A=PR^n where A=future value = $14000 P=present value = $350 n=number of periods = 30 and R=rate of interest So, we need to solve for R 14000=
350R^n take logs log(14000/350)=30log(R) log(R)=log(40)/30=.053402 R=1.1308 ...
Tuesday, April 26, 2011 at 4:55pm by MathMate
Continuous compounding: future value = present value * ert where t=number of periods, and r=rate future value/present value = 2 or ert=2 e0.06t=2 take natural log on both sides, 0.06t = ln(2) t=11.55
years. The rule of 69 ============== In fact, you can apply the rule of 69 ...
Friday, December 10, 2010 at 12:23am by MathMate
Personal Finance
NEED HELP IMMEDIATELY, HAVE UNTIL 10P.M. CENTRAL TO HAVE ANSWERS?? 1.Determining the Future Value of Education. Jenny Franklin estimates that as a result of completing her master’s degree, she will
earn $6,000 a year more for the next 40 years. a.What would be the total amount...
Saturday, January 29, 2011 at 10:14am by Marie
differences between present value and future in time value
Tuesday, November 10, 2009 at 10:39am by soomal
Present value is an absolute. Future value is uncertain.
Tuesday, November 10, 2009 at 10:39am by Ms. Sue
Having troube with java, i am not a regular programmer, if anybody can help me out writting this program: write a program that takes two numbers from the java console representing, respectively, an
investment and an interest rate(you will expect the user to enter a number such...
Thursday, February 4, 2010 at 1:55pm by John
Present Value suppose your bank account will be worth $7000 in one year. The interest rate (discount rate) that the bank pays is 8%. What is the present value of your bank account today? What would
the present value of the account be if the discount rate is only 3%
Monday, July 27, 2009 at 5:16pm by Eloisa
business math
what are the amount and present value of an annuity of $100 paable at the beginning of each quarter fro 15 years if the interest rate is 12% compounded quarterly? Present Value=PMT[(1-(1+i)^-n)/i]
Amount = ?????
Monday, March 12, 2007 at 6:31pm by Rom
(Present Value) What is the present value of an annuity that pays $250,000 in 30 years if interest accumulates at a rate of 7.5% compounded semiannually? (i.e. How do you have to pay NOW for the
policy? You make no payments other than your lump sum payment.)
Tuesday, April 19, 2011 at 9:42am by Page
Use the present value formula to compute the amount that should be set aside today to ensure a future value of $ 2,000 in 1 year if the interest rate is 12% annually, compounded annually. (a) $
1,776.97 (b) $ 1,765.89 (c) $ 1,785.72 (d) $ 1,786.97 (e) $ 1,768.97
Thursday, November 3, 2011 at 8:20pm by Paul
A=PR^6 (calculate future value from present value P) R^6=A/P=3 (triple) R=3^(1/6) (sixth root of 3)
Wednesday, May 4, 2011 at 6:44pm by MathMate
What type of problem is this? amortization future value?present value? sinking fund? formula...
Tuesday, December 3, 2013 at 4:59pm by Lynda
finite math
Split the problem in two parts, 25 years and 18 years. The interest rate is known for the first part (7% p.a. or per month?) and compounded monthly. So the future value after 25 years is determined,
say A. Assuming A<400,000=target, subtract remaining future value B=400,000...
Saturday, April 30, 2011 at 6:18pm by MathMate
Finite Math
Use the compound interest formula for n=number of periods = 5*12 = 60 R=monthly interest rate, 1.01 for 1% P=present value of investment = $2000 F=future value of investment = $3500 Then F=ARn 3500=
2000R60 rearrange Rn = 3500/2000 = 1.75 Take log on both sides 60*log(R)=log(1....
Tuesday, December 7, 2010 at 11:50pm by MathMate
Help math calc
Find the future value of $700 deposited at 3% for 9 years if the account pays simple interest and the account pays compounded annually? The future value of an account that pays simple interest is
Thursday, February 21, 2013 at 9:48am by Tiffany
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Spurious Significance #2 : Granger and Newbold 1974
"Spurious significance" was a phrase used in the title of our GRL article. We regarded this as perhaps the most essential point of the article, but it seems to have gotten lost. This is the second of
a planned series of notes on spurious significance, to give a sense of the statistical background. Granger and Newbold [1974] posted up here is an extremely famous article, which starts off the
modern discussion of the problem of spurious regression. Granger is a recent Nobel laureate in economics.
Granger and Newbold observed that, although the classic spurious regressions (see Spurious #1) had very high R2 statistics, they had very low (under 1.5) Durbin-Watson (DW) statistics. (The DW
statistic measures autocorrelation in the residuals.) Granger and Newbold:
It is very common to see reported in applied econometric literature time series regression equations with an apparently high degree of fit, as measured by the coefficient of multiple correlation
R2 or the corrected coefficient R2, but with an extremely low value for the Durbin-Watson statistic. We find it very curious that whereas virtually every textbook on econometric methodology
contains explicit warnings of the dangers of autocorrelated errors, this phenomenon crops up so frequently in well-respected applied work. Numerous examples could be cited, but doubtless the
reader has met sufficient cases to accept our point. It would, for example, be easy to quote published equations for which R2 = 0.997 and the Durbin-Watson statistic (d) is 0.53. The most extreme
example we have met is an equation for which R2 = 0.99 and d = 0.093.,,
Granger and Newbold moved beyond the framework of curious examples by doing simulations in which they generated series of random walks, regressing one against another. They found that these
regressions consistently had “statistically significant”‘? F-statistics (the F-statistic is related to the R2 statistic) and suggested that the Durbin-Watson (DW) statistic did a good job of
identifying problems. They didn’t argue that a failed DW statistic was a necessary condition of a spurious condition, but they certainly argued that a failed DW statistic was sufficient for a failed
model. Granger and Newbold:
It has been well known for some time now that if one performs a regression and finds the residual series is strongly autocorrelated, then there are serious problems in interpreting the
coefficients of the equation. Despite this, many papers still appear with equations having such symptoms and these equations are presented as though they have some worth. It is possible that
earlier warnings have been stated insufficiently strongly. From our own studies we would conclude that if a regression equation relating economic variables is found to have strongly
autocorrelated residuals, equivalent to a low Durbin-Watson value, the only conclusion that can be reached is that the equation is mis-specified, whatever the value of R2 observed. “⤍
It is not our intention in this paper to go deeply into the problem of how one should estimate equations in econometrics, but rather to point out the difficulties involved. In our opinion the
econometrician can no longer ignore the time series properties of the variables with which he is concerned – except at his peril. The fact that many economic “‘œlevels’ are near random walks or
integrated processes means that considerable care has to be taken in specifying one’s equations”⤍
One cannot propose universal rules about how to analyse a group of time series as it is virtually always possible to find examples that could occur for which the rule would not apply.
In this first systematic article on spurious regression, you can see what appears to me to be the over-riding goal of theoreticians: to find a statistic or statistics which can identify spurious
relationships in an unsupervised way i.e. as some functional of the data and the residuals.
While Granger and Newbold did not propose the DW statistic as a magic bullet for testing spurious regressions, not performing a DW statistic on a regression relating highly autocorrelated series
would be inconceivable for any econometrician after 1974. I’ve seen occasional use of DW statistics in paleoclimate articles, but very few. Given the remarkable autocorrelations in paleoclimate
series, you would think that it would be a very standard test. It’s almost as though paleoclimatologists are afraid to use this test.
I’ll give some examples tomorrow in series that we’ve discussed in the past.
6 Comments
1. Durbin-Watson is part of the standard introductory treatment of econometrics and has been for decades, because it comes up a lot and autocorrelation matters a lot. However DW has a couple of
limitations. It’s got an obscure distribution (but so what, there are tables and Shazam can compute the exact p-value), it’s not valid if there are lagged dependent variables, and it only tests
for AR1. There’s another test that has a fancy-sounding name and is easy to do (2 big advantages, in my view), called the LM test, which is more general and which is steadily getting into the
texts. Or there’s the brute force method of estimating models with ARMA residuals and testing lags for insignificance.
The connection to the term “spurious” is building across these notes, but already a key point is worth stressing. When you do a regression the package mechanically computes the ratio of the
estimated parameter to the estimated standard error and sticks it in a column under the heading “t-statistic”. But that is no guarantee the number therein came from a data generating process that
follows a t-distribution. You have to be able to rule out some influential model misspecification problems. Otherwise you might be comparing your “t-statistic” to the wrong critical values. In
the case of Granger and Newbold they looked at regressing random walks on each other. In that case a “t-stat” of, say, 4.0 does not mean the relationship is significant since the ratio in
question doesn’t follow a t distribution. “Spurious significance” in this sense means comparing your test statistic to the wrong benchmark and concluding you have significance when in reality you
do not.
2. I would go further than this:
not performing a DW statistic on a regression relating highly autocorrelated series would be inconceivable for any econometrician after 1974
Any reasonably knowledgable econometrician would perform such tests on ALL the time series in use PRIOR to any regression, in order to determine the existance of units roots and hence understand
the nature of the data being analysed and the potential statistical pitfalls that could result.
That has always been one of the glaring omissions on all hte time series work on temperature or proxies – some basic unit root test on the data and a description of whether they are stationary or
3. Some of the articles that have interested me the most pertain to situations where the DW statistic does not work. That accounts for much of my present interest in ARMA(1,1) statistics, where Feng
[2005] appears to have used “almost integrated almost white” (ARMA(1,1) processes to explain spurious regressions in Ferson et al [2003]. I’m hoping to get there in these notes without stumbling
too much.
4. Re #2: Individual series need to be tested for unit roots, but this is different from applying a DW test on a regression model. And there certainly are papers that examine geophysical data for
nonstationarity prior to proceeding with trend modeling or other analysis. But the result is a body of literature that is divided on whether temperature is nonstationary or highly autocorrelated;
either way it means the data need to be handled and interpreted carefully.
5. Paul, one of the interesting features of the temperature series is that, modeled as ARMA(1,1), their AR coefficients are >>0.9 (and a negative MA1 coefficient), quite a bit higher than modeled as
ARMA(1,0), which is the more usual comparison. I don’t entirely know where this leads, but I’m going to get to some econometric models raising real issues about spurious significance in this type
of context.
6. #5.
Indeed. Especially for forecasting purposes! (multiple equation models employing lagged endogenous variables exhibiting autocorrelation and all that)
One Trackback
1. [...] in the spurious regression range. It is unacceptable for the hockey Team to simply ignore this. See my discussion of Granger and Newbold [1974] who said over 30 years ago: It is very common
to see reported in applied econometric literature [...]
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Symbolic Integration: The Algorithms
The pioneering paper about symbolic integration is:
Joel Moses:
Symbolic Integration: The Stormy Decade
Communications of the ACM, Vol 14, No 8 August 1971, pp. 548-650
This paper is an updated resumee of Moses' thesis 'Symbolic Integration', which is available at these addresses:
This thesis describes a three stage algorithm whose design is still followed by symbolic integrators of all major computer algebra systems.
A modern text about symbolic integration is:
Manueal Bronstein:
Symbolic Integration I Transcendental Functions
1997 Springer Berlin Heidelberg New York
Fatemans paper about algebraic simplification:
The Integration Algorithm
The following description of the integration algorithm is taken from the ACM article (with substancial omissions):
First Stage
The first stage is a simple test to determine whether derivatives of a subexpression of the integrand divide the rest of the integrand. This test determines whether the integrand is of the form:
∫ c* op(u(x))*u'(x) dx
• c is a constant
• u(x) is some function of x
• u'(x) is its derivative
• op is an elementary function, namely one of
□ sin
□ cos
□ tan
□ cot
□ sec
□ csc
□ asin
□ atan
□ asec
□ log
In addition, op(u(x)) can have the forms
u(x) (op being the identity)
1/u(x) (op being the reciprocal)
u(x)^d where d <> -1
d^u(x) where d is a constant.
For integrands of this form, a table lookup and a substitution are sufficient to answer the integral.
Experience shows that this simple algorithm solves many integrals. The first stage examples demonstrates the power of the algorithm.
Second Stage
The algorithm enters the second stage when the first stage cannot solve the integral. The second stage contains eleven methods which might be applicable to a given problem. A pattern matching routine
determines which methods should be attempted.
Third Stage
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