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Fourier Series for e^x
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Fourier Series for e^x
I've run into a problem with deriving the Fourier series for e^x.
I've got
putting it all together:
But, clearly this can't be valid, since plugging n = 1 into the RHS gives an undefined result for the first term. What can I do about this?
Re: Fourier Series for e^x
Interesting to look at but you do know that Fourier series work best for periodic functions, which e^x is not.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Fourier Series for e^x
So what I have done is nonsense?
Re: Fourier Series for e^x
I did not say that. I will have to look at more. Just do not expect it to be a very good approximation.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Fourier Series for e^x
What if I defined it as having period 2π?
Re: Fourier Series for e^x
In some interval you can possibly fit a non periodic function with a Fourier series. Fourier series are notorious for having problems at the endpoints due to the Runge effect just like polynomial
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Real Member
Re: Fourier Series for e^x
It has a period of 2*i*pi not 2*pi.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Fourier Series for e^x
Yes, he may have used the wrong equations for the coefficients.
This yields:
Plotting that and e^x we get the typical Fourier fit of a function. Notice how the purple line ( fourier ) hugs e^x.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Fourier Series for e^x
So the period is 2iπ since you're thinking about it in the complex plane? (2iπ being one revolution)
Those are the equations I used for the co-efficients...
Re: Fourier Series for e^x
Would you say that is a fairly good fit?
Re: Fourier Series for e^x
No the period of my equations is either -2π to 2π or -π to π. I would never use anything for serious computation that contained i.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Fourier Series for e^x
Oh, okay. I thought it was 2π since a lot of Fourier series tend to be defined that way (f(x) = f(x + 2π)).
Re: Fourier Series for e^x
I would say that is a fairly useless fit. The good point about Fourier fits is that they are orthogonal and least squares at the same time.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Fourier Series for e^x
Where is the orthogonality in the diagram? Also, do you think there are any uses for this Fourier series? For example, finding the Fourier series for x^4 can get you the sum of 1/(n^4) (integer n).
Re: Fourier Series for e^x
One question at a time because you have asked a big one. The orthogonality is in the basis.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Fourier Series for e^x
What do you mean by the basis? Are you referring to the peaks/troughs of the Fourier series, and the curve (e^x) it is approximating? And, why is this orthogonality useful?
Re: Fourier Series for e^x
To understand it as mathematicians do you have to understand basis vectors.
I understand it in a way more general, a geometric way.
We can discuss it all you want but we will have delve into some pretty bizarre concepts.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Fourier Series for e^x
Could you tell me the geometric way? Maybe that could then help me understand basis vectors.
Re: Fourier Series for e^x
You already know about something about basis vectors but I will draw some pictures that will make it clear to you.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Fourier Series for e^x
Okay, thank you.
Re: Fourier Series for e^x
First some background. When I frst met you in here you wanted to end up at Cern. You might find when you get there that they are doing a completely different type of mathematics than you were taught.
You are and were taught bookmath. It is important to understand that it is not real math. Meaning, it is not the real mathematics you might be called upon to do on a day to day basis. Most students
do not even know this and even if you do not believe listen to it anyway.
1) In bookmath the number line is continuous, in practical math the number line has holes in it! These holes are numbers that are left out!
2) No one should ever use the quadratic formula to solve a quadratic equation.
3) We can not really subtract as accurately as we can add!
4) (a^2 - b^2) ≠ (a -b)(a+b)
5) a1 + a2 + a3 +...+an ≠ an +...+ a3 + a2 + a1
6) A matrix has three states, invertible, singular and nearly singular.
7) Newton's iteration is rarely the best way to go and should be rarely used.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Fourier Series for e^x
I don't understand #3-5...
Re: Fourier Series for e^x
It is okay for now, you do not need to for us to get back on track.
Take a look at this drawing.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Fourier Series for e^x
Hmm. So, AB, and CD, are orthogonal. But... I don't understand what's going on on the right-hand side. So you are saying that EF and GH are orthogonal (as you showed with AB and CD) but the bit about
the line joining EG and FH has confused me...
Re: Fourier Series for e^x
Forget orthogonality for a second. It is easy for you to eyeball the intersection of AB and CD. It is well defined.It is a tiny point. GH is on top of EF and therefore it is impossible to pinpoint an
intersection. There are zillions of them. The third one is the interesting case, it is what we call nearly singular. It is close to being on top but it is not. The circle shows how wide the possible
point of intersection is. It is hard to pick it out by eye. Do you follow?
Going to get some food in me, see you in a bit.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Post reply
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Math Forum Discussions - Re: Given an iterative equation to solve an equation
Date: Jan 20, 2013 7:23 AM
Author: Greg Heath
Subject: Re: Given an iterative equation to solve an equation
"Jacob" wrote in message <kdf58n$958$1@newscl01ah.mathworks.com>...
> An iterative equation for solving the equation x^2-x-1=0 is given by
> x(r+1)=1+(1/x(r)) for r=0,1,2,...
> Given x0=2, write a Matlab script to solve the equation. Sufficient accuracy is obtained when abs(x(r+1)-x(r))<.0005.
> I am a new to Matlab and I am having a hard time really even starting this problem. I was thinking that using some sort of loop until the accuracy condition is met would work. Any help would be much appreciated.
> Thanks in advance,
> Jacob
Your plan sounds ok to me. | {"url":"http://mathforum.org/kb/plaintext.jspa?messageID=8114326","timestamp":"2014-04-16T04:31:56Z","content_type":null,"content_length":"1688","record_id":"<urn:uuid:9bd4893e-1e44-4e6b-a934-3c6aeccd7786>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00137-ip-10-147-4-33.ec2.internal.warc.gz"} |
Taking the gamble out of DNA sequencing
The latest news about biotechnologies, biomechanics synthetic biology, genomics, biomediacl engineering...
Posted: Feb 24, 2013
Taking the gamble out of DNA sequencing
(Nanowerk News) Two USC scientists have developed an algorithm that could help make DNA
sequencing affordable enough for clinics – and could be useful to researchers of all
Andrew Smith, a computational biologist at the USC Dornsife College of Letters, Arts and
Sciences, developed the algorithm along with USC graduate student Timothy Daley to help
predict the value of sequencing more DNA, to be published in Nature Methods on February
Extracting information from the DNA means deciding how much to sequence: sequencing too
little and you may not get the answers you are looking for, but sequence too much and you
will waste both time and money. That expensive gamble is a big part of what keeps DNA
sequencing out of the hands of clinicians. But not for long, according to Smith.
"It seems likely that some clinical applications of DNA sequencing will become routine in
the next five to 10 years," Smith said. "For example, diagnostic sequencing to understand
the properties of a tumor will be much more effective if the right mathematical methods
are in place."
The beauty of Smith and Daley's algorithm, which predicts the size and composition of an
unseen population based on a small sample, lies in its broad applicability.
"This is one of those great instances where a specific challenge in our research led us to
uncover a powerful algorithm that has surprisingly broad applications," Smith said.
Think of it: how often do scientists need to predict what they haven't seen based on what
they have? Public health officials could use the algorithm to estimate the population of
HIV positive individuals; astronomers could use it to determine how many exoplanets exist
in our galaxy based on the ones they have already discovered; and biologists could use it
to estimate the diversity of antibodies in an individual.
The mathematical underpinnings of the algorithm rely on a model of sampling from ecology
known as capture-recapture. In this model, individuals are captured and tagged so that a
recapture of the same individual will be known – and the number of times each individual
was captured can be used to make inferences about the population as a whole.
In this way scientists can estimate, for example, the number of gorillas remaining in the
wild. In DNA sequencing, the individuals are the various different genomic molecules in a
sample. However, the mathematical models used for counting gorillas don't work on the
scale of DNA sequencing.
"The basic model has been known for decades, but the way it has been used makes it highly
unstable in most applications. We took a different approach that depends on lots of
computing power and seems to work best in large-scale applications like modern DNA
sequencing," Daley said.
Scientists faced a similar problem in the early days of the human genome sequencing
project. A mathematical solution was provided by Michael Waterman of USC, in 1988, which
found widespread use. Recent advances in sequencing technology, however, require thinking
differently about the mathematical properties of DNA sequencing data.
"Huge data sets required a novel approach. I'm very please it was developed here at USC,"
said Waterman.
Subscribe to a free copy of one of our daily
Nanowerk Newsletter Email Digests
with a compilation of all of the day's news. | {"url":"http://www.nanowerk.com/news2/biotech/newsid=29227.php","timestamp":"2014-04-16T07:15:40Z","content_type":null,"content_length":"37075","record_id":"<urn:uuid:3228b375-7a7d-4b73-a8bf-b205b855f113>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00069-ip-10-147-4-33.ec2.internal.warc.gz"} |
Revista IBRACON de Estruturas e Materiais
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On-line version ISSN 1983-4195
Rev. IBRACON Estrut. Mater. vol.5 no.5 São Paulo Oct. 2012
Punching strength of reinforced concrete flat slabs without shear reinforcement
P. V. P. Sacramento^I; M. P. Ferreira^II; D. R. C. Oliveira^III; G. S. S. A. Melo^IV
^IMestrando em Engenharia Civil, Universidade Federal do Pará, Belém, Pará, Brasil
^IIProfessor, Faculdade de Engenharia Civil, Universidade Federal do Pará, mpina@ufpa.br, Belém, Pará, Brasil
^IIIProfessor, Faculdade de Engenharia Civil, Universidade Federal do Pará, denio@ufpa.br, Belém, Pará, Brasil
^IVProfessor, Departamento de Engenharia Civil e Ambiental, Universidade de Brasília, melog@unb.br, Brasília, Distrito Federal, Brasil
Punching strength is a critical point in the design of flat slabs and due to the lack of a theoretical method capable of explaining this phenomenon, empirical formulations presented by codes of
practice are still the most used method to check the bearing capacity of slab-column connections. This paper discusses relevant aspects of the development of flat slabs, the factors that influence
the punching resistance of slabs without shear reinforcement and makes comparisons between the experimental results organized in a database with 74 slabs carefully selected with theoretical results
using the recommendations of ACI 318, EUROCODE 2 and NBR 6118 and also through the Critical Shear Crack Theory, presented by Muttoni (2008) and incorporated the new fib Model Code (2010).
Keywords: flat slab, punching shear, reinforced concrete, codes.
1. Introduction
Flat slabs are those which are directly supported on columns without capitals. They can be considered as a good option for concrete buildings since they may reduce the construction time due to the
simplification of forms and rebars and especially by attributing greater flexibility in layout of floors. The design of slab-column connection is the most critical point in the design of flat slabs,
because of the concentration of shear stresses in this region that can lead to punching, which is a localized failure mode that can occur without significant warnings and may lead the whole structure
to ruin through the progressive collapse. Figure 1a shows an example of punching failure recorded by Ferreira [1]. One way to ensure local ductility and prevent progressive collapse of flat slabs is
through the use of post-punching reinforcement as those shown in Figure 1b, which must be designed to carry the vertical reaction in the column, and must be detailed in order to ensure that they are
sufficiently anchored beyond the region of the possible punching cone.
Since tests carried by Elstner and Hognestad [2] many other studies have been conducted aiming to understand the behavior and strength of flat slabs. Some theoretical methods were proposed but none
was generally accepted because they were not able to accurately estimate the punching resistance of slab-column connections and at the same time explains the phenomenon with all its variables. Thus,
the design of flat slabs to the punching is normally done using recommendations presented by codes of practice for design of concrete structures, which are essentially empirical.
Recently Muttoni [3] presented a new theoretical approach called Critical Shear Crack Theory (CSCT), which is able not only of predicting the bearing capacity of slab-column connection, but also of
estimating their behavior in service (rotation, displacements and strains). This theory is based on the idea that the punching resistance decreases with increasing rotation of the slab, and has
recently been embodied in the first draft of the new fib Model Code [4,5], which was presented in 2010, and has come to replace the old CEB-FIP MC90 [6]. This paper aims to evaluate this method by
comparing its theoretical results with experimental results of tests of 74 reinforced concrete flat slabs without shear reinforcement carefully selected (see section 6 of article) to form a large
database, with specimens with a significant variation of parameters such as the effective depth, flexural reinforcement ratio and compressive strength of concrete. These experimental results were
also compared with the theoretical results obtained by using the recommendations of ACI 318 [7], EUROCODE 2 [8] and NBR 6118 [9].
2. Historical development of flat slabs
There is controversy about who idealized the flat slabs structural system. Gasparini [10] states that the credit for the development of this system should be given to George M. Hill, an engineer who
reportedly designed and built constructions like filtration plants and storehouses in different regions of the United States between 1899 and 1901. He emphasizes, however, that C A. P. Turner, an
American inventor and engineer, was the one responsible for demonstrating that these slabs were reliable with numerous buildings constructed, the first being Johnson-Bovey building in the city of
Minneapolis in 1906. Turner's "mushroom" slab were characterized by the presence of capitals in the slab-column connection and also by the use a cage comprising bars of 32 mm diameter, working as
Furst and Marti [11] attributed the invention of this system to the Swiss engineer Robert Maillart, most famous for his works with bridges than the development of such structural system. According to
these researchers, Maillart would have designed the system in 1900, but had only completed his tests in 1908, coming to get the patent of the system in 1909. Kierdorf [12] points out that while the
system was developed independently in the United States and Switzerland, with the prohibition of the use of reinforced concrete in Russia in 1905, the engineer Arthur F. Loleit designed and
implemented a factory nearby at the Moscow in 1907 in flat slabs, have been the first of several buildings in slabs without beams made by him in Russia. The author further comments that if his
presentations of "beamless" construction at the regular meeting of cement specialists in Moscow (1912) and to the Russian Society for Materials Research (1913) had been documented, and also if WWI
(1914-1918) had not happened, Loleit would certainly have presented his work to a broader public. Some details on the development of flat slabs can be seen in Figure 2.
Many obstacles had to be translated until the flat slabs could be used safely and economically. Initially there was strong discussion about the theoretical methods for the determination of the forces
on a system without beams and these slabs were used in manner practically empirical, observing significant variations in the amount and arrangement of the flexural reinforcement between the competing
systems. Furst and Marti [11] highlight that the first well founded theory for calculation of forces on floors without beams was published only in 1921, with the work of Westergaard and Slater, whom
by using the method of finite differences were able to treat different load cases, the influence of the stiffness of the columns and capitals.
It was also necessary to establish rules to normalize the use of flat slabs, which became increasingly popular. That was possible only in 1925 with the publication of American code (ACI) for the
design of reinforced concrete structures, which was the first to present recommendations for flat slabs. These first recommendations were based on experimental tests carried out in the USA like those
from Talbot [13], who tested footings in University of Illinois, as shown in Figure 3.
However the footings tested by Talbot [13] were very thick compared to the mushroom slabs at that time, and therefore, these results were not adequate in terms of the punching strength. Trying to
fill this and other gaps, Elstner and Hognestad [2] tested 39 slabs, aiming to evaluate the influence of important variables such as the flexural reinforcement ratio, concrete strength, amount of
compression reinforcement, support conditions, size of columns and amount and distribution of shear reinforcement in the punching strength of flat slabs. They concluded that practically all of these
factors have strong influence on the shear strength of flat slabs, except for the increase in the compression reinforcement ratio, which was considered by them as having a small influence on the
ultimate strength of tested slabs.
Subsequently were published two of the most important papers on punching. Kinnunen and Nylander [14] presented a mechanical model that sought to explain the punching failure mechanism and predict the
strength of slab-column connections. This model was based on experimental observations obtained after performing an extensive experimental program. The model was based on the formation of bending and
shear cracks to divide the slab into segments, and, assuming that the region external to the punching cone presented rigid body rotations around a point away by a distance x (height of the slab's
neutral axis) either vertically and horizontally in relation to the column faces, it related the ultimate punching strength with the compressive strength of an imaginary shell confined between the
column and the critical shear crack. This method was a relevant original contribution, being the first rational theory presented, but at the time his equations were considered complex and the
accuracy observed for the theoretical results did not justify its use over the existing empirical methods.
One year after this publication, Moe [15] published a report of a large series of tests analyzing several variables, including the cases of unbalanced moments in slab-column connections, and his work
remains the basis for the recommendations of ACI 318 [7]. After that, many works have been conducted and many contributions were made to the better understand of the punching shear phenomenon and
also for the definition of the influence of the involved parameters in the ultimate strength of the slabs, as will be shown below.
3. Factors that influence in the punching resistance
Results of several tests indicate that the punching resistance of reinforced concrete flat slabs without shear reinforcement is mainly influenced by the compressive strength of concrete (f[c]), the
tensile flexural reinforcement ratio (ρ), the size and geometry of the column and also the size effect (ξ) which is a coefficient that takes into account the reduction of the nominal shear strength
of the slab by increasing the effective depth (d). The influence of each of these parameters is discussed below based on relevant test results.
3.1 Strength of Concrete
The shear failure of a concrete element without shear reinforcement is governed, among other factors, by the tensile strength of concrete. Establishing the compressive strength of concrete is the
initial step in the design process of a concrete structure and also normative formulations tend to relate the tensile strength of concrete as a function of its compressive strength. These are the
reasons why it is common to observe that experimental researches correlate the shear strength to the compressive strength of concrete.
Graf [16] was among the first to try to assess the influence of concrete strength in the punching resistance, concluding that there was not a linear relationship between the increases of the strength
of a slab-column connection with the increase of concrete strength. Moe [15] proposed that the punching resistance could be expressed with a function proportional to the square root of the
compressive strength of concrete, proposition until today used by the ACI. However, the results of recent research, such as Hallgren [17], which analyzed concrete slabs with high strength concrete,
indicate that in these cases, relating the punching resistance with a function proportional to the square root of the compressive strength of concrete tends to overestimate its influence. For this
reason ACI limits the use its expression for concrete with strengths up to 69 MPa or 10.000 psi.
Marzouk and Hussein [18] analyzed slabs with high strength concrete varying the effective depth of the slab and also the flexural reinforcement ratio, concluding that a function proportional to the
cube root of the concrete strength better represents the trend of the experimental results, what is also recommended by Hawkins et al. [19] and Regan [20]. Figure 4 shows a graph made in order to
evaluate the influence of the concrete strength in the punching resistance of flat slabs. It was compared the trend obtained by using a function proportional to the cube root of the compressive
strength of concrete (as proposed by the equations of Eurocode 2) with experimental results from the database, observing a good correlation between the experimental results and the function
3.2 Flexural Reinforcement Ratio
The flexural reinforcement ratio (ρ) is defined as the ratio between the area of tensile flexural reinforcement (As) and the area of concrete (Ac), which is given by the product of the effective
depth of the slab (d) by a certain width to be considered. In practical cases it is reasonable to consider that only a certain number of bars close to the column area will effectively contribute to
the punching resistance. Considering results of experimental tests, Regan [20] recommends that the effective width to be considered in which the flexural reinforcement will contribute to the punching
resistance should be taken as 3d away from the faces of the column.
The flexural reinforcement ratio influences the punching resistance, especially in cases of slabs without shear reinforcement. Regan [21] explains that increasing the flexural reinforcement ratio
raises the compression zone, reducing cracking in the slab-column connection due to bending, which is beneficial since it facilitates the formation of mechanisms for transmitting shear forces.
Furthermore, the thickness of the bending cracks is reduced, which facilitates the transfer of forces through the interlock of aggregates, what may also increase the dowel effect.
Kinnunen and Nylander [14], testing slabs with a thickness of 150 mm, when varied the flexural reinforcement ratio from 0.8% to 2.1% observed that the punching strength increased about 95%. Marzouk
and Hussein [18], also tests slabs with a thickness of 150 mm, observed that the punching strength increased around 63% when the flexural reinforcement ratio was raised from 0.6% to 2.4%. Long [22]
used results of several authors to conclude that the punching resistance was influenced by the flexural reinforcement ratio with a function proportional to the fourth root. Moreover, Regan and
Braestrup [23] and Sherif and Dilger [24] suggest that the punching resistance is influenced by a function proportional to the cube root of the tensile flexural reinforcement ratio. Figure 5 uses
results of the experimental database to evaluate the contribution of the flexural reinforcement ratio of slabs in its punching resistance.
3.3 Geometry and Dimensions of Columns
The geometry and dimensions of the column also affects the punching resistance of slabs because they influence the distribution of stresses in the slab-column connection. Vanderbilt [25] tested slabs
supported on circular and square columns and monitored the region of the slab at the ends of the columns, and was among the first to check the stress concentration at the corners of square columns.
The author concluded that the stress concentration could justify the fact that slabs supported on square columns presented lower resistance than those supported on circular columns, in which he
observed a uniform distribution of stresses.
In rectangular columns, which are the most commonly used in buildings, the concentration of stresses in the corners may be even greater. Hawkins et al. [26] varied the ratio between the largest and
the smallest sides of the column (c[max]/c[min]) from 2.0 to 4.3 and observed that for ratios greater than two the nominal shear strength decreases with increasing ratios between the column sides.
This research conducted by Hawkins is the basis of the recommendations of ACI for the consideration of the rectangularity index of columns (μ), which can reduce by more than a half the nominal shear
strength around rectangular columns.
OLIVEIRA et al. [27], analyzing slabs tested by Forssel and Holmberg supported on a rectangular column with sides of 300 x 25 mm (c[max]/c[min] = 12) observed that the punching resistance can be well
estimated using the recommendations of CEB-FIP MC90 [6], which does not take into account the relationship c[max]/c[min]. OLIVEIRA et al. [27] believe that this can be explained by the relationship c
[max]/d that for this specific slab is around 2.88·d, value that may be considered small compared to the usual cases. After conducting an experimental program with 16 slabs, Oliveira et al. [27]
concluded that the relationship c[max]/d may be a better parameter than the relationship c[max]/c[min] for determining the punching strength of slabs supported on rectangular columns and proposed a
correction factor λ to refine the recommendations for codes such as ACI 318 [7 ] and CEB-FIP MC90 [6].
3.4 Size-Effect
It is common to use scale factors in the definition of the dimension of specimens used for experimental tests of concrete elements. This is done in order to save material resources but mainly because
testing full-scale structural elements can be a difficulty in most laboratories. For this reason, many of the tests carried out on flat slabs have been made on specimens with reduced dimensions.
Muttoni [3] states that when the current formulation for estimating the punching resistance of slabs presented by ACI was originally developed in the 1960's, only tests in relatively small thickness
slabs were available and therefore, the influence of the size effect was not apparent. But as the punching expressions are also normally used for the verification of both thick slabs and footings,
testing in experimental models thicker have been carried out and this effect became evident.
The first ones that observed that the nominal shear strength could vary in non-proportional way with the thickness of the slabs were Graf [28] and Richart [29]. At the time these authors have
proposed formulas to describe this effect, but they are no longer used. Subsequently, various expressions have been proposed. Regan and Braestrup [23] and Broms [30] suggest that the reduction of the
nominal shear strength with increasing thickness of the element (size effect) can be estimated by (1/d)^1/3. CEB-FIP MC90 [6] and EUROCODE 2 [8] recommend that the size effect should be estimated by
1+(200/d)^1/2, however, Eurocode limits results of this expression to the maximum of 2.0. The effect of this limitation is to reduce the increase in estimates of punching resistance of flat slabs
with effective depth less than 200 mm by limiting the value of ξ. It is noteworthy that a solid experimental basis to justify this limitation is not evident and thus a series of tests seeking to
evaluate the recommendation of Eurocode could be of interest.
Some experimental results that can aid understanding of the variation of the nominal shear strength as a function of effective depth of the slab come from tests made by Li [31] and Birkle [32]. Li
[31] varied the effective depth of his slabs from 100 mm to 500 mm. In slabs with effective depth of 100 mm, 150 mm and 200 mm the flexural reinforcement ratio used was 0.98%, 0.90% and 0.83%
respectively. For slabs with effective depth of 300 mm, 400 mm and 500 mm was used a constant flexural reinforcement ratio of 0.76%. Birkle [32] studied the influence of the thickness for slabs with
shear reinforcement, but in the analysis presented in Figure 6 are going to be considered only results of slabs without shear reinforcement, which had effective depth of 124 mm, 190 mm and 260 mm.
The flexural reinforcement ratio of these slabs was 1.52%, 1.35% and 1.10% respectively. Figure 6 shows the variation of the nominal shear strength for each code as a function of the effective depth
of the slabs. Is possible to notice that by using the equations of Eurocode, in both researches there was an approximately linear reduction in the shear nominal stress, regardless of the effective
depth of the slab, indicating that there is no justification for limiting the ξ as mentioned above. However, using the equations of ACI, is possible to see a change in the behavior of slabs tested by
Li with effective depth exceeding 200 mm.
4. Recommendations from codes of practice
4.1 ACI 318
According to ACI 318 [7] the punching resistance of reinforced concrete flat slabs without shear reinforcement should be verified by checking the shear stresses in a control perimeter d/2 away from
the column faces or the ends of the loaded area, as shown in Figure 7a. The punching strength can be computed using Equation 1.
β[c ]is the ratio between the largest and smaller side of the column;
α[s ]is a coefficient that is taken as 40 for internal columns, 30 for edge columns and 20 for corner columns;
u[1 ]is the length of a control perimeter away d/2 from the column face;
f[c ]is the compressive strength of concrete in MPa (f[c] < 69 MPa);
d is the effective depth of the slab.
4.2 NBR 6118
Recommendations presented by NBR 6118 [9] are based on those from CEB-FIPMC90 [6]. The Brazilian code recommends that the punching strength of slabs without shear reinforcement should be checked in
both: a control perimeter u[0] using Equation 2 to verify the maximum strength of the slab-column connection; and in a control perimeter u1 using Equation 3 to verify the diagonal tensile strength of
the slab-column connection. Figure 7b presents details on the control perimeters of this code.
u[0 ]is the control perimeter.
ρ is the flexural reinforcement ratio expressed by
ρ[x ]and ρ[y] are the flexural reinforcement ratio in two orthogonal directions;
f[c ]is the compressive strength of concrete in MPa (f[c]< 50 MPa);
4.3 EUROCODE 2
EUROCODE 2 [8] also bases its recommendations to estimate the punching resistance of flat slabs in the recommendations of MC90. Thus, it recommendations are similar to the ones from NBR 6118.
However, this code limits the value of the size effect on ξ < 2.0 and also of the flexural reinforcement ratio ρ < 2%, possibly trying to reduce the trends of unsafe results. Thus, punching strength
is taken as the lowest value provided by Equations 4 and 5. Figure 7b shows the control perimeters of this code.
f[c] is the compressive strength of concrete in MPa (f[c]< 90 MPa);
ρ is the flexural reinforcement ratio of the slab taken as
ρ[x] and ρ[y] are the flexural reinforcement ratios in orthogonal directions x and y, considering only bars within a region away 3·d from the faces of column;
u[1] is the length of the control perimeter away 2·d of the faces of column.
5. Critical Shear Crack Theory (CSCT)
The theory presented by Muttoni [3] is based on the idea that the punching resistance decreases with increasing rotation of the slab. This was explained by Muttoni and Schwartz [33] who observed that
the shear strength decreases with the formation of a critical shear crack that propagates along the slab thickness, cutting the compression strut responsible for transmitting shear forces to the
column in a mechanism as shown in Figure 8.
The authors use some experimental evidences to justify this idealization of the behavior of the slab-column connection. They argue that, as shown in several experimental punching tests, the curvature
in the radial direction is concentrated in the region close to the support, so that concentric cracks in the form of rings are only observed in this region. In the rest of the slab only radial cracks
are observed (see Figure 9a). Since shear is not transferred in the tangential direction, the stress state is not affected by such cracks. In the region of the tangential cracks, part of the shear
may be resisted by aggregate interlock on the surface of cracks and another part may be supported by dowel effect of the flexural reinforcement. As the tensile strength of concrete in the tensile
diagonal is reached the tangential cracks (originally caused by bending of the slab) start to spread towards the column.
Also according to reports from several authors, including Ferreira [1], compressive strains in the radial direction nearby the ends of the column, after reaching a certain maximum value at a certain
load level, start to decrease. Just before the punching failure it is possible to observe tensile strains in this area. This phenomenon can be explained by the formation of an elbow shaped strut (see
Figure 9b) with a horizontal tensile member as a result of the advance of the critical shear crack, cutting the compression zone. The opening of this crack reduces the resistance of the compression
strut because it affects the capacity of transferring shear forces by interlock aggregate and can eventually lead to a punching failure. Also according Muttoni and Schwartz [33] the thickness of this
crack is proportional to the product ψ·d (see Figure 8). However, the transmission of shear in the critical crack is directly linked to its roughness, which in turn is a function of maximum aggregate
size. Based on these concepts Muttoni [3] shows that the shear strength provided by the concrete can be estimated according to Equation 6.
u[1] is the length of a control perimeter d/2 away from the faces of the column (see Figure 7c);
f[c] is the compressive strength of concrete;
ψ is the rotation of the slab;
d[g0] is a reference diameter of the aggregate admitted as 16 mm;
d[g] is the maximum diameter of the aggregate used in the concrete of the slab.
The rotation ψ of the slab is expressed by the Equation 7.
r[s] is the distance between the axis of the column and the line of contraflexure of moments;
r[q] is the distance between the axis of the column and the load line;
r[c] is the radius of the circular column or the equivalent radius of a rectangular column;
f[ys] is the yield stress of the tensile flexural reinforcement;
E[s] is the modulus of elasticity of the tensile flexural reinforcement;
V[E ]is the applied force;
With V[E], ψ and V[R,c] is possible to draw a graph with two curves. The first is a curve that expresses the theoretical load-rotation behavior of the slab. The second curve expresses the strength
reduction of the slab due to the increase of rotation. The point of intersection of these two curves express the punching strength of a slab-column connection. Figure 10 illustrates this graph.
6. Evaluation of theoretical methods
Aiming to evaluate the accuracy of the theoretical methods presented in the previous sections, results of tests on 74 flat slabs were taken together in a database. The main criterias for the
formation of this database were the level of reliability of the results, trying to select results with great acceptance within the scientific community, and the range of the database related to the
parameters that influence the punching resistance of flat slabs without shear reinforcement. Were used slabs tested by Elstner and Hognestad [2], Kinunnem and Nylander [14], Moe [15] Regan [20],
Marzouk and Hussein [18], Tomaszewicz [34] and Hallgren [17]. Table 1 shows the characteristics of the slabs of the database. It should be emphasized that slabs in this database partially attend the
limits of design codes. For example, NBR 6118 states that the smallest thickness for a flat slab must be 160 mm, which does not occur in all the slabs in the database. However, it is considered that
scientifically it is important not to stick to these limits, since the interest is to understand the phenomenon as a whole and not just for the most common design situations.
Some criteria were established in order to evaluate results obtained with the theoretical methods used in comparison with the experimental results. In general, it is expected that theoretical methods
meet two basic principles: safety and precision. Primarily, it is desirable that, within a representative range of the design variables of flat slabs or slabs with the loads applied in small areas,
the methods are able to provide safety results, with a minimum of fragile results (unsafe). In this regard, it was established that no more than 5% of unsafe results would be ideal. The accuracy of
obtained results was evaluated according to the average of the ratio P[u]/V[calc], were P[u] is the experimental failure load and V[calc ]is the theoretical resistance estimated by each method. For
the average, it was established that: the method presents a high level of precision if 1.0 < P[u]/V[calc] < 1.10; for to values of 1.10 < P[u]/V[calc] < 1.30 the method has a satisfactory level of
precision; and for P[u]/V[calc] > 1.30 the method is conservative. The coefficient of variation (COV) was also used to evaluate the precision of the methods, but without establishing ranges for the
ideal values of the coefficient of variation, with these results used only in a qualitative way.
Figure 11 shows a comparison between the experimental results with theoretical results obtained with the recommendations of ACI 318 [7]. The solid line in the figures represents the level of the
nominal strength and the dotted line represents the level of the design strength. By varying the parameters fc (compressive strength of concrete) and B/d (equivalent diameter of the column u[0]/π
divided by the effective depth d of slab) it is observed that only 5% of the results are against safety. One of these results, represented by point without filling in the graphs, is below of the
design strength estimated by ACI. It refers to slab HSC 9 from Hallgren [17], in which a small flexural reinforcement ratio was used (0.3%) and, although not specified by the author, is possibly a
slab that failed by flexural.
Figure 12 and Figure 13 show comparisons of experimental results with those obtained using recommendations of NBR 6118 and Eurocode 2, respectively. It is possible to perceive that Eurocode, which
presents recommendations similar to NB1, but with limitations on the value of size effect (ξ < 2.0) and of the flexural reinforcement ratio (ρ < 2%) shows about 11% of unsafe results, but no results
below the line of the design strength. However, NB1 presents average nominal strength close to the experimental results, with no results below the design strength, but is far from meeting the limit
of only 5% of unsafe results. In Figure 14 are shown comparisons with results obtained according to CSCT. It may be noted that 11% of results are below the nominal strength, but no result is below
the design strength. Figure 15 shows graphs with the tendency of the results of codes and CSCT compared with experimental results of 74 slabs from the database. It can be seen that the dispersion of
these results, when using the recommendations of NBR 6118, is very small.
Table 2 summarizes comparisons between the experimental and theoretical results. It is possible to perceive that the recommendations of ACI are conservative and show a high coefficient of variation
if compared to the other methods due to the fact that the only parameter used to estimate the punching strength of flat slabs is the compression strength of concrete. However, this code presented
only 5% of unsafe results, which is suitable for a code of practice. Both Eurocode and CSCT showed satisfactory accuracy with CSCT presenting results slightly more accurate. By correlating the
punching resistance with the flexural behavior, CSCT was more sensitive to variables, presenting a lower coefficient of variation.
Results from the Brazilian code indicate that its recommendations must be reviewed. At the same time that it showed the smallest average (1.01) and lower coefficient of variation (0.11), the
Brazilian code presented about 47% of results below the nominal strength, indicating that its equations need some adjustment. Many proposals could be, but undoubtedly the one that requires lowest
level of changes and that could eliminate this trend of unsafe results would be modifying the coefficient 0.18 in Equation 3 to 0.16. This small change would increase the average to 1.14, same value
as CSCT, it wouldn't change the coefficient of variation, and what is really important, could reduce the percentage of unsafe results from 47.3% to 9.5%, leaving the results of this code similar to
the CSCT.
7. Conclusions
Several aspects of the development of flat slabs and of the parameters that influence their punching resistance were discussed in this paper. Recommendations of ACI 318 [7], EUROCODE 2 [8] and NBR
6118 [9] were also presented as well as those from the Critical Shear Crack Theory , as presented by Muttoni [3], which is the basis of recommendations for punching presented in new fib Model Code
[4.5]. To evaluate the safety and precision of these theoretical methods, a database was formed with experimental results of tests in 74 flat slabs without shear reinforcement.
It was observed that, generally, ACI's recommendations are meant to be safe, but underestimate the punching strength of flat slabs in about 37% for those in the database. This code also presented a
high coefficient of variation (0.16) for this which is the simplest case the design of a slab-column connection. EC2 presented satisfactory and safety results, being registered average results for
the ratio P[u]/V[calc] of 1.19. This code also presented a coefficient of variation of 0.14, below of the American code due to the fact that it takes into account the influence of parameters such as
the flexural reinforcement ratio and size effect, while that the American code considers only the compressive strength of concrete.
The Critical Shear Crack Theory has been widely discussed by the scientific community and some critics are noteworthy. The main one, as pointed out by Ferreira [1], is that according to a scientific
point of view, taking as a fundamental hypothesis that the failure mechanism by punching occurs with only rigid body rotations of the segment of slab outside the punching cone (delimited by critical
crack) contradicts experimental evidence (in the region of failure occurs rotation and sliding) and can lead to inappropriate results, especially in the case of slabs with shear reinforcement
(estimating higher forces in the outer perimeters, which in practice is not observed). From technical point of view, is a significantly more complex method for routine use in design offices and, as
noted, presents results similar to those from Eurocode. It is noteworthy that in this paper it was used CSCT in its most accurate version and if it had been used the version adopted in the new fib
code, results would be practically as conservative as those from ACI (see Ferreira [1]).
The Brazilian code presented average results near to the experimental ones (average 1.01). By not limiting parameters such as flexural reinforcement ratio and size effect, how does Eurocode, NBR 6118
presented a coefficient of variation of 0.11, lower than other codes. However, for 47% of slabs the punching strength estimated according to these equations were unsafe. This indicates that it is
extremely necessary to review its recommendations in order to avoid this inadequate trend. It was showed also that a simple change in the equation of this code could change this trend of unsafe
results, raising the average to 1.14, equal to of the CSCT, but reducing the percentage of unsafe results to only 9.5%.
8. Acknowledgements
The authors would like to acknowledge CNPq and CAPES for financial support in all steps of this research.
9. References
[01] FERREIRA, M. P. (2010). Punção em Lajes Lisas de Concreto Armado com Armaduras de Cisalhamento e Momentos Desbalanceados. Tese de Doutorado em Estruturas e Construção Civil, Publicação E.TD
007 A/10 Departamento de Engenharia Civil e Ambiental, Universidade de Brasília, Brasília, DF, 275p. [ Links ]
[02] ELSTNER, R. C., e HOGNESTAD, E., Shearing Strength of Reinforced Concrete Slabs. Journal of the American Concrete Institute, Proceedings, V. 53, No. 1, Jul. 1956, pp. 29-58. [ Links ]
[03] MUTTONI, A., Punching Shear Strength of Reinforced Concrete Slabs without Transverse Reinforcement, ACI Structural Journal, V. 105, No. 4, July-Aug. 2008, pp. 440-450. [ Links ]
[04] fib Bulletin 55, Model Code 2010 First complete draft, Volume 1,318p., 2010. [ Links ]
[05] fib Bulletin 56, Model Code 2010 First complete draft, Volume 2, 312p., 2010. [ Links ]
[06] Comité Euro-InternationalduBéton. CEB-FIP Model Code 1990. London, Thomas Telford, 1993. [ Links ]
[07] ACI Committee 318, Building Code Requirements for Structural Concrete (ACI 318-08) and Commentary, American Concrete Institute, Farmington Hills, Michigan, 2008. [ Links ]
[08] Eurocode 2, Design of Concrete Structures Part 1-1: General Rules and Rules for Buildings, CEN, EN 1992-1-1, Brussels, Belgium, 2004, 225 pp. [ Links ]
[09] ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 6118 Projeto de Estruturas de Concreto. Rio de Janeiro, 2007. [ Links ]
[10] GASPARINNI D. A., Contributions of C. A. P. Turner to development of reinforced concrete flat slabs 1905 1999. Journal of Structural Engineering, 2002, 128, No. 10, 1243 1252. [ Links ]
[11] FURST, A., MARTI, D., Robert Maillart's design approach for flat slabs. Journal of Structural Engineering, 1997, No. 123(8), 1102 1110. [ Links ]
[12] KIERDORF, A., Early Mushroom Slab Construction in Switzerland, Russia and the U.S.A. - A Study in Parallel Technological Development, In: Proceedings of the Second International Congress on
Construction History, vol II, pp 1793 1807. Cambridge Construction History Society, Cambridge University, 2006. [ Links ]
[13] TALBOT, A. N., Reinforced Concrete Wall Footings and Column Footings. Engineering Experiment Station, University of Illinois, Urbana, Bulletin No. 67, Mar. 1913, 114p. [ Links ]
[14] KINNUNEN, S., NYLANDER, H., Punching of Concrete Slabs Without Shear Reinforcement. Transactions of the Royal Institute of Technology, No. 158, Stockholm, Sweden, 1960, 112 pp. [ Links ]
[15] MOE, J., Shearing Strength of Reinforced Concrete Slabs and Footings Under Concentrated Loads. Development Department Bulletin D47, Portland Cement Association, Skokie, Illinois, Apr. 1961,
129p. [ Links ]
[16] GRAF, O.. Versuche über die Widerstandsfähigkeit von Eisenbetonplatten unter konzentrierter Last nahe einem Auflager. DeutscherAusschuß fürEisenbeton, Heft 73, Berlin, 1933, 16 pp. [ Links ]
[17] HALLGREN, M., Punching Shear Capacity of Reinforced High Strength Concrete Slabs. PhD-Thesis, KTH Stockholm, TRITA-BKN. Bulletin No. 23, 1996, 150p. [ Links ]
[18] MARZOUK, H.; HUSSEIN, A., Experimental Investigation on the Behavior of High-Strength Concrete Slabs. ACI Structural Journal, V. 88, No. 6, Nov.-Dec. 1991, pp. 701-713. [ Links ]
[19] HAWKINS, N.M., CRISWELL, M.E., and ROLL, F., Shear Strength of Slabs Without Shear Reinforcement, ACI Publication, Shear in Reinforced Concrete, V. SP 42, No. 30, 1974, pp. 677-720. [ Links ]
[20] REGAN, P. E.,Symmetric Punching of Reinforced Concrete Slabs. Magazine of Concrete Research, V. 38, No. 136, Sep. 1986, pp 115-128. [ Links ]
[21] REGAN, P. E., Behavior of reinforced concrete flat slabs. Report 89, Construction Industry Research and Information Association (CIRIA); London, Feb. 1981, p 89. [ Links ]
[22] LONG, A. E.,A Two-Phase Approach to the Prediction of Punching Strength of Slabs. Journal of the American Concrete Institute, Proceedings, V. 72, No. 2, Fev. 1975, pp. 37-45. [ Links ]
[23] REGAN, P. E.; BRÆSTRUP, M. W.,Punching Shear in Reinforced Concrete. Comité Euro-International du Béton, Bulletin d Information, No. 168, Jan. 1985, 232 pp. [ Links ]
[24] SHERIF, A. G.; DILGER, W. H.,Punching Failure of a Full Scale High Strength Concrete Flat Slab. International Workshop on Punching Shear Capacity of RC Slabs Proceedings, TRITA-BKN Bulletin
57, Stockholm, Sweden, 2000, pp 235-243. [ Links ]
[25] VANDERBILT, M. D.,Shear Strength of Continuous Plates. Journal of Structural Division, Proceedings, ASCE, V. 98, No. ST5, May 1972, pp. 961-973. [ Links ]
[26] HAWKINS, N.M., FALLSEN, H.B., and HINOJOSA, R.C., Influence of Column Rectangularity on the Behavior of Flat Plate Structures, ACI Publication, Cracking,Deflection, and Ultimate Load of Concrete
Slab Systems, V. SP-30, No. 6, 1971, pp. 127-146. [ Links ]
[27] OLIVEIRA, D. R. C.; REGAN, P. E.; MELO, G. S. S.,Punching Resistance of RC Slabs with Rectangular Columns. Magazine of Concrete Research, Vol. 56, No. 3, London, 2004, pp. 123-138. [ Links ]
[28] GRAF, O., Versuche über die Widerstandsfähigkeit von allseitig aufliegenden dicken Eisenbetonplatten unter Einzellasten, Deutscher Ausschuß für Eisenbeton, Heft 88, Berlin, 1938, 22p. [ Links ]
[29] RICHART, F. E., Reinforced Concrete Wall and Column Footings. ACI Journal, Proceedings, V. 45, No. 10, Oct. 1948, pp. 97-127. [ Links ]
[30] BROMS, C. E., Shear Reinforcement for Deflection Ductility of Flat Plates, ACI Structural Journal, V. 87, No. 6, Nov.-Dec. 1990, pp. 696-705. [ Links ]
[31] LI, K. K. L.,Influence of Size on Punching Shear Strength of Concrete Slabs. M. Eng. Thesis, McGill University, Montreal, Québec, 2000, 78 pp. [ Links ]
[32] BIRKLE, G., Punching of Flat Slabs: The Influence of Slab Thickness and Stud Layout. PhD Thesis. Department of Civil Engineering, University of Calgary, Calgary, Canadá, 2004, 152 pp. [ Links ]
[33] MUTTONI, A., and SCHWARTZ, J., Behavior of Beams and Punching in Slabs without Shear Reinforcement, IABSE Colloquium, V. 62, Zurich, Switzerland, 1991, pp. 703-708. [ Links ]
[34] TOMASZEWICZ, A., High-Strength Concrete. SP2 Plates and Shells. Report 2.3 Punching Shear Capacity of Reinforced Concrete Slabs. Report No. STF70 A93082, SINTEF Structures and Concrete,
Trondheim, 36pp. [ Links ]
Received: 30 Mar 2012
Accepted: 28 Jun 2012
Available Online: 02 Oct 2012 | {"url":"http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1983-41952012000500005&lng=en&nrm=iso&tlng=en","timestamp":"2014-04-16T04:26:35Z","content_type":null,"content_length":"82184","record_id":"<urn:uuid:8a4555dd-9b0c-4d8f-8b22-02edc64a17f5>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00042-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: IEEE 754 vs Fortran arithmetic
Newsgroups: comp.compilers,comp.lang.fortran
From: diamond@tkov50.enet.dec.com (diamond@tkovoa)
Keywords: Fortran
Organization: Digital Equipment Corporation Japan , Tokyo
References: <9010230628.AA22160@admin.ogi.edu> <1990Oct24.162529.20452@zoo.toronto.edu> <1990Oct25.010604.4796@twinsun.com>
Date: Fri, 26 Oct 90 01:06:47 PDT
Bogus: In article <1990Oct25.010604.4796@twinsun.com> eggert@twinsun.com (Paul Eggert) writes:
Bogus: >ANSI X3.9-1978 says that -0.0 and +0.0 must print the same way. IEEE 754-1985
Bogus: >says that -0.0 and +0.0 are distinct, and that printing any number and reading
Bogus: >it back in must yield the original number if the proper precision is used.
Bogus: Unfortunately, it is possible. For example, "proper precision" could be
Bogus: specified as leaving room for +0.0 to print as 0.0 while -0.0 prints as
Bogus: (I can hardly type it)
Bogus: 00.0
Bogus: (oh I feel ill).
Bogus: --
Bogus: Norman Diamond, Nihon DEC diamond@tkov50.enet.dec.com (tkou02 is scheduled for demolition)
Bogus: --
Bogus: Send compilers articles to compilers@esegue.segue.boston.ma.us
Bogus: {ima | spdcc | world}!esegue. Meta-mail to compilers-request@esegue. | {"url":"http://compilers.iecc.com/comparch/article/90-10-115","timestamp":"2014-04-19T17:08:42Z","content_type":null,"content_length":"6457","record_id":"<urn:uuid:b3cd96d7-1da6-48eb-9dc3-bf416aa0e797>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00575-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Recently, several algorithms using number field sieves have been given to factor a number n in heuristic expected time Ln[1/3; c], where Ln[v; c] = exp{(c + o(1))(log n) v (log log n) 1−v}, for n →
∞. In this paper we present an algorithm to solve the discrete logarithm problem for GF (p) with heuristic expected running time Lp[1/3; 3 2/3]. For numbers of a special form, there is an
asymptotically slower but more practical version of the algorithm. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=987515","timestamp":"2014-04-16T16:26:59Z","content_type":null,"content_length":"11875","record_id":"<urn:uuid:41c67ee0-7c03-4202-82fa-8c59317823b7>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00176-ip-10-147-4-33.ec2.internal.warc.gz"} |
Oakland Park, FL Math Tutor
Find an Oakland Park, FL Math Tutor
...The reason for this is that just memorizing tricks will cause you problems in the long run since knowledge is cumulative. If you put in the hard work toward understanding, then it will all be
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...I have a degree in Physics and post-degree studies in Geophysics and Information Systems. My background as a physicist, my experience as a teacher in college and high school, and more than 15
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Wilton Manors, FL Math Tutors | {"url":"http://www.purplemath.com/oakland_park_fl_math_tutors.php","timestamp":"2014-04-20T07:01:47Z","content_type":null,"content_length":"24096","record_id":"<urn:uuid:e825bed0-06cc-45f0-9616-1c397f0b8b74>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00167-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathML; PDF
With a growing community of researchers working on the recognition, parsing and digital exploitation of mathematical formulae, a need has arisen for a set of samples or benchmarks which can be used
to compare, evaluate and help to develop different implementations and algorithms. The benchmark set would have to cover a wide range of mathematics, contain enough information to be able to search
for specific samples and be accessible to the whole community. In this paper, we propose an on-line system and repository where researchers may upload samples of mathematics in various formats such
as scanned images, images directly rendered from born-digital documents, or born-digital document extracts. The system will support community tagging of these samples with attributes about their
syntactic structure, semantic origin, image quality and source. Each sample in the database may then be searched for by any of its associated attributes, and users could download sets of sorted or
random formulae to meet their own requirements. Associated with the system will be freely downloadable tools to assist in extracting and clipping mathematical samples from various kinds of documents
to prepare them for uploading. Additionally, the system will allow users to annotate each sample with their own files, in LaTeX, MathML, OpenMath and other formats. The intention here is that these
annotation files will correspond either to the recognition results of the users’ own systems on the samples, or manually constructed results. We believe that this facility will help to build a
community verified ground truth set, available to anyone accessing the system.
1. Hoos, H.H., Stutzle, T.:
SATLIB: An online resource for research on SAT
. In: Proceedings of the Third Workshop on Satisfiability (SAT 2000), IOS Press (2000) 283–292
2. Sutcliffe, G., Suttner, C.:
The TPTP Problem Library: CNF Release v1.2.1
. Journal of Automated Reasoning 21(2) (1998) 177–203
MR 1646570
Zbl 0910.68197
3. Suzuki, M., Uchida, S., Nomura, A.:
A ground-truthed mathematical character and symbol image database
. In: Proceedings of the Eighth International Conference on Document Analysis and Recognition (ICDAR 2005), IEEE Society Press (2005) 675–679 | {"url":"http://dml.cz/handle/10338.dmlcz/702559","timestamp":"2014-04-18T10:36:04Z","content_type":null,"content_length":"16064","record_id":"<urn:uuid:1ed43bd3-3f26-4ad4-9a0a-2ba54cf955a5>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00365-ip-10-147-4-33.ec2.internal.warc.gz"} |
Determining Big O Notation
1. It iterates 12 times so the exactly growth rate is stable, you "could" say O(12) or O(1)
2. yes, log(n) means you need to raise that number to a certain power to get that number. The smallest way to get log(n) is to multiply the increment by 2 (larger numbers also work and probably have
a more exact answer, but I lump them into log(n)).
3. That is an infinite loop.
4. No idea. It's a typo, I'll fix it.
5. It's only O(n) if it iterates 'n' number of times, where 'n' is the # of elements in the data structure. Since we're multiplying n by itself we now iterate O(n^2) times. | {"url":"http://www.dreamincode.net/forums/topic/125427-determining-big-o-notation/page__st__15__p__975162","timestamp":"2014-04-19T05:24:02Z","content_type":null,"content_length":"139948","record_id":"<urn:uuid:08d46fbc-2335-47dd-8e5e-62d2661a6baa>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00131-ip-10-147-4-33.ec2.internal.warc.gz"} |
Kenmore Precalculus Tutor
Find a Kenmore Precalculus Tutor
...I have taught many students how to play tennis; from beginners to advanced. First divided the class/into what level they thought they were. Many students were beginners; so taught them 1st how
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24 Subjects: including precalculus, English, algebra 1, accounting
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11 Subjects: including precalculus, reading, writing, geometry
...I have always loved to write, and I enjoy using that passion to help others work with language as well. I graduated from Rice University with a B.A. in English, so I've written my fair share of
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35 Subjects: including precalculus, English, reading, writing
Hi My name is George. I graduated from Bergen Community College, NJ, in 2009 with Associate in Science degree in Engineering Science. I earned my Bachelor of Science degree in Mechanical and
Aerospace Engineering from Rutgers University (New Brunswick, NJ) in 2012.
11 Subjects: including precalculus, calculus, algebra 1, algebra 2 | {"url":"http://www.purplemath.com/Kenmore_precalculus_tutors.php","timestamp":"2014-04-21T00:12:21Z","content_type":null,"content_length":"24136","record_id":"<urn:uuid:aa4c083c-e1e3-4782-a830-8b3234b0ca39>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00015-ip-10-147-4-33.ec2.internal.warc.gz"} |
User user02138
bio website user02138.myopenid.com
location Cambridge, MA 02138
visits member for 3 years, 6 months
seen Mar 21 at 13:56
stats profile views 378
My mathematical interests include Topological Quantum Field Theory, Algebraic Topology, Number Theory and Combinatorics. Litterarum radices amarae, fructus dulces. (Bitter are the roots of study, but
how sweet their fruit.) — Cato | {"url":"http://mathoverflow.net/users/10280/user02138?tab=questions","timestamp":"2014-04-21T00:21:43Z","content_type":null,"content_length":"47705","record_id":"<urn:uuid:d329be4c-eaa8-4452-b5ea-de3ae2902d80>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00151-ip-10-147-4-33.ec2.internal.warc.gz"} |
On the Spectrum of a Discrete Non-Hermitian Quantum System
SIGMA 5 (2009), 007, 24 pages arXiv:0901.2916 http://dx.doi.org/10.3842/SIGMA.2009.007
Contribution to the Proceedings of the VIIth Workshop ''Quantum Physics with Non-Hermitian Operators''
On the Spectrum of a Discrete Non-Hermitian Quantum System
Ebru Ergun
Department of Physics, Ankara University, 06100 Tandogan, Ankara, Turkey
Received October 28, 2008, in final form January 13, 2009; Published online January 19, 2009
In this paper, we develop spectral analysis of a discrete non-Hermitian quantum system that is a discrete counterpart of some continuous quantum systems on a complex contour. In particular, simple
conditions for discreteness of the spectrum are established.
Key words: difference operator; non-Hermiticity; spectrum; eigenvalue; eigenvector; completely continuous operator.
pdf (325 kb) ps (206 kb) tex (27 kb)
1. Bender C.M., Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), 947-1018, hep-th/0703096.
2. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT-symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
3. Dorey P., Dunning C., Tateo T., Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics, J. Phys. A: Math. Gen. 34 (2001), 5679-5704, hep-th/
4. Shin K.C., On the reality of the eigenvalues for a class of PT-symmetric oscillators, Comm. Math. Phys. 229 (2002), 543-564, math-ph/0201013.
5. Mostafazadeh A., Pseudo-Hermitian description of PT-symmetric systems defined on a complex contour, J. Phys. A: Math. Gen. 38 (2005), 3213-3234, quant-ph/0410012.
6. Kelley W.G., Peterson A.C., Difference equations. An introduction with applications, Academic Press, Inc., Boston, MA, 1991.
7. Teschl G., Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, Vol. 72, American Mathematical Society, Providence, RI, 2000.
8. Bender C.M., Meisinger P.N., Wang Q., Finite dimensional PT-symmetric Hamiltonians, J. Phys. A: Math. Gen. 36 (2003), 6791-6797, quant-ph/0303174.
9. Weigret S., How to test for digonalizability: the discretized PT-invariant square-well potential, Czechoslovak J. Phys. 55 (2005), 1183-1186.
10. Znojil M., Matching method and exact solvability of discrete PT-symmetric square wells, J. Phys. A: Math. Gen. 39 (2006), 10247-10261, quant-ph/0605209.
11. Znojil M., Maximal couplings in PT-symmetric chain models with the real spectrum of energies, J. Phys. A: Math. Theor. 40 (2007), 4863-4875, math-ph/0703070.
12. Znojil M., Tridiagonal PT-symmetric N by N Hamiltonians and fine-tuning of their obsevability domains in the strongly non-Hermitian regime, J. Phys. A: Math. Theor. 40 (2007), 13131-13148,
13. Jones H.F., Scattering from localized non-Hermitian potentials, Phys. Rev. D 76 (2007), 125003, 5 pages, arXiv:0707.3031.
14. Znojil M., Scattering theory with localized non-Hermiticities, Phys. Rev. D 78 (2008), 025026, 10 pages, arXiv:0805.2800.
15. Ergun E., On the reality of the spectrum of a non-Hermitian discrete Hamiltonian, Rep. Math. Phys. 63 (2009), 75-93.
16. Akhiezer N.I., Glazman I.M., Theory of linear operators in Hilbert space, Vol. 1, Ungar, New York, 1961.
17. Lusternik L.A., Sobolev V.J., Elements of functional analysis, H. Ward Crowley Frederick Ungar Publishing Co., New York, 1961. | {"url":"http://www.emis.de/journals/SIGMA/2009/007/","timestamp":"2014-04-18T08:14:41Z","content_type":null,"content_length":"7776","record_id":"<urn:uuid:7cadf528-2b2a-4ed8-8803-e6bfcf0d86b4>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00269-ip-10-147-4-33.ec2.internal.warc.gz"} |
A pretty hard problem .. please help
I have to solve that equation : 11n-2n=k2 where n,k are from N .
Well by observation n=1, k=3 is one solution. I don't think there are other solutions but I have no way of proving their aren't
Yes , I have also find that . And n can be also =0 and k=0 . So there are only 2 solutions,I think .. but how should I demonstrate ?
"There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers {1, 2, 3, ...}, while for others the term
designates the non-negative integers {0, 1, 2, 3, ...}. The former definition is the traditional one, with the latter definition having first appeared in the 19th century. Some authors use the term
natural number to exclude 0 and whole number to include it; others use whole number in a way that includes both 0 and the negative integers, i.e., as an equivalent of the integer term." See this web
entry. I have a great many different textbooks on set theory. In the vast majority of them use $0\in\mathbb{N}$.
It's one of the things in mathematics that irks me
Ok , firstly , thanks for the explanation . I've read the article about the natural numbers about one year ago I haven't seen that the author had mentioned that the numbers are different from 0 . And
also , in the some problems that i've seen if the numbers aren't null they are included in N* and in the problem the number are only from N . | {"url":"http://mathhelpforum.com/algebra/215074-pretty-hard-problem-please-help.html","timestamp":"2014-04-20T16:44:07Z","content_type":null,"content_length":"47752","record_id":"<urn:uuid:649c9710-ec6c-4fa8-bee0-c8dbe67ccf53>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00152-ip-10-147-4-33.ec2.internal.warc.gz"} |
Gold distribution.
I & my friend met an interesting man. After talking, We shared our food.
I had 6 sandwiches & my friend had 5. All of us shared equally.
After eating, the man gave us 11 gold coins.
How much should each of us get & why?
Last edited by G-man (2011-03-01 16:25:36) | {"url":"http://www.mathisfunforum.com/viewtopic.php?pid=166558","timestamp":"2014-04-20T08:35:07Z","content_type":null,"content_length":"10355","record_id":"<urn:uuid:62cb7728-8472-416b-8b8d-4c566133e709>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00596-ip-10-147-4-33.ec2.internal.warc.gz"} |
Electronics/Basic Concepts
From Wikibooks, open books for an open world
What is Electronics?[edit]
Electronics is the study of flow of electrons in various materials or space subjected to various conditions. In the past, electronics dealt with the study of Vacuum Tubes or Thermionic valves, today
it mainly deals with flow of electrons in semiconductors. However, despite these technological differences, the main focus of electronics remains the controlled flow of electrons through a medium. By
controlling the flow of electrons, we can make them perform special tasks, such as power an induction motor or heat a resistive coil.
Plumbing Analogy A simple way to understand electrical circuits is to think of them as pipes. Let's say you have a simple circuit with a voltage source and a resistor between the positive and
negative terminals on the source. When the circuit is powered, electrons will move from the negative terminal, through the resistor, and into the positive terminal. The resistor is basically a path
of conduction that resists the movement of electrons. This circuit could also be represented as a plumbing network. In the plumbing network, the resistor would be equivalent to a section of pipe,
where the water is forced to move around several barriers to pass through, effectively slowing its flow. If the pipe is level, no water will flow in an organized fashion, since the pressure is equal
throughout the pipe. However, if we tilt the pipe to a vertical position (similar to turning on a voltage source), a pressure difference is created (similar to a voltage difference) and the water
begins flowing through the pipe. This flow of water is similar to the flow of electrons in a circuit.
To understand electronics, you need to understand electricity and what it is. Basically, electricity is the flow of electrons due to a difference in electrical charge between two points. This
difference in charge is created due to a difference in electron density. If you have a point where the electron density is higher than the electron density at another point, the electrons in the area
of higher density will want to balance the charge by migrating towards the area with lower density. This migration is referred to as electrical current. Thus, flow in an electrical circuit is induced
by putting more electrons on one side of the circuit than the other, forcing them to move through the circuit to balance the charge density.
Electric Charge[edit]
In normal conditions all matter has a neutral or has a zero net charge. When an object receives an electron the object becomes negatively charged. When an object gives up an electron the object
becomes positively charged. Each charge possesses electric field lines and charge quantities. A positive charge possesses charge quantities of +Q and has electric field lines going outward. A
negative charge possesses charge quantities of -Q and has electric field lines going inward. In general, like charges will oppose each other and opposite charges will attract each other. Hence, it is
a property of matter.
Coulomb's Law[edit]
The force of attraction between two charges can be calculated by Coulomb's Law. Below would be the calculation between a positive and negative charge.
$F = k \frac{Q+.Q-}{r^2}$
Ampere's Law[edit]
The electric force, F, on a charge, Q, within an electric field, E, are related by Ampere's Law. On an atomic basis, this is the force that gives rise to current.
$F = E Q$
Lorentz's Law[edit]
When a charge in motion passes through a magnetic field. The magnetic field will push a positive charge upward and negative charge downward in the direction perpendicular to the initial direction
traveled. The magnetic force on the charge is calculated by Lorennt's Law
$F = Qv B$
$F = -Qv B$
ElectroMagnetic Force[edit]
The sum of Ampere's Force and Lorentz's Force exert on a charge is called EletroMagnetic Force
$F = F + F = Q E + Q vB = Q (E + vB)$
$F = F + F = Q E - Q vB = Q (E - vB)$
Electricity and Matter[edit]
All matter interacts with Electricity, and are divided into three categories: Conductors, Semi Conductors, and Non Conductors.
Matter that conducts Electricity easily. Metals like Zinc (Zn) and Copper (Cu) conduct electricity very easily. Therefore, they are used to make Conductors.
Matter that does not conduct Electricity at all. Non-Metals like Wood and Rubber do not conduct electricity so easily. Therefore, they are used to make Non-Conductors.
Matter that conducts electricity in a manner between that of Conductors and Non-Conductors. For example, Silicon (Si) and Germanium (Ge) conduct electricity better than non-conductors but worse
than conductors. Therefore, they are used to make Semi Conductors.
Electricity and Conductors[edit]
Normally, all conductors have a zero net charge . If there is an electric force that exerts a pressure on the charges in the conductor to force charges to move in a straight line result in a stream
of electric charge moving in a straight line
The pressure the electric force exert on the charges is called voltage denoted as V measured in Volt (V) and defined as the ratio of Work Done on Charge
$V = \frac{W}{Q}$
$V = \frac{P}{I}$
The moving of straight lines of electric charges in the conductor is called current denoted as I measured in Ampere (A) and defined as Charge flow through an area in a unit of time
$I = \frac{Q}{t}$
Conductance is defined as the ratio of current over voltage denoted as Y measured in mho
$Y = \frac{I}{V}$
Resistance is defined as the ratio of voltage over current denoted as R measured in Ohm
$R = \frac{V}{I}$
Generally, resistance of any conductor is found to increase with increasing temperature
For Conductor
R = Ro(1 + nT)
For Semi Conductor
R = Ro e^nT
When a conductor conducts electricity, it dissipates heat energy into the surrounding . This results in a loss of electric energy transmitted . If the electric supply energy is P[V] and the electric
loss energy is P[R] Then the electric energy delivered:
P = P[V] - P[R]
$P = IV - I^2 R = IV - \frac{V^2}{R}$
$P = I (V - IR) = V (I - \frac{V}{R})$
$P = Cos \theta$
Black Body Radiation[edit]
Further experience with conductors that conduct electricity . It is observed that all conductors that conduct electricity exhibit
1. Change in Temperature
2. Release Radiant Heat Energy into the surrounding
Connect a conductor with an electric source in a closed loop . Plot the value I at different f to have a I - f diagram
for f<fo
Current increasing with increasing f .
Radiant heat is a wave travels at velocity v = λf carries energy E = m v^2 .
for f=fo,
Current stops increasing .
Radiant heat is a wave travels at velocity v = c (speed of Light) carries energy E = hfo .
for f>fo,
Current remains at the value of current at fo .
Radiant heat is a wave travels at velocity v = c (speed of Light) carries energy E = h nfo
1. All conductor that conducts Electricity has a threshold frequency fo
2. The Radiant Heat Energy is a Light Wave of dual Wave Particle characteristic. Sometimes it behaves like Particle, sometimes it behaves like Wave
3. At Frequency f > fo . The energy of the Light is quantized . it can only have the value of multiple integer of fo . E = hf = h nfo | {"url":"https://en.wikibooks.org/wiki/Electronics/Basic_Concepts","timestamp":"2014-04-20T16:46:14Z","content_type":null,"content_length":"41014","record_id":"<urn:uuid:101d7293-a880-4cca-9007-9143ca5f6751>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00007-ip-10-147-4-33.ec2.internal.warc.gz"} |
Logical Operators and Loops
Logical Operators and Looping Structures
Review Question:
Write a program that reads an integer from the user, and prints a
message indicating whether or not the number is divisible by 4.
• Logical operators can be applied only to boolean values
• The result is also a boolean value (type bool) - either True or False
The logical operators:
• p and q is true only when both p and q are true. Otherwise p and q is false.
• p or q is true when at least one of p and q is true. If p and q are both false, then p or q is false.
• not(p) is false if p is true, and not(p) is true if p is false.
i = 14
j = 18
c1 = 'elf'
c2 = 'help'
print (i<j) # output: true
print i < j and i == i # output: true
test = not(i < j) # test is false
test2 = c1 == c2 # test2 is false
test3 = i<j or c1>c2 # test3 is true
test4 = i<j and c1 > c2 # test 4 is false
Important Note about Comparing Floating-Point Numbers
• Floating-point numbers have limited precision.
• Calculations lead to roundoff errors.
Consider the following code:
import math # import math library, which includes sqrt() function
root = math.sqrt(2)
double diff = root*root - 2 # we expect diff to be 0
if diff == 0:
print "sqrt(2)^2 - 2 is 0"
print "sqrt(2)^2 - 2 is ", diff
sqrt(2)^2 - 2 is 4.4408920985e-16
You should compare whether floating-point numbers are "close enough", rather than exactly equal. Roundoff errors are unavoidable.
Exercise: Write a program that reads two floating-point numbers from the keyboard, and prints a message indicating whether they are within epsilon of each other, where epsilon is defined to be
Exercise: Write a program that reads a string from the keyboard, and prints a message indicating whether or not each of the vowels a, e, i, o and u occur in the string. To do this, we will need to
import the string library, and use the find() function from this library.
import string
s = "hello world hello"
print string.find(s, "hello")
Output: 0
Iteration Structures
The while statement
while I'm hungry:
eat another bite
while there are more lines of text in the input file:
read the next line from the file
while condition:
How it works:
A while statement executes a block of code repeatedly, as long as the associated condition is true. The condition is evaluated before each execution of the statements in the while body, and if the
expression evalutes to false, execution of the while loop terminates.
count = 1
while count <= 5:
count = count + 1
print "count is ", count
# initialize variables to use in loop
num = 1
sum = 0
# as long as num is at most 5, add num to the sum
while num <= 5:
sum = sum + num # add the current value of num to the sum
num = num + 1 # add 1 to num - what happens if I omit this statement?
print "The sum of the first 5 positive integers is ", sum
Exercise: Write a program that prints the integers from 1 to 15 to the console window, one number per line.
Exercise: Write a program that asks the user to enter 10 numbers, and prints the sum of those numbers. | {"url":"http://www.cs.utexas.edu/~eberlein/cs303e/lec7.html","timestamp":"2014-04-21T15:20:22Z","content_type":null,"content_length":"9178","record_id":"<urn:uuid:de200c24-01dc-4574-9673-8a947a21760b>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00546-ip-10-147-4-33.ec2.internal.warc.gz"} |
Voices of dissent: statistical hypothesis testing
Glen M. Sizemore gmsizemore2 at yahoo.com
Fri Nov 8 15:36:41 EST 2002
>From the following website:
"If we can control statistical significance simply by changing sample size,
if statistical significance is not equivalent to scientific significance, if
statistical significant testing corrupts the scientific method, and if it
has only questionable relevance to one out of fifteen threats to research
validity, then I believe we should eliminate statistical significance
testing in our research. Such testing is not only useless, it is also
harmful because it is interpreted to mean something it is not" (Carver,
1978, p. 392).
· "The test of statistical significance in psychological research may be
taken as an instance of a kind of essential mindlessness in the conduct of
research" (Morrison & Henkel, 1970, p. 436)
· "Significance tests do not provide the information that scientists need,
and furthermore, they are not the most effective method for analyzing and
summarizing data" (Clark, 1963, pp. 469).
· "The time has arrived for educational researchers to divest themselves of
the yoke of statistical hypothesis testing" (Shulman, 1970, p. 389).
· "The time has arrived to exorcise the null hypothesis" (Cronbach, 1975, p.
· A null hypothesis test is a ritualized exercise of devil's advocacy
(Abelson, 1995, p. 12).
· "It would hardly be exaggeration to describe hypothesis testing as a
method of giving a misleading answer to a question which nobody is asking!"
(Novick & Jackson, 1974, p. 245).
Carver, R.P. (1978). The case against statistical significance. Harvard
Educational Review, 48, 378-399.
Morrison, D.E., & Henkel, R.E. (1970). Significance tests in behavioral
research: Skeptical conclusions and beyond. In D.E. Morrison & R.E. Henkel
(Eds.), The significance testing controversy: A reader. Chicago: Aldine.
Clark, C.A. (1963). Hypothesis testing in relation to statistical
methodology. Review of Educational Research, 33, 455-473.
Shulman, L.S. (1970). Reconstruction of educational research. Review of
Educational Research, 40, 371-393. Task Force on Statistical Inference
Initial report.
Cronbach, L.J. (1975). Beyond the two disciplines of scientific psychology.
American Psychologist, 30, 116-127.
Abelson, R.P. (1995) Statistics as principled argument. Hillsdale, NJ:
Novick, M., & Jackson, P. (1974). Statistical Methods for Educational and
Psychological Research New York: McGraw Hill.
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Portability non-portable
Stability experimental
Maintainer jed@59A2.org
Safe Haskell Safe-Infered
This module exposes an interface to FFTW, the Fastest Fourier Transform in the West.
These bindings present several levels of interface. All the higher level functions (dft, idft, dftN, ...) are easily derived from the general functions (dftG, dftRCG, ...). Only the general functions
let you specify planner flags. The higher levels all set estimate so you should not have to wait through time consuming planning (see below for more).
The simplest interface is the one-dimensional transforms. If you supply a multi-dimensional array, these will only transform the first dimension. These functions only take one argument, the array to
be transformed.
At the next level, we have multi-dimensional transforms where you specify which dimensions to transform in and the array to transform. For instance
b = dftRCN [0,2] a
is the real to complex transform in dimensions 0 and 2 of the array a which must be at least rank 3. The array b will be complex valued with the same extent as a in every dimension except 2. If a had
extent n in dimension 2 then the b will have extent a div 2 + 1 which consists of all non-negative frequency components in this dimension (the negative frequencies are conjugate to the positive
frequencies because of symmetry since a is real valued).
The real to real transforms allow different transform kinds in each transformed dimension. For example,
b = dftRRN [(0,DHT), (1,REDFT10), (2,RODFT11)] a
is a Discrete Hartley Transform in dimension 0, a discrete cosine transform (DCT-2) in dimension 1, and distrete sine transform (DST-4) in dimension 2 where the array a must have rank at least 3.
The general interface is similar to the multi-dimensional interface, takes as its first argument, a bitwise .|. of planning Flags. (In the complex version, the sign of the transform is first.) For
b = dftG DFTBackward (patient .|. destroy_input) [1,2] a
is an inverse DFT in dimensions 1 and 2 of the complex array a which has rank at least 3. It will use the patient planner to generate a (near) optimal transform. If you compute the same type of
transform again, it should be very fast since the plan is cached.
Inverse transforms are typically normalized. The un-normalized inverse transforms are dftGU, dftCRGU and dftCROGU. For example
b = dftCROGU measure [0,1] a
is an un-normalized inverse DFT in dimensions 0 and 1 of the complex array a (representing the non-negative frequencies, where the negative frequencies are conjugate) which has rank at least 2. Here,
dimension 1 is logically odd so if a has extent n in dimension 1, then b will have extent (n - 1) * 2 + 1 in dimension 1. It is more common that the logical dimension is even, in which case we would
use dftCRGU in which case b would have extent (n - 1) * 2 in dimension 1.
The FFTW library separates transforms into two steps. First you compute a plan for a given transform, then you execute it. Often the planning stage is quite time-consuming, but subsequent transforms
of the same size and type will be extremely fast. The planning phase actually computes transforms, so it overwrites its input array. For many C codes, it is reasonable to re-use the same arrays to
compute a given transform on different data. This is not a very useful paradigm from Haskell. Fortunately, FFTW caches its plans so if try to generate a new plan for a transform size which has
already been planned, the planner will return immediately. Unfortunately, it is not possible to consult the cache, so if a plan is cached, we may use more memory than is strictly necessary since we
must allocate a work array which we expect to be overwritten during planning. FFTW can export its cached plans to a string. This is known as wisdom. For high performance work, it is a good idea to
compute plans of the sizes you are interested in, using aggressive options (i.e. patient), use exportWisdomString to get a string representing these plans, and write this to a file. Then for
production runs, you can read this in, then add it to the cache with importWisdomString. Now you can use the estimate planner so the Haskell bindings know that FFTW will not overwrite the input
array, and you will still get a high quality transform (because it has wisdom).
Data types
data Sign Source
Determine which direction of DFT to execute.
data Kind Source
Real to Real transform kinds.
Planner flags
Algorithm restriction flags
destroyInput :: FlagSource
Allows FFTW to overwrite the input array with arbitrary data; this can sometimes allow more efficient algorithms to be employed.
Setting this flag implies that two memory allocations will be done, one for work space, and one for the result. When estimate is not set, we will be doing two memory allocations anyway, so we set
this flag as well (since we don't retain the work array anyway).
preserveInput :: FlagSource
preserveInput specifies that an out-of-place transform must not change its input array. This is ordinarily the default, except for complex to real transforms for which destroyInput is the default. In
the latter cases, passing preserveInput will attempt to use algorithms that do not destroy the input, at the expense of worse performance; for multi-dimensional complex to real transforms, however,
no input-preserving algorithms are implemented so the Haskell bindings will set destroyInput and do a transform with two memory allocations.
Planning rigor flags
estimate :: FlagSource
estimate specifies that, instead of actual measurements of different algorithms, a simple heuristic is used to pick a (probably sub-optimal) plan quickly. With this flag, the input/output arrays are
not overwritten during planning.
This is the only planner flag for which a single memory allocation is possible.
measure :: FlagSource
measure tells FFTW to find an optimized plan by actually computing several FFTs and measuring their execution time. Depending on your machine, this can take some time (often a few seconds). measure
is the default planning option.
patient :: FlagSource
patient is like measure, but considers a wider range of algorithms and often produces a more optimal plan (especially for large transforms), but at the expense of several times longer planning time
(especially for large transforms).
exhaustive :: FlagSource
exhaustive is like patient but considers an even wider range of algorithms, including many that we think are unlikely to be fast, to produce the most optimal plan but with a substantially increased
planning time.
DFT of complex data
DFT in first dimension only
Multi-dimensional transforms
General transform
Un-normalized general transform
DFT of real data
DFT in first dimension only
Multi-dimensional transforms
dftCRN :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i (Complex r) -> CArray i rSource
Multi-dimensional inverse DFT of Hermitian-symmetric data (where only the non-negative frequencies are given).
dftCRON :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i (Complex r) -> CArray i rSource
Multi-dimensional inverse DFT of Hermitian-symmetric data (where only the non-negative frequencies are given) and the last transformed dimension is logically odd.
General transform
dftCRG :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i (Complex r) -> CArray i rSource
Normalized general complex to real DFT where the last transformed dimension is logically even.
dftCROG :: (FFTWReal r, Ix i, Shapable i) => Flag -> [Int] -> CArray i (Complex r) -> CArray i rSource
Normalized general complex to real DFT where the last transformed dimension is logicall odd.
Un-normalized general transform
Real to real transforms (all un-normalized)
Transforms in first dimension only
Multi-dimensional transforms with the same transform type in each dimension
dct2N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource
Multi-dimensional Type 2 discrete cosine transform. This is commonly known as the DCT.
dct3N :: (FFTWReal r, Ix i, Shapable i) => [Int] -> CArray i r -> CArray i rSource
Multi-dimensional Type 3 discrete cosine transform. This is commonly known as the inverse DCT. The result is not normalized.
Multi-dimensional transforms with possibly different transforms in each dimension
General transforms | {"url":"http://hackage.haskell.org/package/fft-0.1.7/docs/Math-FFT.html","timestamp":"2014-04-18T07:10:09Z","content_type":null,"content_length":"80476","record_id":"<urn:uuid:53c2592a-b567-4575-83fb-d01fdcca355c>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00585-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 277.05135
Autor: Bollobás, Béla; Erdös, Paul
Title: On the structure of edge graphs. (In English)
Source: Bull. Lond. Math. Soc. 5, 317-321 (1973).
Review: Let K[r](t) be a graph with r groups of t vertices, each two vertices of which are connected iff they belong to different groups. denote g(n,r, \epsilon) the minimal such t that any graph
with n vertices and m = [((r-2)/2(r-1)+\epsilon)n^2] edges contains a K[r](t), (\epsilon > 0). In this article it is proved that for any r and 0 < \epsilon < {1 over 2}(r-1) there exist constants c
[1],c[2] such that c[1] log n \leq g(n,r, \epsilon) \leq c[2] log n for sufficiently large n and c[2] > 0 if \epsilon > 0.
Reviewer: St.Znám
Classif.: * 05C35 Extremal problems (graph theory)
05C99 Graph theory
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag
│Books │Problems │Set Theory │Combinatorics │Extremal Probl/Ramsey Th. │
│Graph Theory │Add.Number Theory│Mult.Number Theory│Analysis │Geometry │
│Probabability│Personalia │About Paul Erdös │Publication Year│Home Page │ | {"url":"http://www.emis.de/classics/Erdos/cit/27705135.htm","timestamp":"2014-04-16T19:06:15Z","content_type":null,"content_length":"3746","record_id":"<urn:uuid:370483af-8d0b-4e81-8dc7-97844b604db6>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00264-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Patente US7327470 - Spatial and spectral wavefront analysis and measurement
This is a continuation of application No. 09/829,435 filed on Apr. 9, 2001 now U.S. Pat. No. 6,819,435 which claims the benefit thereof and incorporates the same by reference. This application is
based on provisional application U.S. Ser. No. 60/196,862, filed on Apr. 12, 2000.
The present invention relates to wavefront analysis generally and to various applications of wavefront analysis.
The following patents and publications are believed to represent the current state of the art:
U.S. Pat. Nos.:
• 5,969,855; 5,969,853; 5,936,253; 5,870,191; 5,814,815; 5,751,475; 5,619,372; 5,600,440; 5,471,303; 5,446,540; 5,235,587; 4,407,569; 4,190,366;
Non-U.S. patents:
• JP 9230247 (Abstract); JP 9179029 (Abstract); JP 8094936 (Abstract); JP 7261089 (Abstract); JP 7225341 (Abstract); JP 6186504 (Abstract);
Other Publications:
• Phillion D. W. “General methods for generating phase-shifting interferometry algorithms”—Applied Optics, Vol. 36, 8098 (1997).
• Pluta M. “Stray-light problem in phase contrast microscopy or toward highly sensitive phase contrast devices: a review”—Optical Engineering, Vol. 32, 3199 (1993).
• Noda T. and Kawata S. “Separation of phase and absorption images in phase-contrast microscopy”—Journal of the Optical Society of America A, Vol. 9., 924 (1992).
• Creath K. “Phase measurement interferometry techniques”—Progress in Optics XXVI, 348 (1988).
• Greivenkamp J. E. “Generalized data reduction for heterodyne interferometry”—Optical Engineering, Vol. 23, 350 (1984).
• Morgan C. J. “Least-squares estimation in phase-measurement interferometry”—Optics Letters, Vol. 7, 368 (1982).
• Golden L. J. “Zernike test. 1: Analytical aspects”—Applied Optics, Vol. 16, 205 (1977).
• Bruning J. H. et al. “Digital wavefront measuring interferometer for testing optical surfaces and lenses”—Applied Optics, Vol. 13, 2693 (1974).
The present invention seeks to provide methodologies and systems for wavefront analysis as well as for surface mapping, phase change analysis, spectral analysis, object inspection, stored data
retrieval, three-dimensional; imaging and other suitable applications utilizing wavefront analysis.
There is thus provided in accordance with a preferred embodiment of the present invention a method of wavefront analysis. The method includes obtaining a plurality of differently phase changed
transformed wavefronts corresponding to a wavefront being analyzed which has an amplitude and a phase, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts
and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed.
There is also provided in accordance with a preferred embodiment of the present an apparatus for wavefront analysis including a wavefront transformer operating to provide a plurality of differently
phase changed transformed wavefronts corresponding to a wavefront being analyzed which has an amplitude and a phase, an intensity map generator operating to provide a plurality of intensity maps of
the plurality of phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps for providing an output indicating the amplitude and phase of the
wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention the plurality of intensity maps are employed to provide an analytical output indicating the amplitude and phase.
Still further in accordance with a preferred embodiment of the present invention the plurality of differently phase changed transformed wavefronts are obtained by interference of the wavefront being
analyzed along a common optical path.
Additionally in accordance with a preferred embodiment of the present invention the plurality of differently phase changed transformed wavefronts are realized in a manner substantially different from
performing a delta-function phase change to the transformed wavefront.
Further in accordance with a preferred embodiment of the present invention the plurality of intensity maps are employed to obtain an output indicating the phase which is substantially free from halo
and shading off distortions.
Preferably, the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts resulting from at least one of application of spatial phase changes to a transformed
wavefront and transforming of a wavefront following application of spatial phase changes thereto.
Additionally in accordance with a preferred embodiment of the present invention, the step of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform to
the wavefront being analyzed thereby to obtain a transformed wavefront and applying a plurality of different phase changes to the transformed wavefront, thereby to obtain a plurality of differently
phase changed transformed wavefronts. Preferably, the plurality of different phase changes includes spatial phase changes and the plurality of different spatial phase changes are effected by applying
a time-varying spatial phase change to part of the transformed wavefront.
Further in accordance with a preferred embodiment of the present invention the plurality of different spatial phase changes are effected by applying a spatially uniform, time-varying spatial phase
change to part of the transformed wavefront. Preferably, the transform applied to the wavefront being analyzed is a Fourier transform and wherein the step of obtaining a plurality of intensity maps
of the plurality of phase changed transformed wavefronts includes applying a Fourier transform to the plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the transform applied to the wavefront being analyzed is a Fourier transform and the plurality of different spatial phase
changes includes at least three different phase changes. Preferably, the plurality of intensity maps includes at least three intensity maps and the step of employing the plurality of intensity maps
to obtain an output indicating the amplitude and phase of the wavefront being analyzed includes: expressing the wavefront being analyzed as a first complex function which has an amplitude and phase
identical to the amplitude and phase of the wavefront being analyzed, expressing the plurality of intensity maps as a function of the first complex function and of a spatial function governing the
spatially uniform, time-varying spatial phase change, defining a second complex function, having an absolute value and a phase, as a convolution of the first complex function and of a Fourier
transform of the spatial function governing the spatially uniform, time-varying spatial phase change. Expressing each of the plurality of intensity maps as a third function of: the amplitude of the
wavefront being analyzed, the absolute value of the second complex function, a difference between the phase of the wavefront being analyzed and the phase of the second complex function and a known
phase delay produced by one of the at least three different phase changes corresponding to one of the at least three intensity maps, solving the third function to obtain the amplitude of the
wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function, solving
the second complex function to obtain the phase of the second complex function and obtaining the phase of the wavefront being analyzed by adding the phase of the second complex function to the
difference between the phase of the wavefront being analyzed and the phase of the second complex function.
Further in accordance with a preferred embodiment of the present invention the absolute value of the second complex function is obtained by approximating the absolute value to a polynomial of a given
Still further in accordance with a preferred embodiment of the present invention the second complex function is obtained by expressing the second complex function as an eigen-value problem where the
complex function is an eigen-vector obtained by an iterative process.
Preferably the second complex function is obtained by: approximating the Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change to a polynomial
and approximating the second complex function to a polynomial.
Preferably, the wavefront being analyzed, the absolute value of the second complex function, and the difference between the phase of the second complex function and the phase of the wavefront being
analyzed, are obtained by a least-square method, which has increased accuracy as the number of the plurality of intensity maps increases.
Further in accordance with a preferred embodiment of the present invention the plurality of different phase changes includes at least four different phase changes, the plurality of intensity maps
includes at least four intensity maps and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed and includes: expressing
each of the plurality of intensity maps as a third function of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function, a difference between the phase of the
wavefront being analyzed and the phase of the second complex function, a known phase delay produced by one of the at least four different phase changes in which each corresponds to one of the at
least four intensity maps and at least one additional unknown relating to the wavefront analysis, where the number of the at least one additional unknown is no greater than the number by which the
plurality intensity maps exceeds three and solving the third function to obtain the amplitude of the wavefront being analyzed, the absolute value of the second complex function, the difference
between the phase of the wavefront being analyzed and the phase of the second complex function and the additional unknown.
Further in accordance with a preferred embodiment of the present invention the phase changes are chosen as to maximize contrast in the intensity maps and to minimize effects of noise on the phase of
the wavefront being analyzed.
Preferably, expressing each of the plurality of intensity maps as a third function of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function, a difference
between the phase of the wavefront being analyzed and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes which corresponds
to one of the at least three intensity maps and includes: defining fourth, fifth and sixth complex functions, none of which being a function of any of the plurality of intensity maps or of the
time-varying spatial phase change, each of the fourth, fifth and sixth complex functions being a function of the amplitude of the wavefront being analyzed, the absolute value of the second complex
function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function and expressing each of the plurality of intensity maps as a sum of the
fourth complex function, the fifth complex function multiplied by the sine of the known phase delay corresponding to each one of the plurality of intensity maps and the sixth complex function
multiplied by the cosine of the known phase delay corresponding to each one of the plurality of intensity maps.
Preferably, the step of solving the third function to obtain the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of
the wavefront being analyzed and the phase of the second complex function includes: obtaining two solutions for each of the amplitude of the wavefront being analyzed, the absolute value of the second
complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function, the two solutions being a higher value solution and a lower value
solution, combining the two solutions into an enhanced absolute value solution for the absolute value of the second complex function, by choosing at each spatial location either the higher value
solution or the lower value solution of the two solutions in a way that the enhanced absolute value solution satisfies the second complex function. Preferably, combining the two solutions of the
amplitude of the wavefront being analyzed into enhanced amplitude solution, by choosing at each spatial location the higher value solution or the lower value solution of the two solutions of the
amplitude in the way that at each location where the higher value solution is chosen for the absolute value solution, the higher value solution is chosen for the amplitude solution and at each
location where the lower value solution is chosen for the absolute value solution, the lower value solution is chosen for the amplitude solution, combining the two solutions of the difference between
the phase of the wavefront being analyzed and the phase of the second complex function into an enhanced difference solution, by choosing at each spatial location the higher value solution or the
lower value solution of the two solutions of the difference in the way that at each location where the higher value solution is chosen for the absolute value solution, the higher value solution is
chosen for the difference solution and at each location where the lower value solution is chosen for the absolute value solution, the lower value solution is chosen for the difference solution.
Further in accordance with a preferred embodiment of the present invention the spatially uniform, time-varying spatial phase change is applied to a spatially central part of the transformed
Preferably, the transform applied to the wavefront being analyzed is a Fourier transform and wherein the step of obtaining a plurality of intensity maps of the plurality of phase changed transformed
wavefronts includes applying a Fourier transform to the plurality of differently phase changed transformed wavefronts.
Still further in accordance with a preferred embodiment of the present invention the method also includes adding a phase component including relatively high frequency components to the wavefront
being analyzed prior to applying the transform thereto in order to increase the high-frequency content of the transformed wavefront prior to the applying the spatially uniform, time-varying spatial
phase change to part of the transformed wavefront.
Preferably, the spatially uniform, time-varying spatial phase change is applied to a spatially centered generally circular region of the transformed wavefront and the spatially uniform, time-varying
spatial phase change is applied to approximately one half of the transformed wavefront.
Additionally in accordance with a preferred embodiment of the present invention the transformed wavefront includes a DC region and a non-DC region and the spatially uniform, time-varying spatial
phase change is applied to at least part of both the DC region and the non-DC region.
Further in accordance with a preferred embodiment of the present invention the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts whose phase has been
changed by employing an at least time varying phase change function. Alternatively, the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts whose phase has
been changed by applying an at least time varying phase change function to the wavefront being analyzed.
Preferably, the at least time varying phase change function is applied to the wavefront being analyzed prior to transforming thereof. Alternatively, the at least time varying phase change function is
applied to the wavefront being analyzed subsequent to transforming thereof.
Further in accordance with a preferred embodiment of the present invention the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts whose phase has been
changed by employing an at least time varying phase change function.
Additionally or alternatively, the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts whose phase has been changed by applying an at least time varying
phase change function to the wavefront to be analyzed.
Preferably, the at least time varying phase change function is a spatially uniform spatial function.
Additionally in accordance with a preferred embodiment of the present invention the transformed wavefront includes a plurality of different wavelength components and the plurality of different
spatial phase changes are effected by applying a phase change to the plurality of different wavelength components of the transformed wavefront.
Preferably, the phase change applied to the plurality of different wavelength components of the transformed wavefront is a time-varying spatial phase change.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components of the transformed wavefront is effected by
passing the transformed wavefront through an object, at least one of whose thickness and refractive index varies spatially.
Still further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components of the transformed wavefront is effected
by reflecting the transformed wavefront from a spatially varying surface.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components of the transformed wavefront is selected to be
different to a predetermined extent for at least some of the plurality of different wavelength components.
Additionally in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components of the transformed wavefront is identical
for at least some of the plurality of different wavelength components.
Further in accordance with a preferred embodiment of the present invention the wavefront being analyzed includes a plurality of different wavelength components.
Preferably, the plurality of differently phase changed transformed wavefronts are obtained by applying a phase change to the plurality of different wavelength components of the wavefront being
Preferably, the phase change is applied to the plurality of different wavelength components of the wavefront being analyzed prior to transforming thereof.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components is effected by passing the wavefront being
analyzed through an object, at least one of whose thickness and refractive index varies spatially.
Further in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of intensity maps is performed simultaneously for all of the plurality of different
wavelength components and obtaining a plurality of intensity maps includes dividing the plurality of phase changed transformed wavefronts into separate wavelength components.
Still further in accordance with a preferred embodiment of the present invention the step of dividing the plurality of phase changed transformed wavefronts is effected by passing the plurality of
phase changed transformed wavefronts through a dispersion element.
Additionally in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components is effected by passing the wavefront being
analyzed through an object, at least one of whose thickness and refractive index varies spatially, following transforming of the wavefront being analyzed.
Preferably, the phase change which is applied to the plurality of different wavelength components is effected by reflecting the wavefront being analyzed from a spatially varying surface, following
transforming of the wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components is selected to be different to a predetermined
extent for at least some of the plurality of different wavelength components. Preferably, the phase change which is applied to the plurality of different wavelength components is identical for at
least some of the plurality of different wavelength components.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components is effected by passing the wavefront being
analyzed through a plurality of objects, each characterized in that at least one of its thickness and refractive index varies spatially.
Preferably, the phase change applied to the plurality of different wavelength components is effected by passing the wavefront being analyzed through a plurality of objects, each characterized in that
at least one of its thickness and refractive index varies spatially, following transforming of the wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention the wavefront being analyzed includes a plurality of different polarization components and the plurality of differently
phase changed transformed wavefronts are obtained by applying a phase change to the plurality of different polarization components of the wavefront being analyzed prior to transforming thereof.
Still further in accordance with a preferred embodiment of the present invention the transformed wavefront includes a plurality of different polarization components and the plurality of different
spatial phase changes are effected by applying a phase change to the plurality of different polarization components of the transformed wavefront.
Additionally in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different polarization components of the transformed wavefront is
different for at least some of the plurality of different polarization components.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different polarization components of the transformed wavefront is identical for
at least some of the plurality of different polarization components.
Additionally in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts includes
applying a transform to the plurality of differently phase changed transformed wavefronts.
Preferably, the plurality of phase changed transformed wavefronts are reflected from a reflecting surface so that the transform applied to the plurality of differently phase changed transformed
wavefronts is identical to the transform applied to the wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention the transform applied to the wavefront being analyzed is a Fourier transform.
Still further in accordance with a preferred embodiment of the present invention the plurality of intensity maps are obtained by reflecting the plurality of differently phase changed transformed
wavefronts from a reflecting surface so as to transform the plurality of differently phase changed transformed wavefronts.
Additionally in accordance with a preferred embodiment of the present invention the method of obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts includes
applying a transform to the plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the method of employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the
wavefront being analyzed includes expressing the plurality of intensity maps as at least one mathematical function of phase and amplitude of the wavefront being analyzed and employing the at least
one mathematical function to obtain an output indicating the phase and amplitude.
Preferably, the method of employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed includes expressing the plurality of
intensity maps as at least one mathematical function of phase and amplitude of the wavefront being analyzed and of the plurality of different phase changes, wherein the phase and amplitude are
unknowns and the plurality of different phase changes are known and employing the at least one mathematical function to obtain an output indicating the phase and amplitude.
Further in accordance with a preferred embodiment of the present invention the plurality of intensity maps includes at least four intensity maps and employing the plurality of intensity maps to
obtain an output indicating the amplitude and phase of the wavefront being analyzed, includes employing a plurality of combinations, each of at least three of the plurality of intensity maps, to
provide a plurality of indications of the amplitude and phase of the wavefront being analyzed.
Preferably, the method also includes employing the plurality of indications of the amplitude and phase of the wavefront being analyzed to provide an enhanced indication of the amplitude and phase of
the wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention at least some of the plurality of indications of the amplitude and phase are at least second order indications of the
amplitude and phase of the wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform to the
wavefront being analyzed, thereby obtaining a transformed wavefront and applying a plurality of different phase and amplitude changes to the transformed wavefront, thereby obtaining a plurality of
differently phase and amplitude changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the plurality of different phase and amplitude changes includes at least three different phase and intensity changes, the
plurality of different phase and amplitude changes are effected by applying at least one of a spatially uniform, time-varying spatial phase change and a spatially uniform, time-varying spatial
amplitude change to at least part of the transformed wavefront, the plurality of intensity maps includes at least three intensity maps. Preferably, the step of employing the plurality of intensity
maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed includes: expressing the wavefront being analyzed as a first complex function which has an amplitude and
phase identical to the amplitude and phase of the wavefront being analyzed, expressing the plurality of intensity maps as a function of the first complex function and of a spatial function governing
at least one of a spatially uniform, time-varying spatial phase change and a spatially uniform, time-varying spatial amplitude change, defining a second complex function having an absolute value and
a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change, expressing each of the
plurality of intensity maps as a third function of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function and a difference between the phase of the
wavefront being analyzed and the phase of the second complex function. Preferably, the spatial function governing at least one of a spatially uniform, time-varying spatial phase change and a
spatially uniform, time-varying spatial amplitude change includes: defining fourth, fifth, sixth and seventh complex functions, none of which is a function of any of the plurality of intensity maps
or of the time-varying spatial phase change. Preferably, each of the fourth, fifth, sixth and seventh complex functions being a function of at least one of: the amplitude of the wavefront being
analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function, defining an eighth
function of a phase delay and of an amplitude change, both produced by one of the at least three different phase and amplitude changes, corresponding to the at least three intensity maps and
expressing each of the plurality of intensity maps as a sum of the fourth complex function, the fifth complex function multiplied by the absolute value squared of the eighth function, the sixth
complex function multiplied by the eighth function and the seventh complex function multiplied by the complex conjugate of the eighth function, solving the third function to obtain the amplitude of
the wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function,
solving the second complex function to obtain the phase of the second complex function and obtaining the phase of the wavefront being analyzed by adding the phase of the second complex function to
the difference between the phase of the wavefront being analyzed and phase of the second complex function.
Further in accordance with a preferred embodiment of the present invention the wavefront being analyzed includes at least two wavelength components. Preferably, the step of obtaining a plurality of
intensity maps also includes dividing the phase changed transformed wavefronts according to the at least two wavelength components in order to obtain at least two wavelength components of the phase
changed transformed wavefronts and in order to obtain at least two sets of intensity maps, each set corresponding to a different one of the at least two wavelength components of the phase changed
transformed wavefronts and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed, obtaining an output indicative of the
phase of the wavefront being analyzed from each of the at least two sets of intensity maps and combining the outputs to provide an enhanced indication of phase of the wavefront being analyzed, in
which enhanced indication, there is no 2π ambiguity.
Additionally in accordance with a preferred embodiment of the present invention the wavefront being analyzed is an acoustic radiation wavefront.
Still further in accordance with a preferred embodiment of the present invention the wavefront being analyzed includes at least one one-dimensional component, the transform applied to the wavefront
being analyzed is a one-dimensional Fourier transform, performed in a dimension perpendicular to a direction of propagation of the wavefront being analyzed, thereby to obtain at least one
one-dimensional component of the transformed wavefront in the dimension perpendicular to the direction of propagation. The plurality of differently phase changed transformed wavefronts are obtained
by applying the plurality of different phase changes to each of the at least one one-dimensional component, thereby obtaining at least one one-dimensional component of the plurality of phase changed
transformed wavefronts and the plurality of intensity maps are employed to obtain an output indicating amplitude and phase of the at least one one-dimensional component of the wavefront being
Preferably, the plurality of different phase changes is applied to each of the one-dimensional component by providing a relative movement between the wavefront being analyzed and an element.
Preferably, the element generates spatially varying, time-constant phase changes, the relative movement being in an additional dimension which is perpendicular both to the direction of propagation
and to the dimension perpendicular to the direction of propagation.
Further in accordance with a preferred embodiment of the present invention the wavefront being analyzed includes a plurality of different wavelength components, the plurality of different phase
changes are applied to the plurality of different wavelength components of each of the plurality of one-dimensional components of the wavefront being analyzed and the step of obtaining a plurality of
intensity maps includes dividing the plurality of one-dimensional components of the plurality of phase changed transformed wavefronts into separate wavelength components.
Still further in accordance with a preferred embodiment of the present invention dividing the plurality of one-dimensional components of the plurality of phase changed transformed wavefronts into
separate wavelength components is achieved by passing the plurality of phase changed transformed wavefronts through a dispersion element.
Further in accordance with a preferred embodiment of the present invention the transform applied to the wavefront being analyzed includes an additional Fourier transform to minimize cross-talk
between different one-dimensional components of the wavefront being analyzed.
There is provided in accordance with another preferred embodiment of the present invention a method of surface mapping. The method includes obtaining a surface mapping wavefront having an amplitude
and a phase, by reflecting radiation from a surface and analyzing the surface mapping wavefront by: obtaining a plurality of differently phase changed transformed wavefronts corresponding to the
surface mapping wavefront, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity maps to obtain an output
indicating the amplitude and phase of the surface mapping wavefront.
There is further provided in accordance with a preferred embodiment of the present invention an apparatus for surface mapping. The apparatus includes a wavefront obtainer operating to obtain a
surface mapping wavefront having an amplitude and a phase, by reflecting radiation from a surface, a wavefront analyzer, analyzing the surface mapping wavefront and including a wavefront transformer
operating to provide a plurality of differently phase changed transformed wavefronts corresponding to the surface mapping wavefront, an intensity map generator operating to provide a plurality of
intensity maps of the plurality of phase changed transformed wavefronts and an intensity map utilizer, the plurality of intensity maps provide an output indicating the amplitude and phase of the
surface mapping wavefront.
Further in accordance with a preferred embodiment of the present invention the radiation reflected from the surface has a narrow band about a given wavelength, causing the phase of the surface
mapping wavefront to be proportional to geometrical variations in the surface, the proportion being an inverse linear function of the wavelength.
Still further in accordance with a preferred embodiment of the present invention the radiation reflected from the surface has at least two narrow bands, each centered about a different wavelength,
providing at least two wavelength components in the surface mapping wavefront and at least two indications of the phase of the surface mapping wavefront, thereby enabling an enhanced mapping of the
surface to be obtained by avoiding an ambiguity in the mapping which exceeds the larger of the different wavelengths about which the two narrow bands are centered.
Additionally in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform to
the surface mapping wavefront, thereby to obtain a transformed wavefront and applying a plurality of different phase changes, including spatial phase changes, to the transformed wavefront, thereby to
obtain a plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the transform applied to the surface mapping wavefront is a Fourier transform, the plurality of different phase changes
includes at least three different phase changes, effected by applying a spatially uniform, time-varying spatial phase change to part of the transformed wavefront, the plurality of intensity maps
includes at least three intensity maps. Preferably, the step of employing the plurality of intensity maps to obtain an output indicates the amplitude and phase of the surface mapping wavefront and
includes expressing the surface mapping wavefront as a first complex function which has an amplitude and phase identical to the amplitude and phase of the surface mapping wavefront, expressing the
plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change, defining a second complex
function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial
phase change. Preferably, expressing each of the plurality of intensity maps as a third function of: the amplitude of the surface mapping wavefront, the absolute value of the second complex function,
a difference between the phase of the surface mapping wavefront and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes,
corresponding to one of the at least three intensity maps, solving the third function to obtain the amplitude of the surface mapping wavefront, the absolute value of the second complex function and
the difference between the phase of the surface mapping wavefront and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex function
and obtaining the phase of the surface mapping wavefront by adding the phase of the second complex function to the difference between the phase of the surface mapping wavefront and phase of the
second complex function.
Preferably, the surface mapping wavefront includes a plurality of different wavelength components. The plurality of differently phase changed transformed wavefronts are preferably obtained by:
transforming the surface mapping wavefront thereby obtaining a transformed wavefront including a plurality of different wavelength components and applying a phase change to the plurality of different
wavelength components of the transformed wavefront by passing the transformed wavefront through an object, at least one of whose thickness and refractive index varies spatially.
There is also provided in accordance with yet another preferred embodiment of the present invention a method of inspecting an object. The method includes obtaining an object inspection wavefront
which has an amplitude and a phase, by transmitting radiation through the object and analyzing the object inspection wavefront by: obtaining a plurality of differently phase changed transformed
wavefronts corresponding to the object inspection wavefront, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity
maps to obtain an output indicating the amplitude and phase of the object inspection wavefront.
There is further provided in accordance with a preferred embodiment of the present invention an apparatus for inspecting an object. The apparatus includes a wavefront obtainer operating to obtain an
object inspection wavefront which has an amplitude and a phase, by transmitting radiation through the object, a wavefront analyzer, analyzing the object inspection wavefront and including a wavefront
transformer operating to provide a plurality of differently phase changed transformed wavefronts corresponding to the object inspection wavefront, an intensity map generator operating to provide a
plurality of intensity maps of the plurality of phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps to provide an output indicating the
amplitude and phase of the object inspection wavefront.
Preferably, when the object is substantially uniform in material and other optical properties, the phase of the object inspection wavefront is proportional to the object thickness.
Additionally, when the object is substantially uniform in thickness, the phase of the object inspection wavefront is proportional to optical properties of the object.
Further in accordance with a preferred embodiment of the present invention the radiation has at least two narrow bands, each centered about a different wavelength, providing at least two wavelength
components in the object inspection wavefront and at least two indications of the phase of the object inspection wavefront, thereby enabling an enhanced mapping of thickness of the object to be
inspected by avoiding an ambiguity in the mapping which exceeds the larger of the different wavelengths about which the two narrow bands are centered.
Still further in accordance with a preferred embodiment of the present invention the method of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform
to the object inspection wavefront, thereby obtaining a transformed wavefront and applying a plurality of different phase changes, including spatial phase changes, to the transformed wavefront,
thereby obtaining a plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the transform applied to the object inspection wavefront is a Fourier transform, the plurality of different phase changes
includes at least three different phase changes, effected by applying a spatially uniform, time-varying spatial phase change to part of the transformed wavefront. Preferably, the plurality of
intensity maps includes at least three intensity maps and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the object inspection wavefront and
includes: expressing the object inspection wavefront as a first complex function which has an amplitude and phase identical to the amplitude and phase of the object inspection wavefront, expressing
the plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change, defining a second complex
function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial
phase change, expressing each of the plurality of intensity maps as a third function of: the amplitude of the object inspection wavefront, the absolute value of the second complex function, a
difference between the phase of the object inspection wavefront and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes,
corresponding to one of the at least three intensity maps, solving the third function to obtain the amplitude of the object inspection wavefront, the absolute value of the second complex function and
the difference between the phase of the object inspection wavefront and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex
function and obtaining the phase of the object inspection wavefront by adding the phase of the second complex function to the difference between the phase of the object inspection wavefront and phase
of the second complex function.
Still further in accordance with a preferred embodiment of the present invention the object inspection wavefront includes a plurality of different wavelength components. The plurality of differently
phase changed transformed wavefronts are preferably obtained by: transforming the object inspection wavefront thereby obtaining a transformed wavefront including a plurality of different wavelength
components and applying a phase change to the plurality of different wavelength components of the transformed wavefront by reflecting the transformed wavefront from a spatially varying surface.
There is also provided in accordance with yet another preferred embodiment of the present invention a method of spectral analysis. The method includes obtaining a spectral analysis wavefront having
an amplitude and a phase, by causing radiation to impinge on an object, analyzing the spectral analysis wavefront by: obtaining a plurality of differently phase changed transformed wavefronts
corresponding to the spectral analysis wavefront which has an amplitude and a phase, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the
plurality of intensity maps to obtain an output indicating the amplitude and phase of the spectral analysis wavefront and employing the output indicating the amplitude and phase to obtain an output
indicating spectral content of the radiation.
There is provided in accordance with a further preferred embodiment of the present invention an apparatus for spectral analysis. The apparatus includes a wavefront obtainer operating to obtain a
spectral analysis wavefront having an amplitude and a phase, by causing radiation to impinge on an object, a wavefront analyzer, analyzing the spectral analysis wavefront, including a wavefront
transformer operating to provide a plurality of differently phase changed transformed wavefronts corresponding to the spectral analysis wavefront which has an amplitude and a phase, an intensity map
generator operating to provide a plurality of intensity maps of the plurality of phase changed transformed wavefronts, an intensity map utilizer, employing the plurality of intensity maps to provide
an output indicating the amplitude and phase of the spectral analysis wavefront and a phase and amplitude utilizer, employing the output indicating the amplitude and phase to obtain an output
indicating spectral content of the radiation.
Further in accordance with a preferred embodiment of the present invention and wherein obtaining the spectral analysis wavefront is effected by reflecting the radiation from the object.
Still further in accordance with a preferred embodiment of the present invention and wherein obtaining the spectral analysis wavefront is effected by transmitting the radiation through the object.
Additionally in accordance with a preferred embodiment of the present invention the radiation is substantially of a single wavelength, the phase of the spectral analysis wavefront is inversely
proportional to the single wavelength, and is related to at least one of a surface characteristic and thickness of the impinged object.
Still further in accordance with a preferred embodiment of the present invention the step of employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the
spectral analysis wavefront includes: expressing the plurality of intensity maps as at least one mathematical function of phase and amplitude of the spectral analysis wavefront and of the plurality
of different phase changes, wherein at least the phase is unknown and a function generating the plurality of phase changed transformed wavefronts is known and employing the at least one mathematical
function to obtain an output indicating at least the phase.
Additionally in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform to
the spectral analysis wavefront, thereby obtaining a transformed wavefront and applying a plurality of different phase changes, including spatial phase changes, to the transformed wavefront, thereby
obtaining a plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the transform applied to the spectral analysis wavefront is a Fourier transform, the plurality of different phase changes
includes at least three different phase changes, effected by applying a spatially uniform, time-varying spatial phase change to part of the transformed wavefront. Preferably, the plurality of
intensity maps includes at least three intensity maps and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the spectral analysis wavefront includes:
expressing the spectral analysis wavefront as a first complex function which has an amplitude and phase identical to the amplitude and phase of the spectral analysis wavefront, expressing the
plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change, defining a second complex
function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial
phase change, expressing each of the plurality of intensity maps as a third function of: the amplitude of the spectral analysis wavefront, the absolute value of the second complex function, a
difference between the phase of the spectral analysis wavefront and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes,
corresponding to one of the at least three intensity maps, solving the third function to obtain the amplitude of the spectral analysis wavefront, the absolute value of the second complex function and
the difference between the phase of the spectral analysis wavefront and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex
function and obtaining the phase of the spectral analysis wavefront by adding the phase of the second complex function to the difference between the phase of the spectral analysis wavefront and phase
of the second complex function.
Further in accordance with a preferred embodiment of the present invention the spectral analysis wavefront includes a plurality of different wavelength components and the plurality of differently
phase changed transformed wavefronts are obtained by applying a phase change to the plurality of different wavelength components of the spectral analysis wavefront.
There is further provided in accordance with a preferred embodiment of the present invention a method of phase change analysis. The method includes obtaining a phase change analysis wavefront which
has an amplitude and a phase, applying a transform to the phase change analysis wavefront thereby to obtain a transformed wavefront, applying a plurality of different phase changes to the transformed
wavefront, thereby to obtain a plurality of differently phase changed transformed wavefronts, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and
employing the plurality of intensity maps to obtain an output indication of differences between the plurality of different phase changes applied to the transformed phase change analysis wavefront.
There is also provided in accordance with yet another preferred embodiment of the present invention an apparatus for phase change analysis. The apparatus includes a wavefront obtainer, operating to
obtain a phase change analysis wavefront which has an amplitude and a phase, a transform applier, applying a transform to the phase change analysis wavefront thereby to obtain a transformed
wavefront, a phase change applier, applying at least one phase change to the transformed wavefront, thereby to obtain at least one phase changed transformed wavefront, an intensity map generator
operating to provide at least one intensity map of the phase changed transformed wavefront and an intensity map utilizer, employing the plurality of intensity maps to provide an output indication of
differences between the plurality of different phase changes applied to the transformed phase change analysis wavefront.
Typically, when lateral shifts appear in the plurality of different phase changes, corresponding changes appear in the plurality of intensity maps and the step of employing the plurality of intensity
maps results in obtaining an indication of the lateral shifts.
Still further in accordance with a preferred embodiment of the present invention the step of employing the plurality of intensity maps to obtain an output indication of differences between the
plurality of different phase changes applied to the transformed phase change analysis wavefront includes: expressing the plurality of intensity maps as at least one mathematical function of phase and
amplitude of the phase change analysis wavefront and of the plurality of different phase changes, where at least the phase and amplitude are known and the plurality of different phase changes are
unknown and employing the mathematical function to obtain an output indicating the differences between the plurality of different phase changes.
There is further provided in accordance with yet a further preferred embodiment of the present invention a method of phase change analysis. The method includes obtaining a phase change analysis
wavefront which has an amplitude and a phase, applying a transform to the phase change analysis wavefront thereby to obtain a transformed wavefront, applying at least one phase change to the
transformed wavefront, thereby to obtain at least one phase changed transformed wavefront, obtaining at least one intensity map of the at least one phase changed transformed wavefront and employing
the intensity map to obtain an output indication of the at least one phase change applied to the transformed phase change analysis wavefront.
There is also provided in accordance with yet another preferred embodiment of the present invention an apparatus for phase change analysis. The apparatus includes a wavefront obtainer, operating to
obtain a phase change analysis wavefront which has an amplitude and a phase, a transform applier, applying a transform to the phase change analysis wavefront thereby to obtain a transformed
wavefront, a phase change applier, applying at least one phase change to the transformed wavefront, thereby to obtain at least one phase changed transformed wavefront, an intensity map generator
operating to provide at least one intensity map of the phase changed transformed wavefront and an intensity map utilizer, employing the intensity map to provide an output indication of the phase
change applied to the transformed phase change analysis wavefront.
Preferably, the phase change is a phase delay, having a value selected from a plurality of pre-determined values, and the output indication of the phase change includes the value of the phase delay.
There is also provided in accordance with a preferred embodiment of the present invention a method of stored data retrieval. The method includes obtaining a stored data retrieval wavefront which has
an amplitude and a phase, by reflecting radiation from the media in which information is encoded, by selecting the height of the media at each of a multiplicity of different locations on the media.
Preferably, analyzing the stored data retrieval wavefront by: obtaining a plurality of differently phase changed transformed wavefronts corresponding to the stored data retrieval wavefront, obtaining
a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity maps to obtain an indication of the amplitude and phase of the stored
data retrieval wavefront and employing the indication of the amplitude and phase to obtain the information.
There is further provided in accordance with yet another preferred embodiment of the present invention an apparatus for stored data retrieval. The apparatus includes a wavefront obtainer operating to
obtain a stored data retrieval wavefront which has an amplitude and a phase, by reflecting radiation from the media in which information is encoded by selecting the height of the media at each of a
multiplicity of different locations on the media, a wavefront analyzer, analyzing the stored data retrieval wavefront and including a wavefront transformer operating to provide a plurality of
differently phase changed transformed wavefronts corresponding to the stored data retrieval wavefront, an intensity map generator operating to obtain a plurality of intensity maps of the plurality of
phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps to provide an indication of the amplitude and phase of the stored data retrieval
wavefront and a phase and amplitude utilizer, employing the indication of the amplitude and phase to provide the information.
Preferably, the step of obtaining a plurality of differently phase changed transformed wavefronts includes: applying a transform to the stored data retrieval wavefront thereby to obtain a transformed
wavefront and applying a plurality of different phase changes to the transformed wavefront, thereby to obtain a plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the transform applied to the stored data retrieval wavefront is a Fourier transform, the plurality of different phase
changes includes at least three different phase changes, effected by applying a spatially uniform, time-varying spatial phase change to part of the transformed wavefront, the plurality of intensity
maps includes at least three intensity maps and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the stored data retrieval wavefront includes:
expressing the stored data retrieval wavefront as a first complex function which has an amplitude and phase identical to the amplitude and phase of the stored data retrieval wavefront, expressing the
plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change, defining a second complex
function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial
phase change, expressing each of the plurality of intensity maps as a third function of: the amplitude of the stored data retrieval wavefront, the absolute value of the second complex function, a
difference between the phase of the stored data retrieval wavefront and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes,
corresponding to one of the at least three intensity maps, solving the third function to obtain the amplitude of the stored data retrieval wavefront, the absolute value of the second complex function
and the difference between the phase of the stored data retrieval wavefront and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex
function and obtaining the phase of the stored data retrieval wavefront by adding the phase of the second complex function to the difference between the phase of the stored data retrieval wavefront
and phase of the second complex function.
Still further in accordance with a preferred embodiment of the present invention the stored data retrieval wavefront includes at least one one-dimensional component, the transform applied to the data
retrieval wavefront is a one-dimensional Fourier transform, performed in a dimension perpendicular to a direction of propagation of the data retrieval wavefront, thereby to obtain at least one
one-dimensional component of the transformed wavefront in the dimension perpendicular to the direction of propagation, the plurality of differently phase changed transformed wavefronts are obtained
by applying the plurality of different phase changes to each of the one-dimensional component, thereby obtaining at least one one-dimensional component of the plurality of phase changed transformed
wavefronts and the plurality of intensity maps are employed to obtain an output indicating amplitude and phase of the one-dimensional component of the data retrieval wavefront.
Preferably, the plurality of different phase changes is applied to each of the at least one one-dimensional component by providing a relative movement between the media and a component generating
spatially varying, time-constant phase changes, the relative movement being in a dimension perpendicular to the direction of propagation and to the dimension perpendicular to the direction of
Additionally in accordance with a preferred embodiment of the present invention the information is encoded on the media whereby: an intensity value is realized by reflection of light from each
location on the media to lie within a predetermined range of values, the range corresponding an element of the information stored at the location and by employing the plurality of intensity maps,
multiple intensity values are realized for each location, providing multiple elements of information for each location on the media.
Preferably, the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts whose phase has been changed by applying an at least time varying phase change function
to the stored data retrieval wavefront.
Further in accordance with a preferred embodiment of the present invention the stored data retrieval wavefront includes a plurality of different wavelength components and the plurality of differently
phase changed transformed wavefronts are obtained by applying at least one phase change to the plurality of different wavelength components of the stored data retrieval wavefront.
Further in accordance with a preferred embodiment of the present invention the radiation which is reflected from the media includes a plurality of different wavelength components, resulting in the
stored data retrieval wavefront including a plurality of different wavelength components and the plurality of differently phase changed transformed wavefronts are obtained by applying a phase change
to the plurality of different wavelength components of the stored data retrieval wavefront.
Still further in accordance with a preferred embodiment of the present invention the information encoded by selecting the height of the media at each of a multiplicity of different locations on the
media is also encoded by selecting the reflectivity of the media at each of a plurality of different locations on the media and employing the indication of the amplitude and phase to obtain the
information includes employing the indication of the phase to obtain the information encoded by selecting the height of the media and employing the indication of the amplitude to obtain the
information encoded by selecting the reflectivity of the media.
There is provided in accordance with another preferred embodiment of the present invention a method of 3-dimensional imaging. The method includes obtaining a 3-dimensional imaging wavefront, which
has an amplitude and a phase, by reflecting radiation from an object to be viewed and analyzing the 3-dimensional imaging wavefront by: obtaining a plurality of differently phase changed transformed
wavefronts corresponding to the 3-dimensional imaging wavefront, obtaining a plurality of intensity maps of the plurality of differently phase changed transformed wavefronts and employing the
plurality of intensity maps to obtain an output indicating the amplitude and phase of the 3-dimensional imaging wavefront.
There is further provided in accordance with a preferred embodiment of the present invention an apparatus for 3-dimensional imaging. The apparatus includes a wavefront obtainer operating to obtain a
3-dimensional imaging wavefront, which has an amplitude and a phase, by reflecting radiation from an object to be viewed, a wavefront analyzer, analyzing the 3-dimensional imaging wavefront including
a wavefront transformer operative to provide a plurality of differently phase changed transformed wavefronts corresponding to the 3-dimensional imaging wavefront, an intensity map generator operative
to provide a plurality of intensity maps of the plurality of differently phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps to provide an
output indicating the amplitude and phase of the 3-dimensional imaging wavefront.
Further in accordance with a preferred embodiment of the present invention the radiation reflected from the object has a narrow band about a given wavelength, causing the phase of the 3-dimensional
imaging wavefront to be proportional to geometrical variations in the object, the proportion being an inverse linear function of the wavelength.
Additionally in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform to
the 3-dimensional imaging wavefront, thereby to obtain a transformed wavefront and applying a plurality of different phase changes, including spatial phase changes, to the transformed wavefront,
thereby to obtain a plurality of differently phase changed transformed wavefronts.
Still further in accordance with a preferred embodiment of the present invention the 3-dimensional imaging wavefront includes a plurality of different wavelength components and the plurality of
differently phase changed transformed wavefronts are obtained by: transforming the 3-dimensional imaging wavefront, thereby obtaining a transformed wavefront including a plurality of different
wavelength components and applying phase changes to the plurality of different wavelength components of the transformed wavefront by passing the transformed wavefront through an object, at least one
of whose thickness and refractive index varies spatially.
There is also provided in accordance with yet another preferred embodiment of the present invention a method of wavefront analysis. The method includes obtaining a plurality of differently phase
changed transformed wavefronts corresponding to a wavefront being analyzed, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the
plurality of intensity maps to obtain an output indicating at least the phase of the wavefront being analyzed by combining the plurality of intensity maps into a second plurality of combined
intensity maps, the second plurality being less than the first plurality, obtaining at least an output indicative of the phase of the wavefront being analyzed from each of the second plurality of
combined intensity maps and combining the outputs to provide at least an enhanced indication of phase of the wavefront being analyzed.
There is also provided in accordance with yet another preferred embodiment of the present invention an apparatus wavefront analysis. The apparatus includes a wavefront transformer operating to
provide a plurality of differently phase changed transformed wavefronts corresponding to a wavefront being analyzed, an intensity map generator operating to obtain a plurality of intensity maps of
the plurality of phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps to obtain an output indicating at least amplitude of the wavefront being
analyzed and including an intensity combiner operating to combine the plurality of intensity maps into a second plurality of combined intensity maps, the second plurality being less than the first
plurality, an indication provider operating to provide at least an output indicative of the amplitude of the wavefront being analyzed from each of the second plurality of combined intensity maps and
an enhanced indication provider, combining the outputs to provide at least an enhanced indication of amplitude of the wavefront being analyzed.
There is provided in accordance with a further preferred embodiment of the present invention a method of wavefront analysis. The method includes obtaining a plurality of differently phase changed
transformed wavefronts corresponding to a wavefront being analyzed, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefront and employing the plurality of
intensity maps to obtain an output indicating at least amplitude of the wavefront being analyzed by combining the plurality of intensity maps into a second plurality of combined intensity maps, the
second plurality being less than the first plurality, obtaining at least an output indicative of the amplitude of the wavefront being analyzed from each of the second plurality of combined intensity
maps and combining the outputs to provide at least an enhanced indication of amplitude of the wavefront being analyzed.
There is provided in accordance with a preferred embodiment of the present invention an apparatus for wavefront analysis. The apparatus includes a wavefront transformer operating to provide a
plurality of differently phase changed transformed wavefronts corresponding to a wavefront being analyzed, an intensity map generator operating to provide a plurality of intensity maps of the
plurality of phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps to provide an output indicating at least the phase of the wavefront being
analyzed. Preferably, the apparatus also includes an intensity map expresser, expressing the plurality of intensity maps as a function of: amplitude of the wavefront being analyzed, phase of the
wavefront being analyzed and a phase change function characterizing the plurality of differently phase changed transformed wavefronts, a complex function definer, defining a complex function of: the
amplitude of the wavefront being analyzed, the phase of the wavefront being analyzed and the phase change function characterizing the plurality of differently phase changed transformed wavefronts,
the complex function being characterized in that the intensity at each location in the plurality of intensity maps is a function predominantly of a value of the complex function at the location and
of the amplitude and the phase of the wavefront being analyzed at the location. The apparatus also typically, includes complex function expresser, expressing the complex function as a function of the
plurality of intensity maps and a phase obtainer, obtaining values for the phase by employing the complex function expressed as a function of the plurality of intensity maps.
There is also provided in accordance with another preferred embodiment of the present invention a method of wavefront analysis. The method includes obtaining a plurality of differently phase changed
transformed wavefronts corresponding to a wavefront being analyzed, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of
intensity maps to provide an output indicating at least the phase of the wavefront being analyzed by: expressing the plurality of intensity maps as a function of: amplitude of the wavefront being
analyzed, phase of the wavefront being analyzed and a phase change function characterizing the plurality of differently phase changed transformed wavefronts. Additionally, defining a complex function
of: the amplitude of the wavefront being analyzed, the phase of the wavefront being analyzed and the phase change function characterizing the plurality of differently phase changed transformed
wavefronts, the complex function being characterized in that the intensity at each location in the plurality of intensity maps is a function predominantly of a value of the complex function at the
location and of the amplitude and the phase of the wavefront being analyzed at the location, expressing the complex function as a function of the plurality of intensity maps and obtaining values for
the phase by employing the complex function expressed as a function of the plurality of intensity maps.
There is further provided in accordance with yet a further preferred embodiment of the present invention a method of wavefront analysis. The method includes applying a Fourier transform to a
wavefront being analyzed which has an amplitude and a phase, thereby obtaining a transformed wavefront, applying a spatially uniform time-varying spatial phase change to part of the transformed
wavefront, thereby to obtain at least three differently phase changed transformed wavefronts, applying a second Fourier transform to obtain at least three intensity maps of the at least three phase
changed transformed wavefronts and employing the at least three intensity maps to obtain an output indicating at least one of the phase and the amplitude of the wavefront being analyzed by:
expressing the wavefront being analyzed as a first complex function which has an amplitude and phase identical to the amplitude and phase of the wavefront being analyzed, expressing the plurality of
intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change, defining a second complex function having an
absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change,
expressing each of the plurality of intensity maps as a third function of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function, a difference between the
phase of the wavefront being analyzed and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes, which each correspond to one
of the at least three intensity maps, solving the third function to obtain the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between
the phase of the wavefront being analyzed and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex function and obtaining the phase
of the wavefront being analyzed by adding the phase of the second complex function to the difference between the phase of the wavefront being analyzed and phase of the second complex function.
There is further provided in accordance with yet a further preferred embodiment of the present invention an apparatus for wavefront analysis. The apparatus includes a first transform applier,
applying a Fourier transform to a wavefront being analyzed which has an amplitude and a phase thereby to obtain a transformed wavefront, a phase change applier, applying a spatially uniform
time-varying spatial phase change to part of the transformed wavefront, thereby obtaining at least three differently phase changed transformed wavefronts, a second transform applier, applying a
second Fourier transform to the at least three differently phase changed transformed wavefronts, thereby obtaining at least three intensity maps. The apparatus also typically includes an intensity
map utilizer, employing the at least three intensity maps to provide an output indicating the phase and the amplitude of the wavefront being analyzed and a wavefront expresser, expressing the
wavefront being analyzed as a first complex function which has an amplitude and phase identical to the amplitude and phase of the wavefront being analyzed, a first intensity map expresser, expressing
the plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change. Preferably, the apparatus
also includes a complex function definer, defining a second complex function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the
spatial function governing the spatially uniform, time-varying spatial phase change, a second intensity map expresser, expressing each of the plurality of intensity maps as a third function of: the
amplitude of the wavefront being analyzed, the absolute value of the second complex function, a difference between the phase of the wavefront being analyzed and the phase of the second complex
function and a known phase delay produced by one of the at least three different phase changes, which each correspond to one of the at least three intensity maps. The apparatus further typically
includes a first function solver, solving the third function to obtain the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between the
phase of the wavefront being analyzed and the phase of the second complex function, a second function solver, solving the second complex function to obtain the phase of the second complex function
and a phase obtainer, obtaining the phase of the wavefront being analyzed by adding the phase of the second complex function to the difference between the phase of the wavefront being analyzed and
the phase of the second complex function.
The present invention will be understood and appreciated more fully from the following detailed description, taken in conjunction with the drawings in which:
FIG. 1A is a simplified partially schematic, partially pictorial illustration of wavefront analysis functionality operative in accordance with a preferred embodiment of the present invention;
FIG. 1B is a simplified partially schematic, partially block diagram illustration of a wavefront analysis system suitable for carrying out the functionality of FIG. 1A in accordance with a preferred
embodiment of the present invention;
FIG. 2 is a simplified functional block diagram illustration of the functionality of FIG. 1A where time-varying phase changes are applied to a transformed wavefront;
FIG. 3 is a simplified functional block diagram illustration of the functionality of FIG. 1A where time-varying phase changes are applied to a wavefront prior to transforming thereof;
FIG. 4 is a simplified functional block diagram illustration of the functionality of FIG. 2 where time-varying, non-spatially varying spatial phase changes are applied to a transformed wavefront;
FIG. 5 is a simplified functional block diagram illustration of the functionality of FIG. 3 where time-varying, non-spatially varying spatial phase changes are applied to a wavefront prior to
transforming thereof;
FIG. 6 is a simplified functional block diagram illustration of the functionality of FIG. 1A where phase changes are applied to a plurality of different wavelength components of a transformed
FIG. 7 is a simplified functional block diagram illustration of the functionality of FIG. 1A where phase changes are applied to a plurality of different wavelength components of a wavefront prior to
transforming thereof;
FIG. 8 is a simplified functional block diagram illustration of the functionality of FIG. 1A where phase changes are applied to a plurality of different polarization components of a transformed
FIG. 9 is a simplified functional block diagram illustration of the functionality of FIG. 1A where phase changes are applied to a plurality of different polarization components of a wavefront prior
to transforming thereof;
FIG. 10A is a simplified functional block diagram illustration of the functionality of FIG. 1A where a wavefront being analyzed comprises at least one one-dimensional component;
FIG. 10B is a simplified partially schematic, partially pictorial illustration of a wavefront analysis system suitable for carrying out the functionality of FIG. 10A in accordance with a preferred
embodiment of the present invention;
FIG. 11 is a simplified functional block diagram illustration of the functionality of FIG. 1A where an additional transform is applied following the application of spatial phase changes;
FIG. 12 is a simplified functional block diagram illustration of the functionality of FIG. 1A, wherein intensity maps are employed to provide information about a wavefront being analyzed, such as
indications of amplitude and phase of the wavefront;
FIG. 13 is a simplified functional block diagram illustration of part of the functionality of FIG. 1A, wherein the transform applied to the wavefront being analyzed is a Fourier transform, wherein at
least three different spatial phase changes are applied to a transformed wavefront, and wherein at least three intensity maps are employed to obtain indications of at least the phase of a wavefront;
FIG. 14 is a simplified partially schematic, partially pictorial illustration of part of one preferred embodiment of a wavefront analysis system of the type shown in FIG. 1B;
FIG. 15 is a simplified partially schematic, partially pictorial illustration of a system for surface mapping employing the functionality and structure of FIGS. 1A and 1B;
FIG. 16 is a simplified partially schematic, partially pictorial illustration of a system for object inspection employing the functionality and structure of FIGS. 1A and 1B;
FIG. 17 is a simplified partially schematic, partially pictorial illustration of a system for spectral analysis employing the functionality and structure of FIGS. 1A and 1B;
FIG. 18 is a simplified partially schematic, partially pictorial illustration of a system for phase-change analysis employing the functionality and structure of FIGS. 1A and 1B;
FIG. 19 is a simplified partially schematic, partially pictorial illustration of a system for stored data retrieval employing the functionality and structure of FIGS. 1A and 1B;
FIG. 20 is a simplified partially schematic, partially pictorial illustration of a system for 3-dimensional imaging employing the functionality and structure of FIGS. 1A and 1B;
FIG. 21A is a simplified partially schematic, partially pictorial illustration of wavefront analysis functionality operative in accordance with another preferred embodiment of the present invention;
FIG. 21B is a simplified partially schematic, partially block diagram illustration of a wavefront analysis system suitable for carrying out the functionality of FIG. 21A in accordance with another
preferred embodiment of the present invention; and
FIG. 22 is a simplified partially schematic, partially pictorial illustration of a system for surface mapping employing the functionality and structure of FIGS. 21A and 21B.
Reference is now made to FIG. 1A, which is a simplified partially schematic, partially pictorial illustration of wavefront analysis functionality operative in accordance with a preferred embodiment
of the present invention. The functionality of FIG. 1A can be summarized as including the following sub-functionalities:
• A. obtaining a plurality of differently phase changed transformed wavefronts corresponding to a wavefront being analyzed, which has an amplitude and a phase;
• B. obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts; and
• C. employing the plurality of intensity maps to obtain an output indicating at least one and possibly both of the phase and the amplitude of the wavefront being analyzed.
As seen in FIG. 1A, the first sub-functionality, designated “A” may be realized by the following functionalities:
A wavefront, which may be represented by a plurality of point sources of light, is generally designated by reference numeral 100. Wavefront 100 has a phase characteristic which is typically spatially
non-uniform, shown as a solid line and indicated generally by reference numeral 102. Wavefront 100 also has an amplitude characteristic which is also typically spatially non-uniform, shown as a
dashed line and indicated generally by reference numeral 103. Such a wavefront may be obtained in a conventional manner by receiving light from any object, such as by reading an optical disk, for
example a DVD or compact disk 104.
A principal purpose of the present invention is to measure the phase characteristic, such as that indicated by reference numeral 102, which is not readily measured. Another purpose of the present
invention is to measure the amplitude characteristic, such as that indicated by reference numeral 103 in an enhanced manner. A further purpose of the present invention is to measure both the phase
characteristic 102 and the amplitude characteristic 103. While there exist various techniques for carrying out such measurements, the present invention provides a methodology which is believed to be
superior to those presently known, inter alia due to its relative insensitivity to noise.
A transform, indicated here symbolically by reference numeral 106, is applied to the wavefront being analyzed 100, thereby to obtain a transformed wavefront. A preferred transform is a Fourier
transform. The resulting transformed wavefront is symbolically indicated by reference numeral 108.
A plurality of different phase changes, preferably spatial phase changes, represented by optical path delays 110, 112 and 114 are applied to the transformed wavefront 108, thereby to obtain a
plurality of differently phase changed transformed wavefronts, represented by reference numerals 120, 122 and 124 respectively. It is appreciated that the illustrated difference between the
individual ones of the plurality of differently phase changed transformed wavefronts is that portions of the transformed wavefront are delayed differently relative to the remainder thereof. The
difference in the phase changes, which are applied to the transformed wavefront 108, is represented in FIG. 1A by the change in thickness of the optical path delays 110, 112 and 114.
As seen in FIG. 1A, the second sub-functionality, designated “B”, may be realized by applying a transform, preferably a Fourier transform, to the plurality of differently phase changed transformed
wavefronts. Alternatively, the sub-functionality B may be realized without the use of a Fourier transform, such as by propagation of the differently phase changed transformed wavefronts over an
extended space. Finally, functionality B requires detection of the intensity characteristics of plurality of differently phase changed transformed wavefronts. The outputs of such detection are the
intensity maps, examples of which are designated by reference numerals 130, 132 and 134.
As seen in FIG. 1A, the third sub-functionality, designated “C” may be realized by the following functionalities:
□ expressing, such as by employing a computer 136, the plurality of intensity maps, such as maps 130, 132 and 134, as at least one mathematical function of phase and amplitude of the wavefront
being analyzed and of the plurality of different phase changes, wherein at least one and possibly both of the phase and the amplitude are unknown and the plurality of different phase changes,
typically represented by optical path delays 110, 112 and 114 to the transformed wavefront 108, are known; and
□ employing, such as by means of the computer 136, the at least one mathematical function to obtain an indication of at least one and possibly both of the phase and the amplitude of the
wavefront being analyzed, here represented by the phase function designated by reference numeral 138 and the amplitude function designated by reference numeral 139, which, as can be seen,
respectively represent the phase characteristics 102 and the amplitude characteristics 103 of the wavefront 100. In this example, wavefront 100 may represent the information contained in the
compact disk or DVD 104.
In accordance with an embodiment of the present invention, the plurality of intensity maps comprises at least four intensity maps. In such a case, employing the plurality of intensity maps to obtain
an output indicating at least the phase of the wavefront being analyzed includes employing a plurality of combinations, each of at least three of the plurality of intensity maps, to provide a
plurality of indications at least of the phase of the wavefront being analyzed.
Preferably, the methodology also includes employing the plurality of indications of at least the phase of the wavefront being analyzed to provide an enhanced indication at least of the phase of the
wavefront being analyzed.
Also in accordance with an embodiment of the present invention, the plurality of intensity maps comprises at least four intensity maps. In such a case, employing the plurality of intensity maps to
obtain an output indicating at least the amplitude of the wavefront being analyzed includes employing a plurality of combinations, each of at least three of the plurality of intensity maps, to
provide a plurality of indications at least of the amplitude of the wavefront being analyzed.
Preferably, the methodology also includes employing the plurality of indications of at least the amplitude of the wavefront being analyzed to provide an enhanced indication at least of the amplitude
of the wavefront being analyzed.
It is appreciated that in this manner, enhanced indications of both phase and amplitude of the wavefront may be obtained.
In accordance with a preferred embodiment of the present invention, at least some of the plurality of indications of the amplitude and phase are at least second order indications of the amplitude and
phase of the wavefront being analyzed.
In accordance with one preferred embodiment of the present invention, the plurality of intensity maps are employed to provide an analytical output indicating the amplitude and phase.
Preferably, the phase changed transformed wavefronts are obtained by interference of the wavefront being analyzed along a common optical path.
In accordance with one preferred embodiment of the present invention, the plurality of differently phase changed transformed wavefronts are realized in a manner substantially different from
performing a delta-function phase change to the transformed wavefront, whereby a delta-function phase change is applying a uniform phase delay to a small spatial region, having the characteristics of
a delta-function, of the transformed wavefront.
In accordance with another preferred embodiment of the present invention, the plurality of intensity maps are employed to obtain an output indicating the phase of the wavefront being analyzed, which
is substantially free from halo and shading off distortions, which are characteristic of many of the existing ‘phase-contrast’ methods.
In accordance with another embodiment of the present invention the output indicating the phase of the wavefront being analyzed may be processed to obtain the polarization mode of the wavefront being
In accordance with still another embodiment of the present invention, the plurality of intensity maps may be employed to obtain an output indicating the phase of the wavefront being analyzed by
combining the plurality of intensity maps into a second plurality of combined intensity maps, the second plurality being less than the first plurality, obtaining at least an output indicative of the
phase of the wavefront being analyzed from each of the second plurality of combined intensity maps and combining the outputs to provide an enhanced indication of the phase of the wavefront being
In accordance with yet another embodiment of the present invention, the plurality of intensity maps may be employed to obtain an output indicating amplitude of the wavefront being analyzed by
combining the plurality of intensity maps into a second plurality of combined intensity maps, the second plurality being less than the first plurality, obtaining at least an output indicative of the
amplitude of the wavefront being analyzed from each of the second plurality of combined intensity maps and combining the outputs to provide an enhanced indication of the amplitude of the wavefront
being analyzed.
Additionally in accordance with a preferred embodiment of the present invention, the foregoing methodology may be employed for obtaining a plurality of differently phase changed transformed
wavefronts corresponding to a wavefront being analyzed, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity maps
to obtain an output of an at least second order indication of phase of the wavefront being analyzed.
Additionally or alternatively in accordance with a preferred embodiment of the present invention, the foregoing methodology may be employed for obtaining a plurality of differently phase changed
transformed wavefronts corresponding to a wavefront being analyzed, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of
intensity maps to obtain an output of an at least second order indication of amplitude of the wavefront being analyzed.
In accordance with yet another embodiment of the present invention, the obtaining of the plurality of differently phase changed transformed wavefronts comprises applying a transform to the wavefront
being analyzed, thereby to obtain a transformed wavefront, and then applying a plurality of different phase and amplitude changes to the transformed wavefront, where each of these changes can be a
phase change, an amplitude change or a combined phase and amplitude change, thereby to obtain a plurality of differently phase and amplitude changed transformed wavefronts.
In accordance with yet another embodiment of the present invention, a wavefront being analyzed comprises at least two wavelength components. In such a case, obtaining a plurality of intensity maps
also includes dividing the phase changed transformed wavefronts according to the at least two wavelength components in order to obtain at least two wavelength components of the phase changed
transformed wavefronts and in order to obtain at least two sets of intensity maps, each set corresponding to a different one of the at least two wavelength components of the phase changed transformed
Subsequently, the plurality of intensity maps are employed to provide an output indicating the amplitude and phase of the wavefront being analyzed by obtaining an output indicative of the phase of
the wavefront being analyzed from each of the at least two sets of intensity maps and combining the outputs to provide an enhanced indication of phase of the wavefront being analyzed. In the enhanced
indication, there is no 2π ambiguity once the value of the phase exceeds 2π, which conventionally results when detecting a phase of a single wavelength wavefront.
It is appreciated that the wavefront being analyzed may be an acoustic radiation wavefront.
It is also appreciated that the wavefront being analyzed may be an electromagnetic radiation wavefront, of any suitable wavelength, such as visible light, infrared, ultra-violet and X-ray radiation.
It is further appreciated that wavefront 100 may be represented by a relatively small number of point sources and defined over a relatively small spatial region. In such a case, the detection of the
intensity characteristics of the plurality of differently phase changed transformed wavefronts may be performed by a detector comprising only a single detection pixel or several detection pixels.
Additionally, the output indicating at least one and possibly both of the phase and amplitude of the wavefront being analyzed, may be provided by computer 136 in a straight-forward manner.
Reference is now made to FIG. 1B, which is a simplified partially schematic, partially block diagram illustration of a wavefront analysis system suitable for carrying out the functionality of FIG. 1A
in accordance with a preferred embodiment of the present invention. As seen in FIG. 1B, a wavefront, here designated by reference numeral 150 is focused, as by a lens 152, onto a phase manipulator
154, which is preferably located at the focal plane of lens 152. The phase manipulator 154 generates phase changes, and may be, for example, a spatial light modulator or a series of different
transparent, spatially non-uniform objects.
A second lens 156 is arranged so as to image wavefront 150 onto a detector 158, such as a CCD detector. Preferably the second lens 156 is arranged such that the detector 158 lies in its focal plane.
The output of detector 158 is preferably supplied to data storage and processing circuitry 160, which preferably carries out functionality “C” described hereinabove with reference to FIG. 1A.
Reference is now made to FIG. 2, which is a simplified functional block diagram illustration of the functionality of FIG. 1A where time-varying phase changes are applied to a transformed wavefront.
As seen in FIG. 2, and as explained hereinabove with reference to FIG. 1A, a wavefront 200 is preferably transformed to provide a transformed wavefront 208.
A first phase change, preferably a spatial phase change, is applied to the transformed wavefront 208 at a first time T1, as indicated by reference numeral 210, thereby producing a phase changed
transformed wavefront 212 at time T1. This phase changed transformed wavefront 212 is detected, as by detector 158 (FIG. 1B), producing an intensity map, an example of which is designated by
reference numeral 214, which map is stored as by circuitry 160 (FIG. 1B).
Thereafter, a second phase change, preferably a spatial phase change, is applied to the transformed wavefront 208 at a second time T2, as indicated by reference numeral 220, thereby producing a phase
changed transformed wavefront 222 at time T2. This phase changed transformed wavefront 222 is detected, as by detector 158 (FIG. 1B), producing an intensity map, an example of which is designated by
reference numeral 224, which map is stored as by circuitry 160 (FIG. 1B).
Thereafter, a third phase change, preferably a spatial phase change, is applied to the transformed wavefront 208 at a third time T3, as indicated by reference numeral 230, thereby producing a phase
changed transformed wavefront 232 at time T3. This phase changed transformed wavefront 232 is detected, as by detector 158 (FIG. 1B), producing an intensity map, an example of which is designated by
reference numeral 234, which map is stored as by circuitry 160 (FIG. 1B).
It is appreciated that any suitable number of spatial phase changes may be made at successive times and stored for use in accordance with the present invention.
In accordance with a preferred embodiment of the present invention, at least some of the phase changes 210, 220 and 230, are spatial phase changes effected by applying a spatial phase change to part
of the transformed wavefront 208.
In accordance with another preferred embodiment of the present invention, at least some of the phase changes 210, 220 and 230, are spatial phase changes, effected by applying a time-varying spatial
phase change to part of the transformed wavefront 208.
In accordance with another preferred embodiment of the present invention, at least some of the phase changes 210, 220 and 230, are spatial phase changes, effected by applying a non time-varying
spatial phase change to part of transformed wavefront 208, producing spatially phase changed transformed wavefronts 212, 222 and 232, which subsequently produce spatially varying intensity maps 214,
224 and 234 respectively.
Reference is now made to FIG. 3, which is a simplified functional block diagram illustration of the functionality of FIG. 1A where time-varying phase changes are applied to a wavefront prior to
transforming thereof. As seen in FIG. 3, a first phase change, preferably a spatial phase change, is applied to a wavefront 300 at a first time T1, as indicated by reference numeral 310. Following
application of the first phase change to wavefront 300, a transform, preferably a Fourier transform, is applied thereto, thereby producing a phase changed transformed wavefront 312 at time T1. This
phase changed transformed wavefront 312 is detected, as by detector 158 (FIG. 1B), producing an intensity map, an example of which is designated by reference numeral 314, which map is stored as by
circuitry 160 (FIG. 1B).
Thereafter, a second phase change, preferably a spatial phase change, is applied to wavefront 300 at a second time T2, as indicated by reference numeral 320. Following application of the second phase
change to wavefront 300, a transform, preferably a Fourier transform, is applied thereto, thereby producing a phase changed transformed wavefront 322 at time T2. This phase changed transformed
wavefront 322 is detected, as by detector 158 (FIG. 1B), producing an intensity map, an example of which is designated by reference numeral 324, which map is stored as by circuitry 160 (FIG. 1B).
Thereafter, a third phase change, preferably a spatial phase change, is applied to wavefront 300 at a third time T3, as indicated by reference numeral 330. Following application of the third phase
change to wavefront 300, a transform, preferably a Fourier transform, is applied thereto, thereby producing a phase changed transformed wavefront 332 at time T3. This phase changed transformed
wavefront 332 is detected, as by detector 158 (FIG. 1B), producing an intensity map, an example of which is designated by reference numeral 334, which map is stored as by circuitry 160 (FIG. 1B).
It is appreciated that any suitable number of spatial phase changes may be made at successive times and stored for use in accordance with the present invention.
In accordance with a preferred embodiment of the present invention, at least some of the phase changes 310, 320 and 330, are spatial phase changes effected by applying a spatial phase change to part
of wavefront 300.
In accordance with another preferred embodiment of the present invention, at least some of the phase changes 310, 320 and 330, are spatial phase changes, effected by applying a time-varying spatial
phase change to part of wavefront 300.
In accordance with another preferred embodiment of the present invention, at least some of the phase changes 310, 320 and 330, are spatial phase changes, effected by applying a non time-varying
spatial phase change to part of wavefront 300, producing spatially phase changed transformed wavefronts 312, 322 and 332, which subsequently produce spatially varying intensity maps 314, 324 and 334
Reference is now made to FIG. 4, which is a simplified functional block diagram illustration of the functionality of FIG. 2, specifically in a case where time-varying, non-spatially varying, spatial
phase changes are applied to a transformed wavefront. As seen in FIG. 4, and as explained hereinabove with reference to FIG. 1A, a wavefront 400 is preferably transformed to provide a transformed
wavefront 408. A preferred transform is a Fourier transform.
A first spatial phase change is applied to the transformed wavefront 408 at a first time T1, as indicated by reference numeral 410. This phase change preferably is effected by applying a spatially
uniform spatial phase delay D, designated by reference ‘D=D1’, to a given spatial region of the transformed wavefront 408. Thus, at the given spatial region of the transformed wavefront, the value of
the phase delay at time T1 is D1, while at the remainder of the transformed wavefront, where no phase delay is applied, the value of the phase delay is D=0.
The first spatial phase change 410 thereby produces a spatially phase changed transformed wavefront 412 at time T1. This spatially phase changed transformed wavefront 412 is detected, as by detector
158 (FIG. 1B), producing a spatially varying intensity map, an example of which is designated by reference numeral 414, which map is stored as by circuitry 160 (FIG. 1B).
Thereafter, a second spatial phase change is applied to the transformed wavefront 408 at a second time T2, as indicated by reference numeral 420. This phase change preferably is effected by applying
a spatially uniform spatial phase delay D, designated by reference ‘D=D2’, to a given spatial region of the transformed wavefront 408. Thus, at the given spatial region of the transformed wavefront,
the value of the phase delay at time T2 is D2, while at the remainder of the transformed wavefront, where no phase delay is applied, the value of the phase delay is D=0.
The second spatial phase change 420 thereby produces a spatially phase changed transformed wavefront 422 at time T2. This spatially phase changed transformed wavefront 422 is detected, as by detector
158 (FIG. 1B), producing a spatially varying intensity map, an example of which is designated by reference numeral 424, which map is stored as by circuitry 160 (FIG. 1B).
Thereafter, a third spatial phase change is applied to the transformed wavefront 408 at a third time T3, as indicated by reference numeral 430. This phase change preferably is effected by applying a
spatially uniform spatial phase delay D, designated by reference ‘D=D3’, to a given spatial region of the transformed wavefront 408. Thus, at the given spatial region of the transformed wavefront,
the value of the phase delay at time T3 is D3, while at the remainder of the transformed wavefront, where no phase delay is applied, the value of the phase delay is D=0.
The third spatial phase change 430 thereby produces a spatially phase changed transformed wavefront 432 at time T3. This spatially phase changed transformed wavefront 432 is detected, as by detector
158 (FIG. 1B), producing a spatially varying intensity map, an example of which is designated by reference numeral 434, which map is stored as by circuitry 160 (FIG. 1B).
It is appreciated that any suitable number of spatial phase changes may be made at successive times and stored for use in accordance with the present invention.
In accordance with a preferred embodiment of the present invention, the transform applied to the wavefront 400 is a Fourier transform, thereby providing a Fourier-transformed wavefront 408. In
addition, the plurality of phase changed transformed wavefronts 412, 422 and 432 may be further transformed, preferably by a Fourier transform, prior to detection thereof.
In accordance with a preferred embodiment of the present invention, the spatial region of the transformed wavefront 408 to which the spatially uniform, spatial phase delays D1, D2 and D3 are applied
at times T1, T2 and T3 respectively is a spatially central region of the transformed wavefront 408.
In accordance with an embodiment of the present invention, a phase component comprising relatively high frequency components may be added to the wavefront 400 prior to applying the transform thereto,
in order to increase the high-frequency content of the transformed wavefront 408 prior to applying the spatially uniform, spatial phase delays to a spatial region thereof.
Additionally, in accordance with a preferred embodiment of the present invention, the spatial region of the transformed wavefront 408 to which the spatially uniform, spatial phase delays D1, D2 and D
3 are applied at times T1, T2 and T3 respectively is a spatially central region of the transformed wavefront 408, the transform applied to the wavefront 400 is a Fourier transform, and the plurality
of phase changed transformed wavefronts 412, 422 and 432 are Fourier transformed prior to detection thereof.
In accordance with another embodiment of the present invention, the region of the transformed wavefront 408 to which the spatially uniform, spatial phase delays D1, D2 and D3 are applied at times T1,
T2 and T3 respectively is a spatially centered generally circular region of the transformed wavefront 408.
In accordance with yet another embodiment of the present invention, the region of the transformed wavefront 408 to which the spatially uniform, spatial phase delays D1, D2 and D3 are applied at times
T1, T2 and T3 respectively is a region covering approximately one half of the entire region in which transformed wavefront 408 is defined.
In accordance with a preferred embodiment of the present invention, the transformed wavefront 408 includes a non-spatially modulated region, termed a DC region, which represents an image of a light
source generating the wavefront 400, and a non-DC region. The region of the transformed wavefront 408 to which the spatially uniform, spatial phase delays D1, D2 and D3 are applied at times T1, T2
and T3 respectively includes at least parts of both the DC region and the non-DC region.
Reference is now made to FIG. 5, which is a simplified functional block diagram illustration of the functionality of FIG. 3, where time-varying, non-spatially varying, spatial phase changes are
applied to a wavefront prior to transforming thereof.
As seen in FIG. 5, a first spatial phase change is applied to a wavefront 500 at a first time T1, as indicated by reference numeral 510. This phase change preferably is effected by applying a
spatially uniform spatial phase delay D, designated by reference ‘D=D1’, to a given spatial region of the wavefront 500. Thus, at the given spatial region of the wavefront, the value of the phase
delay at time T1 is D1, while at the remainder of the wavefront, where no phase delay is applied, the value of the phase delay is D=0.
Following application of the first spatial phase change to wavefront 500, a transform, preferably a Fourier transform, is applied thereto, thereby producing a spatially phase changed transformed
wavefront 512 at time T1. This spatially phase changed transformed wavefront 512 is detected, as by detector 158 (FIG. 1B), producing a spatially varying intensity map, an example of which is
designated by reference numeral 514, which map is stored as by circuitry 160 (FIG. 1B).
Thereafter, a second spatial phase change is applied to wavefront 500 at a second time T2, as indicated by reference numeral 520. This phase change preferably is effected by applying a spatially
uniform spatial phase delay D, designated by reference ‘D=D2’, to a given spatial region of the wavefront 500. Thus, at the given spatial region of the wavefront, the value of the phase delay at time
T2 is D2, while at the remainder of the wavefront, where no phase delay is applied, the value of the phase delay is D=0.
Following application of the second spatial phase change to wavefront 500, a transform, preferably a Fourier transform, is applied thereto, thereby producing a spatially phase changed transformed
wavefront 522 at time T2. This spatially phase changed transformed wavefront 522 is detected, as by detector 158 (FIG. 1B), producing a spatially varying intensity map, an example of which is
designated by reference numeral 524, which map is stored as by circuitry 160 (FIG. 1B).
Thereafter, a third spatial phase change is applied to wavefront 500 at a third time T3, as indicated by reference numeral 530. This phase change preferably is effected by applying a spatially
uniform spatial phase delay D, designated by reference ‘D=D3’, to a given spatial region of the wavefront 500. Thus, at the given spatial region of the wavefront, the value of the phase delay at time
T3 is D3, while at the remainder of the wavefront, where no phase delay is applied, the value of the phase delay is D=0.
Following application of the third spatial phase change to wavefront 500, a transform, preferably a Fourier transform, is applied thereto, thereby producing a spatially phase changed transformed
wavefront 532 at time T3. This spatially phase changed transformed wavefront 532 is detected, as by detector 158 (FIG. 1B), producing a spatially varying intensity map, an example of which is
designated by reference numeral 534, which map is stored as by circuitry 160 (FIG. 1B).
It is appreciated that any suitable number of spatial phase changes may be made at successive times and stored for use in accordance with the present invention.
In accordance with a preferred embodiment of the present invention, the spatial region of the wavefront 500 to which the spatially uniform, spatial phase delays D1, D2 and D3 are applied at times T1,
T2 and T3 respectively is a spatially central region of the wavefront 500.
In accordance with an embodiment of the present invention, a phase component comprising relatively high frequency components may be added to the wavefront 500 prior to applying the spatial phase
changes thereto, in order to increase the high-frequency content of the wavefront 500.
Additionally, in accordance with a preferred embodiment of the present invention, the spatial region of the wavefront 500 to which the spatially uniform, spatial phase delays D1, D2 and D3 are
applied at times T1, T2 and T3 respectively is a spatially central region of the wavefront 500, the transforms are Fourier transforms, and the plurality of phase changed transformed wavefronts 512,
522 and 532 are Fourier transformed prior to detection thereof.
In accordance with another embodiment of the present invention, the region of the wavefront 500 to which the spatially uniform, spatial phase delays D1, D2 and D3 are applied at times T1, T2 and T3
respectively is a spatially centered generally circular region of the wavefront 500.
In accordance with yet another embodiment of the present invention, the region of the wavefront 500 to which the spatially uniform, spatial phase delays D1, D2 and D3 are applied at times T1, T2 and
T3 respectively is a region covering approximately one half of the entire region in which wavefront 500 is defined.
In accordance with a preferred embodiment of the present invention, the wavefront 500 includes a non-spatially modulated region, termed a DC region, which represents an image of a light source
generating the wavefront 500, and a non-DC region. The region of the wavefront 500 to which the spatially uniform, spatial phase delays D1, D2 and D3 are applied at times T1, T2 and T3 respectively
includes at least parts of both the DC region and the non-DC region.
Reference is now made to FIG. 6, which is a simplified functional block diagram illustration of the functionality of FIG. 1A where phase changes are applied to a plurality of different wavelength
components of a transformed wavefront. As seen in FIG. 6, a wavefront 600, which comprises a plurality of different wavelength components, is preferably transformed to obtain a transformed wavefront
602. The transform is preferably a Fourier transform.
Similarly to wavefront 600, the transformed wavefront 602 also includes a plurality of different wavelength components, represented by reference numerals 604, 606 and 608. It is appreciated that both
the wavefront 600 and the transformed wavefront 602 can include any suitable number of wavelength components.
A plurality of phase changes, preferably spatial phase changes, represented by reference numerals 610, 612 and 614 are applied to respective wavelength components 604, 606 and 608 of the transformed
wavefront, thereby providing a plurality of differently phase changed transformed wavefront components, represented by reference numerals 620, 622 and 624 respectively.
The phase changed transformed wavefront components 620, 622, and 624 may be transformed, preferably by a Fourier transform, and are subsequently detected, as by detector 158 (FIG. 1B), producing
spatially varying intensity maps, examples of which are designated by reference numerals 630, 632 and 634 respectively. These intensity maps are subsequently stored as by circuitry 160 (FIG. 1B).
In accordance with an embodiment of the present invention, phase changes 610, 612 and 614 are effected by passing the transformed wavefront 602 through an object, at least one of whose thickness and
refractive index varies spatially, thereby applying a different spatial phase delay to each of the wavelength components 604, 606 and 608 of the transformed wavefront.
In accordance with another embodiment of the present invention, the phase changes 610, 612 and 614 are effected by reflecting the transformed wavefront 602 from a spatially varying surface, thereby
applying a different spatial phase delay to each of the wavelength components 604, 606 and 608 of the transformed wavefront.
In accordance with yet another embodiment of the present invention, the phase changes 610, 612 and 614 are realized by passing the transformed wavefront 602 through a plurality of objects, each
characterized in that at least one of its thickness and refractive index varies spatially. The spatial variance of the thickness or of the refractive index of the plurality of objects is selected in
a way such that the phase changes 610, 612 and 614 differ to a selected predetermined extent for at least some of the plurality of different wavelength components 604, 606 and 608.
Alternatively, the spatial variance of the thickness or refractive index of the plurality of objects is selected in a way such that the phase changes 610, 612 and 614 are identical for at least some
of the plurality of different wavelength components 604, 606 and 608.
Additionally, in accordance with an embodiment of the present invention, the phase changes 610, 612 and 614 are time-varying spatial phase changes. In such a case, the plurality of phase changed
transformed wavefront components 620, 622 and 624 include a plurality of differently phase changed transformed wavefronts for each wavelength component thereof, and the intensity maps 630, 632 and
634 include a time-varying intensity map for each such wavelength component.
In accordance with an embodiment of the present invention, termed a “white light” embodiment, all the wavelength components may be detected by a single detector, resulting in a time-varying intensity
map representing several wavelength components.
In accordance with another embodiment of the present invention, the plurality of phase changed transformed wavefront components 620, 622 and 624 are broken down into separate wavelength components,
such as by a spatial separation effected, for example, by passing the phase changed transformed wavefront components through a dispersion element. In such a case, the intensity maps 630, 632 and 634
are provided simultaneously for all of the plurality of different wavelength components.
Reference is now made to FIG. 7, which is a simplified functional block diagram illustration of the functionality of FIG. 1A where phase changes are applied to a plurality of different wavelength
components of a wavefront, prior to transforming thereof As seen in FIG. 7, a wavefront 700 comprises a plurality of different wavelength components 704, 706 and 708. It is appreciated that the
wavefront can include any suitable number of wavelength components.
A plurality of phase changes, preferably spatial phase changes, represented by reference numerals 710, 712 and 714, are applied to the respective wavelength components 704, 706 and 708 of the
Following application of the spatial phase changes to wavefront components 704, 706 and 708, a transform, preferably a Fourier transform, is applied thereto, thereby providing a plurality of
different phase changed transformed wavefront components, represented by reference numerals 720, 722 and 724 respectively.
These phase changed transformed wavefront components 720, 722 and 724 are subsequently detected, as by detector 158 (FIG. 1B), producing spatially varying intensity maps, examples of which are
designated by reference numerals 730, 732 and 734. These intensity maps are subsequently stored as by circuitry 160 (FIG. 1B).
In accordance with an embodiment of the present invention, phase changes 710, 712 and 714 are effected by passing the wavefront 700 through an object, at least one of whose thickness and refractive
index varies spatially, thereby applying a different spatial phase delay to each of the wavelength components 704, 706 and 708 of the wavefront.
In accordance with another embodiment of the present invention, the phase changes 710, 712 and 714 are effected by reflecting the wavefront 700 from a spatially varying surface, thereby applying a
different spatial phase delay to each of the wavelength components 704, 706 and 708 of the wavefront.
In accordance with yet another embodiment of the present invention phase changes 710, 712 and 714 are realized by passing the wavefront 700 through a plurality of objects, each characterized in that
at least one of its thickness and refractive index varies spatially. The spatial variance of the thickness or refractive index of these objects is selected in a way such that the phase changes 710,
712 and 714 differ to a selected predetermined extent for at least some of the plurality of different wavelength components 704, 706 and 708.
Alternatively, the spatial variance of the thickness or refractive index of these objects is selected in a way that the phase changes 710, 712 and 714 are identical for at least some of the plurality
of different wavelength components 704, 706 and 708.
Reference is now made to FIG. 8, which is a simplified functional block diagram illustration of the functionality of FIG. 1A where phase changes are applied to a plurality of different polarization
components of a transformed wavefront. As seen in FIG. 8, a wavefront 800, which comprises a plurality of different polarization components, is preferably transformed to obtain a transformed
wavefront 802. The transform is preferably a Fourier transform. Similarly to wavefront 800, the transformed wavefront 802 also includes a plurality of different polarization components, represented
by reference numerals 804 and 806. It is appreciated that the polarization components 804 and 806 can be either spatially different or spatially identical, but are each of different polarization. It
is further appreciated that both the wavefront 800 and the transformed wavefront 802 preferably each include two polarization components but can include any suitable number of polarization
A plurality of phase changes, preferably spatial phase changes, represented by reference numerals 810 and 812, are applied to the respective polarization components 804 and 806 of the transformed
wavefront 802, thereby providing a plurality of differently phase changed transformed wavefront components, represented by reference numerals 820 and 822 respectively.
It is appreciated that phase changes 810 and 812 can be different for at least some of the plurality of different polarization components 804 and 806. Alternatively, phase changes 810 and 812 can be
identical for at least some of the plurality of different polarization components 804 and 806.
The phase changed transformed wavefront components 820 and 822 are detected, as by detector 158 (FIG. 1B), producing spatially varying intensity maps, examples of which are designated by reference
numerals 830 and 832. These intensity maps are subsequently stored as by circuitry 160 (FIG. 1B).
Reference is now made to FIG. 9, which is a simplified functional block diagram illustration of the functionality of FIG. 1A where phase changes are applied to a plurality of different polarization
components of a wavefront prior to transforming thereof. As seen in FIG. 9, a wavefront 900 comprises a plurality of different polarization components 904 and 906. It is appreciated that the
wavefront preferably includes two polarization components but can include any suitable number of polarization components.
A plurality of phase changes, preferably spatial phase changes, represented by reference numerals 910 and 912, are applied to the respective polarization components 904 and 906 of the wavefront.
It is appreciated that phase changes 910 and 912 can be different for at least some of the plurality of different polarization components 904 and 906. Alternatively, phase changes 910 and 912 can be
set to be identical for at least some of the plurality of different polarization components 904 and 906.
Following application of the spatial phase changes to wavefront components 904 and 906, a transform, preferably a Fourier transform, is applied thereto, thereby providing a plurality of different
phase changed transformed wavefront components, designated by reference numerals 920 and 922 respectively.
Phase changed transformed wavefront components 920 and 922 are subsequently detected, as by detector 158 (FIG. 1B), producing spatially varying intensity maps, examples of which are designated by
reference numeral 930 and 932. These intensity maps are subsequently stored as by circuitry 160 (FIG. 1B).
Reference is now made to FIG. 10A, which is a simplified functional block diagram illustration of the functionality of FIG. 1A, where a wavefront being analyzed comprises at least one one-dimensional
component. In the embodiment of FIG. 10A, a one-dimensional Fourier transform is applied to the wavefront. Preferably, the transform is performed in a dimension perpendicular to a direction of
propagation of the wavefront being analyzed, thereby to obtain at least one one-dimensional component of the transformed wavefront in the dimension perpendicular to the direction of propagation.
A plurality of different phase changes are applied to each of the at least one one-dimensional components, thereby obtaining at least one one-dimensional component of the plurality of phase changed
transformed wavefronts.
A plurality of intensity maps are employed to obtain an output indicating amplitude and phase of the at least one one-dimensional component of the wavefront being analyzed.
As seen in FIG. 10A, a plurality of different phase changes are applied to at least one one-dimensional component of a transformed wavefront. In the illustrated embodiment, typically five
one-dimensional components of a wavefront are shown and designated by reference numerals 1001, 1002, 1003, 1004 and 1005. The wavefront is transformed, preferably by a Fourier transform. It is thus
appreciated that due to transform of the wavefront, the five one-dimensional components 1001, 1002, 1003, 1004 and 1005 are transformed into five corresponding one-dimensional components of the
transformed wavefront, respectively designated by reference numerals 1006, 1007, 1008, 1009 and 1010.
Three phase changes, respectively designated 1011, 1012 & 1013 are each applied to the one-dimensional components 1006, 1007, 1008, 1009 and 1010 of transformed wavefront to produce three phase
changed transformed wavefronts, designated generally by reference numerals 1016, 1018 and 1020.
In the illustrated embodiment, phase changed transformed wavefront 1016 includes five one-dimensional components, respectively designated by reference numerals 1021, 1022, 1023, 1024 and 1025.
In the illustrated embodiment, phase changed transformed wavefront 1018 includes five one-dimensional components, respectively designated by reference numerals 1031, 1032, 1033, 1034 and 1035.
In the illustrated embodiment, phase changed transformed wavefront 1020 includes five one-dimensional components, respectively designated by reference numerals 1041, 1042, 1043, 1044 and 1045.
The phase changed transformed wavefronts 1016, 1018 and 1020 are detected, as by detector 158 (FIG. 1B), producing three intensity maps, designated generally by reference numerals 1046, 1048 and 1050
In the illustrated embodiment, intensity map 1046 includes five one-dimensional intensity map components, respectively designated by reference numerals 1051, 1052, 1053, 1054 and 1055.
In the illustrated embodiment, intensity map 1048 includes five one-dimensional intensity map components, respectively designated by reference numerals 1061, 1062, 1063, 1064 and 1065.
In the illustrated embodiment, intensity map 1050 includes five one-dimensional intensity map components, respectively designated by reference numerals 1071, 1072, 1073, 1074 and 1075.
The intensity maps 1046, 1048 and 1050 are stored as by circuitry 160 (FIG. 1B).
In accordance with an embodiment of the present invention, the wavefront being analyzed, illustrated in FIG. 10A by the one-dimensional components 1001, 1002, 1003, 1004 and 1005, may comprise a
plurality of different wavelength components and the plurality of different phase changes, 1011, 1012 and 1013, are applied to the plurality of different wavelength components of each of the
plurality of one-dimensional components of the wavefront being analyzed. Preferably, obtaining a plurality of intensity maps 1046, 1048 and 1050, includes dividing the plurality of one-dimensional
components of the plurality of phase changed transformed wavefronts 1016, 1018 and 1020 into separate wavelength components.
Preferably, dividing the plurality of one-dimensional components of the plurality of phase changed transformed wavefronts into separate wavelength components is achieved by passing the plurality of
phase changed transformed wavefronts 1016, 1018 and 1020 through a dispersion element.
Reference is now made to FIG. 10B, which is a simplified partially schematic, partially pictorial illustration of a wavefront analysis system suitable for carrying out the functionality of FIG. 10A
in accordance with a preferred embodiment of the present invention.
As seen in FIG. 10B, a wavefront, here designated by reference numeral 1080, and here including five one-dimensional components 1081, 1082, 1083, 1084 and 1085 is focused, as by a cylindrical lens
1086 onto a single axis displaceable phase manipulator 1087, which is preferably located at the focal plane of lens 1086. Lens 1086 preferably produces a one-dimensional Fourier transform of each of
the one-dimensional wavefront components 1081, 1082, 1083, 1084 and 1085 along the Y-axis.
As seen in FIG. 10B, the phase manipulator 1087 preferably comprises a multiple local phase delay element, such as a spatially non-uniform transparent object, typically including five different phase
delay regions, each arranged to apply a phase delay to one of the one-dimensional components at a given position of the object along an axis, here designated as the X-axis, extending perpendicularly
to the direction of propagation of the wavefront along a Z-axis and perpendicular to the axis of the transform produced by lens 1086, here designated as the Y-axis.
A second lens 1088, preferably a cylindrical lens, is arranged so as to image the one-dimensional components 1081, 1082, 1083, 1084 and 1085 onto a detector 1089, such as a CCD detector. Preferably
the second lens 1088 is arranged such that the detector 1089 lies in its focal plane. The output of detector 1089 is preferably supplied to data storage and processing circuitry 1090, which
preferably carries out functionality “C” described hereinabove with reference to FIG. 1A.
There is provided relative movement between the optical system comprising phase manipulator 1087, lenses 1086 and 1088 and detector 1089 and the one-dimensional wavefront components 1081, 1082, 1083,
1084 and 1085 along the X-axis. This relative movement sequentially matches different phase delay regions with different wavefront components, such that preferably each wavefront component passes
through each phase delay region of the phase manipulator 1087.
It is a particular feature of the embodiment of FIGS. 10A and 10B, that each of the one dimensional components of the wavefront is separately processed. Thus, in the context of FIG. 10B, it can be
seen that the five one-dimensional wavefront components 1081, 1082, 1083, 1084 and 1085 are each focused by a separate portion of the cylindrical lens 1086, are each imaged by a corresponding
separate portion of the cylindrical lens 1088 and each pass through a distinct region of the phase manipulator 1087. The images of each of the five one-dimensional wavefront components 1081, 1082,
1083, 1084 and 1085 at detector 1089 are thus seen to be separate and distinct images, as designated respectively by reference numerals 1091, 1092, 1093, 1094 and 1095. It is appreciated that these
images may appear on separate detectors together constituting detector 1089 instead of on a monolithic detector.
In accordance with an embodiment of the present invention, the transform applied to the wavefront includes an additional Fourier transform. This additional Fourier transform may be performed by lens
1086 or by an additional lens and is operative to minimize cross-talk between different one-dimensional components of the wavefront. In such a case, preferably a further transform is applied to the
phase changed transformed wavefront. This further transform may be performed by lens 1088 or by an additional lens.
Reference is now made to FIG. 11, which is a simplified functional block diagram illustration of the functionality of FIG. 1A, where an additional transform is applied following the application of
spatial phase changes. As seen in FIG. 11, and as explained hereinabove with reference to FIG. 1A, a wavefront 1100 is transformed, preferably by a Fourier transform and a plurality of phase changes
are applied to the transformed wavefront, thereby to provide a plurality of differently phased changed transformed wavefronts, represented by reference numerals 1120, 1122, and 1124.
The phase changed transformed wavefronts are subsequently transformed, preferably by a Fourier transform, and then detected, as by detector 158 (FIG. 1B), producing spatially varying intensity maps,
examples of which are designated by reference numerals 1130, 1132 and 1134. These intensity maps are subsequently stored as by circuitry 160 (FIG. 1B).
It is appreciated that any suitable number of differently phased changed transformed wavefronts can be obtained, and subsequently transformed to a corresponding plurality of intensity maps to be
stored for use in accordance with the present invention.
Reference in now made to FIG. 12, which is a simplified functional block diagram illustration of the functionality of FIG. 1A, wherein intensity maps are employed to provide information about a
wavefront being analyzed, such as indications of amplitude and phase of the wavefront. As seen in FIG. 12, and as explained hereinabove with reference to FIG. 1A, a wavefront 1200 is transformed,
preferably by a Fourier transform, and phase changed by a phase-change function to obtain several, preferably at least three, differently phase-changed transformed wavefronts, respectively designated
by reference numerals 1210, 1212 and 1214. The phase changed transformed wavefronts 1210, 1212 and 1214 are subsequently detected, as by detector 158 (FIG. 1B), producing spatially varying intensity
maps, examples of which are designated by reference numerals 1220, 1222 and 1224.
In parallel to producing the plurality of intensity maps, such as intensity maps 1220, 1222 and 1224, the expected intensity maps are expressed as a first function of the amplitude of wavefront 1200,
of the phase of wavefront 1200, and of the phase change function characterizing the differently phase changed transformed wavefronts 1210, 1212 and 1214, as indicated at reference numeral 1230.
In accordance with a preferred embodiment of the present invention, at least one of the phase and the amplitude of the wavefront is unknown or both the phase and the amplitude are unknown. The
phase-change function is known.
The first function of the phase and amplitude of the wavefront and of the phase change function is subsequently solved as indicated at reference numeral 1235, such as by means of a computer 136 (FIG.
1A), resulting in an expression of at least one and possibly both of the amplitude and phase of wavefront 1200 as a second function of the intensity maps 1220, 1222 and 1224, as indicated at
reference numeral 1240.
The second function is then processed together with the intensity maps 1220, 1222 and 1224 as indicated at reference numeral 1242. As part of this processing, detected intensity maps 1220, 1222 and
1224 are substituted into the second function. The processing may be carried out by means of a computer 136 (FIG. 1A) and provides information regarding wavefront 1200, such as indications of at
least one and possibly both of the amplitude and the phase of the wavefront.
In accordance with a further embodiment of the present invention, the plurality of intensity maps comprises at least four intensity maps. In such a case, employing the plurality of intensity maps to
obtain an indication of at least one of the phase and the amplitude of the wavefront 1200 includes employing a plurality of combinations, each of the combinations being a combination of at least
three of the plurality of intensity maps, to provide a plurality of indications of at least one of the phase and the amplitude of wavefront 1200. Preferably, this methodology also includes employing
the plurality of indications of at least one of the phase and the amplitude of the wavefront 1200 to provide an enhanced indication at least one of the phase and the amplitude of the wavefront 1200.
In accordance with a preferred embodiment of the present invention, at least some of the plurality of indications of the amplitude and phase are at least second order indications of the amplitude and
phase of the wavefront 1200.
In accordance with another embodiment of the present invention, the first function may be solved as a function of some unknowns to obtain the second function by expressing, as indicated by reference
numeral 1240, some unknowns, such as at least one of the amplitude and phase of wavefront 1200, as a second function of the intensity maps.
Accordingly, solving the first function may include:
□ defining a complex function of the amplitude of wavefront 1200, of the phase of wavefront 1200, and of the phase change function characterizing the differently phase changed transformed
wavefronts 1210, 1212 and 1214. This complex function is characterized in that intensity at each location in the plurality of intensity maps is a function predominantly of a value of the
complex function at that location and of the amplitude and the phase of wavefront 1200 at the same location;
□ expressing the complex function as a third function of the plurality of intensity maps 1220, 1222 and 1224; and
□ obtaining values for the unknowns, such as at least one of phase and amplitude of wavefront 1200, by employing the complex function expressed as a function of the plurality of intensity maps.
In accordance with this embodiment, preferably the complex function is a convolution of another complex function, which has an amplitude and phase identical to the amplitude and phase of wavefront
1200, and of a Fourier transform of the phase change function characterizing the differently phase changed transformed wavefronts 1210, 1212 and 1214.
Reference in now made to FIG. 13, which is a simplified functional block diagram illustration of part of the functionality of FIG. 1A, wherein the transform applied to the wavefront being analyzed is
a Fourier transform, wherein at least three different spatial phase changes are applied to the thus transformed wavefront, and wherein at least three intensity maps are employed to obtain indications
of at least one of the phase and the amplitude of the wavefront.
As explained hereinabove with reference to FIG. 1A, a wavefront 100 (FIG. 1A) being analyzed, is transformed and phase changed by at least three different spatial phase changes, all governed by a
spatial function, to obtain at least three differently phase-changed transformed wavefronts, represented by reference numerals 120, 122 and 124 (FIG. 1A) which are subsequently detected, as by
detector 158 (FIG. 1B), producing spatially varying intensity maps, examples of which are designated by reference numerals 130, 132 and 134 (FIG. 1A). As seen in FIG. 13, and designated as
sub-functionality “C” hereinabove with reference in FIG. 1A, the intensity maps are employed to obtain an output indication of at least one and possibly both of the phase and the amplitude of the
wavefront being analyzed.
Turning to FIG. 13, it is seen that the wavefront being analyzed is expressed as a first complex function ƒ(x)=A(x)[e] ^iφ(x), where ‘x’ is a general indication of a spatial location. The complex
function has an amplitude distribution A(x) and a phase distribution φ(x) identical to the amplitude and phase of the wavefront being analyzed. The first complex function ƒ(x)=A(x)[e] ^iφ(x) is
indicated by reference numeral 1300.
As noted hereinabove with reference to FIG. 1A, each of the plurality of different spatial phase changes is applied to the transformed wavefront preferably by applying a spatially uniform spatial
phase delay having a known value to a given spatial region of the transformed wavefront. As seen in FIG. 13, the spatial function governing these different phase changes is designated by ‘G’ and an
example of which, for a phase delay value of θ, is designated by reference numeral 1304.
Function ‘G’ is a spatial function of the phase change applied in each spatial location of the transformed wavefront. In the specific example designated by reference numeral 1304, the spatially
uniform spatial phase delay, having a value of θ, is applied to a spatially central region of the transformed wavefront, as indicated by the central part of the function having a value of θ, which is
greater than the value of the function elsewhere.
A plurality of expected intensity maps, indicated by spatial functions I[1](x), I[2](x) and I[3](x), are each expressed as a function of the first complex function ƒ(x) and of the spatial function G,
as indicated by reference numeral 1308.
Subsequently, a second complex function S(x), which has an absolute value |S(x)| and a phase α(x), is defined as a convolution of the first complex function ƒ(x) and of a Fourier transform of the
spatial function ‘G’. This second complex function, designated by reference numeral 1312, is indicated by the equation S(x)=ƒ(x)*ℑ(G)=|S(x)|[e] ^iα(x), where the symbol ‘*’ indicates convolution and
ℑ(G) is the Fourier transform of the function ‘G’.
The difference between φ(x), the phase of the wavefront, and α(x), the phase of the second complex function, is indicated by ψ(x), as designated by reference numeral 1316.
The expression of each of the expected intensity maps as a function of ƒ(x) and G, as indicated by reference numeral 1308, the definition of the absolute value and the phase of S(x), as indicated by
reference numeral 1312 and the definition of ψ(x), as indicated by reference numeral 1316, enables expression of each of the expected intensity maps as a third function of the amplitude of the
wavefront A(x), the absolute value of the second complex function |S(x)|, the difference between the phase of the wavefront and the phase of the second complex function ψ(x), and the known phase
delay produced by one of the at least three different phase changes which each correspond to one of the at least three intensity maps.
This third function is designated by reference numeral 1320 and includes three functions, each preferably having the general form
$I n ( x ) = | A ( x ) + ( ⅇ ⅈ θ n - 1 ) | S ( x ) | ⅇ - ⅈψ ( x ) | 2$
where I[n](x) are the expected intensity maps and n=1,2 or 3. In the three functions, θ[1], θ[2 ]and θ[3 ]are the known values of the uniform spatial phase delays, each applied to a spatial region of
the transformed wavefront, thus effecting the plurality of different spatial phase changes which produce the intensity maps I[1](x), I[2](x) and I[3](x), respectively.
It is appreciated that preferably the third function at any given spatial location x[0 ]is a function of A, ψ and |S| only at the same spatial location x[0].
The intensity maps are designated by reference numeral 1324.
The third function is solved for each of the specific spatial locations x[0], by solving at least three equations, relating to at least three intensity values I[1](x[0]), I[2](x[0]) and I[3](x[0]) at
at least three different phase delays θ[1], θ[2 ]and θ[3], thereby to obtain at least part of three unknowns A(x[0]), |S(x[0])| and ψ(x[0]). This process is typically repeated for all spatial
locations and results in obtaining the amplitude of the wavefront A(x), the absolute value of the second complex function |S(x)| and the difference between the phase of the wavefront and the phase of
the second complex function ψ(x), as indicated by reference numeral 1328.
Thereafter, once A(x), |S(x)| and ψ(x) are known, the equation defining the second complex function, represented by reference numeral 1312, is typically solved globally for a substantial number of
spatial locations ‘x’ to obtain a(x), the phase of the second complex function, as designated by reference numeral 1332.
Finally, the phase φ(x) of the wavefront being analyzed is obtained by adding the phase α(x) of the second complex function to the difference ψ(x) between the phase of the wavefront and the phase of
the second complex function, as indicated by reference numeral 1336.
In accordance with an embodiment of the present invention, the absolute value |S| of the second complex function is obtained preferably for every specific spatial location x[0 ]by approximating the
absolute value to a polynomial of a given degree in the spatial location x.
In accordance with another preferred embodiment of the present invention, the phase α(x) of the second complex function is obtained by expressing the second complex function S(x) as an eigen-value
problem, such as S=S·M where M is a matrix, and the complex function is an eigen-vector of the matrix obtained by an iterative process. An example of such an iterative process is S[0]=|S|, S[n+1]=S
[n]M/∥S[n]M∥, where n is the iterative step number.
In accordance with yet another preferred embodiment of the present invention, the phase α(x) of the second complex function is obtained by approximating the Fourier transform of the spatial function
‘G’, governing the spatial phase change, to a polynomial in the location x, by approximating the second complex function S(x) to a polynomial in the location x, and by solving, according to these
approximations, the equation defining the second complex function:
$S ( x ) = ( A ( x ) ⅇ ⅈψ ( x ) | S ( x ) | S ( x ) ) * ?? [ G ] ,$
where the function
$A ( x ) ⅇ ⅈψ ( x ) | S ( x ) |$
is known.
In accordance with still another preferred embodiment of the present invention, at any location x the amplitude A(x) of the wavefront being analyzed, the absolute value |S(x)| of the second complex
function, and the difference ψ(x) between the phase of the second complex function and the phase of the wavefront are obtained by a best-fit method, such as a least-square method, preferably a linear
least-square method, from the values of the intensity maps at this location I[n](x), where n=1,2, . . . ,N and N is the number of intensity maps. The accuracy of this process increases as the number
N of the plurality of intensity maps increases.
In accordance with one preferred embodiment of the present invention, the plurality of different phase changes comprises at least four different phase changes, the plurality of intensity maps
comprises at least four intensity maps, and the function designated by reference numeral 1320 can express each of the expected intensity maps as a third function of:
□ the amplitude of the wavefront A(x),
□ the absolute value of the second complex function |S(x)|;
□ the difference between the phase of the wavefront and the phase of the second complex function ψ(x);
□ the known phase delay produced by one of the at least four different phase changes each of which corresponds to one of the at least four intensity maps; and
□ at least one additional unknown relating to the wavefront analysis, where the number of the at least one additional unknown is no greater than the number by which the plurality intensity maps
exceeds three.
The third function 1320, is then solved by solving at least four equations, resulting from at least four intensity values at at least four different phase delays, thereby to obtain the amplitude of
the wavefront being analyzed, the absolute value of the second complex function, the difference between the phase of the wavefront and the phase of the second complex function and the at least one
additional unknown.
In accordance with another preferred embodiment of the present invention, the values of the uniform spatial phase delays θ[1], θ[2], . . . , θ[N ]applied to a spatial region of the transformed
wavefront, thus effecting the plurality of different spatial phase changes, producing the intensity maps I[1](x), I[2](x), . . . , I[N](x) respectively, are chosen as to maximize contrast in the
intensity maps and to minimize effects of noise on the phase of the wavefront being analyzed.
In accordance with one more preferred embodiment of the present invention, the function designated by reference numeral 1320, expressing each of the expected intensity maps as a third function of the
amplitude of the wavefront A(x), the absolute value of the second complex function |S(x)|, the difference between the phase of the wavefront and the phase of the second complex function ψ(x), and the
known phase delay θ[i ]produced by one of the at least three different phase changes which each correspond to one of the at least three intensity maps, comprises several functionalities:
□ defining fourth, fifth and sixth complex functions, designated as β[0](x), β[s](x) and β[c](x) respectively, none of which is a function of any of the plurality of intensity maps or of the
spatial function ‘G’ governing the phase change. Each of the fourth, fifth and sixth complex functions is preferably a function of the amplitude of the wavefront A(x), the absolute value of
the second complex function |S(x)|, the difference between the phase of the wavefront and the phase of the second complex function ψ(x); and
□ expressing each of the plurality of intensity maps I[n](x) as I[n](x)=β[0](x)+β[C](x)cos(θ[n])+β[S](x)sin(θ[n]), where θ[n ]is the value of the phase delay corresponding to intensity map I[n]
(x). Each intensity map I[n](x), where n=1,2, . . . N, preferably expressed as
$I n ( x ) = | A ( x ) + ( ⅇ ⅈ θ n - 1 ) | S ( x ) | ⅇ - ⅈψ ( x ) | 2 ,$
can be subsequently expressed as I[n](x)=β[0](x)+β[C](x)cos(θ[n])+β[S](x)sin(θ[n]), where
${ β 0 ( x ) = A ( x ) 2 + 2 | S ( x ) | 2 - 2 A ( x ) | S ( x ) | cos ( ψ ) β C ( x ) = 2 A ( x ) | S ( x ) | cos ( ψ ) - 2 | S ( x ) | 2 β S ( x ) = 2 A
( x ) | S ( x ) | sin ( ψ )$
Preferably the foregoing methodology also includes solving the third function 1320 by using a linear least-square method to compute from the different intensities I(θ[1]), . . . ,I(θ[N]), the values
of β[0], β[c ]and β[s ]best fitting to I(θ[n])=β[0]+β[c ]cos θ[n]+β[s ]sin θ[n]. Subsequently the amplitude A(x) is found by
$A ( x ) = β 0 ( x ) + β c ( x ) ,$
the absolute value |S(x)| of the second complex function is found by solving the second degree equation
$ S ( x ) 4 - β 0 ( x ) S ( x ) 2 + β C ( x ) 2 + β S ( x ) 2 4 = 0$
for |S(x)|^2, and ψ(x) is found by ψ(x)=arg(β[C](x)+2|S(x)|^2+iβ[S](x))
In accordance with yet another preferred embodiment of the present invention, solving of the third function, designated by reference numeral 1320, to obtain, as designated by reference numeral 1328,
the amplitude of the wavefront A(x), the absolute value of the second complex function |S(x)| and the difference between the phase of the wavefront and the phase of the second complex function ψ(x),
includes several functionalities:
□ obtaining two solutions for the absolute value |S(x)| of the second complex function, these two solutions, being designated by |S[h](x)| and |S[l](x)|, namely a higher value solution and a
lower value solution respectively; and
□ combining the two solutions into an enhanced absolute value solution |S(x)| for the absolute value of the second complex function, by choosing at each spatial location ‘x[0]’ either the
higher value solution |S[h](x[0])| or the lower value solution |S[l](x[0])| such that the enhanced absolute value solution satisfies the second complex function, designated by reference
numeral 1312.
Preferably the methodology also includes:
□ obtaining two solutions for each of the amplitude A(x) of the wavefront being analyzed and the difference ψ(x) between the phase of the wavefront and the phase of the second complex function,
these two solutions being higher value solutions A[h](x) and ψ[h](x) and lower value solutions A[l](x) and ψ[l](x); and
□ combining the two solutions A[h](x) and A[l](x) for the amplitude into an enhanced amplitude solution A(x) by choosing at each spatial location ‘x[0]’ either the higher value solution A[h](x
[0]) or the lower value solution A[l](x[0]) in a way that at each spatial location ‘x[0]’ if |S[h](x[0])| is chosen for the absolute value solution, then A[h](x[0]) is chosen for the
amplitude solution and at each location ‘x[1]’ if |S[l](x[1])| is chosen for the absolute value solution, then A[l](x[1]) is chosen for the amplitude solution; and
□ combining the two solutions ψ[h](X) and ψ[l](x) of the difference between the phase of the wavefront and the phase of the second complex function into an enhanced difference solution ψ(x), by
choosing at each spatial location ‘x[0]’ either the higher value solution ψ[h](x[0]) or the lower value solution ψ[l](x[0]) in a way that at each spatial location ‘x[0]’ if |S[h](x[0])| is
chosen for the absolute value solution, then ψ[h](x[0]) is chosen for the difference solution and at each location ‘x[1]’ if |S[l](x[1])| is chosen for the absolute value solution, then ψ[l]
(x[1]) is chosen for the difference solution.
Additionally, in accordance with an embodiment of the present invention, the plurality of different phase changes applied to the transformed wavefront, thereby to obtain a plurality of differently
phase changed transformed wavefronts, also include amplitude changes, resulting in a plurality of differently phase and amplitude changed transformed wavefronts. These amplitude changes are
preferably known amplitude attenuations applied to the same spatial region of the transformed wavefront to which the uniform phase delays θ[1], θ[2], . . . , θ[N], are applied, the spatial region
being defined by the spatial function ‘G’.
The amplitude attenuations are designated by σ[1], σ[2], . . . , σ[N], where the n-th change, where n=1,2, . . . N, applied to the transformed wavefront includes a phase change θ[n ]and an amplitude
attenuation σ[n]. It is appreciated that some of the phase changes may be equal to zero, indicating no phase-change and that some of the amplitude attenuations may be equal to unity, indicating no
amplitude attenuation.
In this embodiment, the function designated by reference numeral 1320, expressing each of the expected intensity maps I[n](x) as a third function of the amplitude of the wavefront A(x), the absolute
value of the second complex function |S(x)|, the difference between the phase of the wavefront and the phase of the second complex function ψ(x), and the phase delay θ[n], also expresses each of the
expected intensity maps also as a function of the amplitude attenuation σ[n ]and comprises several functionalities:
□ defining fourth, fifth, sixth and seventh complex functions, designated by β[0](x), β[1](x), β[2](x) and β[3](x) respectively, none of which is a function of any of the plurality of intensity
maps or of the spatial function ‘G’ governing the phase and amplitude changes. Each of the fourth, fifth, sixth and seventh complex functions is preferably a function of the amplitude of the
wavefront A(x), the absolute value of the second complex function |S(x)|, the difference between the phase of the wavefront and the phase of the second complex function ψ(x);
□ defining an eighth function, designated μ, as a combination of the phase delay and of the amplitude attenuation, where for the n-th change applied to the transformed wavefront, including a
phase change θ[n ]and an amplitude attenuation σ[n], this eighth function is designated by μ[n]. Preferably the combination μ[n ]is defined by μ[n]=σ[n]e^iθ ^ n −1; and
□ expressing each of the plurality of intensity maps I[n](x) as
$I n ( x ) = β 0 ( x ) + β 1 ( x ) μ n 2 + β 2 ( x ) μ n + β 3 ( x ) μ _ n , where β 0 ( x ) = A 2 ( x ) ; β 1 ( x ) = S ( x ) 2 ; β 2 ( x ) = A ( x )
S ( x ) ⅇ - ⅈψ ( x ) and β 3 ( x ) = A ( x ) S ( x ) ⅇ ⅈψ ( x ) .$
Preferably the foregoing methodology also includes solving the third function 1320 by computing from the different intensities I[n](x), the values of β[0](x), β[1](x), β[2](x) and β[3](x) best
fitting to the equation I[n](x)=β[0](x)+β[1](x)|μ[n]|^2+β[2](x)μ[n]+β[3](x) μ [n]. Subsequently the amplitude A(x) is found by
$A ( x ) = β 0 ( x ) ,$
the absolute value |S(x)| of the second complex function is found by
$ S ( x ) = β 1 ( x )$
and ψ(x) is found by solving e^iψ(x)=angle(β[3](x)).
It is appreciated that the amplitude attenuations σ[1], σ[2], . . . , σ[N], may be unknown. In such a case, additional intensity maps are obtained, where the number of the unknowns is no greater than
the number by which the plurality of intensity maps exceeds three. The unknowns are obtained in a manner similar to that described hereinabove, where there exists at least one unknown relating to the
wavefront analysis.
Reference is now made to FIG. 14, which is a simplified partially schematic, partially pictorial illustration of part of one preferred embodiment of a wavefront analysis system of the type shown in
FIG. 1B. As seen in FIG. 14, a wavefront, here designated by reference numeral 1400 is partially transmitted through a beam splitter 1402 and subsequently focused, as by a lens 1404 onto a phase
manipulator 1406, which is preferably located at the focal plane of lens 1404. The phase manipulator 1406 may be, for example, a spatial light modulator or a series of different transparent,
spatially non-uniform objects.
A reflecting surface 1408 is arranged so as to reflect wavefront 1400 after it passes through the phase manipulator 1406. The reflected wavefront is imaged by lens 1404 onto a detector 1410, such as
a CCD detector via beam splitter 1402. Preferably the beam splitter 1402 and the detector 1410 are arranged such that the detector 1410 lies in the focal plane of lens 1404. The output of detector
1410 is preferably supplied to data storage and processing circuitry 1412, which preferably carries out functionality “C” described hereinabove with reference to FIG. 1A.
It is appreciated that adding the reflecting surface 1408 to an imaging system, doubles the phase delay generated by phase manipulator 1406, enables imaging with a single lens 1404, and generally
enables realization of a more compact system.
Reference is now made to FIG. 15, which is a simplified partially schematic, partially pictorial illustration of a system for surface mapping employing the functionality and structure of FIGS. 1A and
1B. As seen in FIG. 15, a beam of radiation, such as light or acoustic energy, is supplied from a radiation source 1500, optionally via a beam expander 1502, onto a beam splitter 1504, which reflects
at least part of the radiation onto a surface 1506 to be inspected. The radiation reflected from the inspected surface 1506, is a surface mapping wavefront, which has an amplitude and a phase, and
which contains information about the surface 1506. At least part of the radiation incident on surface 1506 is reflected from the surface 1506 and transmitted via the beam splitter 1504 and focused
via a focusing lens 1508 onto a phase manipulator 1510, which is preferably located at the image plane of radiation source 1500.
The phase manipulator 1510 may be, for example, a spatial light modulator or a series of different transparent, spatially non-uniform objects. It is appreciated that phase manipulator 1510 can be
configured such that a substantial part of the radiation focused thereonto is reflected therefrom. Alternatively the phase manipulator 1510 can be configured such that a substantial part of the
radiation focused thereonto is transmitted therethrough.
A second lens 1512 is arranged so as to image surface 1506 onto a detector 1514, such as a CCD detector. Preferably the second lens 1512 is arranged such that the detector 1514 lies in its focal
plane. The output of detector 1514, an example of which is a set of intensity maps designated by reference numeral 1515, is preferably supplied to data storage and processing circuitry 1516, which
preferably carries out functionality “C” described hereinabove with reference to FIG. 1A, providing an output indicating at least one and possibly both of the phase and the amplitude of the surface
mapping wavefront. This output is preferably further processed to obtain information about the surface 1506, such as geometrical variations and reflectivity of the surface.
In accordance with a preferred embodiment of the present invention, the beam of radiation supplied from radiation source 1500 has a narrow wavelength band about a given central wavelength, causing
the phase of the radiation reflected from surface 1506 to be proportional to geometrical variations in the surface 1506, the proportion being an inverse linear function of the central wavelength of
the radiation.
In accordance with another preferred embodiment of the present invention, the beam of radiation supplied from radiation source 1500 has at least two narrow wavelength bands, each centered about a
different wavelength, designated λ[1], . . . , λ[n]. In such a case, the radiation reflected from the surface 1506 has at least two wavelength components, each centered around a wavelength λ[1], . .
. , λ[n ]and at least two indications of the phase of the surface mapping wavefront are obtained. Each such indication corresponds to a different wavelength component of the reflected radiation.
These at least two indications may be subsequently combined to enable enhanced mapping of the surface 1506, by avoiding ambiguity in the mapping, known as 2π ambiguity, when the value of the mapping
at a given spatial location in the surface exceeds the value of the mapping at a different spatial location in the surface by the largest of the different wavelengths λ[1], . . . , λ[n]. A proper
choice, of the wavelengths λ[1], . . . , λ[n], may lead to elimination of this ambiguity when the difference in values of the mapping at different locations is smaller than the multiplication product
of all the wavelengths.
In accordance with still another preferred embodiment of the present invention, the phase manipulator 1510 applies a plurality of different spatial phase changes to the radiation wavefront reflected
from surface 1506 and Fourier transformed by lens 1508. Application of the plurality of different spatial phase changes provides a plurality of differently phase changed transformed wavefronts which
may be subsequently detected by detector 1514.
In accordance with yet another preferred embodiment of the present invention, at least three different spatial phase changes are applied by phase manipulator 1510, resulting in at least three
different intensity maps 1515. The at least three intensity maps are employed by the data storage and processing circuitry 1516 to obtain an output indicating at least the phase of the surface
mapping wavefront. In such a case, the data storage and processing circuitry 1516, carries out functionality “C” described hereinabove with reference to FIG. 1A, preferably in a manner described
hereinabove with reference to FIG. 13, where the wavefront being analyzed (FIG. 13) is the surface mapping wavefront.
Additionally, in accordance with a preferred embodiment of the present invention, the beam of radiation supplied from radiation source 1500 comprises a plurality of different wavelength components,
thereby providing a plurality of wavelength components in the surface mapping wavefront and subsequently in the transformed wavefront impinging on phase manipulator 1510. In this case the phase
manipulator may be an object, at least one of whose thickness, refractive index and surface geometry varies spatially. This spatial variance of the phase manipulator generates a different spatial
phase change for each of the wavelength components, thereby providing a plurality of differently phase changed transformed wavefronts to be subsequently detected by detector 1514.
Reference is now made to FIG. 16, which is a simplified partially schematic, partially pictorial illustration of a system for object inspection employing the functionality and structure of FIGS. 1A
and 1B. As seen in FIG. 16, a beam of radiation, such as light or acoustic energy, is supplied from a radiation source 1600, optionally via a beam expander, onto at least partially transparent object
to be inspected 1602. The radiation transmitted through the inspected object 1602, is an object inspection wavefront, which has an amplitude and a phase, and which contains information about the
object 1602. At least part of the radiation transmitted through object 1602 is focused via a focusing lens 1604 onto a phase manipulator 1606, which is preferably located at the image plane of
radiation source 1600.
The phase manipulator 1606 may be, for example, a spatial light modulator or a series of different transparent, spatially non-uniform objects. It is appreciated that phase manipulator 1606 can be
configured such that a substantial part of the radiation focused thereonto is reflected therefrom. Alternatively the phase manipulator 1606 can be configured such that a substantial part of the
radiation focused thereonto is transmitted therethrough.
A second lens 1608 is arranged so as to image object 1602 onto a detector 1610, such as a CCD detector. Preferably, the second lens 1608 is arranged such that the detector 1610 lies in its focal
plane. The output of detector 1610, an example of which is a set of intensity maps designated by reference numeral 1612, is preferably supplied to data storage and processing circuitry 1614, which
preferably carries out functionality “C” described hereinabove with reference to FIG. 1A, providing an output indicating at least one and possibly both of the phase and the amplitude of the object
inspection wavefront. This output is preferably further processed to obtain information about the object 1602, such as a mapping of the object's thickness, refractive index or transmission.
In accordance with one preferred embodiment of the present invention, the beam of radiation supplied from radiation source 1600 has a narrow wavelength band about a given central wavelength, and the
object 1602 is substantially uniform in material and other optical properties, causing the phase of the radiation transmitted through object 1602 to be proportional to thickness of the object 1602.
In accordance with one more preferred embodiment of the present invention, the beam of radiation supplied from radiation source 1600 has a narrow wavelength band about a given central wavelength, and
the object 1602 is substantially uniform in thickness, causing the phase of the radiation transmitted through object 1602 to be proportional to optical properties, such as refraction index or
density, of the object 1602. It is appreciated that object 1602 may be any optical conduction element, such as an optical fiber.
In accordance with another preferred embodiment of the present invention, the beam of radiation supplied from radiation source 1600 has at least two narrow wavelength bands, each centered about a
different wavelength, designated λ[1], . . . , λ[n]. In such a case, the radiation transmitted through object 1602 has at least two wavelength components, each centered around a wavelength λ[1], . .
. , λ[n ]and at least two indications of the phase of the object inspection wavefront are obtained. Each such indication corresponds to a different wavelength component of the transmitted radiation.
These at least two indications may be subsequently combined to enable enhanced mapping of the properties, such as thickness, of object 1602, by avoiding ambiguity in the mapping, known as 2π
ambiguity, when the value of the mapping at a given spatial location in the object exceeds the value of the mapping at a different spatial location in the object by the largest of the different
wavelengths λ[1], . . . , λ[n]. A proper choice of the wavelengths λ[1], . . . , λ[n], may lead to elimination of this ambiguity when the difference in values of the mapping at different locations is
smaller than the multiplication product of all the wavelengths.
In accordance with still another preferred embodiment of the present invention, the phase manipulator 1606 applies a plurality of different spatial phase changes to the radiation wavefront
transmitted through object 1602 and Fourier transformed by lens 1604. Application of the plurality of different spatial phase changes produces a plurality of differently phase changed transformed
wavefronts which may be subsequently detected by detector 1610.
In accordance with yet another preferred embodiment of the present invention, at least three different spatial phase changes are applied by phase manipulator 1606, resulting in at least three
different intensity maps 1612. The at least three intensity maps 1612 are employed by the data storage and processing circuitry 1614 to obtain an output indicating at least the phase of the object
inspection wavefront. In such a case, the data storage and processing circuitry 1614, carries out functionality “C” described hereinabove with reference to FIG. 1A, preferably in a manner described
hereinabove with reference to FIG. 13, where the wavefront being analyzed (FIG. 13) is the object inspection wavefront.
Additionally, in accordance with a preferred embodiment of the present invention, the beam of radiation supplied from radiation source 1600 comprises a plurality of different wavelength components,
thereby providing a plurality of wavelength components in the object inspection wavefront and subsequently in the transformed wavefront impinging on phase manipulator 1606. In this case the phase
manipulator 1606 may be an object, at least one of whose thickness, refractive index and surface geometry varies spatially. This spatial variance of the phase manipulator generates a different
spatial phase change for each of the wavelength components, thereby providing a plurality of differently phase changed transformed wavefronts to be subsequently detected by detector 1610.
Reference is now made to FIG. 17, which is a simplified partially schematic, partially pictorial illustration of a system for spectral analysis employing the functionality and structure of FIGS. 1A
and 1B. As seen in FIG. 17, a beam of radiation, such as light or acoustic energy, is supplied from a radiation source to be tested 1700, optionally via a beam expander, onto a known element 1702,
such as an Etalon or a plurality of Etalons. Element 1702 is intended to generate an input wavefront, having at least varying phase or intensity. The radiation transmitted through the element 1702,
is a spectral analysis wavefront, which has an amplitude and a phase, and which contains information about the spectrum of the radiation source 1700. At least part of the radiation transmitted
through element 1702 is focused via a focusing lens 1704 onto a phase manipulator 1706, which is preferably located at the image plane of radiation source 1700.
The phase manipulator 1706 may be, for example, a spatial light modulator or a series of different transparent, spatially non-uniform objects. It is appreciated that phase manipulator 1706 can be
configured such that a substantial part of the radiation focused thereonto is reflected therefrom. Alternatively the phase manipulator 1706 can be configured such that a substantial part of the
radiation focused thereonto is transmitted therethrough.
A second lens 1708 is arranged so as to image element 1702 onto a detector 1710, such as a CCD detector. Preferably, the second lens 1708 is arranged such that the detector 1710 lies in its focal
plane. The output of detector 1710, an example of which is a set of intensity maps designated by reference numeral 1712, is preferably supplied to data storage and processing circuitry 1714, which
preferably carries out functionality “C” described hereinabove with reference to FIG. 1A, providing an output indicating at least one and possibly both of the phase and the amplitude of the spectral
analysis wavefront. This output is preferably further processed to obtain information about the radiation source 1700, such as the spectrum of the radiation supplied from radiation source 1700.
In accordance with a preferred embodiment of the present invention, the spectral analysis wavefront is obtained by reflecting the radiation supplied from radiation source 1700 from element 1702.
In accordance with another preferred embodiment of the present invention, the spectral analysis wavefront is obtained by transmitting the radiation supplied from radiation source 1700 through element
In accordance with one more preferred embodiment of the present invention, the beam of radiation supplied from radiation source 1700 has a narrow wavelength band about a central wavelength, causing
the phase of the radiation impinged on the object 1702 to be inversely proportional to the central wavelength supplied from radiation source 1700 and related to at least one of a surface
characteristic and thickness of element 1702.
In accordance with another preferred embodiment of the present invention, the plurality of intensity maps 1712 are employed by the data storage and processing circuitry 1714, to obtain an output
indicating at least one and possibly both of the phase and amplitude of the spectral analysis wavefront by expressing the plurality of intensity maps as at least one mathematical function of phase
and amplitude of the spectral analysis wavefront and of plurality of different phase changes applied by phase manipulator 1706, wherein at least one and possibly both of the phase and amplitude is
unknown and a function generating the different phase changes is known. This at least one mathematical function is subsequently employed to obtain an output indicating at least the phase of the
spectral analysis wavefront.
In accordance with still another preferred embodiment of the present invention, the phase manipulator 1706 applies a plurality of different spatial phase changes to the radiation wavefront
transmitted through element 1702 and Fourier transformed by lens 1704. Application of the plurality of different spatial phase changes produces a plurality of differently phase changed transformed
wavefronts which may be subsequently detected by detector 1710.
In accordance with yet another preferred embodiment of the present invention, at least three different spatial phase changes are applied by phase manipulator 1706, resulting in at least three
different intensity maps 1712. The at least three intensity maps are employed by the data storage and processing circuitry 1714 to obtain an output indicating at least the phase of the spectral
analysis wavefront. In such a case, the data storage and processing circuitry 1714, carries out functionality “C” described hereinabove with reference to FIG. 1A, preferably in a manner described
hereinabove with reference to FIG. 13, where the wavefront being analyzed (FIG. 13) is the spectral analysis wavefront.
Additionally, in accordance with a preferred embodiment of the present invention, the beam of radiation supplied from radiation source 1700 comprises a plurality of different wavelength components,
thereby providing a plurality of wavelength components in the spectral analysis wavefront and subsequently in the transformed wavefront impinging on phase manipulator 1706. In this case the phase
manipulator may be an object, at least one of whose thickness, refractive index and surface geometry varies spatially. This spatial variance of the phase manipulator generates a different spatial
phase change for each of the wavelength components, thereby providing a plurality of differently phase changed transformed wavefronts to be subsequently detected by detector 1710.
In accordance with an embodiment of the present invention, the phase manipulator 1706 comprises a plurality of objects, each characterized in that at least one of its thickness and refractive index
varies spatially. The spatial variance of the thickness or of the refractive index of the plurality of objects may be selected in a way such that the phase changes applied by phase manipulator 1706
differ to a selected predetermined extent for at least some of the wavelength components supplied by radiation source 1700.
A specific selection of the objects is such that the phase change applied to an expected wavelength of radiation source differs substantially from the phase change applied to an actual wavelength of
the radiation source. Alternatively, the spatial variance of the thickness or refractive index of the plurality of objects may be selected in a way such that the phase changes applied by phase
manipulator 1706 are identical for at least some of the plurality of different wavelength components wavelength components supplied by radiation source 1700.
In accordance with another embodiment of the present invention, the known element 1702 comprises a plurality of objects, each characterized in that at least one of its thickness and refractive index
varies spatially. The spatial variance of the thickness or of the refractive index of the plurality of objects may be selected in a way such that the wavelength components of the input wavefront,
generated by passing the wavelength components of the radiation supplied by radiation source 1700 through the element 1702, differ to a selected predetermined extent for at least some of the
wavelength components supplied by radiation source 1700.
A specific selection of the objects is such that the wavelength component of the input wavefront generated by an expected wavelength of radiation source differs substantially from the wavelength
component of the input wavefront generated by an actual wavelength of the radiation source. Alternatively, the spatial variance of the thickness or refractive index of the plurality of objects may be
selected in a way such that the wavelength components of the input wavefront, generated by passing the wavelength components of the radiation supplied by radiation source 1700 through the element
1702, are identical for at least some of the wavelength components supplied by radiation source 1700.
Reference is now made to FIG. 18, which is a simplified partially schematic, partially pictorial illustration of a system for phase-change analysis employing the functionality and structure of FIGS.
1A and 1B. As seen in FIG. 18, a known wavefront 1800, which is a phase change analysis wavefront, having an amplitude and a phase, is focused via a focusing lens 1802, preferably performing a
Fourier transform to wavefront 1800, onto a phase manipulator 1804, which is preferably located at the focal plane of lens 1802. The phase manipulator applies a plurality of different phase changes
to the transformed phase change analysis wavefront.
The phase manipulator 1804 may be, for example, a spatial light modulator or a series of different transparent, spatially non-uniform objects. It is appreciated that phase manipulator 1804 can be
configured such that a substantial part of the radiation focused thereonto is reflected therefrom. Alternatively the phase manipulator 1804 can be configured such that a substantial part of the
radiation focused thereonto is transmitted therethrough.
A second lens 1806 is arranged so as to image wavefront 1800 onto a detector 1808, such as a CCD detector. Preferably, the second lens 1806 is arranged such that the detector 1808 lies in its focal
plane. The output of detector 1808, an example of which is a set of intensity maps designated by reference numeral 1810, is preferably supplied to data storage and processing circuitry 1812, which
employs the plurality of intensity maps to obtain an output indication of differences between the plurality of different phase changes applied by the phase manipulator 1804.
In accordance with one preferred embodiment of the present invention, lateral shifts appear in the plurality of different phase changes. These may be produced, for example, by vibrations of the phase
manipulator or by impurities in the phase manipulator. Consequently, corresponding changes appear in the plurality of intensity maps 1810, and result in obtaining an indication of these lateral
In accordance with another preferred embodiment of the present invention, the plurality of intensity maps 1810 are employed by the data storage and processing circuitry 1812 to obtain an output
indicating the differences between the plurality of different phase changes applied by the phase manipulator 1804, by expressing the plurality of intensity maps as at least one mathematical function
of phase and amplitude of the phase change analysis wavefront and of the plurality of different phase changes applied by phase manipulator 1804, where at least the phase and amplitude of the
wavefront 1800 are known and the plurality of different phase changes are unknown. This at least one mathematical function is subsequently employed to obtain an output indicating at least the
differences between the plurality of different phase changes.
In accordance with still another preferred embodiment of the present invention, the phase manipulator 1804 applies a plurality of different spatial phase changes to the wavefront 1800 Fourier
transformed by lens 1802. Application of the plurality of different spatial phase changes provides a plurality of differently phase changed transformed wavefronts which may be subsequently detected
by detector 1808.
In accordance with yet another preferred embodiment of the present invention, at least three different spatial phase changes are applied by phase manipulator 1804, resulting in at least three
different intensity maps 1810. The at least three intensity maps are employed by the data storage and processing circuitry 1812 to obtain an output indicating at least the differences between the
plurality of different phase changes. In such a case, the data storage and processing circuitry 1814, carries out functionality “C” described hereinabove with reference to FIG. 1A, preferably in a
manner similar to the manner described hereinabove with reference to FIG. 13, where the wavefront being analyzed (FIG. 13) is the known phase change analysis wavefront, and the spatial phase changes
applied by phase manipulator 1804 are unknown.
Additionally, in accordance with a preferred embodiment of the present invention, the wavefront 1800 comprises a plurality of different wavelength components, thereby providing a plurality of
wavelength components in the transformed wavefront impinging on phase manipulator 1804. In this case the phase manipulator may be an object, at least one of whose thickness, refractive index and
surface geometry varies spatially. This spatial variance of the phase manipulator generates a different spatial phase change for each of the wavelength components, thereby providing a plurality of
differently phase changed transformed wavefronts to be subsequently detected by detector 1808.
Additionally, in accordance with another embodiment of the present invention, phase manipulator 1804 applies one phase change to the radiation focused onto each spatial location thereon, resulting in
one intensity map 1810, as an output of detector 1808. In such a case, the data storage and processing circuitry 1812 employs the intensity map and the known wavefront 1800 to obtain at least an
output indicating the phase change applied by phase manipulator 1804.
In accordance with the foregoing methodology, the phase change applied by the phase manipulator may be a phase delay, having a value selected from one of a plurality of pre-determined values,
including a possible value of zero phase delay and the output indication of the phase change obtained by data storage and processing circuitry 1812 is the value of the phase delay. In such a case,
the phase manipulator may be media which stores information by different values of the phase delays at each of a multiplicity of different locations thereon, where the value of the phase delay
constitutes the stored information. The stored information, encoded in the different values of the phase delays, is retrieved by data storage and processing circuitry 1812. It is appreciated that in
such a case, wavefront 1800 may also comprise a plurality of different, wavelength components, resulting in a plurality of intensity maps and consequently in an increase of the information encoded on
the phase manipulator at each of the multiplicity of different locations.
Reference is now made to FIG. 19, which is a simplified partially schematic, partially pictorial illustration of a system for stored data retrieval employing the functionality and structure of FIGS.
1A and 1B. As seen in FIG. 19, optical storage media 1900, such as a DVD or compact disk, has information encoded thereon by selecting the height of the media at each of a multiplicity of different
locations thereon, as shown in enlargement and designated by reference numeral 1902. At each location on the media, the height of the media can be one of several given heights or levels. The specific
level of the media at that location determines the information stored at that location.
A beam of radiation, such as light or acoustic energy, is supplied from a radiation source 1904, such as a laser or a LED, optionally via a beam expander, onto a beam splitter 1906, which reflects at
least part of the radiation onto the surface of the media 1900. The radiation reflected from an area 1908 on the media, onto which the radiation impinges, is a stored data retrieval wavefront, which
has an amplitude and a phase, and which contains information stored in area 1908. At least part of the radiation incident on area 1908 is reflected from the area 1908 and transmitted via the beam
splitter 1906 onto an imaging system 1910, which may include a phase manipulator or other device which generates a varying phase function.
Imaging system 1910 preferably carries out functionalities “A” and “B” described hereinabove with reference to FIG. 1A, obtaining a plurality of differently phase changed transformed wavefronts
corresponding to the stored data retrieval wavefront and obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts.
Preferably, imaging system 1910 comprises a first lens 1508 (FIG. 15), a phase manipulator 1510 (FIG. 15), a second lens 1512 (FIG. 15) and a detector 1514 (FIG. 15). The outputs of imaging system
1910 are supplied to data storage and processing circuitry 1912, which preferably carries out functionality “C” described hereinabove with reference to FIG. 1A, providing an output indicating at
least one and possibly both of the phase and amplitude of the stored data retrieval wavefront. This output is preferably further processed to read out the information encoded in area 1908 of media
1900 and displayed on display unit 1914.
In accordance with a preferred embodiment of the present invention, the beam of radiation supplied from radiation source 1904 has a narrow wavelength band about a given central wavelength, causing
the phase of the radiation reflected from media 1900 to be proportional to geometrical variations in the media 1900, which contain the encoded information, the proportion being an inverse linear
function of the central wavelength of the radiation.
In accordance with another preferred embodiment of the present invention, the beam of radiation supplied from radiation source 1904 has at least two narrow wavelength bands, each centered about a
different wavelength, designated λ[1], . . . , λ[n]. In such a case, the radiation reflected from area 1908 in media 1900 has at least two wavelength components, each centered around a wavelength λ
[1], . . . , λ[n].
At least two indications of the phase of the stored data retrieval wavefront are obtained, each such indication corresponding to a different wavelength component of the reflected radiation. These at
least two indications may be subsequently combined to enhance mapping of the surface of area 1908 of media 1900 and therefore enhance retrieval of the information, by avoiding an ambiguity in the
mapping, known as 2π ambiguity, when the value of the height of the media at a given location exceeds the largest of the different wavelengths λ[1], . . . , λ[n].
In such a case, the range of possible heights at each location in area 1908 can exceed the value of the largest of the different wavelengths, without ambiguity in the reading of the heights. This
extended dynamic range enables storing more information on media 1900 than would otherwise be possible. A proper choice of the wavelengths λ[1], . . . , λ[n], may lead to elimination of this
ambiguity when the difference of heights of the media in area 1908 at different locations is smaller than the multiplication product of all the wavelengths.
In accordance with still another preferred embodiment of the present invention, a phase manipulator incorporated in imaging system 1910 applies a plurality of different spatial phase changes to the
radiation wavefront reflected from media 1900 and Fourier transformed by a lens, also incorporated in imaging system 1910. Application of the plurality of different spatial phase changes provides a
plurality of differently phase changed transformed wavefronts which may be subsequently detected by a detector incorporated in imaging system 1910.
In accordance with yet another preferred embodiment of the present invention, at least three different spatial phase changes are applied by a phase manipulator incorporated in imaging system 1910,
resulting in an output from imaging system 1910 of at least three different intensity maps. The at least three intensity maps are employed by the data storage and processing circuitry 1912 to obtain
an output indicating at least the phase of the stored data retrieval wavefront. In such a case, the data storage and processing circuitry 1912, carries out functionality “C” described hereinabove
with reference to FIG. 1A, preferably in a manner described hereinabove with reference to FIG. 13, where the wavefront being analyzed (FIG. 13) is the stored data retrieval wavefront.
Additionally, in accordance with an embodiment of the present invention, the beam of radiation supplied from radiation source 1904 comprises a plurality of different wavelength components, thereby
providing a plurality of wavelength components in the stored data retrieval wavefront and subsequently in the transformed wavefront impinging on a phase manipulator incorporated into imaging system
1910. In this case the phase manipulator may be an object, at least one of whose thickness, refractive index and surface geometry varies spatially. This spatial variance of the phase manipulator
generates a different spatial phase change for each of the wavelength components, thereby providing a plurality of differently phase changed transformed wavefronts to be subsequently detected by a
detector incorporated in imaging system 1910.
In accordance with another embodiment of the present invention, information is encoded on media 1900 by selecting the height of the media at each given location to be such that the intensity value of
the intensity map resulting from light reflected from the location and passing through imaging system 1910 lies within a predetermined range of values. This range corresponds to an element of the
information stored at the location. By employing the plurality of intensity maps, multiple intensity values are realized for each location, each intensity value lying within a specific range of
values. The resulting plurality of ranges of intensity values provide multiple elements of information for each location on the media 1900.
It is appreciated that in such a case, retrieving the information stored at area 1908 on the media from the outputs of imaging system 1910 may be performed by data storage and processing circuitry
1912 in a straight-forward manner, as by mapping each of the resulting intensity values at each location to their corresponding ranges, and subsequently to the information stored at the location.
Preferably, the foregoing methodology also includes use of a phase manipulator incorporated in imaging system 1910, that applies an at least time-varying phase change function to the transformed data
retrieval wavefront impinging thereon. This time-varying phase change function provides the plurality of intensity maps.
Alternatively or additionally, the beam of radiation supplied from radiation source 1904 comprises a plurality of different wavelength components, thereby providing a plurality of wavelength
components in the stored data retrieval wavefront. The plurality of differently phase changed transformed wavefronts are obtained in imaging system 1910 by applying at least one phase change to the
plurality of different wavelength components of the stored data retrieval wavefront. The phase changed transformed stored data retrieval wavefront can be detected by a single detector or
alternatively separated, as by a dispersion element, into its separate plurality of different wavelength components, each component being detected by a different detector.
In accordance with yet another embodiment of the present invention, media 1900 may have different reflectivity coefficients for the radiation supplied from light source 1904 at each of a multiplicity
of different locations on the media. At each location on the media, a different percentage of the radiation may be reflected. The percentage may have one of several given values, where the specific
value may at least partially determine the information stored at that location.
In such a case, the information encoded on media 1900 may be encoded by selecting the height of the media at each of a multiplicity of different locations on the media and by selecting the
reflectivity of the media at each of a multiplicity of different locations on the media. In such a case, more information can be stored at each location on the media, than could otherwise be stored.
Furthermore, in such a case, employing an indication of the amplitude and phase of the stored data retrieval wavefront to obtain the encoded information includes employing the indication of the phase
to obtain the information encoded by selecting the height of the media and employing the indication of the amplitude to obtain said information encoded by selecting the reflectivity.
In accordance with still another embodiment of the present invention, the information is encoded onto media 1900 at several layers in the media. The information is encoded on the media by selecting
the height of the media at each of multiplicity of different locations on each layer of the media. Each of these layers, placed one on top of the other in media 1900, is partially reflecting and
partially transmitting. The beam of radiation from source 1904 impinging onto media 1900 is partially reflected from the top, first layer of the media, and partially transmitted to the layers lying
therebelow. The energy transmitted by the second layer is partially reflected and partially transmitted to the layers lying therebelow, and so on, until the radiation transmitted through all the
layers is partially reflected from the undermost layer. In such a case, radiation source 1904 preferably includes a focusing system that focuses the radiation onto each one of the layers of media
1900 in order to retrieve the information stored on that layer. Alternatively or additionally, imaging system 1910 may include confocal microscopy elements operative to differentiate between the
different layers.
It is appreciated that area 1908 of media 1900 may be a relatively small area, comprising a single location on which information is encoded and possibly also neighboring locations. In such a case,
the detector incorporated in imaging system 1910 may define only a single or several detection pixels. Additionally, the output indicating at least one and possibly both of the phase and amplitude of
the stored data retrieval wavefront provided by circuitry 1912, includes at least one and possibly both of the height of the media and the reflectivity of the media at the location or locations
covered by area 1908.
In accordance with yet another embodiment of the present invention, the stored data retrieval wavefront comprises at least one one-dimensional component, corresponding to at least one one-dimensional
part of area 1908 on media 1900. In such a case, the imaging system 1910 is preferably similar to the imaging system described hereinabove with reference to FIG. 10B. It preferably includes a first
lens, such as cylindrical lens 1086 (FIG. 10B).
The first lens preferably produces a one-dimensional Fourier transform, performed along an axis extending perpendicularly to the direction of propagation of the data retrieval wavefront, thereby
providing at least one one-dimensional component of the transformed wavefront in a dimension perpendicular to direction of propagation. The first lens, such as lens 1086, focuses the stored data
retrieval wavefront onto a phase manipulator, such as a single axis displaceable phase manipulator 1087 (FIG. 10B), which is preferably located at the focal plane of lens 1086. The phase manipulator
1087 applies a plurality of different phase changes to each of the at least one one-dimensional components of the transformed wavefront, thereby obtaining at least one one-dimensional component of
the plurality of phase changed transformed wave fronts.
Additionally the imaging system may include a second lens, such as cylindrical lens 1088 (FIG. 10B), arranged so as to image the at least one one-dimensional component of the stored data retrieval
wavefront onto a detector 1089, such as a CCD detector. Additionally the plurality of intensity maps are employed by circuitry 1912 to obtain an output indicating al least one and possibly both of
the amplitude and phase of the at least one one-dimensional component of the data retrieval wavefront.
Additionally, in accordance with the foregoing methodology, and as described hereinabove with reference to FIG. 10B, the phase manipulator 1087 preferably comprises a multiple local phase delay
element, such as a spatially non-uniform transparent object, typically including several different phase delay regions, each arranged to apply a phase delay to one of the at least one one-dimensional
component at a given position of the object along a phase manipulator axis, extending perpendicularly to the direction of propagation of the wavefront and perpendicular to the axis of the transform
produced by lens 1086.
In such a case, there is provided relative movement between the imaging system 1910 and the media 1900 along the phase manipulator axis. This relative movement sequentially matches different phase
delay regions with different wavefront components, corresponding to different parts of area 1908 on media 1900, such that preferably each wavefront component passes through each phase delay region of
the phase manipulator.
It is appreciated that the relative movement between the imaging system 1910 and the at least one one-dimensional wavefront component can be obtained by the rotation of media 1900 about its axis,
while the imaging system is not moving.
It is a particular feature of this embodiment, that each of the at least one one-dimensional component of the wavefront is separately processed. Thus, each of the at least one one-dimensional
wavefront component, corresponding to a one-dimensional part of area 1908, is focused by a separate portion of the first cylindrical lens of imaging system 1910, is imaged by a corresponding separate
portion of the second cylindrical lens and passes through a distinct region of the phase manipulator. The images of each of the one-dimensional parts of area 1908 at the detector incorporated in
imaging system 1910 are thus separate and distinct images. It is appreciated that these images may appear on separate detectors or on a monolithic detector.
In accordance with an embodiment of the present invention, the transform applied to the stored data retrieval wavefront includes an additional Fourier transform. This additional Fourier transform may
be performed by the first cylindrical lens of imaging system 1910 or by an additional lens and is operative to minimize cross-talk between different one-dimensional components of the wavefront. In
such a case, preferably an additional transform, such as that provided by an additional lens adjacent to the second cylindrical lens, is applied to the phase changed transformed wavefront. In such a
case, preferably a further transform is applied to the phase changed transformed wavefront. This further transform may be performed by the second cylindrical lens of imaging system 1910 or by an
additional lens.
Reference is now made to FIG. 20, which is a simplified partially schematic, partially pictorial illustration of a system for 3-dimensional imaging employing the functionality and structure of FIGS.
1A and 1B. As seen in FIG. 20, a beam of radiation, such as light or acoustic energy, is supplied from a radiation source 2000, optionally via a beam expander, onto a beam splitter 2004, which
reflects at least part of the radiation onto a 3-dimensional object 2006 to be imaged. The radiation reflected from the object 2006, is a 3-dimensional imaging wavefront, which has an amplitude and a
phase, and which contains information about the object 2006. At least part of the radiation incident on the surface of object 2006 is reflected from the object 2006 and transmitted via the beam
splitter 2004 and focused via a focusing lens 2008 onto a phase manipulator 2010, which is preferably located at the image plane of radiation source 2000.
The phase manipulator 2010 may be, for example, a spatial light modulator or a series of different transparent, spatially non-uniform objects. It is appreciated that phase manipulator 2010 can be
configured such that a substantial part of the radiation focused thereonto is reflected therefrom. Alternatively the phase manipulator 2010 can be configured such that a substantial part of the
radiation focused thereonto is transmitted therethrough.
A second lens 2012 is arranged so as to image object 2006 onto a detector 2014, such as a CCD detector. Preferably the second lens 2012 is arranged such that the detector 2014 lies in its focal
plane. The output of detector 2014, an example of which is a set of intensity maps designated by reference numeral 2015, is preferably supplied to data storage and processing circuitry 2016, which
preferably carries out functionality “C” described hereinabove with reference to FIG. 1A, providing an output indicating at least one and possibly both of the phase and amplitude of the 3-dimensional
imaging wavefront. This output is preferably further processed to obtain information about the object 2006, such as the 3-dimensional shape of the object.
In accordance with a preferred embodiment of the present invention, the beam of radiation supplied from radiation source 2000 has a narrow wavelength band about a given central wavelength, causing
the phase of the radiation reflected from object 2006 to be proportional to geometrical variations in the surface 2006, the proportion being an inverse linear function of the central wavelength of
the radiation.
In accordance with another preferred embodiment of the present invention, the beam of radiation supplied from radiation source 2000 has at least two narrow wavelength bands, each centered about a
different wavelength, designated λ[1], . . . , λ[n]. In such a case, the radiation reflected from the object 2006 has at least two wavelength components, each centered around a wavelength λ[1], . . .
, λ[n ]and at least two indications of the phase of the 3-dimensional imaging wavefront are obtained. Each such indication corresponds to a different wavelength component of the reflected radiation.
These at least two indications may be subsequently combined to enable enhanced imaging of the object 2006, by avoiding 2π ambiguity in the 3-dimensional imaging.
In accordance with still another preferred embodiment of the present invention, the phase manipulator 2010 applies a plurality of different spatial phase changes to the radiation wavefront reflected
from surface 2006 and Fourier transformed by lens 2008. Application of the plurality of different spatial phase changes provides a plurality of differently phase changed transformed wavefronts which
may be subsequently detected by detector 2014.
In accordance with yet another preferred embodiment of the present invention, at least three different spatial phase changes are applied by phase manipulator 2010, resulting in at least three
different intensity maps 2015. The at least three intensity maps are employed by the data storage and processing circuitry 2016 to obtain an output indicating at least the phase of the 3-dimensional
imaging wavefront. In such a case, the data storage and processing circuitry 2016, carries out functionality “C” described hereinabove with reference to FIG. 1A, preferably in a manner described
hereinabove with reference to FIG. 13, where the wavefront being analyzed (FIG. 13) is the 3-dimensional imaging wavefront.
Additionally, in accordance with a preferred embodiment of the present invention, the beam of radiation supplied from radiation source 2000 comprises a plurality of different wavelength components,
thereby providing a plurality of wavelength components in the 3-dimensional imaging wavefront and subsequently in the transformed wavefront impinging on phase manipulator 2010. In this case the phase
manipulator 2010 may be an object, at least one of whose thickness, refractive index and surface geometry varies spatially. This spatial variance of the phase manipulator generates a different
spatial phase change for each of the wavelength components, thereby providing a plurality of differently phase changed transformed wavefronts to be subsequently detected by detector 2014.
Reference is now made to FIG. 21A, which is a simplified partially schematic, partially pictorial illustration of wavefront analysis functionality operative in accordance with another preferred
embodiment of the present invention. The functionality of FIG. 21A can be summarized as including the following sub-functionalities:
• A. obtaining a plurality of differently amplitude changed transformed wavefronts corresponding to a wavefront being analyzed, which has an amplitude and a phase;
• B. obtaining a plurality of intensity maps of the plurality of amplitude changed transformed wavefronts; and
• C. employing the plurality of intensity maps to obtain an output indicating at least one and possibly both of the phase and the amplitude of the wavefront being analyzed.
As seen in FIG. 21A, the first sub-functionality, designated “A” may be realized by the following functionalities:
A wavefront, which may be represented by a plurality of point sources of light, is generally designated by reference numeral 2100. Wavefront 2100 has a phase characteristic which is typically
spatially non-uniform, shown as a solid line and indicated generally by reference numeral 2102. Wavefront 2100 also has an amplitude characteristic which is typically spatially non-uniform, shown as
a dashed line and indicated generally by reference numeral 2103. Such a wavefront may be obtained in a conventional manner by receiving light from any suitable object, such as by reading an optical
disk, for example, a DVD or compact disk 2104.
A principal purpose of the present invention is to measure the phase characteristic, such as that indicated by reference numeral 2102, which is not readily measured. Another purpose of the present
invention is to measure the amplitude characteristic, such as that indicated by reference numeral 2103 in an enhanced manner. A further purpose of the present invention is to measure both the phase
characteristic 2102 and the amplitude characteristic 2103. While there exist various techniques for carrying out such measurements, the present invention provides a methodology which is believed to
be superior to those presently known, inter alia due to its relative insensitivity to noise.
A transform, indicated here symbolically by reference numeral 2106, is applied to the wavefront being analyzed 2100, thereby to obtain a transformed wavefront. A preferred transform is a Fourier
transform. The resulting transformed wavefront is symbolically indicated by reference numeral 2108.
A plurality of different amplitude changes, preferably spatial amplitude changes, represented by optical attenuation components 2110, 2112 and 2114 are applied to the transformed wavefront 2108,
thereby to obtain a plurality of differently amplitude changed transformed wavefronts, represented by reference numerals 2120, 2122 and 2124 respectively. It is appreciated that the illustrated
difference between the individual ones of the plurality of differently amplitude changed transformed wavefronts is that portions of the transformed wavefront are attenuated differently relative to
the remainder thereof.
As seen in FIG. 21A, the second sub-functionality, designated “B”, may be realized by applying a transform, preferably a Fourier transform, to the plurality of differently amplitude changed
transformed wavefronts. Alternatively, the sub-functionality B may be realized without the use of a Fourier transform, such as by propagation of the differently amplitude changed transformed
wavefronts over an extended space. Finally, functionality B requires detection of the intensity characteristics of plurality of differently amplitude changed transformed wavefronts. The outputs of
such detection are the intensity maps, examples of which are designated by reference numerals 2130, 2132 and 2134.
As seen in FIG. 21A, the third sub-functionality, designated “C” may be realized by the following functionalities:
□ expressing, such as by employing a computer 2136, the plurality of intensity maps, such as maps 2130, 2132 and 2134, as at least one mathematical function of phase and amplitude of the
wavefront being analyzed and of the plurality of different amplitude changes, wherein at least one and possibly both of the phase and the amplitude are unknown and the plurality of different
amplitude changes, typically represented by optical attenuation components 2110, 2112 and 2114 applied to the transformed wavefront 2108, are known; and
□ employing, such as by means of the computer 2136, the at least one mathematical function to obtain an indication of at least one and possibly both of the phase and the amplitude of the
wavefront being analyzed, here represented by the phase function designated by reference numeral 2138 and the amplitude function designated by reference numeral 2139, which, as can be seen,
respectively represent the phase characteristics 2102 and the amplitude characteristics 2103 of the wavefront 2100. In this example, wavefront 2100 may represent the information contained in
the compact disk or DVD 2104.
In accordance with an embodiment of the present invention, the plurality of intensity maps comprises at least four intensity maps. In such a case, employing the plurality of intensity maps to obtain
an output indicating at least the phase of the wavefront being analyzed includes employing a plurality of combinations, each of at least three of the plurality of intensity maps, to provide a
plurality of indications at least of the phase of the wavefront being analyzed.
Preferably, the methodology also includes employing the plurality of indications of at least the phase of the wavefront being analyzed to provide an enhanced indication at least of the phase of the
wavefront being analyzed.
Also in accordance with an embodiment of the present invention, the plurality of intensity maps comprises at least four intensity maps. In such a case, employing the plurality of intensity maps to
obtain an output indicating at least the amplitude of the wavefront being analyzed includes employing a plurality of combinations, each of at least three of the plurality of intensity maps, to
provide a plurality of indications at least of the amplitude of the wavefront being analyzed.
Preferably, the methodology also includes employing the plurality of indications of at least the amplitude of the wavefront being analyzed to provide an enhanced indication at least of the amplitude
of the wavefront being analyzed.
It is appreciated that in this manner, enhanced indications of both phase and amplitude of the wavefront may be obtained.
In accordance with a preferred embodiment of the present invention, at least some of the plurality of indications of the amplitude and phase are at least second order indications of the amplitude and
phase of the wavefront being analyzed.
In accordance with one preferred embodiment of the present invention, the plurality of intensity maps are employed to provide an analytical output indicating the amplitude and phase.
Preferably, the amplitude changed transformed wavefronts are obtained by interference of the wavefront being analyzed along a common optical path.
In accordance with another preferred embodiment of the present invention, the plurality of intensity maps are employed to obtain an output indicating the phase of the wavefront being analyzed, which
is substantially free from halo and shading off distortions, which are characteristic of many of the existing ‘phase-contrast’ methods.
In accordance with still another embodiment of the present invention, the plurality of intensity maps may be employed to obtain an output indicating the phase of the wavefront being analyzed by
combining the plurality of intensity maps into a second plurality of combined intensity maps, the second plurality being less than the first plurality, obtaining at least an output indicative of the
phase of the wavefront being analyzed from each of the second plurality of combined intensity maps and combining the outputs to provide an enhanced indication of the phase of the wavefront being
In accordance with yet another embodiment of the present invention, the plurality of intensity maps may be employed to obtain an output indicating amplitude of the wavefront being analyzed by
combining the plurality of intensity maps into a second plurality of combined intensity maps, the second plurality being less than the first plurality, obtaining at least an output indicative of the
amplitude of the wavefront being analyzed from each of the second plurality of combined intensity maps and combining the outputs to provide an enhanced indication of the amplitude of the wavefront
being analyzed.
Additionally in accordance with a preferred embodiment of the present invention, the foregoing methodology may be employed for obtaining a plurality of differently amplitude changed transformed
wavefronts corresponding to a wavefront being analyzed, obtaining a plurality of intensity maps of the plurality of amplitude changed transformed wavefronts and employing the plurality of intensity
maps to obtain an output of an at least second order indication of phase of the wavefront being analyzed.
Additionally or alternatively in accordance with a preferred embodiment of the present invention, the foregoing methodology may be employed for obtaining a plurality of differently amplitude changed
transformed wavefronts corresponding to a wavefront being analyzed, obtaining a plurality of intensity maps of the plurality of amplitude changed transformed wavefronts and employing the plurality of
intensity maps to obtain an output of an at least second order indication of amplitude of the wavefront being analyzed.
In accordance with yet another embodiment of the present invention, the obtaining of the plurality of differently amplitude changed transformed wavefronts comprises applying a transform to the
wavefront being analyzed, thereby to obtain a transformed wavefront, and then applying a plurality of different phase and amplitude changes to the transformed wavefront, where each of these changes
can be a phase change, an amplitude change or a combined phase and amplitude change, thereby to obtain a plurality of differently phase and amplitude changed transformed wavefronts.
In accordance with yet another embodiment of the present invention, a wavefront being analyzed comprises at least two wavelength components. In such a case, obtaining a plurality of intensity maps
also includes dividing the amplitude changed transformed wavefronts according to the at least two wavelength components in order to obtain at least two wavelength components of the amplitude changed
transformed wavefronts and in order to obtain at least two sets of intensity maps, each set corresponding to a different one of the at least two wavelength components of the amplitude changed
transformed wavefronts.
Subsequently, the plurality of intensity maps are employed to provide an output indicating the amplitude and phase of the wavefront being analyzed by obtaining an output indicative of the phase of
the wavefront being analyzed from each of the at least two sets of intensity maps and combining the outputs to provide an enhanced indication of phase of the wavefront being analyzed. In the enhanced
indication, there is no 2π ambiguity once the value of the phase exceeds 2π, which conventionally results when detecting phase of a single wavelength wavefront.
It is appreciated that the wavefront being analyzed may be an acoustic radiation wavefront.
It is also appreciated that the wavefront being analyzed may be an electromagnetic radiation wavefront, of any suitable wavelength, such as visible light, infrared, ultra-violet and X-ray radiation.
It is further appreciated that wavefront 2100 may be represented by a relatively small number of point sources and defined over a relatively small spatial region. In such a case, the detection of the
intensity characteristics of the plurality of differently amplitude changed transformed wavefronts may be performed by a detector comprising only a single detection pixel or several detection pixels.
Additionally, the output indicating at least one and possibly both of the phase and amplitude of the wavefront being analyzed may be provided by computer 2136 in a straight-forward manner.
In accordance with an embodiment of the present invention, the plurality of different amplitude changes 2110, 2112 and 2114, preferably spatial amplitude changes, are effected by applying a
time-varying spatial amplitude change to part of the transformed wavefront 2108.
In accordance with a preferred embodiment of the present invention, the plurality of different amplitude changes 2110, 2112 and 2114 are effected by applying a spatially uniform, time-varying spatial
amplitude change to part of the transformed wavefront 2108.
In accordance with an embodiment of the present invention, each of the wavefront 2100 and the transformed wavefront 2108 comprises a plurality of different wavelength components. In such a case, the
plurality of different spatial amplitude changes may be effected by applying an amplitude change to each of the plurality of different wavelength components of the transformed wavefront. It is
appreciated that the amplitude changes may be spatially different or that the amplitude may be differently attenuated for each different wavelength component.
In accordance with another embodiment of the present invention, each of the wavefront 2100 and the transformed wavefront 2108 comprises a plurality of different polarization components. In such a
case, the plurality of different spatial amplitude changes may be effected by applying an amplitude change to each of the plurality of different polarization components of the transformed wavefront.
It is appreciated that the amplitude changes may be spatially different or that the amplitude may be differently attenuated for each different polarization component.
In accordance with another embodiment of the present invention, the transform 2106 applied to the wavefront 2100 is a Fourier transform, the plurality of different spatial amplitude changes comprise
at least three different amplitude changes, effected by applying a spatially uniform, time-varying spatial amplitude attenuation to part of the transformed wavefront 2108, and the plurality of
intensity maps 2130, 2132 and 2134 comprises at least three intensity maps. In such a case, employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the
wavefront being analyzed preferably includes:
□ expressing the wavefront being analyzed 2100 as a first complex function which has an amplitude and phase identical to the amplitude and phase of the wavefront being analyzed;
□ expressing the plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial amplitude change;
□ defining a second complex function, having an absolute value and a phase, as a convolution of the first complex function and of a Fourier transform of the spatial function governing the
spatially uniform, time-varying spatial amplitude attenuation;
□ expressing each of the plurality of intensity maps as a third function of:
☆ the amplitude of the wavefront being analyzed;
☆ the absolute value of the second complex function;
☆ a difference between the phase of the wavefront being analyzed and the phase of the second complex function; and
☆ a known amplitude attenuation produced by one of the at least three different amplitude changes, to each of which one of the at least three intensity maps corresponds;
□ solving the third function to obtain the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront
being analyzed and the phase of the second complex function;
□ solving the second complex function to obtain the phase of the second complex function; and
□ obtaining the phase of the wavefront being analyzed by adding the phase of the second complex function to the difference between the phase of the wavefront being analyzed and the phase of the
second complex function.
Reference is now made to FIG. 21B, which is a simplified partially schematic, partially block diagram illustration of a wavefront analysis system suitable for carrying out the functionality of FIG.
21A in accordance with a preferred embodiment of the present invention. As seen in FIG. 21B, a wavefront, here designated by reference numeral 2150 is focused, as by a lens 2152, onto an amplitude
attenuator 2154, which is preferably located at the focal plane of lens 2152. The amplitude attenuator 2154 generates amplitude changes, such as amplitude attenuation, and may be, for example, a
spatial light modulator or a series of different partially transparent objects.
A second lens 2156 is arranged so as to image wavefront 2150 onto a detector 2158, such as a CCD detector. Preferably the second lens 2156 is arranged such that the detector 2158 lies in its focal
plane. The output of detector 2158 is preferably supplied to data storage and processing circuitry 2160, which preferably carries out functionality “C” described hereinabove with reference to FIG.
Reference is now made to FIG. 22, which is a simplified partially schematic, partially pictorial illustration of a system for surface mapping employing the functionality and structure of FIGS. 21A
and 21B. As seen in FIG. 22, a beam of radiation, such as light or acoustic energy, is supplied from a radiation source 2200, optionally via a beam expander 2202, onto a beam splitter 2204, which
reflects at least part of the radiation onto a surface 2206 to be inspected. The radiation reflected from the inspected surface, is a surface mapping wavefront, which has an amplitude and a phase,
and which contains information about the surface 2206. At least part of the radiation incident on surface 2206 is reflected from the surface 2206 and transmitted via the beam splitter 2204 and
focused via a focusing lens 2208 onto an amplitude attenuator 2210, which is preferably located at the image plane of radiation source 2200.
The amplitude attenuator 2210 may be, for example, a spatial light modulator or a series of different partially transparent non-spatially uniform objects. It is appreciated that amplitude attenuator
2210 can be configured such that a substantial part of the radiation focused thereonto is reflected therefrom. Alternatively the amplitude attenuator 2210 can be configured such that a substantial
part of the radiation focused thereonto is transmitted therethrough.
A second lens 2212 is arranged so as to image surface 2206 onto a detector 2214, such as a CCD detector. Preferably the second lens 2212 is arranged such that the detector 2214 lies in its focal
plane. The output of detector 2214, an example of which is a set of intensity maps designated by reference numeral 2215, is preferably supplied to data storage and processing circuitry 2216, which
preferably carries out functionality “C” described hereinabove with reference to FIG. 21A, providing an output indicating at least one and possibly both of the phase and the amplitude of the surface
mapping wavefront. This output is preferably further processed to obtain information about the surface 2206, such as geometrical variations and reflectivity of the surface.
In accordance with a preferred embodiment of the present invention, the beam of radiation supplied from radiation source 2200 has a narrow wavelength band about a given central wavelength, causing
the phase of the radiation reflected from surface 2206 to be proportional to geometrical variations in the surface 2206, the proportion being an inverse linear function of the central wavelength of
the radiation.
In accordance with an embodiment of the present invention, the beam of radiation supplied from radiation source 2200 has at least two narrow wavelength bands, each centered about a different
wavelength, designated λ[1], . . . , λ[n]. In such a case, the radiation reflected from the surface 2206 has at least two wavelength components, each centered around a wavelength λ[1], . . . , λ[n].
At least two indications of the phase of the surface mapping wavefront are obtained. Each such indication corresponds to a different wavelength component of the reflected radiation. These at least
two indications may be subsequently combined to enable enhanced mapping of the surface 2206, by avoiding ambiguity in the mapping, known as 27 i ambiguity, when the value of the mapping at a given
spatial location in the surface exceeds the value of the mapping at a different spatial location in the surface by the largest of the different wavelengths λ[1], . . . , λ[n]. A proper choice of the
wavelengths λ[1], . . . , λ[n], may lead to elimination of this ambiguity when the difference in values of the mapping at different locations is smaller than the multiplication product of all the
In accordance with a preferred embodiment of the present invention, the amplitude attenuator 2210 applies a plurality of different spatial amplitude changes to the radiation wavefront reflected from
surface 2206 and Fourier transformed by lens 2208. Application of the plurality of different spatial amplitude changes provides a plurality of differently amplitude changed transformed wavefronts
which may be subsequently detected by detector 2214.
In accordance with yet another preferred embodiment of the present invention, at least three different spatial amplitude changes are applied by amplitude attenuator 2210, resulting in at least three
different intensity maps 2215. The at least three intensity maps are employed by the data storage and processing circuitry 2216 to obtain an output indicating at least one and possibly both of the
phase and amplitude of the surface mapping wavefront. In such a case, the data storage and processing circuitry 2216, carries out functionality “C” described hereinabove with reference to FIG. 21A,
where the wavefront being analyzed (FIG. 21A) is the surface mapping wavefront.
Additionally, in accordance with a preferred embodiment of the present invention, the beam of radiation supplied from radiation source 2200 comprises a plurality of different wavelength components,
thereby providing a plurality of wavelength components in the surface mapping wavefront and subsequently in the transformed wavefront impinging on amplitude attenuator 2210. In this case the
amplitude attenuator may be an object, at least one of whose reflection and transmission varies spatially. This spatial variance of the amplitude attenuator generates a different spatial amplitude
change for each of the wavelength components, thereby providing a plurality of differently amplitude changed transformed wavefronts to be subsequently detected by detector 2214. It is appreciated
that the amplitude attenuation generated by attenuator 2210 may be different for each of the different wavelength components.
In accordance with an embodiment of the present invention, the surface 2206 is a surface of media in which information is encoded by selecting the height of the media at each of a multiplicity of
different locations on the media. In such a case, the indications of the amplitude and phase of the surface mapping wavefront provided by data storage and processing circuitry 2216 are employed to
obtain the information encoded on the media.
It is appreciated that other applications, such as those described hereinabove with respect to FIGS. 16-20 may also be provided in accordance with the present invention wherein amplitude attenuation
is performed instead of phase manipulation. It is further appreciated that all of the applications described hereinabove with reference to FIGS. 15-20 may also be provided in accordance with the
present invention wherein both amplitude attenuation and phase manipulation are performed.
It will be appreciated by persons skilled in the art that the present invention is not limited by what has been particularly shown and described hereinabove. Rather the present invention includes
both combinations and subcombinations of features described hereinabove as well as modifications and variations of such features which would occur to a person of ordinary skill in the art upon
reading the foregoing description and which are not in the prior art. | {"url":"http://www.google.es/patents/US7327470?dq=flatulence","timestamp":"2014-04-21T04:34:31Z","content_type":null,"content_length":"353676","record_id":"<urn:uuid:dd83274a-c016-4200-bc90-52ba226f38f5>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00122-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Tools Discussion: All Topics, failure functions
Discussion: All Topics
Topic: failure functions
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Subject: failure functions
Author: deva
Date: Jan 11 2009
I would like to introduce myself as a non-academic mathematician
with two published papers to my credit: 1)"Euler's generalisation of Fermat's
theorem- a further generalisation" and 2) "Minimum universal exponent
generalisation of Fermat's theorem".
Object of this mail: To discuss "failure functions". What I propose to do
is to a) define them b) furnish ne=umerical examples
and c)furnish proofs.
In the next mail I will commence with definitions-trust it will be
found interesting.
A.K. Devaraj
Reply to this message Quote this message when replying?
yes no
Post a new topic to the General Discussion discussion
Discussion Help | {"url":"http://mathforum.org/mathtools/discuss.html?context=all&do=r&msg=71906","timestamp":"2014-04-17T17:10:31Z","content_type":null,"content_length":"15519","record_id":"<urn:uuid:5ed97292-45f7-4324-b659-a0dcd32cfcae>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00447-ip-10-147-4-33.ec2.internal.warc.gz"} |
So this is what was happening
Today we were in our Maths Class. The teacher asked us what is congruence. Most of the people of our class could not answer but I was able to give the definition of it. Now he started asking us what
is the total number of congruence criteria in triangles.
This is what I said:
1. Side-Side-Side
2. Side-Angle-Side
3. Angle-Angle-Side
4. Right Angle- Hypotenuse-Side
Then he said:
You are missing a basic point
I said:
Is it Angle-Side-Angle
He said:
I said:
But that's only a special case of Angle-Angle-Side
He said:
The basic criteria is Angle-Side-Angle.
However, its application is Angle-Angle-Side
*I found it difficult to continue arguing*
My question is whether he is correct.
If he is what does he mean?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: So this is what was happening
I'm just considering this. Give me 5 mins
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Re: So this is what was happening
In school we learned it without the Right Angle- Hypotenuse-Side and with Angle-Side-Angle.
EDIT: Btw,SAS and SSA are two different criteria.I don't think either is a special case of the other one.
Last edited by anonimnystefy (2012-06-21 19:59:00)
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: So this is what was happening
hi Agnishom
angle-side-angle and angle-angle-side amount to the same thing as you can always work out the third angle anyway.
I'm assuming when you say side-angle-side you are putting the angle between the two known sides.
That distinquishes it from side-side-angle where the angle isn't between the known sides.
That case needs to be considered separately because SAS has a unique construction but SSA may not, depending on the measurements.
I'll work on a diagram to illustrate this
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Re: So this is what was happening
I am talking about AAS and ASA
I want to know is he correct when he says:
AAS is the application of ASA
Last edited by Agnishom (2012-06-21 20:08:03)
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: So this is what was happening
AAS and ASA are the same thing. But you are missing SSA in your list. SSA is the generalization of RHS.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: So this is what was happening
Does SSA exist?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: So this is what was happening
Of course it does! As I said,it is a generalization of the Right Angle-Hypotenuse-Side rule.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: So this is what was happening
You need to look at post #11 as well to make sense of this.
Say AB = 8.31
AC = 3.81
ABC = 23.3 degrees
There are two possible places for C (second labelled E)
Therefore having SS and the non included angle may lead to a congruence or it may not.
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Re: So this is what was happening
Anyway, I want to know why did he tell me that I am missing out "A-S-A"
I think he shouldn't have told it bcause I had already mentioned A-A-S
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: So this is what was happening
Yes, I agree with you there. When you know two angles, you also know the third, so AAS and ASA are the same thing.
SSA case: Now I've got this clear in my head. Hope I can make it clear for you.
SSA definitely exists. Diagram below shows an example.
Because AB is shorter than AC the point C is uniquely constructed ( draw AB, then the angle, then the curve to pinpoint C).
But the earlier example in post #9 shows that you don't always get a pair of congruent triangles with the same SSA, so you have to be careful.
Do you know the SINE RULE because it all becomes clear once you use that.
ps. What has construction got to do with congruence?
Just this: If I give two people some measurements for a triangle and ask them to draw it, will they land up with congruent triangles? A test for congruency cases is "When you try to construct a
triangle with these measurements, is the solution unique?" This assumes that rotations and reflections are considered to be the same and that it doesn't matter what you use as letter labels for the
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Re: So this is what was happening
Do you mean this?:
For three sides a, b, c and angles alpha,beta and gama
sine(a)/alpha = sine(b)/beta = sine(c)/gama
I have only seen it but I don' have any Idea how to use it
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: So this is what was happening
Yes, that's it.
I have edited post #11 and 9 by the way.
If you know the right information about a triangle you can substitute in the values into the sine rule and so work out all the missing sides and angles.
There is a direct link between some of the congruency cases and the sine rule.
eg. If you have AAS you can use the sine rule to work out the unknown sides. Thus a pair of triangles with the same AAS will also be SSS once you've worked out the missing sides.
But when you try the same trick with SSA you will have to do an inv sine to get a second angle. Because there is an obtuse angle that has the same sine as every acute angle {sin (180 - x) = sinx} you
end up with two triangles with the same SSA . This is what happens in post #9
The cases that aren't covered by the sine rule are covered by the cosine rule
So rather neatly, congruence, constructability and advanced trig. all lead to the same results.
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Re: So this is what was happening
Ya, but what actually is happening is that I can't tell my Maths Teacher that because we don't have yet learnt that law in school
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: So this is what was happening
hi Agnishom
Well you'll have to decide what to do. If it were me, I wouldn't mind that one of my students had done some independent research.
I'd probably get him in front of the class to explain it.
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Re: So this is what was happening
Argument against SSA
See The Image
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: So this is what was happening
Hi Agnishom
I have just realized that the SSA rule is invalid, but not for the reason provided in your post. In your post one triangle has the angle opposing the 5cm side and in the other it is opposing the 4cm
But there are in fact two triangles where the angles stay in he same places with respect to the sides.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: So this is what was happening
I thought I covered this in posts #9 and #11.
Provided you know which cases are valid then SSA works for those cases.
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Re: So this is what was happening
I have just remembered how the SSA rule truly goes."Two triangles are congruent if they have two pairs of equal sides and equal angles opposite one of them and all other angles are of the same type
(meaning either acute, right or obtuse).
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: So this is what was happening
That sounds right to me.
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Re: So this is what was happening
Yes, I remembered it now. That is how we learned it.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: So this is what was happening
Arrhhh............ the ideal student
Me, I always have to work everything out from first principles as I'm hopeless at remembering anything. I cannot even remember a phone number in the time from seeing it written down to the time when
my fingers are on the buttons!
That's why it takes me longer to sort out someone's "help me" problem.
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Re: So this is what was happening
Do you derive phone numbers from the first principal?
Last edited by Agnishom (2012-07-10 23:00:53)
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: So this is what was happening
bob bundy wrote:
Arrhhh............ the ideal student
Me, I always have to work everything out from first principles as I'm hopeless at remembering anything. I cannot even remember a phone number in the time from seeing it written down to the time
when my fingers are on the buttons!
That's why it takes me longer to sort out someone's "help me" problem.
I don't agree with the first part. I mostly remember things I learned in mthematics, but since I self-taught and bobbym-taught myself most of the stuff, it really isn't hard for me to remember things
from class.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: So this is what was happening
Do you derive phone numbers from the first principal?
You may well ask! If I could, then I would not have such trouble punching them in. For this reason the only one I can manage easily is
The person who has this number is getting fed up with my calls.
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei | {"url":"http://www.mathisfunforum.com/viewtopic.php?id=17884&p=1","timestamp":"2014-04-16T13:05:18Z","content_type":null,"content_length":"48157","record_id":"<urn:uuid:95f8ad19-c03c-473f-9eba-317027330b4f>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00053-ip-10-147-4-33.ec2.internal.warc.gz"} |
Unbalanced Forces Acting On Wishbones
Unbalanced Forces cause an object to accelerate (change speed and/or direction).
The three diagrams represent balanced forces and unbalanced forces acting on the ends of a wishbone. These are forces applied by the hands of the two contestants in the Wishbone Breaking Contest.
Diagram 1: Balanced Forces expressed by the equation: A + B
When the two forces are equal and in opposite directions, the net sum of the two forces is ZERO.This means the wishbone doesn’t move. This is the desired result.
Diagram 2: Unbalanced Forces expressed by the equations: A < B; B > A
This diagram show that the wishbone has moved to the right, in the direction of Force B. Unbalanced Forces are needed to cause a stationary object to move. The object moves (accelerates) in the
direction of the larger force, which is the pulling force of hand B.
Diagram 3. Unbalanced Forces expressed by the equations: A > B; B < A
This diagram show that the wishbone has moved to the left, in the direction of Force A. Unbalanced Forces are needed to cause a stationary object to move. The object moves (accelerates) in the
direction of the larger force, which is the pulling force of hand A.
Janice VanCleave’s Engineering for Every Kid: Easy Activities That Make Learning Science Fun | {"url":"http://scienceprojectideasforkids.com/2010/unbalanced-forces-on-wishbones/","timestamp":"2014-04-19T04:19:18Z","content_type":null,"content_length":"21651","record_id":"<urn:uuid:4e33dd48-faad-4296-9e45-32565c586574>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00086-ip-10-147-4-33.ec2.internal.warc.gz"} |
Graphing Radical Equations using a Table - Concept
Solving rational equations is substantially easier with like denominators. When solving rational equations, first multiply every term in the equation by the common denominator so the equation is
"cleared" of fractions. Next, use an appropriate technique for solving for the variable.
Whenever you're asked to graph anything in Math class, the most basic way to create a graph is using an xy table. That's what we're going to look at today in graphing rational functions. Excuse me
radical functions, we're not doing rationals. Radicals. It's a big difference.
Okay, so first thing you want I want to remind you guys is of domain and range. The domain is the set of all possible input values. That's really important when it comes to square roots because you
guys know the square root of a negative number is not a real value, is not a real solution. So in order to find the domain, the radicand must be greater than or equal to zero. Let me show you what
I'm talking about. If I want to take the square root of something, this thing whatever it is, I'm drawing a little cloud. That cloud or the radicand has to be greater than or equal to zero. That's
how you find the domain of a radical function.
Like in this problem for example, we're going to look at the parent function. In order to find the domain or my x values that I'm going to put in my table, I'm going to start by setting my radicand
greater than or equal to zero. Radicand is whatever's under the square root. All I have was x so all I need is x is greater than or equal to zero. That tells me when I'm setting up my table that I
want to use x numbers that are zero or bigger. I don't want to use any negative values because that would be non-real solutions in this function.
So let's go through and plug in these x numbers one at a time and find their corresponding y numbers. If I stick in the square root of zero, if I just put in zero there, square root of zero is zero.
Square root of one is one, square root of two, I'm going to approximate the decimal with 1.41, you can check that on your calculator. Square root of three when I stick that in there, I get 1.7
something, 1.73 I think, and then I'm going to stick in 4. Square root of 4 you guys know, is 2.
Okay. So now I have a good number of points. I'm going to get these guys on the graph and you'll see the shape that all radical functions have. Here we go. I first start with 0 0, then I had 1 1, 2
1.4, that's like almost one and a half. I have to approximate a little bit. 3 is 1.7 and then 4 was 2.
Okay. So you can kind of see what this is looking like. It's not a straight line. What this is, is like half of a parabola on it's side. This curve continues forever and ever. It goes out in this
direction out forever and ever, but notice how it stops right there at the end. That's because my domain was only x numbers that were bigger than or equal to zero. My graph doesn't continue in that
directions. Please don't put an arrow on that side because it doesn't continue. It just stops right there at zero and then it heads out in this direction.
So all of the graphs that you're going to be doing by making a table or any time you graph a radical function, it's going to have this half parabola shape. It's going to be having like a dead end on
one on one side and then an arrow on the other side that continues out forever and ever.
When you're making your table, be sure to be clever about what x values you choose. Use the domain to tell you what x numbers are possible. That is take the radicand, set it greater than or equal to
zero and then solve for x in order to find what x number should go to your table.
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A theory of dimension, K-theory 11
Results 1 - 10 of 62
- Commun. Math. Phys
"... Dedicated to John E. Roberts on the occasion of his sixtieth birthday We describe the structure of the inclusions of factors A(E) ⊂A(E ′ ) ′ associated with multi-intervals E ⊂ R for a local
irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. I ..."
Cited by 63 (26 self)
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Dedicated to John E. Roberts on the occasion of his sixtieth birthday We describe the structure of the inclusions of factors A(E) ⊂A(E ′ ) ′ associated with multi-intervals E ⊂ R for a local
irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion
associated with two separated intervals is isomorphic to the Longo-Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a
consequence, the index of A(E) ⊂A(E ′ ) ′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the
representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold
true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.
- J. Pure Appl. Alg , 2003
"... We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on
both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground f ..."
Cited by 52 (6 self)
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We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both
subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The
central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = F-Vect, where F is a field. An object X ∈ A with two-sided dual X gives rise to a
Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing
the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted
A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A
≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗-categories) have equal dimensions and
give rise the same state sum invariant of closed oriented 3-manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple
Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion
paper math.CT/0111205. 1
- FIELDS INST. COMMUN. AMER. MATH. SOC., PROVIDENCE, RI , 2003
"... ... inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-product we obtain results about the Frobenius-Schur indicator in
sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the d ..."
Cited by 50 (14 self)
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... inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-product we obtain results about the Frobenius-Schur indicator in sovereign
tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present
illustrative examples. We show that when the module category is tensor, then it gives rise to a NIM-rep of the fusion rules, and discuss a possible relation with the representation theory of vertex
operator algebras.
- Adv. Math , 2000
"... Given a braided tensor ∗-category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a
tensor ∗-category with conjugates and an irreducible unit. (A ∗-category is a category enriched over VectC ..."
Cited by 29 (6 self)
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Given a braided tensor ∗-category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a
tensor ∗-category with conjugates and an irreducible unit. (A ∗-category is a category enriched over VectC with positive ∗-operation.) A Galois correspondence is established between intermediate
categories sitting between C and C ⋊S and closed subgroups of the Galois group Gal(C⋊S/C) = AutC(C⋊S) of C, the latter being isomorphic to the compact group associated to S by the duality theorem of
Doplicher and Roberts. Denoting by D ⊂ C the full subcategory of degenerate objects, i.e. objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C⋊S
iff S ⊂ D. Under this condition C⋊S has no non-trivial degenerate objects iff S = D. If the original category C is rational (i.e. has only finitely many isomorphism classes of irreducible objects)
then the same holds for the new one. The category C ≡ C ⋊ D is called the modular closure of C since in the rational case it is modular, i.e. gives rise to a unitary representation of the modular
group SL(2, Z). (In passing we prove that every braided tensor ∗-category with conjugates automatically is a ribbon category, i.e. has a twist.) If all simple objects of S have dimension one the
structure of the category C ⋊ S can be clarified quite explicitly in terms of group cohomology. 1
- In preparation
"... ..."
"... We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the
pairs of A-D2n-E6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We f ..."
Cited by 27 (13 self)
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We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs
of A-D2n-E6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c<1 with
certain coset nets. Then, by using the classification of modular invariants for the minimal models by Cappelli-Itzykson-Zuber and the method of α-induction in subfactor theory, we classify all local
irreducible extensions of the Virasoro nets for c<1 and infer our main classification result. As an application, we identify in our classification list certain concrete coset nets studied in the
- Commun. Math. Phys
"... Let A be a local conformal net of factors on S 1 with the split property. We provide a topological construction of soliton representations of the n-fold tensor product A ⊗ · · · ⊗ A, that
restrict to true representations of the cyclic orbifold (A ⊗ · · · ⊗ A) Zn. We prove a quantum index the ..."
Cited by 27 (14 self)
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Let A be a local conformal net of factors on S 1 with the split property. We provide a topological construction of soliton representations of the n-fold tensor product A ⊗ · · · ⊗ A, that restrict to
true representations of the cyclic orbifold (A ⊗ · · · ⊗ A) Zn. We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational
iff the symmetrized tensor product (A ⊗ A) flip has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector
then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the µ-index of A is finite then A
turns out to be strongly additive. By [31], if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is non-degenerate. In interesting
cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.
- J. Funct. Anal
"... Let X be a finite graph, with edges colored and possibly oriented, such that an oriented edge and a non-oriented one cannot have same color. The universal Hopf algebra H(X) coacting on X is in
general non commutative, infinite dimensional, bigger than the algebra of functions on the usual symmetry g ..."
Cited by 24 (11 self)
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Let X be a finite graph, with edges colored and possibly oriented, such that an oriented edge and a non-oriented one cannot have same color. The universal Hopf algebra H(X) coacting on X is in
general non commutative, infinite dimensional, bigger than the algebra of functions on the usual symmetry group G(X). For a graph with no edges Tannakian duality makes H(X) correspond to a
Temperley-Lieb algebra. We study some versions of this correspondence.
- Math. Ann , 1999
"... Abstract. If B is C ∗-algebra of dimension 4 ≤ n < ∞ then the finite dimensional irreducible representations of the compact quantum automorphism group of B, say Gaut ( ̂B), have the same fusion
rules as the ones of SO(3). As consequences, we get (1) a structure result for Gaut ( ̂B) in the case ..."
Cited by 23 (18 self)
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Abstract. If B is C ∗-algebra of dimension 4 ≤ n < ∞ then the finite dimensional irreducible representations of the compact quantum automorphism group of B, say Gaut ( ̂B), have the same fusion rules
as the ones of SO(3). As consequences, we get (1) a structure result for Gaut ( ̂B) in the case where B is a matrix algebra (2) if n ≥ 5 then the dual ̂Gaut ( ̂B) is not amenable (3) if n ≥ 4 then the
fixed point subfactor | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=2699702","timestamp":"2014-04-23T20:01:15Z","content_type":null,"content_length":"37219","record_id":"<urn:uuid:20443ad1-1c90-49f0-a715-f67d776ed2a4>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00166-ip-10-147-4-33.ec2.internal.warc.gz"} |
Penrose's Philosophical Error
Mathematics Department
King's College London
in Concepts for Neural Networks, L.J.Landau and J.G.Taylor eds. (Springer 1997) ISBN 3-540-76163-2
This is a somewhat modified HTML version.
1. Can Computers Think?
The United States Department of Energy's Sandia National Laboratory and Intel Corporation have built a supercomputer that on December 17, 1996 reached the one trillion-operations-per-second mark.^1
It weighs 44 tons, has 573 gigabytes of memory and 2.25 trillion bytes of disk space. In the time it takes you to blink an eye, the computer will complete 40 billion calculations. It is only 25 years
since Intel introduced the first microprocessor, which could carry out 60,000 operations per second. Who can say what breakthroughs will occur in the next 25 years or 250 years? Computers with
tremendous power and speed will surely be developed. Will these computers be able to think? Will they be conscious? Is there today anyone who would say that no matter what breakthroughs occur in
microelectronics, no matter what developments occur in computer programming, such as learning and random elements as in neural networks, no matter how powerful computers become, a computer will never
think? One such person is Roger Penrose, who has written in his book Shadows of the Mind [14] that:
there must be more to human thinking than can ever be achieved by a computer. ... Consciousness, in its particular manifestation in the human quality of understanding, is doing something that
mere computation cannot. I make clear that the term `computation' includes both `top-down' systems, which act according to specific well-understood algorithmic procedures, and `bottom-up'
systems, which are more loosely programmed in ways that allow them to learn by experience.
Figure1: Roger Penrose, 1996. © Steve Green. Reproduced with permission.
The question at issue is not a dualistic distinction between the conscious mind and the physical brain. Penrose accepts that the conscious mind arises as a functioning of the physical brain, but he
does not believe this functioning could be simulated by a large number of silicon chips operating as they do in a modern computer.
John Lucas
In the 1930's Alan Turing (1912-1954) proved a theorem showing that certain types of idealized computers, operating according to fixed rules, do have limitations. Turing's theorem is of a general
type first proved by Kurt Gödel (1906-1978) concerning limitations to formal mathematical systems. Penrose bases his argument that the mind is not computational on these limitations uncovered by
Turing and Gödel. His use of Gödel's theorem is similar to that of J.R.Lucas [13]. Lucas's article begins like this:
Gödel's theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines.
It is interesting, however, that Lucas doesn't rule out the construction of intelligent computers! At the end of his article, Lucas says:
When we increase the complexity of our machines there may, perhaps, be surprises in store for us. [Turing] draws a parallel with a fission pile. Below a certain ``critical'' size, nothing much
happens: but above the critical size, the sparks begin to fly. So too, perhaps, with brains and machines.... This may be so. Complexity often does introduce qualitative differences. Although it
sounds implausible, it might turn out that above a certain level of complexity, a machine ceased to be predictable, even in principle, and started doing things on its own account, or, to use a
very revealing phrase, it might begin to have a mind of its own. It would begin to have a mind of its own when it was no longer entirely predictable and entirely docile, but was capable of doing
things which we recognized as intelligent.... But then it would cease to be a machine, within the meaning of the act.
So for Lucas, a machine cannot think by definition of what a machine is. Penrose certainly does not agree with Lucas's view here. On the contrary, Penrose claims that no matter how complex they
become, computers will not have the capacity of human understanding.
Hilary Putnam
The results of Turing and Gödel belong to an area of mathematics known as mathematical logic. How do mathematical logicians react to Penrose's argument?^2 In his book review [16] of Shadows of the
Mind, Hilary Putnam^3 states:
In 1961 John Lucas - an Oxford philosopher well known for espousing controversial views - published a paper in which he purported to show that a famous theorem of mathematical logic known as
Gödel's Second Incompleteness Theorem implies that human intelligence cannot be simulated by a computer. Roger Penrose is perhaps the only well-known present-day thinker to be convinced by
Lucas's argument.... The right moral to draw from Lucas's argument, according to Penrose, is that noncomputational processes do somehow go on in the brain, and we need a new kind of physics to
account for them.
Shadows of the mind will be hailed as a ``controversial'' book, and it will no doubt sell very well even though it includes explanations of difficult concepts from quantum mechanics and
computational science. And yet this reviewer regards its appearance as a sad episode in our current intellectual life. Roger Penrose ... is convinced by - and has produced this book as well as
the earlier The emperor's new mind to defend - an argument that all experts in mathematical logic have long rejected as fallacious. The fact that the experts all reject Lucas's infamous argument
counts for nothing in Penrose's eyes. He mistakenly believes that he has a philosophical disagreement with the logical community, when in fact this is a straightforward case of a mathematical
© 1994 by The New York Times Co. Reprinted by Permission.
Kevin Warwick
How do the robotics experts react to Penrose's argument? The leader of the cybernetics group at Reading University is Kevin Warwick, who has just published a book [28] subtitled Why the New Race of
Robots will Rule the World, in which he states:
I believe that in the next ten to twenty years some machines will become more intelligent than humans.
We need to go to an appropriate level of modelling the functioning of a human brain in order to obtain a very good approximation.... Is it down, much further than the neuron level, to the atomic
level? Should we look at quantum physics... as Penrose thinks, generating an atomic model of the brain's functioning that we do not yet have?... In reality I feel that we already do have
sufficient basic modelling blocks.... What are those blocks? They are the neurons, the fundamental cells in the brain which cause it to operate as it does.
One major problem with consciousness is that it is a state which we ourselves feel that we are in, and through this we are aware of ourselves. It can, therefore, be extremely difficult, when
thinking about ourselves, to realise that this is simply a result of a specific state of the neurons in our brain. These `private' thoughts and feelings we have, how can they possibly be simply a
state of mind, realised by a state of the neurons? Well, that is exactly what they are.
Alan Turing
Figure2: Alan Turing, 1951. By courtesy of the National Portrait Gallery, London.
Since part of Penrose's argument is based on a theorem of Turing concerning a limitation on what can be achieved by computers, it is interesting to know Turing's thoughts on these issues. Turing
wrote an article [24]^4 in which he gives his opinion on whether or not a computer will ever be able to think. In the article Turing states his belief that his own fundamental theorem concerning the
limitations of computing machines is not an obstacle to the creation of intelligent computers.
Turing begins his article with the sentence:
I propose to consider the question, `Can machines think?'
The question in this form is, he believes, too vague and he restricts the machine to being a computer. He then replaces the question whether a computer can think like a human with the question
whether a computer can behave like a human, and more specifically whether a computer can answer questions posed by an interrogator so that the interrogator cannot distinguish the replies of the
computer from those of a human. (He calls this the imitation game. It is now known as the Turing test^5.)
Turing states clearly that he believes the computer will do well at imitating human behavior and gives 50 years as the time scale when computers will have become sufficiently powerful to begin doing
this. He was writing in 1950 and so we are just coming to his predicted time period. (Remember the new Sandia-Intel supercomputer!) About his beliefs he says:
I believe further that no useful purpose is served by concealing these beliefs. The popular view that scientists proceed inexorably from well-established fact to well-established fact, never
being influenced by any unproved conjecture, is quite mistaken. Provided it is made clear which are proved facts and which are conjectures, no harm can result. Conjectures are of great importance
since they suggest useful lines of research.
To construct intelligent computers, Turing proposes to use the technique of computer learning:
Instead of trying to produce a programme to simulate the adult mind, why not rather try to produce one which simulates the child's? If this were then subjected to an appropriate course of
education one would obtain the adult brain.... Our hope is that there is so little mechanism in the child-brain that something like it can be easily programmed. The amount of work in the
education we can assume, as a first approximation, to be much the same as for the human child.
Turing considers, and rejects, contrary views to his own. He considers in particular the mathematical objection based on Turing's theorem, the issue we will be considering in this chapter. We give
his discussion of this point in its entirety:
There are a number of results of mathematical logic which can be used to show that there are limitations to the powers of discrete-state machines. The best known of these results is known as
Gödel's theorem, and shows that in any sufficiently powerful logical system statements can be formulated which can neither be proved nor disproved within the system, unless possibly the system
itself is inconsistent. There are other, in some respects similar, results due to Church, Kleene, Rosser, and Turing. The latter result is the most convenient to consider, since it refers
directly to machines, whereas the others can only be used in a comparatively indirect argument: for instance if Gödel's theorem is to be used we need in addition to have some means of describing
logical systems in terms of machines, and machines in terms of logical systems. The result in question refers to a type of machine which is essentially a digital computer with an infinite
capacity. It states that there are certain things that such a machine cannot do. If it is rigged up to give answers to questions as in the imitation game, there will be some questions to which it
will either give a wrong answer, or fail to give an answer at all however much time is allowed for a reply. There may, of course, be many such questions, and questions which cannot be answered by
one machine may be satisfactorily answered by another. We are of course supposing for the present that the questions are of the kind to which an answer `Yes' or `No' is appropriate, rather than
questions such as `What do you think of Picasso?' The questions that we know the machines must fail on are of this type, ``Consider the machine specified as follows... Will this machine ever
answer `Yes' to any question?'' The dots are to be replaced by a description of some machine in a standard form, which could be something like that used in 5. When the machine described bears a
certain comparatively simple relation to the machine which is under interrogation, it can be shown that the answer is either wrong or not forthcoming. This is the mathematical result: it is
argued that it proves a disability of machines to which the human intellect is not subject.
The short answer to this argument is that although it is established that there are limitations to the powers of any particular machine, it has only been stated, without any sort of proof, that
no such limitations apply to the human intellect. But I do not think this view can be dismissed quite so lightly. Whenever one of these machines is asked the appropriate critical question, and
gives a definite answer, we know that this answer must be wrong, and this gives us a certain feeling of superiority. Is this feeling illusory? It is no doubt quite genuine, but I do not think too
much importance should be attached to it. We too often give wrong answers to questions ourselves to be justified in being very pleased at such evidence of the fallibility on the part of the
machines. Further, our superiority can only be felt on such an occasion in relation to the one machine over which we have scored our petty triumph. There would be no question of triumphing
simultaneously over all machines. In short, then, there might be men cleverer than any given machine, but then again there might be other machines cleverer again, and so on.
Turing ends his article looking towards the future:
We may hope that machines will eventually compete with men in all purely intellectual fields. But which are the best ones to start with? Even this is a difficult decision. Many people think that
a very abstract activity, like the playing of chess^6, would be best. It can also be maintained that it is best to provide the machine with the best sense organs that money can buy, and then
teach it to understand and speak English. This process could follow the normal teaching of a child. Things would be pointed out and named, etc. Again I do not know what the right answer is, but I
think both approaches should be tried.
We can only see a short distance ahead, but we can see plenty there that needs to be done.
Penrose's Error
We have seen that Penrose's argument, its basis and implications, is rejected by experts in the fields which it touches. So why expend effort studying it? Firstly, a discussion of Penrose's argument
gives us the opportunity to consider the mathematical theorems on computation and formal systems which are interesting in their own right. Second, the conscious mind remains shrouded in mystery. If a
precise mathematical theorem (and the bedrock of scientific precision is mathematics) can shed even a glimmer of understanding on the nature of consciousness, then it is worth careful study indeed.
Perhaps you don't have the opportunity to invest the time and effort required to understand fully the point of Penrose's argument. As Alwyn Scott says [21] in answer to the question why he has such a
stark difference of opinion with Penrose:
One answer to this question might be that I am unable - because of my intellectual limitations, interests in other matters, general laziness, or some combination of all three - to follow
Penrose's arguments through their many details. Perhaps he is right, and I just don't get it.
To understand Penrose's mathematical argument against a computational mind, it will be necessary to disentangle all the various ingredients: Turing's theorem on limitations to computations, Gödel's
theorem on limitations to formal systems, the formal system used to analyze a computation, computational models, mathematical truths, and mathematical beliefs. We shall uncover in section 6 a
category-mistake in Penrose's reasoning, a confusion of the deduction of H believes X with the deduction of X itself. By supposing that the beliefs such as X are theorems, Penrose has made the
consistency and correctness of his deductive system dependent on the consistency and correctness of H's beliefs, and this cannot be right. The beliefs and other thoughts of a mind cannot be theorems
of the deductive system composing our theory of mind. The basic structure of the theory must be consistent whether or not the beliefs of a particular mind are consistent. Indeed, even in our everyday
reasoning about other people's beliefs, those beliefs do not become part of our reasoning. And if those beliefs are contradictory, it does not follow that our reasoning about those beliefs is
contradictory. We should be able, for example, to consistently and correctly deduce that someone will perform a particular action as a result of the incorrect beliefs he holds.
But let us begin by discussing the basic mathematical results, unencumbered by possible applications to a theory of mind.
2. Solving Problems in Arithmetic
Suppose I give you a piece of paper with this written on it:
You might say, `That's easy; 123.' Or you might say `What do you want me to do with this piece of paper?', or `Que voulez-vous que je fasse de ce papier?' if you're French. The point is that before
you can respond to something I do you have to understand what I'm doing. We would have had to establish a means of communication so you will understand that I'm asking a question and understand what
the question is before you can answer it. All sorts of conventions have been established as you have grown up and studied at school so that the initial communication hurdle is not an obstacle.
Suppose you understand that I'm asking a question but you're not sure what some of the symbols on the paper represent. You might ask me `What is that symbol 27?' Perhaps I might show you a bowl with
27 marbles in it. But you might not know what aspect of what I showed you corresponded to the 27. So I might then show you a bowl with 27 oranges. In case you still do not understand I might show you
a bowl with 27 mice. You might then get the idea that 27 is the common characteristic of all the things I showed you: It must be a bowl! No, I'm sure you would understand that 27 represents the
quantity of marbles, or oranges. In fact to keep everything as concrete as possible and avoid unnecessary abstractions, no harm will result if we agree that we are always going to talk about marbles.
So 27 means 27 marbles. Going back to what was written on that piece of paper, we will now suppose that you understand that I'm asking you a question about quantities of marbles. There are two
quantities mentioned: 27 and 96. You might guess that the symbol + represents some operation you are to perform on the two quantities of marbles, but which operation? I'm sure you would say `That's
easy, it means to add the two quantities of marbles together.' It's only easy because you were taught how to do addition and the symbol which represents it in school. We must start somewhere, with
some things understood^7 (for example the meaning of 27 marbles and certain basic operations which could be carried out on quantities of marbles). Then the meaning of more and more complicated
operations on marbles could be defined. For example, I might hand you a piece of paper with this written on it:
You might say, `That's easy; 8487.'
In order to solve problems in arithmetic (which for concreteness we will take to mean questions about quantities of marbles) we must first agree on a certain set of basic instructions, or operations
on the marbles, from which a rule may be given to perform more complicated operations. The rule is a program to be executed by performing the basic instructions one after the other. We might ask
questions about the result of executing a program starting from some initial collection of bowls with marbles in them. If the program represents addition then we are asking a question about addition.
If the program represents multiplication then we are asking a multiplication question. And so on.
Why do we choose to formulate arithmetic in terms of bowls of marbles? Because in this way arithmetic corresponds to concrete actions on concrete objects (marbles, or coins, etc.). This is how
arithmetic originally developed, as a concrete activity rather than an abstract mental construct. At this stage there is no logic, no deduction or proof, just concrete actions carried out according
to instructions which define addition, multiplication, and other more complicated arithmetical procedures. At a later stage we may reason about these arithmetical procedures, using rules of deduction
which seem to us to be sound.
The Basic Instructions
There are a number of ways of selecting basic instructions out of which all arithmetical operations can be performed. Turing took one approach and arrived at Turing machines. We'll use a different
approach here. The basic instructions are simple operations on bowls of marbles (for concreteness). We start with a sufficient supply of bowls, each of which is labeled with a symbol 0,1,2,¼. Each
operation which we will define will require a particular number of bowls, but we won't specify an upper limit to the number of bowls so that arbitrary arithmetical procedures can be carried out. So
we have bowls R[0],R[1],R[2],¼ which are referred to as registers. The number r[k] of marbles in the bowl (register) R[k] is referred to as the content of the register. An arithmetical procedure or
program consists of a finite sequence of instructions I[0],I[1],I[2],¼,I[b]. Only four types of instructions are used:
Zero instructions.
For each k = 0,1,2,¼ there is a zero instruction Z(k). The meaning of the instruction Z(k) is to change the content r[k] of the register R[k] to 0, leaving all other registers unaltered. Thus, to
carry out the instruction Z(k) simply empty the k^th bowl.
Successor instructions.
For each k = 0,1,2,¼ there is a successor instruction S(k). The meaning of the instruction S(k) is to change the content r[k] of the register R[k] to r[k]+1, leaving all other registers
unaltered. Thus to carry out the instruction S(k) simply add a marble to the k^th bowl.
Transfer instructions.
For each j = 0,1,2,¼ and k = 0,1,2,¼ there is a transfer instruction T(j,k). The meaning of the instruction T(j,k) is to replace the content r[k] of the register R[k] with the number r[j]
contained in R[j], leaving all other registers unaltered. Thus to carry out the instruction T(j,k) take a spare bowl and put into it a quantity of marbles equal to the quantity of marbles in the
j^th bowl. Then replace the marbles in the k^th bowl with the marbles in the spare bowl.
Jump instructions.
The preceding three instructions are called arithmetical instructions, and they manipulate the quantity of marbles contained in the bowls. The final type of instruction is of a different sort and
determines how you process the program. For each j = 0,1,2,¼, k = 0,1,2,¼, and l = 0,1,2,¼ there is a jump instruction J(j,k,l). The meaning of the instruction J(j,k,l) is as follows^8:
1. if r[j] = r[k] proceed to the l^th instruction in the program
2. if r[j] ¹ r[k] proceed to the next instruction in the program
None of the contents of the registers is altered by a jump instruction.
That's all there is to it. The program P = I[0],I[1],I[2],¼,I[b] is executed by carrying out each instruction, one after the other, except in response to a jump instruction which may cause you to
jump to another instruction in the program. The execution of the program P ends when the required next instruction does not exist. For example, having executed the last instruction I[b] you proceed
to the next instruction I[b+1], but there is no such instruction. Alternatively, a jump instruction J(j,k,b+1) may indicate a jump to the instruction I[b+1], which does not exist.
Note: A program, containing only finitely many instructions, can refer to only finitely many registers^9, called the working registers. The contents of the remaining registers will not be changed
during the execution of the program, and the values in these registers will not affect the values in the working registers.
The setup we have been considering, consisting of registers (bowls) R[k] containing numbers (of marbles) r[k] and programs of instructions of the previously discussed type, is called an Unlimited
Register Machine or URM for short.^10 Using this approach you can compute any function which can be computed in any other way. In particular, you can compute the same functions as Turing machines
A program P has a finite specification by the instructions I[0],¼,I[b]. You can however execute the program with infinitely many possible initializations of the registers. The program thus represents
a uniform way for you to deal with the infinitely many cases you may be presented with.
Let's make the remark that a program P has no `purpose'and does not `ascertain' anything. It's just a list of instructions which you carry out if you decide to execute that program. It is we who give
the program a purpose according to the use we put the result we get when we complete the program.
A Sample Program
Let's show how to write a program P to add any two numbers (quantities of marbles).
P = J(2,1,4),S(0),S(2),J(0,0,0),Z(1),Z(2) (1)
The registers are initially all set to 0 except R[0] which contains m (marbles) and R[1] which contains n (marbles). The numbers r[0] and r[2] will be increased by 1 until r[1] = r[2], whereupon
registers R[1] and R[2] will be emptied (set to 0) and the program completed. (Note that the jump instruction J(0,0,0) has the effect of causing you to jump to the first (0^th) instruction, since r
[0] is always equal to r[0].) You will wind up with m+n (marbles) in the register R[0], and 0 (marbles) in all other registers. The way the computation goes adding 3 and 2 is shown in Figure 3. At
each step of the program, the state of the computation is described by the configuration of the registers^11 and the next instruction to be carried out. The computation halts when the next
instruction is I[6], since there is no such instruction.
R[0] R[1] R[2] R[3] next instruction
Figure3: The computation 3+2.
Another Sample Program
The only comparison of two numbers n and m which we use as part of our basic instructions is equality^12, n = m: whether or not n is the same as m. Another useful comparison is n £ m: whether or not
n is less than or equal to m. Here is a program to determine if n £ m. Initialize the registers to r[0] = n, r[1] = m, and all other registers initialized to 0. When the program is completed, the
register R[0] will contain 1 if n is less than or equal to m, or 0 if n is greater than m. All other registers will contain 0:
T(0,2),T(1,3),J(1,2,7),J(0,3,10),S(2),S(3),J(0,0,2), (2)
Z(0),S(0),J(0,0,11),Z(0),Z(1),Z(2),Z(3) (3)
The program works by first transferring a copy of n and m into registers R[2] and R[3]. Then the contents of those registers are successively increased until either n is equal to some successor of m,
in which case n > m, or m is equal to some successor of n, in which case n < m.^13 The register R[0] is then set appropriately to 0 or 1, and finally all other registers are set to 0.
Numbering All Programs
Programs are specified by giving their lists of instructions. When reasoning about programs and what can be achieved by using them, it is useful to specify the programs in another way, by giving a
rule to encode all the details of the program into a single number. Thus, if I give you a list of instructions making up a program, you will be able to compute a single number which will represent
that program. Conversely, if I give you a single number which represents a program, you will be able to decode the number and find the program to which it refers. There are many ways this could be
done, but whatever way I choose I must explain it to you, so you will understand which program I am referring to if I say `Execute program number 7.' Here is one way to number the programs: in order
to assign numbers to programs, we first assign numbers to instructions. The first step is to assign numbers to the instructions of each of the four sorts:
Zero instructions.
Assign numbers to the zero instructions by associating the instruction Z(k) with the number k.
Successor instructions.
Assign numbers to the successor instructions by associating the instruction S(k) with the number k.
Transfer instructions.
In order to assign numbers to the transfer instructions we must be able to encode pairs of numbers (j,k) by a single number. We associate^14 the pair of numbers (j,k) with the number n = f(j,k) =
2^j(2k+1)-1. Conversely, we can find j and k from n by j = f[1](n), k = f[2](n) where f[1](n) is the largest power j such that 2^j divides n+1, and:
é ù
f[2](n) = 1 2 ê n+1 2^f[1](n) -1 ú
ë û
Then the transfer instruction T(j,k) is associated with the number n = f(j,k).
Jump instructions.
In order to assign numbers to the jump instructions we must be able to encode triples of numbers (j,k,l) by a single number. In fact, once we have encoded pairs of numbers, we can encode triples,
quadruples, etc. We associate to the triple of numbers (j,k,l) the number m = g(j,k,l) = f(f(j,k),l). From m we can find j,k,l by l = f[2](m), j = f[1](f[1](m)), and k = f[2](f[1](m)).
Then the jump instruction J(j,k,l) is associated with the number m = g(j,k,l).
The next step is to combine the above encoding into a single encoding of all instructions:
Zero instructions.
The zero instruction Z(k) is associated with the number 4k.
Successor instructions.
The successor instruction S(k) is associated with the number 4k+1.
Transfer instructions.
The transfer instruction T(j,k) is associated with the number 4f(j,k)+2.
Jump instructions.
The jump instruction J(j,k,l) is associated with the number 4g(j,k,l)+3.
In this way given any instruction I we know how to find the corresponding number n, and conversely, given any number n we know how to find the corresponding instruction I.
In the final step we assign numbers to the programs P = I[0],I[1],¼,I[b]. Since we have assigned numbers to the individual instructions, we can associate with P the sequence of numbers m[0],m[1],¼,m
[b], where m[j] is the number assigned to the instruction I[j]. Thus we must encode all such sequences of numbers into a single number. One way to do this is to associate to the numbers m[0],m[1],¼,m
[b] the number:
This works because every number ³ 1 may be uniquely expressed as a sum of powers of 2 (which just gives the binary expression for the number). Hence if m[j] is the number associated with the
instruction I[j] in the program P, then P is associated with the number h(m[0],¼,m[b]).
We may explain formula (4) this way. Given the program P and the corresponding Gödel numbers m[0],m[1],¼,m[b] we compute h using the following prescription:
h(m[0],m[1],¼,m[b])[binary] = 1 1¼1 1 -1 (5)
0¼0m[b] 0¼0m[1] 0¼0m[0]
Here h is expressed in binary form. Conversely, given h(m[0],m[1],¼,m[b]) we express h(m[0],m[1],¼,m[b])+1 in binary form and read off m[0],m[1],¼,m[b] from the number of zeros between successive
Computing Gödel Numbers
The numbers associated with instructions and programs are called Gödel numbers, since Gödel first used the coding of non-numerical quantities by numbers. Today such coding is used all the time. For
example, a chess playing computer is running a program which codes the chess pieces, their positions on the board, and their allowed moves into numerical values and relations which can then be
manipulated as the game progresses.
Let's find the programs corresponding to some Gödel numbers:
1. The program with Gödel number 0: Since 0 = 2^0-1 we see from (2) that b = 0 and m[0] = 0. Thus the program has only one instruction and that instruction has number 0. The zero instruction Z(k)
has the number 4k, hence the instruction Z(0) has number 0. In conclusion, the program with Gödel number 0 is P[0] = Z(0). What would you accomplish by executing the program P[0]? Simply empty
the bowl R[0].
2. The program with Gödel number 1: Since 1 = 2^1-1 we see from (2) that b = 0 and m[0] = 1. The successor instruction S(k) has the number 4k+1, hence the instruction S(0) has number 1. In
conclusion, the program with Gödel number 1 is P[1] = S(0). What would you accomplish by executing the program P[1]? Simply add a marble to bowl R[0].
3. The program with Gödel number 2: Since 2 = 2^0+2^1-1 we see from (2) that b = 1 and m[0] = 0,m[1] = 0. Then P[2] = Z(0),Z(0). You will obtain precisely the same result by executing P[2] or P[0]
(but you empty bowl R[0] twice!).
4. The program with Gödel number 7: Since 7 = 2^3-1 we see that b = 0 and m[0] = 3. Thus the program consists of only one instruction and that instruction has number 3. This is the jump instruction
J(0,0,0). So P[7] = J(0,0,0). If you execute this program you will never halt. You will keep repeating the first instruction (which is to perform the first instruction) but you will make no
change to the contents of the registers!
Let's find the Gödel number of the program P (1) which adds any two numbers. First write the Gödel number of each instruction in the program:
Zero instructions.
Since the Gödel number of Z(k) is 4k, the Gödel number of Z(1) is 4 and the Gödel number of Z(2) is 8.
Successor instructions.
Since the Gödel number of S(k) is 4k+1, the Gödel number of S(0) is 1 and the Gödel number of S(2) is 9.
Jump instructions.
Since the Gödel number of J(j,k,l) is
4g(j,k,l)+3 = 4f(f(j,k),l)+3 ,
we compute the Gödel number of J(0,0,0) as 4f(f(0,0),0)+3. Since f(0,0) = 2^0(2·0+1)-1 = 0, we get 4f(0,0)+3 = 4·0+3 = 3. So the Gödel number of J(0,0,0) is 3. Similarly, the Gödel number of J
(2,1,4) is 4g(2,1,4)+3 = 4f(f(2,1),4)+3. Now f(2,1) = 11, so g(2,1,4) = f(f(2,1),4) = 2^11·9-1 = 2048·9-1 = 18431. So the Gödel number of J(2,1,4) is 73727.
The sequence of numbers corresponding to the instructions in P is thus:
Hence the Gödel number of the program P is:
h(73727,1,9,3) = 2^73727+2^73729+2^73739+2^73743+2^73748+2^73757-1
A very large number! Gödel numbers can be extremely large, but we use them as a theoretical device rather than a practical one.
The Marrakech Game
Let's pause for a moment and consider an analog of this process of Gödel numbering, where non-numerical quantities are coded into numbers. Here is an amusing way to code the game tic-tac-toe (noughts
and crosses) into numbers. First assign the numbers from 1 to 9 to squares of the board as shown in Figure 4.
Figure4: Assigning numbers to the squares of a tic-tac-toe board.
This is a magic square, where the numbers add to 15 along any row, column, or diagonal.
The first player begins the game by putting an X in one of the squares on the board. This is the same as choosing a number from 1 to 9. The second player now puts an O in one of the remaining
squares. This is the same as choosing one of the remaining numbers. Play proceeds like this until one player has marked all the squares in a row, column, or diagonal. This is the same as one player
having three numbers which add to 15.
We have thus reformulated the game of tic-tac-toe as an equivalent game involving a purely numerical procedure of choosing numbers from 1 to 9, the game being won when a player gets three numbers
which add to 15. It is described in Karl Fulves' Self-Working Number Magic [9] where it is called the marrakech game.
Of course, as numbers do not carry any meaning beyond their place in the succession of numbers, the marrakech game no longer has the same `meaning' as the original game. (The marrakech game is harder
to play than tic-tac-toe because the geometrical symmetries of the tic-tac-toe board and the geometry of the game play are not present in the marrakech game.)
3. The Barber Paradox
Imagine a small town with one barber who shaves all those and only those who do not shave themselves. Can there be such a barber? No, there can't. To see this consider whether or not the barber
shaves himself. If he doesn't shave himself then he must shave himself, and if he shaves himself then he must not shave himself. This paradox can be translated into mathematics and leads to a version
of Gödel's incompleteness theorem. A nice discussion of the application of this and other paradoxes to obtain limitations on what can be achieved using programs is given by Gregory Chaitin [5].
We have seen that all programs can be numbered: P[0],P[1],P[2],¼. Let's say that a number m is produced by a program P[n] if there is an initial value r[0] of register R[0] such that when all other
registers are initialized to 0 (r[k] = 0 for k ¹ 0) and the program executed, the program will be completed with m in register R[0] (and any numbers in the other registers). We will be interested in
the family of statements S[n] of the form `n is not produced by P[n]'. Which statements S[n] can be shown to be true? For some values of n, by reasoning about the sequence of instructions in P[n] you
might find a way of demonstrating that S[n] is true. For other values of n you might be able to demonstrate that S[n] is false. For still other values of n you might have no idea whether or not S[n]
is true (or even whether or not it is meaningful to ask if it is true). For example:
1. The program P[0] = Z(0). Hence no matter what the value r[0] initially in register R[0] this program will be completed after one step and the content of register R[0] will be 0. Thus 0 is the
only number produced by P[0]. Since 0 is produced by P[0] the statement S[0] is false.
2. The program P[1] = S(0). Hence if r[0] is initially in register R[0] this program will be completed after one step and the content of register R[0] will be r[0]+1. Thus all the numbers 1,2,3,¼
will be produced by P[1]. Since 1 is produced by P[1] the statement S[1] is false.
3. The program P[2] = Z(0),Z(0). This program produces only the number 0. Since 2 is not produced by P[2], the statement S[2] is true.
4. Let a be the number 2^73727+2^73729+2^73739+2^73743+2^73748+2^73757-1. Then P[a] is the program (1) which adds any two numbers. Hence if we set r[0] to any initial value and all other registers
to 0 (in particular r[1] = 0) then the result of executing P[a] will be r[0]+r[1] = r[0] in register R[0]. Thus P[a] produces all numbers. Since a is produced by P[a], the statement S[a] is
Perhaps if we rummage around among our programs P[0],P[1],¼, we might come across one which just happens to produce precisely the numbers n such that S[n] is true! Is this possible? Let's call a
program sound if all the numbers n that it produces are numbers of true statements S[n]. So might there be a sound program P[l] which actually produces the numbers of all the true statements? No,
there is no such program, and the reason could be thought of as coming out of the barber paradox.
To see that no such P[l] exists, suppose P[l] produces the number l. Then, given that P[l] is sound, it follows that S[l] is true, i.e. P[l] does not produce l. This contradiction means that a sound
program cannot produce its own Gödel number. Thus, given that P[l] is sound, it follows that P[l] does not produce l. Therefore S[l] is true. Hence the sound program P[l] cannot produce the number l
of the true statement S[l]. Hence we have our first main result:
│There is no sound program which can produce the numbers n of all true statements S[n].│ (6)
Our reasoning showed more, giving a particular true statement S[l] whose number is not produced by P[l]. This is our second main result:
│If the program P[l] is sound,then the statement S[l] is true and l is not produced by P[l].In this way, we can go beyond any program which is known to be sound,and obtain a true statement not│ (7)
│produced by the program. │
But it is not just the one statement S[l] which is missed by P[l], but infinitely many statements. This can be seen by observing that given the sound program P[l], we can find a program P[l^¢] which
produces precisely the numbers produced by P[l] together with the number l. We illustrate how to do this in the flow diagram, Figure 5:
Figure5: Flow diagram for the program P[l^¢ ].
The registers are initialized to 0 except R[0] which is set to r[0] = m. First test if m = 0 by comparing r[0] and r[1] (since r[1] = 0). If m = 0, add^15 l to r[0] and halt. If m ¹ 0, subtract^16 1
from r[0] and then execute the program P[l].
Consequently the program P[l^¢] is also sound and furthermore produces the number l, a number not produced by P[l]. But the previous argument applied to P[l^¢] shows that S[l^¢] is true but not
produced by P[l^¢] (and hence also not by P[l]). This is our third main result:
│For each sound program P[l], there is anotherprogram P[l^¢ ]which is also sound. The statements produced astrue by P[l^¢ ]are the statements produced as true by P[l]together with S[l],which │ (8)
│was not produced by P[l]. The true statement S[l^¢ ]is not produced by P[l^¢ ](nor by P[l]). │
We may now apply the preceding argument to P[l^¢] to obtain another program P[l^¢¢] which is again sound and which gives all the true statements given by P[l^¢] together with S[l^¢]. We may repeat
this process as often as desired, leading to our fourth main result:
│For each sound program P[l], there isa sequence P[l[1]],P[l[2]],¼ of programs, each of which is soundand each P[l[j+1]] outputs the same as the precedingprogram P[l[j]] together with l[j]. │ (9)
│For each P[l[j]] there remaininfinitely many true statements not obtained by P[l[j]]. │
In all of this discussion it is absolutely essential that the initial program P[l] is sound. Is there a method of obtaining all the sound programs P[l]? That is, is there a program P[k] which
produces precisely the numbers of all sound programs? No, there isn't. Indeed, if P[k] is such a program then if P[k] produces l then it follows from the preceding discussion that S[l] is true. Thus
P[k] is sound. It follows that P[k] doesn't produce k and hence P[k] cannot produce the numbers of all sound programs. Hence no such program exists. So we have obtained our fifth and final main
│There is no program P[k] which produces precisely the numbers of all sound programs.│ (10)
Given a program P, we can execute the instructions and determine one by one the numbers produced by P. Note that the numbers produced by P depend only on the instructions listed by P. If we choose
some Gödel numbering of the programs then the program P will be assigned some number, say 7, so P is P[7]. Now if P doesn't produce the number 7 then S[7] is true. But suppose we choose some other
Gödel numbering. Perhaps now some other program P^¢ is assigned the number 7, so P^¢ is now P[7]. If P^¢ produces the number 7 then S[7] is false. So whether or not the statement S[7] is true depends
on our choice of Gödel numbering. Consequently, whether or not a program is sound also depends on our choice of Gödel numbering. This shows again that programs do not have an inherent `purpose'. They
just produce numbers. It is we who interpret and utilize the numbers as we wish.
4. Gödel's Theorem
Figure6: Kurt Gödel, 1939. Reproduced by permission of the Institute for Advanced Study, Princeton, USA
In section 2 we defined the basic instructions and the programs composed of these instructions. I can explain to you very simply what you are to do when following each instruction and you may then
execute the program. It is all very concrete and involves moving marbles in a specific way. A chimpanzee could be trained to execute some programs. A mechanical contraption could be built which would
execute the programs, depending on how various levers were set. A present-day computer could be set up to execute any program if you type the instructions on the keyboard. There is no `logic' or
reasoning which forms part of the process of executing a program. On the other hand, in section 3 we have been reasoning about programs. This involves something different from the programs
themselves. What constitutes valid reasoning? Can we be absolutely precise as to what shall consistute an acceptable proof? In what language shall the reasoning take place?^17 It was with the
objective in mind of setting down a precise, consistent framework in which to conduct mathematical reasoning that formal mathematical systems were developed. A formal mathematical system is
formulated with a specified language, which consists of an alphabet of symbols, together with rules for writing down formulas. In addition, there are rules^18 which determine when a finite sequence
of formulas constitutes a proof, the final formula of the proof being the theorem which is proved. The most characteristic property of such systems is the mechanical nature of proof-checking, a
property emphasized by David Hilbert (1862-1943). Let's decide on a method^192 to encode formulas and finite sequences of formulas into (Gödel) numbers. Then the mechanical nature of proof-checking
means that there is a program P[l] which will produce precisely the Gödel numbers of proofs, and a program P[m] which will produce the Gödel numbers of the last formula in each proof. Every proof
will eventually be produced by P[l] and hence every theorem will eventually be produced by P[m].
When we use a formal mathematical system we have a particular interpretation in mind for the symbols and formulas. However, someone else may use the same formal system with a different
interpretation.^20 The formal system gives rules for writing symbols on paper. It has no intrinsic meaning and does not come with an interpretation. For this reason, Bertrand Russell once said
(quoted in chapter XI of [12]):
Mathematics may be defined as the subject in which we never know what we are talking about nor whether what we are saying is true.
and Henri Poincarè said (quoted in chapter XII of [12]):
Mathematics is the art of giving the same name to different things.
Once an interpretation is settled upon we can inquire as to whether or not a particular formula is true. You may use any interpretation you wish provided all the axioms of the formal system are true
in your interpretation. The formal system is said to be sound if its theorems are true no matter which interpretation you are using.^21
So suppose we have settled on a particular formal system to make deductions about the programs we have been considering. We interpret^22 the formulas as statements about programs. The statement S[n],
discussed in the preceding section, will be expressed by a formula f[n]. Then if S[n] is true we will say that the formula f[n] is true, and if S[n] is false we will say that the formula f[n] is
From the program P[m] which produces the Gödel numbers of all the theorems of the formal system, we can write down a program P[j] which produces the number n if and only if f[n] is a theorem. Here's
a prescription to do this. For given input, execute P[m]. If P[m] halts and produces the number k, then we know k is the Gödel number of a theorem. To find out if k corresponds to one of the formulas
f[n], we need to use the program P[M] which, given input n, produces the Gödel number of f[n]. We execute P[M] repeatedly with input 0, then 1, then 2, etc. comparing the number produced with k. If k
is not the Gödel number of some f[n] the program will not halt. If k is the Gödel number of f[n] then the program produces n as output. This prescription in words must be translated into a program P
[j]. This is easy to do once we have the programs P[m] and P[M]. A flow diagram for P[j] is shown^237 in Figure 7. Summarizing, from the program P[m] which produces the Gödel numbers of theorems of
our formal system, and the program P[M] which from input n produces the Gödel number of f[n], we can write down a program P[j] which produces the numbers n such that f[n] is a theorem.
Figure7: Flow diagram for the program P[j].
Let us now suppose that our formal system is sound, i.e. the theorems are true. Thus P[m] will produce only Gödel numbers of true formulas, and hence P[j] will produce only numbers n such that S[n]
is true. Thus P[j] is sound. It follows by (5) that the number j is not produced by P[j] and S[j] is true. Thus f[j] is true and not produced by P[m], i.e. f[j] is true but is not a theorem of the
formal system. This conclusion holds for any sound formal mathematical system with a language sufficiently broad to express the statements S[n]. We thus have the following version of Gödel's theorem:
│For any sufficiently broad sound formal mathematical system, we canexplicitly exhibit a true formula f which cannot be proved by the formalsystem. Furthermore, the negation of f is falseand │ (11)
│so also cannot be proved. │
Recall that the formula which we have exhibited is f[j], which expresses the statement S[j]: `j is not produced by P[j]'. But by the way P[j] was constructed, this is equivalent to the statement `f
[j] is not a theorem of F'. Thus the formula f[j] says of itself that it is not a theorem of F. And this is true!
How to Understand Gödel's Theorem
The version of Gödel's theorem (9) which we have discussed above says that the sort of formal system F (supposed sound) which mathematicians use to derive truths (say, about arithmetic) will not be
able to deduce all the true statements. Furthermore we can explicitly exhibit a true formula f which cannot be proved in the formal system F. How are we to understand this limitation to formal
systems? There are a number of viewpoints which can lead to insights into the significance of Gödel's theorem and even to new theorems concerning formal systems. Here are some ideas to think about
concerning Gödel's theorem:
1. We can explicitly exhibit a formula f[j] which is not a theorem of the formal system but is nevertheless true in our interpretation. However in another interpretation f[j] will be false.^24
An interpretation gives meaning to the formal system and this meaning leads us to truths not derivable from the formal system itself. This must be expected if a formal system can be given
different interpretations, and so different meanings and truths.
2. Gödel's theorem says that we cannot deduce as theorems all true formulas, from the limited set of true formulas which constitute the axioms. The axioms constitute a set of truths of limited
complexity and we cannot expect to derive truths of unlimited complexity from them. Chaitin (see [5]) puts it this way:
The complexity of the formal system ¼ is a measure of the amount of information the system contains, and hence the amount of information that can be derived from it. ¼ Gödel's theorem does not
appear to give cause for depression. Instead it seems simply to suggest that in order to progress, mathematicians, like investigators in other sciences, must search for new axioms.
If one has ten pounds of axioms and a twenty pound theorem, then that theorem cannot be derived from those axioms.
3. Using a formal system to deduce theorems (which are truths if the formal system is sound), you are acting as an oracle producing statements. Gödel's theorem describes a limitation which applies
not only to computational oracles (programs or formal systems) but to any oracle, regardless of its nature. An oracle which is known to be sound can never tell us everything we can ascertain.
Thus the Gödel phenomenon cannot be overcome by searching for a non-computational oracle. We will show how this limitation arises by analogy with our previous discussion.
An oracle in the form of a `black box', the interior of which we do not know, produces statements. The oracle is sound if all the statements are true. Now define^25 a device D which does the
following: if the oracle produces the statement D does not produce anything, then D produces the number 0 in response. If the oracle produces any other statement, D does not produce anything in
We may easily deduce that if the oracle is sound, then it will not produce the statement D does not produce anything, and furthermore it is true that D does not produce anything. Thus if we know
the oracle is sound, then we know a truth not produced by the oracle.
5. Penrose
We will rephrase Penrose's argument using the particular mathematical results we have developed in sections 3 and 4. We will try to adhere to the meaning of Penrose's argument, and thus follow the
moral of the following story related in section 2.10 of Shadows of the Mind:
I am reminded of a story concerning the great American physicist Richard Feynman. Apparently Feynman was explaining some idea to a student, but mis-stated it. When the student expressed
puzzlement, Feynman replied: `Don't listen to what I say; listen to what I mean!'
In Chapter 2, The Gödelian case, and Chapter 3, The case for non-computability in mathematical thought, Penrose argues that the human mind cannot be simulated by a computer:
I shall shortly be giving (in Chapters 2 and 3) some very strong reasons for believing that effects of (certain kinds of) understanding cannot be properly simulated in any kind of computational
terms... Thus, the human faculty of being able to `understand' is something that must be achieved by some non-computational activity of the brain or mind... The term `non-computational' here
refers to something beyond any kind of effective simulation by means of any computer based on the logical principles that underlie all the electronic or mechanical calculating devices of today.
Penrose puts forward his main argument in section 2.5 of Shadows of the Mind, and then deals with various counter-arguments in Chapters 2 and 3:
The argument I shall present in the next chapter (section 2.5) provides what I believe to be a very clear-cut argument for a non-computational ingredient in our conscious thinking... In due
course (in Chapters 2 and 3), I shall be addressing, in detail, all the different counter-arguments that have come to my attention.
The Gödelian Case
Consider again the problem of determining the true statements S[n], that is, the numbers n such that the n^th program does not produce n. According to the discussion in section 3 there is no sound
program which will produce the numbers n of all the true statements S[n]. Can the human intellect succeed in producing all the true statements S[n]? There is no evidence of this, so how good is the
human intellect at finding the true statements S[n]? Say that a program P[l] encapsulates human understanding if every number n of true statements S[n] that human mathematicians can produce is also
produced by P[l].
Question: Can human mathematical understanding concerning the statements S[n] be encapsulated in a sound program?
According to (5) if the l^th program is sound, then S[l] is true and l is not produced by P[l]. Does this mean that the answer to the question is no: human mathematical understanding concerning the
statements S[n] cannot be encapsulated in a program? Not quite! In order to go beyond the program P[l] we must know that P[l] is sound. If we know a program is sound call it knowably sound. Then we
can give an answer to the question as follows:
│Answer: Human mathematical understanding concerning the statementsS[n] cannot be encapsulated in a knowably sound program.│ (12)
This answer is given^26 by Penrose in his statement G in section 2.5 of Shadows of the Mind. This is basically all there is in section 2.5. Penrose would probably have preferred the answer:
Penrose's Preferred Answer: Human mathematical understanding concerning the statements S[n] cannot be encapsulated in a sound program.
but this has not been shown. Here is what Putnam says [16] about Penrose's argument so far:
What Penrose has shown is quite compatible with the claim that a computer program could in principle successfully simulate our mathematical capacities. The possibility exists that each of the
rules that a human mathematician explicitly relies on, or can be rationally persuaded to rely on, can be known to be sound and that the program generates all and only these rules but that the
program itself cannot be rendered sufficiently ``perspicuous'' for us to know that that is what it does. Actual programs sometimes consist of thousands of lines of code, and it can happen that by
the time a program has been tinkered with and debugged no one is really able to explain exactly how it works. A program which simulated the brain of an idealized mathematician might well consist
of hundreds of thousands (or millions or billions) of lines of code. Imagine it given in the form of a volume the size of the New York telephone book. Then we might not be able to appreciate it
in a perfectly conscious way, in the sense of understanding it or of being able to say whether it is plausible or implausible that it should output correct mathematical proofs and only correct
mathematical proofs.
© 1994 by The New York Times Co. Reprinted by permission.
In short, there may be a program P[l] which is sound and which produces precisely the same numbers n of true statements S[n] that human mathematicians can produce. But for this it is necessary that
human mathematicians are unable to ascertain that that is what P[l] does.
Do Mathematicians Use Unsound Reasoning?
Are the statements S[l] produced by human mathematicians as true really true? Might mathematicians claim that P[n] does not produce n when it actually does? If P[n] produces n this fact can in
principle be determined, simply by executing P[n] with all possible values for r[0], one after another, until n is produced.^27 Thus human mathematicians could eventually determine that they are
making mistakes. In Chapter 3 of Shadows of the Mind Penrose argues that this possibility is implausible:
I cannot really see that it is plausible that mathematicians are really using an unsound formal system F as the basis of their mathematical understandings and beliefs. I hope the reader will
indeed agree with me that whether or not such a consideration is possible, it is certainly not at all plausible.
But in fact there has been a loss of certainty in the soundness and completeness of mathematics, as Penrose acknowledges and discusses in sections 2.10 and 3.4 of Shadows of the Mind. Here are the
comments of several influential mathematicians:
I have told the story of this controversy in such detail, because I think that it constitutes the best caution against taking the immovable rigour of mathematics too much for granted. This
happened in our own lifetime, and I know myself how humiliatingly easy my own views regarding the absolute mathematical truth changed during this episode, and how they changed three times in
succession!... It is hardly possible to believe in the existence of an absolute, immutable concept of mathematical rigor, dissociated from all human experience.
John von Neumann (from [
Mathematics may be likened to a Promethean labor, full of life, energy and great wonder, yet containing the seed of an overwhelming self-doubt. It is good that only rarely do we pause to review
the situation and to set down our thoughts on these deepest questions. During the rest of our mathematical lives we watch and perhaps partake in the glorious procession.... This is our fate, to
live with doubts, to pursue a subject whose absoluteness we are not certain of, in short to realize that the only ``true'' science is itself of the same mortal, perhaps empirical, nature as all
other human undertakings.
I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many
mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of
mathematics, with more solid foundations than those that had hitherto been thought secure.... After some twenty years of very arduous toil, I came to the conclusion that there was nothing more
that I could do in the way of making mathematical knowledge indubitable.[19]... The splendid certainty which I had always hoped to find in mathematics was lost in a bewildering maze.[20]
Bertrand Russell
Only he who recognizes that he has nothing, that he cannot possess anything, that absolute certainty is unattainable, who completely resigns himself and sacrifices all, who gives everything, who
does not know anything, does not want anything and does not want to know anything, who abandons and neglects everything, he will receive all; to him the world of freedom opens, the world of
painless contemplation and of - nothing.
Unassailable Mathematical Beliefs
In Chapter 3 of Shadows of the Mind, Penrose considers beliefs rather than truths. Consider for example the result (5). If P[l] is sound then S[l] is true and P[l] does not produce l. Thus we know a
true statement S[l] not produced by the program P[l]. But this holds only if we know that P[l] is sound. We may very well not know for sure that P[l] is sound. Then we don't know for sure that S[l]
is true. On the other hand, if we believe that P[l] is sound (but we're not absolutely sure) then we believe (with the same level of confidence, on the basis of (5)) that S[l] is true and l is not
produced by P[l]. Then we will believe (with the same level of confidence, on the basis of (7)) that all the programs P[l],P[l[1]],P[l[2]],¼ are sound. Thus we may construct ever more comprehensive
programs which we believe to be sound with the same level of confidence that we believe P[l] to be sound. In summary:
We can go beyond any program P[l] believed to be sound, and obtain a statement S[l] which we believe to be true and not produced by P[l].
A similar argument can be given concerning formal mathematical systems. Take for example a formal system F of the sort we have considered earlier. If F is sound, so all its theorems are true, then we
can exhibit a formula f which is true (in our standard interpretation) but which is not a theorem of F. But again, we can only do this if we know for sure that F is sound. Perhaps we are not certain
that F is sound, but we believe it to be. (If we are using F to deduce mathematical truths we certainly will believe that we are using a sound formal system.) Then we believe that f is true (with the
same level of confidence, on the basis of (9)) and hence we can add f as a new axiom to the formal system F, obtaining a new formal system F[1]. We will then believe that F[1] is sound with the same
level of confidence that we believe F to be sound. We have not lost anything by broadening our formal system with the addition of the new axiom f. We may repeat this procedure, exhibiting a formula f
[1] which we believe to be true with the same level of confidence that we believe F[1] and F to be sound. Proceeding in this way we obtain a sequence of formal systems F,F[1],F[2],¼ each more
comprehensive than the preceding one (having an additional axiom which is not a theorem of the preceding formal system). We believe each of the formal systems F[j] to be sound with the same level of
confidence that we believe F is sound. This procedure was first studied by Turing in a paper [25], Systems of logic based on ordinals, published in 1939. It is still a subject of research. In
We can go beyond any sufficiently broad formal system F believed to be sound and obtain a formula which is believed true (in our standard interpretation) and not a theorem of F.
Penrose uses [14,15] this result as the basis for his argument against the computational modelling of mathematical understanding. But we will show in section 6 that his argument is mistaken.
6. A Computational Model for Thought?
Do the mathematical theorems on computation and formal systems have implications concerning the computational modelling of the human intellect? Here, we mean computational modelling of the sort that
is done when modelling a wide range of physical phenomena. Consider for example the steps involved in the computational modelling of the motion of a projectile:
1. Decide on which aspects of the phenomenon are to be modelled: in the case of the projectile, the position of the projectile at various moments of time.
2. Set up measuring instruments to code those aspects of the phenomenon into numerical quantities. (The meter readings serve as `Gödel numbers' which code those aspects of the phenomenon into
numbers.) In the case of the projectile, we set up rulers and clocks.
3. Set up a mathematical theory (in this case, based on Newton's Laws of Motion) to compute relationships between the instrument readings. In the case of the projectile, write a program such that
with the registers appropriately initialized, the program is completed with the register R[0] containing a number which should, if the computational model is a good one, be very close to the
number obtained by reading the ruler.
Let's invent a toy model, a naive computational model for the way the brain works, and let's see if any of the mathematical theorems we have discussed could be used to rule out such a model. We
follow the basic ideas set out in Chapters 1 and 2:
1. The functioning of the brain is described by the connections between neurons. The state of the brain at any time is described by the state of firing or non-firing of each of the neurons.
2. A mathematical model of the brain is constructed by associating with the j^th neuron a mathematical variable f[j](t) which takes the value 0 or 1, describing the state of the neuron at time t
(the time being measured in discrete steps). The value 1 corresponds to the neuron firing, and the value 0 to non-firing. The configuration of all the neurons at a given time is thus modelled by
all the values f[0](t),f[1](t),¼,f[N](t). The values f[0](t+1),f[1](t+1),¼,f[N](t+1) are determined by the values f[0](t),f[1](t),¼,f[N](t) together with a computational rule which models the
dynamical behaviour of the neural network. That is, we give a mathematical rule which enables us to calculate each f[j](t+1) given all the values f[0](t),¼,f[N](t). One such rule is described in
Chapters 1 and 2, in terms of weights which correspond to neural connections. The weights w[ji] are introduced, which take positive or negative integer values^28, and model the strength of the
connection from the i^th neuron to the j^th neuron. Then the computational rule governing the behavior of the mathematical model is:
where M[j] is a positive integer corresponding to the threshold of the j^th neuron.
Remark. External input can be incorporated into the model by `plugging in by hand' the values f[j](t) corresponding to the sensory neurons.
3. The preceding computational rule may be expressed^29 by a program P[l]. The program will be constructed so that by executing the program we perform the computations expressed in (11). We
initialize the registers with r[j] = f[j](t) (all others initialized to 0) and execute the program P[l]. When we complete the program the registers will contain r[j]^¢ = f[j](t+1), for j = 0,¼,N.
In this way by executing the program P[l] we may compute the values f[0](t),¼,f[N](t) for all (discrete) times t, and consequently the sequence of states of the neural network.
A Computational Model for the Mind?
The computational model of the brain discussed above may or may not give a good approximation to the actual behavior of a biological neural network. Assuming we could construct a good computational
model of the excitation states of the neurons in the brain, what would this model tell us about the thoughts and emotions experienced by the brain? In other words, what does the model tell us about
the mind? This depends on the nature of mental representation in the brain. In his book [8] Daniel Dennett begins Chapter 3, Brain Writing and Mind Reading, this way:
What are we to make of the popular notion that our brains are somehow libraries of our thoughts and beliefs? Is it in principle possible that brain scientists might one day know enough about the
workings of our brains to be able to ``crack the cerebral code'' and read our minds?
Let's imagine that there is a `dictionary' which can translate from our description (f[0](t),¼,f[N](t)) of the neuronal state of a brain to the beliefs which the mind holds at time t. Denote^30 these
beliefs by B[0](t),¼B[M(t)](t). These beliefs could be expressed in English (together with mathematical symbols), or in French, or in any formal language. Define some Gödel numbering so that each
belief B[j](t) is assigned a number and the collection of all beliefs at time t is assigned the number b(t). So, we will be able to decode the number b(t) and obtain all the beliefs B[0](t),¼, B[M
(t)](t). Suppose the dictionary is computational in the sense that there is a program P[d] such that if we initialize the registers to r[k] = f[k](t) then when we finish executing the program P[d]
the register r[0] will contain the Gödel belief number b(t).
Hence to compute the beliefs held by a brain at time t initialize the registers to r[k] = f[k](0), k = 0,¼,N. Then execute the program P[l]. When you complete the program you will have r[k]^¢ = f[k]
(1) in the registers. Continue these computations until you have obtained f[0](t),¼,f[N](t). Now execute the dictionary program P[d] with the registers initialized to r[k] = f[k](t). When you
complete the program P[d] you will obtain a number b(t) which encodes all the beliefs held by the brain at time t. The number b(t) could be decoded to yield all the beliefs B[0](t),¼B[N](t) at time t
stated in English. Combining the above steps, we can write a program P so that, initializing the registers to f[0](0),¼,f[N](0),t, then upon completing the program P the Gödel belief number b(t) will
be in the register R[0].
Gödel's Objection?
The above computational model of the brain is as straightforward and naive as one could imagine. But if there is no way to rule out, on the basis of the mathematical results discussed earlier, such a
simple model then, ipso facto, there is no way to rule out more complex and sophisticated computational models on the basis of those mathematical results. Let's use the term `computational mind' to
denote a mind whose beliefs could be computed in the above fashion.^31 We will show:
The Gödel and Turing theorems have absolutely nothing to say about the beliefs which may be held by a computational mind. Claims to the contrary are based on a mistaken application of those theorems.
Before discussing the details, consider the fact that people often hold contradictory beliefs without realizing it. Or they may believe completely foolish things, such as `The moon is made of green
cheese.' Why should such beliefs affect the consistency and correctness of your computation of those beliefs? Some mathematicians accept the logical principle of the law of the excluded middle,
others do not. Some may base their mathematical beliefs on modal logic, or on non-constructive methods, or on intuitionistic philosophy. Why should these varying sorts of beliefs affect the way the
brain works at the neuronal (or any other) level or how you compute those beliefs? They don't! Concerning the disagreements among mathematicians on a sound basis for developing mathematical beliefs,
consider this quotation from E.T. Bell (quoted in chapter XI of [12]):
Experience has taught most mathematicians that much that looks solid and satisfactory to one mathematical generation stands a fair chance of dissolving into cobwebs under the steadier scrutiny of
the next... Knowledge in any sense of a reasonably common agreement on the fundamentals of mathematics seems to be non-existent... The bald exhibition of the facts should suffice to establish the
one point of human significance, namely, that equally competent experts have disagreed and do now disagree on the simplest aspects of any reasoning which makes the slightest claim, implicit or
explicit, to universality, generality, or cogency.
The consistency, correctness, or otherwise of a person's beliefs has nothing whatever to do with the consistency or correctness of the formal system used to deduce that person's beliefs. And what
about thoughts in general? Think of something crazy. Something you believe is not, and could not possibly be, true. Should that crazy thing be a part of the deductive system forming a theory of mind?
Or is it only the things you believe to be true that should be theorems? Surely all thoughts and other aspects of mind should be treated in the same way. You can think about anything you like; it
won't affect the consistency or soundness of the theory of mind.
The Mistaken Application of Gödel's Theorem
Penrose uses Gödel's theorem in various arguments against the computational modelling of mathematical understanding throughout Chapter 3 of Shadows of the Mind, in particular in sections 3.2, 3.3,
3.14, and 3.16. All his arguments are subject to the criticism levelled at the argument below. Here is Penrose's reasoning about Gödel's theorem in a nutshell.
Suppose a human, who we call H, has a computational mind (associated with the program P), and that H is made aware of the program P which we are using to compute the properties of his brain. We may
use a formal mathematical system F to logically deduce the behavior of P, and hence the behavior, beliefs, and other thoughts of H. We may say that the system F encapsulates all the knowledge and
beliefs of H. Penrose argues that if H believes statement X, then since F encapsulates all of H's beliefs, it must be possible to prove X in the system F. Penrose then reasons that H will surely
believe that the system F is sound. Consequently, H will believe (by Gödel's theorem) that the formula f is true. But since, again by Gödel's theorem, f is not a theorem of F, it follows that F
cannot after all encapsulate all of H's beliefs.^32 By this contradiction Penrose concludes that H cannot in fact have a computational mind.
Penrose's confusion in this line of reasoning is italicized above. For if H believes X (on day 1) then it is necessary that F deduces that H believes X on day 1. It is not necessary that F deduces X.
If on day 2 H changes his mind and believes the negation of X, ØX, then it is necessary that F deduces that H believes ØX on day 2. It is not necessary that F deduces ØX. If on day 3 H goes crazy
then it is necessary that F deduces that H is crazy on day 3. It is not necessary that F deduces all sorts of crazy formulas.
In short, although F cannot deduce f, there is no reason why F cannot deduce that H believes f. Gödel's theorem doesn't say anything about what can be proved concerning the state of H's mind!
The essential point is that H's beliefs do not form part of the deductive system F. If H believes two contradictory statements X and Y, these statements are not theorems of F and so it does not
follow that F is inconsistent, as it would be if X and Y were theorems.^33
Let's explain in detail the distinction between X and H believes X, using our `toy' computational model. The statement X is one of H's beliefs, say B[k](t), and as such is encoded along with H's
other beliefs in the Gödel belief number b(t). Using the formal system F we can deduce a theorem which states that when the program P is run with registers initialized to r[k] = f[k](0) then when the
program is completed r[0] = b(t). There is a world of difference between the theorem r[0] = b(t) and the statements B[0](t),¼,B[N](t) encoded in the number b(t). The confusion undoubtedly arose
because the mathematical beliefs of H could be expressed in the same language used in the formal system F. No one would think of incorporating a non-mathematical belief, such as `It will rain
tomorrow', into the formal system F !
Perhaps you do not believe that a `mind reading' program can be found which will translate brain states into the beliefs (and other thoughts) of that brain. Consider any means which H may use to
communicate his beliefs to others. He may write them down with pencil and paper, or he may say them. Does Gödel's theorem impose any restrictions on what H may write down or say? No! By computing the
state of the motor neurons, using our program P[l], we can determine the hand movements which H will carry out, and hence the geometrical pattern of marks on paper which will result. Any pattern
whatsoever can be drawn by H. And if that pattern may be interpreted as a mathematical formula, that formula does not thereby become a theorem of the formal system F associated with P[l]. Similarly,
the way H's mouth and vocal cords move, which could be computed using P[l], are not restricted by Gödel's theorem. The theorem which would be proved using F is of the form r[j](t) = 1, indicating a
marble in the bowl R[j] and hence, by our modelling of H's brain, that the j^th neuron is firing. Translating the marks on paper into, say, an English sentence, that sentence plays the role of a term
, not the role of a theorem.
Consider a present day computer. What sort of expressions might it print out? Consider the formal system F associated with the program it is running. Suppose the formula f is not provable in F. Is
there any reason why the formula f cannot be printed out by the computer? No! Again, what comes out of the printer is not a theorem of F.
All this becomes clear when it is realized that what is printed out, or drawn, etc., need not satisfy any grammatical rules, which are required of theorems of F.
Related Ideas in the Literature
Our criticism of Penrose's use of Gödel's theorem rests on the idea that beliefs and other thoughts of a mind are not theorems of the theory of that mind. This idea can be found in the existing
6.0.1 John von Neumann
In August 1955 John von Neumann was diagnosed as having bone cancer. In April 1956 he was admitted to Walter Reed Hospital which he did not leave until his death on February 8, 1957. He took with him
to the hospital the manuscript of the Silliman Lectures (Yale University), which he hoped to be able to present. The title of the lectures was to be The Computer and the Brain. The unfinished
manuscript has been published as a book [27]. The last section of the book is entitled The Language of the Brain Not the Language of Mathematics, and the last lines of that section and of the book
are these:
When we talk mathematics, we may be discussing a secondary language, built on the primary language truly used by the central nervous system. Thus the outward forms of our mathematics are not
absolutely relevant from the point of view of evaluating what the mathematical or logical language truly used by the central nervous system is. However, the above remarks about reliability and
logical and arithmetical depth prove that whatever the system is, it cannot fail to differ considerably from what we consciously and explicitly consider as mathematics.
We may interpret this quotation as stating the view that the mathematical procedures and beliefs of a mind are of a different category from the theorems of the formal system F associated with a
computational model of the mind at the neuronal level. In short, mathematical beliefs are not theorems of F, which is the point of our criticism of Penrose's argument.
Robert Kirk
Robert Kirk, a philosopher from Nottingham University, published an article [11] in 1986 entitled Mental Machinery and Gödel. He levels the same criticism at Lucas's use of Gödel's theorem that we
have levelled at Penrose; reproduced by kind permission of Kluwer Academic Publishers:
If there could be adequate mechanistic accounts which represented a person by means of a formal system, yet did not correlate beliefs, thoughts or statements with that system's theorems or
`outputs', then the argument from Gödel fails. In fact, there is an approach which appears to satisfy this requirement. It deals in terms of the organism's physical states, inputs, and outputs,
all at some rather low level of specification - perhaps the neurophysiological - and in possible transitions from state to state through time.... Certainly mechanists will maintain that those
tokens of mathematical sentences which Alf produces are produced `mechanically'. However, in maintaining this they need not - and should not - say that the token sentences themselves correspond
to theorems of a formal system which adequately represents Alf and his part of the world.
7. Can Computers Think?
Let's finally return to our original question and see if the preceding considerations have gotten us any closer to an answer. We have seen that Penrose's argument against the computational modelling
of mental processes does not work, and that there is no obstruction to a computational mind, of the sort discussed in section 6, arising from the mathematical theorems on computation and formal
systems. We conclude that these theorems have not, in fact, gotten us any closer to an answer to the question:
Will computers of the future be intelligent?
Certainly present-day computers aren't intelligent.^34 We have quoted in section 1 several views that intelligent computers are a likely development in the not too distant future. Here is the vision
of the mathematical physicist David Ruelle, which he paints at the beginning of his book [18]:
Supercomputers will some day soon start competing with mathematicians and may well put them forever out of work.... Of course, present-day machines are useful mostly for rather repetitious and
somewhat stupid tasks. But there is no reason why they could not become more flexible and versatile, mimicking the intellectual processes of man, with tremendously greater speed and accuracy. In
this way, within fifty or a hundred years (or maybe two hundred), not only will computers help mathematicians with their work, but we shall see them take the initiative, introduce new and
fruitful definitions, make conjectures, and then obtain proofs of theorems far beyond human intellectual capabilities.
© 1991 by Princeton University Press. Reprinted with permission.
Some Thoughts about Thinking
We close with some thoughts about thinking, which you might like to think about. Consider again our computational model for the trajectory of a projectile. By executing the program to compute the
numbers read on the ruler, we are not ourselves moving along the trajectory of the projectile! And if a computer is set up to execute the program, it has not itself become a projectile. Although this
is painfully obvious, the implication in connection with models of the mind is perhaps that by setting up a computer to execute a program which constitutes a `computational model of the mind' (in the
sense considered in section 6) the computer is not thereby thinking ! Dennett, in Chapter 11 of his book [8], puts it this way:
It is never to the point in computer simulation that one's model be indistinguishable from the modelled. Consider, for instance, a good computer simulation of a hurricane, as might be devised by
meteorologists. One would not expect to get wet or wind-blown in its presence. That ludicrous expectation would be akin to a use-mention error, like cowering before the word ``lion''.
And Searle [22] puts it this way:
No one would suppose that we could produce milk and sugar by running a computer simulation of the formal sequences in lactation and photosynthesis, but where the mind is concerned many people are
willing to believe in such a miracle because of a deep and abiding dualism: the mind they suppose is a matter of formal processes and is independent of quite specific material causes in the way
that milk and sugar are not.
How might we build a computer which does think?
How to Build a Robot
Let's try to build a thinking robot using our URM. Our robot will have:
• eyes made of video cameras
• ears made of microphones
• a speaker for a voice box
• chemical sensors for a nose
• mechanical legs for moving about
• mechanical arms for touching and moving things
Have we left anything out? Oh yes, a brain. That's the crucial bit!
To make the brain, we'll use the URM. Take all the bowls and marbles, and your program P[l] which defines a good computational model of some particular brain, and sit down inside the robot. Now
suppose that when certain bowls have a marble in them (modelling an excited neuron) an action is performed by the robot (a small arm movement, for example). Suppose also that input from the video
cameras (or microphones, etc.) leads to certain other bowls having a marble dropped in them (corresponding to the excitation of sensory neurons). You now execute the program P[l]. Then, during each
time step t® t+1, the marbles in the bowls change (modelling the varying excitation states of the neurons in the brain) causing the robot to react to external stimuli, to speak through the speaker,
and so on. Of course, you must imagine yourself working faster and faster, and the bowls of marbles very small. In this way, the collection of bowls of marbles with you executing the program P[l]
becomes the brain of the robot.
The Mind Demon
Let's summarize the above discussion this way. A mechanical robot has a brain composed of a very large number of very small bowls containing marbles^35, with a tiny very fast working `mind demon'
executing a program P[l], changing the numbers of marbles in the bowls. The demon knows only how to execute the basic instructions of the URM. He understands nothing else.
Among the bowls are `input bowls' whose contents reflect properties of the outside world, and `output bowls' whose contents determine actions by the robot.
The sequence of bowl contents r[j](t) at times t = 0,1,2,¼, for j = 0,¼,N corresponds to the sequence of neuron firings of the j^th neuron in the brain which is modelled by the program P[l]. The
program P[l] corresponds to the connectivity of the neural network (through the weights w[ij]). The demon, executing the program P[l], corresponds to the electro-chemical forces which induce the
temporal development of the neural network (its varying neuronal firings).
Question. You know that you think^36; how close does this robot come to doing the same?
Searle's Chinese Room
John Searle [22] argues that although such a robot may listen to questions and stories and make sensible replies, there is no understanding:
Whatever purely formal principles you put into the computer, they will not be sufficient for understanding, since a human will be able to follow the formal principles without understanding
anything.... I will argue that in the literal sense the programmed computer understands what the car and the adding machine understand, namely, exactly nothing. The computer understanding is not
just (like my understanding of German) partial or incomplete; it is zero.
For `human' in this quote, substitute our `demon', whom we have agreed does not understand anything except the basic instructions required in the execution of the program P[l]. Remembering that we
said the demon corresponds^37 to the electro-chemical forces responsible for the temporal development of the neural network, we certainly would not ascribe `understanding' to those forces, nor to the
demon. But although the demon does not understand anything of the outside world, does it follow that the robot has no understanding?
Imagine the robot in China, and the program P[l] giving a good computational model of the brain of some Chinese person. The robot will function in China in the same way that the Chinese person does.
It will make replies, in Chinese, to questions asked of it by Chinese people. Searle imagines himself taking the part of the demon. He then compares his excellent understanding of English with his
total lack of understanding of Chinese, and concludes that also the robot could not `understand' Chinese. But again, the demon, or Searle-in-the-robot, is not where understanding is expected to
reside. Searle's reference to the understanding of English which Searle-in-the-robot possesses involves a category-mistake, as pointed out by Margaret Boden [1]:
Searle's description of the robot's pseudo-brain (that is, of Searle-in-the-robot) as understanding English involves a category-mistake comparable to treating the brain as the bearer - as opposed
to the causal basis - of intelligence.
The `Feel' of Thinking
Can our robot with its `bowls-of-marbles + demon' brain see a sunset the same way that we see it? Does it really think and understand in the same way that we do? As we don't expect to build a brain
in this way (where would we find the demon?) the question is not very important. Suppose, instead of the bowls of marbles, we use silicon transistors, and instead of the demon, we use electromagnetic
forces as they are exerted on the transistors of a silicon chip. Suppose, in other words, that we make a copy of a brain `in silicon', in such a way that the sequence of patterns of `on/off'
transistors induced by electromagnetic forces is the same as the sequence of patterns of `firing/non-firing' neurons induced by electrochemical forces. Would our robot, with such a silicon chip
central processing unit for a brain, see and think in the same way that we do? How would it be for the robot: Would it `feel' the same? Searle [22] doesn't think so:
The problem with the brain simulator is that it is simulating the wrong things about the brain. As long as it simulates only the formal structure of the sequence of neurone firings at the
synapses, it won't have simulated what matters about the brain, namely its causal properties, its ability to produce intentional states.
But recall Turing's test based on behavior. The physicist Niels Bohr echoes Turing's ideas this way [10]:
I am quite prepared to talk of the spiritual life of an electronic computer; to say that it is considering or that it is in a bad mood. What really matters is the unambiguous description of its
behaviour, which is what we observe. The question as to whether the machine really feels, or whether it merely looks as though it did, is absolutely as meaningless as to ask whether light is ``in
reality'' waves or particles. We must never forget that ``reality'' too is a human word just like ``wave'' or ``consciousness.'' Our task is to learn to use these words correctly - that is,
unambiguously and consistently.
Be that as it may, the `feel' of the functioning brain could very well depend on the detailed structure of the individual neurons, much as the tone of a violin depends on the quality of wood used and
the manufacturing techniques employed. So a computer's silicon brain might `feel' different from our biological brain, but nevertheless, it would be a thinking computer.
M.A. Boden, Escaping from the Chinese Room, in The Philosophy of Artificial Intelligence, ed. M.A. Boden, (Oxford University Press, 1990), pp.40-66
L. Boltzmann and B. McGuinness, Theoretical Physics and Philosophical Problems, vol.5 (D. Reidel Publishing Company, 1974) p.57
L.E.J. Brouwer, Life, Art and Mysticism 1905 exerpts; in Collected Works 1, ed. A. Heyting, (North-Holland, 1975)
J.N. Crossley, et.al., What is Mathematical Logic?, (Oxford University Press, 1978)
G. Chaitin, Gödel's Theorem and Information, International Journal of Theoretical Physics 22 (Plenum Publishing Corp., 1982) pp.941-954; reprinted in G.J. Chaitin, Information, Randomness, and
Incompleteness (World Scientific,1987)
P.J. Cohen, Comments on the Foundations of Set Theory, in Proceedings of the Symposia in Pure Mathematics Volume XIII Part I, (American Mathematical Society, 1971)
N.J. Cutland, Computability, (Cambridge University Press, 1980)
D.C. Dennett, Brainstorms, (Harvester Wheatsheaf, 1981)
K. Fulves, Self-Working Number Magic (Dover, 1983)
J. Kalckar, Niels Bohr and his youngest disciples; in S. Rozental, ed., Niels Bohr: His life and work as seen by his friends and colleagues, (North-Holland, 1967)
R. Kirk, Mental Machinery and Gödel, Synthese 66 (D. Reidel Publishing Company, 1986) pp.437-452
M. Kline, Mathematics: The Loss of Certainty, (Oxford University Press, 1982)
J. Lucas, Minds, Machines and Gödel, Philosophy, Vol.XXXVI (Cambridge University Press, 1961); reprinted in Minds and Machines, ed. A.R. Anderson, (Prentice Hall, 1964)
R. Penrose, Shadows of the Mind, (Vintage, 1995)
R. Penrose, Psyche, Volume 2, Number 1 (1996) pp.89-129
H. Putnam, The Best of all Possible Brains (Review of Shadows of the Mind by R. Penrose), New York Times, 20 November 1994.
H. Putnam, Mathematics without foundations, Journal of Philosophy 64 (Journal of Philosophy Inc., 1967) pp.5-22; reprinted in Philosophy of Mathematics, edited by P. Benacerraf and H. Putnam,
(Cambridge University Press, 1985)
D. Ruelle, Chance and Chaos, (Princeton University Press, 1991)
B. Russell, Portraits from Memory, Bertrand Russell Peace Foundation (George Allen and Unwin, 1956)
B. Russell, My Philosophical Development (Unwin, 1975).
A. Scott, Stairway to the Mind, (Copernicus, 1995)
J.R. Searle, Minds, Brains, and Programs, Behavioral and Brain Sciences 3 (Cambridge University Press, 1980) 417-24, reprinted in The Philosophy of Artificial Intelligence, ed. M.A. Boden,
(Oxford University Press, 1990), pp.67-88
J.C. Shepherdson and H.E. Sturgis, `Computability of Recursive Functions', J. Assoc. Computing Machinery 10 (1963) 217-255
A. Turing, Computing Machinery and Intelligence, Mind 59 No.236 (Oxford University Press, 1950) 433-460, reprinted in The Philosophy of Artificial Intelligence, ed. M.A. Boden, (Oxford University
Press, 1990), pp.40-66
A. Turing, Systems of Logic Based on Ordinals, P. Lond. Math. Soc. (2) 45 (1939) 161-228
J. von Neumann, The Mathematician. ``Works of the Mind.'' in Collected Works Volume 1, ed. A.H. Taub, (Pergamon Press, 1961)
J. von Neumann, The Computer and the Brain, (Yale University Press, 1958)
K. Warwick, March of the Machines, (Century, 1997)
^1 For more details see Intel's website
http://www.intel.com/pressroom/archive/releases /cn121796.htm
and Sandia's website
^2 In addition to Putnam's review, discussed here, an on-line version of Soloman Feferman's review of Shadows of the Mind can be found at
You can find various other criticisms of Penrose's approach, together with replies by Penrose, at
A large bibliography on the subject of `mind and metamathematics' can be found at
http://ling.ucsc.edu/ ~ chalmers/biblio4.html .2
which is Part 4: Philosophy of Artificial Intelligence of the Contemporary Philosophy of Mind: An Annotated Bibliography, at
http://ling.ucsc.edu/ ~ chalmers/biblio.html
^3 Putnam's own view on the nature of human mentality is indicated by the following quote from [17]:
Why should all truths, even all empirical truths, be discoverable by probabilistic automata (which is what I suspect we are) using a finite amount of observational material?
^4 All quotations from Turing's article in this chapter are reprinted by permission of Oxford University Press.
^5 In 1990 Dr. Hugh Loebner pledged a Grand Prize of $100,000 for the first computer whose responses were indistinguishable from a human's. Each year a prize of $2000 is awarded to the most human
computer, which this year was won on May 16, 1997 by the program Converse, entered by the University of Sheffield. The homepage of the Loebner prize is
http://acm.org/ ~ loebner/loebner-prize.htmlx
^6 On May 11, 1997 the IBM computer Deep Blue won a six game match against the world chess champion Garry Kasparov. For more details see
^7 The difficulty in defining each and every concept one uses is nicely put [2] by Ludwig Boltzmann (1844-1906); reprinted by kind permission of Kluwer Academic Publishers:
Let me begin by defining my position by way of a true story. While I was still at high school my brother (long since dead) tried often and in vain to convince me how absurd was my ideal of a
philosophy that clearly defined each concept at the time of introducing it. At last he succeeded as follows: during a lesson a certain philosophic work (I think by Hume) had been highly
recommended to us for its outstanding consistency. At once I asked for it at the library, where my brother accompanied me. The book was available only in the original English. I was taken aback,
since I knew not a word of English, but my brother objected at once: ``if the work fulfils your expectations, the language surely cannot matter, for then each word must in any case be clearly
defined before it is used.''
^8 To determine if the number of marbles in two bowls is the same, remove one marble from each bowl, then another marble from each bowl, and so on. If both bowls become empty at the same step then
they contained the same number of marbles. If only one bowl is empty then they contained different numbers. (Then replace the marbles in their respective bowls.)
^9 We do not allow instructions of the sort, for example, S(r[1]), which adds a marble to the bowl with number equal to the content of register R[1]. In fact, although we are using numbers as names
for the bowls, any symbols could be used for these names. They do not have the same quantitative significance as the content of a bowl. We also do not allow instructions of the sort J(1,3,r[0]) which
may cause a jump to the instruction with number equal to the content of register R[0]. Again, an instruction number and a number of marbles are two different types of things. One is an ordinal
number, representing an ordering (first, second, etc.), the other a cardinal number, representing an amount.
^10 The URM was invented by J.C.Shepherdson and H.E.Sturgis [23]. A nice discussion of the URM is given by N.J.Cutland [7].
^11 The working registers are R[0],R[1],R[2]; the others remain unchanged and do not influence the computation.
^12 Used explicitly as part of the jump instructions and implicitly as part of the transfer instructions.
^13 The case n = m is dealt with by the first application of J(1,2,7).
^14 See Cutland [7] for further discussion of this and other assignments.
^15 Here is a program to add l to any number (which is placed in register R[0], all other registers initialized to 0): S(0),S(0),¼,S(0) [l copies].
^16 Here is a program to subtract 1 from any number m (which is placed in register R[0], all other registers initialized to 0): S(1),J(0,1,5),S(1),S(2),J(0,0,1),T(2,0),Z(1),Z(2). If m ³ 1, when you
complete the execution of this program there will be m-1 in register R[0], all other registers containing 0. If m = 0, this program will not halt; i.e. you will never complete the program.
^17 Even if you could carry out your reasoning without language, you could not communicate to others how you arrived at your conclusions unless a common language is agreed.
^18 What rules of deduction should be included in the formal system? Is there only one correct logical system? Consider other areas of mathematics. What is the correct geometrical theory? The
classical geometry of Euclid was considered the only correct geometry for thousands of years, but now we consider the correct geometry to be an empirical question. The curved space-time of Einstein's
general relativity theory is considered a better mathematical model than the flat Euclidean geometry. What is the correct algebraic theory? The algebraic structure of a theory of observable
quantities had been supposed to satisfy xy = yx (commutativity) and x(y+z) = xy+xz (distributivity). But now it is considered that quantum theory, involving non-commutative observables, is a better
mathematical model of atomic phenomena. Quantum theory still keeps distributivity, but if the empirical data require it, then a further modification of the algebraic structure of the theory could be
made. What is the correct logic? Perhaps there is no one correct logic. It may also depend on the properties of the system being modelled. We have empirical geometry, empirical algebra, why not
empirical logic?
^19 We could use a method similar to the one used to encode programs, since programs are finite sequences of instructions and formulas are finite sequences of symbols - and proofs are finite
sequences of formulas. So first give numbers to each symbol in the alphabet of L. If f is the formula a[1]¼a[b], where m[j] is the Gödel number of the symbol a[j], then f is assigned the number h(m
[0],¼,m[b]), where the function h is given in (4). Similarly, if f[0],¼,f[b] is a finite sequence of formulas where m[j] is the Gödel number of the formula f[j], then the sequence of formulas is
assigned the number h(m[0],¼,m[b]).
^20 Different interpretations are often given to the same English sentence by different people. They may think they have a disagreement as to which statements are true, but actually they may simply
have assigned a different meaning to the same words! This is called a problem of semantics.
^21 Consequently, if the formal system is sound and if there is a formula f which is true in one interpretation and false in another, then neither f nor its negation can be a theorem of the formal
^22 Someone else may interpret the formulas in a completely different way, as statements about some other activity or physical process.
^23 We suppose that when P[m] halts, all its working registers other than R[0] contain 0. Similarly for P[M]. This can always be arranged. In Figure we take the working registers of P[M] to be R[k]
for k £ N[M].
^24 In our interpretation f[j] expresses the statement S[j], that the j^th program does not produce the number j. However, in another interpretation f[j] no longer has this meaning. The relation
between f[j] and S[j] is lost.
^25 The element of self-reference in the definition is inevitable. The oracle must be able to refer to itself or to something dependent on the oracle. As we have seen in the case of formal systems, a
formula can be written down which may be interpreted as stating that the self-same formula cannot be proved by the formal system.
^26 Recall the Feynman story above.
^27 On the other hand, if P[n] does not produce n but a mathematician claims that it does (without any indication as to what input would produce n), it may not be possible to prove him wrong. There
may be no way to determine the truth or falsity of S[n] and the law of the excluded middle (every statement is either true or false) may not be applicable.
^28 Positive values of w[ji] corresponds to exitation and negative values to inhibition. Given rational values for the weights, we can multiply through by a sufficiently large integer to rewrite the
equations with integer-valued weights.
^29 Negative integers need not enter directly into the computation. Use may be made, for example, of the difference function Dif(n,m) = n-m if n ³ m and = 0 if n < m.
^30 We shall naturally suppose that a (finite) brain can have only finitely many beliefs at any one time.
^31 What could be more computational than that?
^32 It may be that the formal system F does not satisfy the requirements to deduce Gödel's theorem for F. But the point of the argument is that there are statements which H believes to be true but
which are not theorems of the formal system F. This will undoubtedly be the case for mathematical statements of a sort which cannot even be expressed in the particular formal system F, and even more
so for non-mathematical beliefs.
^33 If H's beliefs are not theorems of F, what are they? The beliefs are encoded into the Gödel belief number b(t), which is a term of the formal language.
^34 They only think they are.
^35 To the reader with such a brain: `Don't lose your marbles!'
^36 Well, maybe.
^37 According to the construction of present-day computers, we could also say that the demon corresponds to the CPU (central processing unit) of the computer, which, while executing a program,
manipulates the numbers (on/off transistor patterns) stored in its registers and RAM memory locations.
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Patent application title: FULL SPATIAL DIMENSION EXTRACTION FOR IMPLICIT BEAMFORMING
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Techniques are provided to compute downlink beamforming weights for beamforming multiple spatial streams to a wireless device when that wireless device does not transmit with a maximum number of
spatial streams, and thus when the full dimensional knowledge of the wireless channel to that wireless device needs to be implicitly derived. Uplink signals are received at a plurality of antennas of
a first wireless device that are transmitted via a plurality of antennas of a second wireless device. The first wireless device derives values at a plurality of subcarriers of the received signals
across the plurality of antennas of the first wireless device. Downlink beamforming weights are computed from values of consecutive subcarriers across the plurality of antennas of the first wireless
device. The first wireless device applies the downlink beamforming weights at respective subcarriers to a number of spatial streams to be transmitted to the second wireless device.
A method comprising: receiving uplink signals at a plurality of antennas of a first wireless device that are transmitted via a plurality of antennas of a second wireless device; deriving values at a
plurality of subcarriers of the received uplink signals across the plurality of antennas of the first wireless device; computing downlink beamforming weights from values of consecutive subcarriers
across the plurality of antennas of the first wireless device; and applying the downlink beamforming weights at respective subcarriers to a number of spatial streams in a downlink transmission to be
transmitted to the second wireless device.
The method of claim 1, wherein receiving comprises receiving signals from the second wireless device that comprise a number of spatial streams less than a number of the plurality of antennas at the
second wireless device.
The method of claim 1, wherein computing comprises deriving full spatial information for a wireless channel between the first wireless device and the second wireless device when the second wireless
device uses a spatial expansion matrix to transmit spatial streams to the first wireless device.
The method of claim 1, wherein computing comprises computing the downlink beamforming weights at two consecutive subcarriers based on the values at the two consecutive subcarriers of the uplink
signals received at the plurality of antennas of the first wireless device.
The method of claim 1, wherein computing the downlink beamforming weights comprises computing a covariance matrix at each of three adjacent subcarriers, and computing the downlink beamforming weights
from the covariance matrices computed at the three adjacent subcarriers.
The method of claim 1, wherein computing the downlink beamforming weights comprises computing a covariance matrix at each of four adjacent subcarriers associated with a received uplink signals
transmitted using a cyclic shift diversity scheme with four antennas, and computing the downlink beamforming weights from the covariance matrices computed at the four adjacent subcarriers.
The method of claim 1, wherein computing the downlink beamforming weights is based further on a predetermined cyclic shift diversity amount to represent an amount of cyclic shift diversity performed
at the second wireless device.
The method of claim 1, and further comprising estimating cyclic shift diversity performed at the second wireless device based on values of the received uplink signals across the plurality of antennas
at the first wireless device at consecutive subcarriers.
The method of claim 8, and further comprising receiving a training pattern of known values from the second wireless device, and wherein estimating cyclic shift diversity comprises computing an
average with respect to values of the received uplink signals and the known values of the training pattern over the plurality of antennas of the first wireless device at the consecutive subcarriers.
The method of claim 9, wherein when the second wireless device has two antennas, receiving the training pattern comprises receiving three consecutive subcarriers of the received training pattern
across the plurality of antennas of the first wireless device, and computing a cyclic shift diversity angle θ from an average[(r
)], wherein s
, s
and s
are known values at three consecutive subcarriers of the training pattern, and r
, r
and r
are vectors representing the received values at the three consecutive subcarriers across the plurality of antennas of the first wireless device.
The method of claim 10, wherein receiving the training pattern comprises receiving a high-throughput long training field in accordance with the IEEE
11n wireless communication standard and amendments of the IEEE
11 wireless communication standard.
The method of claim 1, and further comprising sending the downlink transmission without applying the downlink beamforming weights when the number of spatial streams contained in the uplink signals
cannot be accurately estimated.
An apparatus comprising: a plurality of antennas; a receiver coupled to the plurality of antennas and configured to downconvert uplink signals sent by a wireless device and detected by the plurality
of antennas; a modem configured to derive values at a plurality of subcarriers of the received signals across the plurality of antennas and to obtain channel information at the subcarriers; a
processor coupled to the modem and configured to compute downlink beamforming weights from values of consecutive subcarriers across the plurality of antennas; wherein the modem is configured to apply
the downlink beamforming weights at respective subcarriers to a number of spatial streams in a downlink transmission to be transmitted to the wireless device.
The apparatus of claim 13, wherein the processor is configured to compute the downlink beamforming weights at two consecutive subcarriers based on the values at the two consecutive subcarriers of the
uplink signals received at the plurality of antennas.
The apparatus of claim 13, wherein the processor is configured to compute a covariance matrix at each of three adjacent subcarriers, and compute the downlink beamforming weights from the covariance
matrices computed at the three adjacent subcarriers.
The apparatus of claim 13, wherein the processor is configured to compute the downlink beamforming weights based further on a predetermined cyclic shift diversity amount to represent any cyclic shift
diversity performed at the wireless device.
The apparatus of claim 13, wherein the processor is configured to estimate cyclic shift diversity performed at the wireless device based on values of the received uplink signals across the plurality
of antennas at consecutive subcarriers.
The apparatus of claim 17, wherein the processor is configured to estimate cyclic shift diversity by computing an average with respect to values of the received signal and known values of a received
training pattern over the plurality of antennas at the consecutive subcarriers.
One or more computer readable storage media storing instructions that, when executed by a processor, are operable to: derive values at a plurality of subcarriers across a plurality of antennas from
received uplink signals from a wireless device; compute downlink beamforming weights from values of consecutive subcarriers across the plurality of antennas; and apply the downlink beamforming
weights at respective subcarriers to a number of spatial streams in a downlink transmission to be transmitted to the wireless device.
The computer readable storage media of claim 19, wherein the instructions operable to compute comprise instructions operable to compute the downlink beamforming weights at two consecutive subcarriers
based on the values at the two consecutive subcarriers of the uplink signals received at the plurality of antennas.
The computer readable storage media of claim 19, wherein the instructions operable to compute comprise instructions operable to compute a covariance matrix at each of three adjacent subcarriers, and
compute the downlink beamforming weights from the covariance matrices computed at the three adjacent subcarriers.
The computer readable storage media of claim 19, wherein the instructions operable to compute comprise instructions operable to compute the downlink beamforming weights based further on a
predetermined cyclic shift diversity amount to represent any cyclic shift diversity performed at the wireless device.
The computer readable storage media of claim 19, and further comprising instructions operable to estimate cyclic shift diversity performed at the wireless device based on values of the uplink signals
received across the plurality of antennas at consecutive subcarriers.
The computer readable storage media of claim 23, wherein the instructions operable to estimate the cyclic shift diversity comprise instructions operable to compute an average with respect to values
of the received uplink signals and known values of a received training pattern over the plurality of antennas at the consecutive subcarriers.
TECHNICAL FIELD [0001]
The present disclosure relates to multiple-input multiple-output (MIMO) wireless communication systems.
BACKGROUND [0002]
In MIMO wireless communication systems, multiple spatial streams are simultaneously transmitted via multiple antennas of a first wireless device to a second wireless device. When applying beamforming
weights to the signals that form the multiple spatial streams to be transmitted, knowledge of the channel information with respect to all of the antennas of the second wireless device is needed for
accurate computation of the beamforming weights.
One technique is to implicitly estimate beamforming weights from received signals. Another technique is to use a dedicated and separate training transmission sequence. Implicitly determining
beamforming weights is advantageous because it does not increase overhead and latency, and has minimal negative impact on channel coherency.
In order to implicitly determine downlink beamforming weights, it is necessary to receive as many uplink spatial streams as will be transmitted in the downlink to that destination device. There are
various occasions in which a destination device may not use its' full uplink spatial capability, e.g., when using cyclic shift diversity, making it more difficult to extract the full dimensional
spatial information needed.
BRIEF DESCRIPTION OF THE DRAWINGS [0005]
FIG. 1 is a block diagram depicting an example of a MIMO wireless communication system in which a first wireless device is configured to extract full MIMO wireless channel information from received
uplink streams transmitted by a second wireless device.
FIG. 2 is a block diagram showing an example of the first wireless device, e.g., a wireless access point device, configured to extract full MIMO wireless channel information from received uplink
streams from the second wireless device.
FIG. 3 is a flow chart illustrating in more detail examples of operations performed to compute the downlink beamforming weights.
FIG. 4 is a diagram depicting operations to compute downlink beamforming weights from two consecutive subcarriers.
FIG. 5 is a diagram depicting operations to compute downlink beamforming weights from three consecutive subcarriers.
FIG. 6 is a diagram depicting further details associated with the computations of the downlink beamforming weights according to the operations depicted in FIG. 4.
DESCRIPTION OF EXAMPLE EMBODIMENTS Overview [0011]
Techniques are provided to compute downlink beamforming weights for beamforming multiple spatial streams to a wireless device when that wireless device does not transmit with a maximum number of
spatial streams, and thus when the full dimensional knowledge of the wireless channel to that wireless device needs to be implicitly derived. Uplink signals are received at a plurality of antennas of
a first wireless device that are transmitted via a plurality of antennas of a second wireless device. The first wireless device derives values at a plurality of subcarriers of the received signals
across the plurality of antennas of the first wireless device. Downlink beamforming weights are computed from values of consecutive subcarriers across the plurality of antennas of the first wireless
device. The first wireless device applies the downlink beamforming weights at respective subcarriers to a number of spatial streams in a downlink transmission to be transmitted to the second wireless
Example Embodiments [0012]
Referring first to FIG. 1, a wireless communication system is shown generally at reference numeral 5. The system 5 comprises a first wireless communication device, e.g., an access point (AP) 10, and
at least one second wireless communication devices, e.g., a client device or mobile station (STA) 20. In practical deployments, the AP 10 serves a plurality of client devices but for simplicity only
a single client is shown in FIG. 1. The AP 10 comprises a plurality of antennas 12(1)-12(M), and the client device comprises a plurality of antennas 22(1)-22(P). The AP 10 may connect to other wired
data network facilities (not shown) and in that sense serves as a gateway or access point through which client device 20 has access to those data network facilities.
The AP 10 may wirelessly communicate with the client device 20 using a wireless communication protocol. An example of such a wireless communication protocol is the IEEE 802.11n communication
standard, known commercially as the WiFi® communication protocol. Another example of a suitable wireless communication protocol is the WiMAX® communication protocol. The IEEE 802.11n standard, for
example, employs multiple-input multiple-output (MIMO) wireless communication techniques in which the AP 10, for downlink transmissions to the client device 10, transmits M spatial streams
simultaneously via the M plurality of antennas 12(1)-12(M) using beamforming weights applied to the signal streams across the plurality of antennas. Similarly, for uplink transmissions to the AP 10,
the client device 20 may transmit N spatial streams simultaneously via its P plurality of antennas 22(1)-22(P). It is more common for the AP 10 to send multiple spatial streams to the client device
20 to improve the throughput to the client device.
In order to transmit multiple spatial streams to the client device, the AP 10 needs to determine the proper downlink beamforming weights for weighting the spatial streams across the plurality of
antennas 12(1)-12(M). The AP 10 derives information about the wireless channel between its antennas 12(1)-12(M) and the antennas 22(1)-22(P) of the client device based on uplink transmissions
received from the client device 20.
To directly derive sufficient channel information to transmit M spatial streams using MIMO techniques, the client needs to send the same number of spatial streams to the AP that the AP uses for
downlink transmissions to the client. This is not always the case for several reasons. The client device may not be able to support as many spatial streams on the uplink as the AP can support on the
downlink. Even if the client has the ability to support the same number of uplink streams as the number of downlink streams, at any given period of time, the client may select to transmit less
streams in the uplink for various reasons (lack of uplink data to send, reduced transmit power, poor channel conditions, etc.). When this occurs, the AP 10 cannot directly obtain the channel
information it needs to compute the beamforming weights for downlink transmissions of M spatial streams (more than the number of uplink spatial streams).
Accordingly, techniques are provided to enable the AP to extract the full dimensional channel information needed for beamforming M downlink data streams when the AP receives less than M uplink
streams from the client. This scheme will allow the AP to derive sufficient information for downlink beamforming transmission of a plurality of spatial streams when any number, N, of streams is sent
on the uplink. The AP extracts full channel information from received N uplink streams for use in beamforming M downlink streams.
For example, even when the client device sends one uplink spatial stream, the full spatial information can be derived from the received uplink transmission when the client uses all of its antennas
for sending the one uplink spatial stream, for example, when the client device employs a spatial expansion matrix.
Reference is now made to FIG. 2 that shows an example of a block diagram of AP 10 that is configured to perform the implicit beamforming weight derivation techniques for downlink transmissions of
multiple spatial streams. As shown in FIG. 2, the AP 10 comprises a radio receiver 14, a radio transmitter 16, a modem 18, a controller 20 and a memory 22. The radio receiver 14 downconverts signals
detected by the plurality of antennas 12(1)-12(M) and supplies antenna-specific receive signals to the modem 18. The receiver 14 may comprise a plurality of individual receiver circuits, each for a
corresponding one of a plurality of antennas 12(1)-12(M) and which outputs a receive signal associated with a signal detected by a respective one of the plurality of antennas 12(1)-12(M). For
simplicity, these individual receiver circuits are not shown. Similarly, the transmitter 16 upconverts antenna-specific baseband transmit signals (weighted from application of beamforming weights) to
corresponding ones of the plurality of antennas 12(1)-12(M) for transmission. Likewise, the transmitter 16 may comprise individual transmitter circuits that supply respective upconverted signals to
corresponding ones of a plurality of antennas 12(1)-12(M) for transmission. For simplicity, these individual transmitter circuits are not shown in FIG. 2.
The controller 20 supplies data to the modem 18 to be transmitted and processes data recovered by the modem 18 from received signals. In addition, the controller 18 performs other transmit and
receive control functionality. It should be understood that there are analog-to-digital converters (ADCs) and digital-to-analog converters (DACs) in the various signal paths to convert between analog
and digital signals.
The memory 22 stores data used for the techniques described herein. The memory 22 may be separate or part of the controller 20. In addition, instructions for implicit beamforming weight computation
process logic 100 may be stored in the memory 22 for execution by the controller 20. The controller 20 supplies the beamforming weights to the modem 18 and the modem 18 applies the beamforming
weights signal streams to be transmitted to produce a plurality of weighted antenna-specific transmit signals that are upconverted by the transmitter 16 for transmission by corresponding ones of the
plurality of antennas 12(1)-12(M).
The memory 22 is a memory device that may comprise read only memory (ROM), random access memory (RAM), magnetic disk storage media devices, optical storage media devices, flash memory devices,
electrical, optical, or other physical/tangible (non-transitory) memory storage devices. The controller 20 is, for example, a microprocessor or microcontroller that executes instructions for the
process logic 100 stored in memory 22. Thus, in general, the memory 22 may comprise one or more computer readable storage media (e.g., a memory device) encoded with software comprising computer
executable instructions and when the software is executed (by the controller 20) it is operable to perform the operations described herein in connection with process logic 100.
The functions of the controller 20 may be implemented by logic encoded in one or more tangible media (e.g., embedded logic such as an application specific integrated circuit, digital signal processor
instructions, software that is executed by a processor, etc.), wherein the memory 22 stores data used for the computations described herein (and/or to store software or processor instructions that
are executed to carry out the computations described herein). Thus, the process logic 100 may be implemented with fixed logic or programmable logic (e.g., software/computer instructions executed by a
processor) and the controller 22 may be a programmable processor, programmable digital logic (e.g., field programmable gate array) or an application specific integrated circuit (ASIC) that comprises
fixed digital logic, or a combination thereof. Some or all of the controller functions described herein, such as those in connection with the process logic 100, may be implemented in the modem 18.
Reference is now made to FIG. 3 for a general description of the implicit downlink beamforming derivation techniques. Channel estimates (channel state information) from received Orthogonal Frequency
Division Multiplexed (OFDM) symbols of the received uplink signals are used to derive the downlink beamforming weights. At 110, signals in an uplink transmission from P plurality of antennas of the
client device are received at the M plurality of antennas of the AP. The uplink signals received at 110 may comprise signals for a number of spatial streams less than a number of antennas at the
client device (and thus less than the number of spatial streams that the AP could send to the client device).
At 120, values at a plurality of subcarriers of the received signals across the plurality of antennas of the first wireless device are derived from the received signals. At 130, the channel
information at each subcarrier (across the M plurality of antennas) is obtained using any of a variety of direct, extrapolation and interpolation techniques known in the art.
At 140, downlink beamforming weights are computed from values of consecutive subcarriers across the plurality of antennas of the AP. Examples of techniques useful for the computations made at 140 are
described hereinafter in connection with FIGS. 4 and 5.
At 150, the downlink beamforming weights are applied to a number of spatial streams in a downlink transmission to be transmitted to the client device.
Reference is now made to FIG. 4 for a description of a first technique for computing the downlink beamforming weights from values at two consecutive subcarriers of a receive signal across the
antennas of the AP. According to this technique, the downlink beamforming weights are determined based on the values at two consecutive subcarriers of the uplink signals received at the plurality of
antennas of the AP. The vector quantity r
denotes the value of a receive signal at tone i across the plurality of antennas of the AP. Thus, the dimension of vector r
is equal to M (the number of antennas of the AP). The vector quantity r
+1 denotes a vector for a tone that is consecutively adjacent to tone i.
The received signals for any two consecutive tones (in the case of single stream transmission) using cyclic shift diversity (CSD) are: r
and r
+1, where r
is the received column vector (size M) at tone i to all receive antennas, h
is a column channel vector with respect to the transmit antenna i of the client device (h
has size M), α is the CSD rotation and s
and s
+1 are the transmitted values at subcarriers i and i+1 respectively.
The following is defined:
and λ
are respectively an eigenvector and an eigenvalue of Coν
and λ
are respectively an eigenvector and an eigenvalue of Coν
+1, where Coν is the covariance matrix operation.
The channel vectors h
and h
may be derived from the following set of equations:
[ λ 1 w 1 = h ^ 1 + h ^ 2 λ 2 w 2 = h ^ 1 + α h ^ 2 ##EQU00001##
where w[1]
; w
jθ, θ=angle (ν
) and h
and h
are estimated channel vectors. These equations are the linear estimation of the channel vectors via eigenvectors.
An angle with respect to the signals at the two consecutive tones is defined as: ang_r=angle(r
+1), wherein ' denotes the conjugate operation and * denotes the multiplication operation. The goal is to obtain a covariance matrix G whose singular value decomposition (SVD) vectors are the sought
after downlink beamforming weights G=(h
'). Estimated channel vectors h
and h
can be computed as h
)/(1-e.sup.(jπ/4)) and h
)/(1-e.sup.(-jπ/4)). The covariance matrix at each tone is denoted Coν
, Coν
. The eigenvectors and eigenvalues of the covariance matrices are easy to derive because each covariance matrix is a rank deficient matrix (made by one vector). Specifically, the eigenvector of this
matrix can be obtained without a SVD computation. The eigenvector is one normalized column of the respective covariance matrix. Thus, there is no need to calculate all elements of each covariance
matrix and its eigenvalue is the norm of the vector.
According to this method, the estimated channel vectors h
are computed and the beamforming vectors are derived from these vectors using, for example, a Gram-Schmidt orthogonalizing vector. In this way there is no need to calculate matrix G and to perform a
SVD process to derive a downlink beamforming weight vector.
Reference is now made to FIG. 5 for a description of a second technique for computing the downlink beamforming weight vectors from values at three consecutive subcarriers of a received signal across
the antennas of the AP. In this technique, a covariance matrix is computed at each of three adjacent subcarriers, and the downlink beamforming weights are computed from the covariance matrix computed
at the three adjacent subcarriers. The three subcarriers are arbitrarily denoted i, i+1 and i+2. The received signals for any three consecutive subcarriers are:
[i] r[i]
The covariance matrix for each subcarrier is computed as:
.su- p.H+h
[1]^H Co
h.su- b.2
[1]^H Co
h.su- b.2
.s- up.H
where h[i]
is, as described above, a column vector from the transmit antenna i of the client device.
These three equations may be rewritten as:
where A
[2]^H B
[1]^H G
[2]^H [0038]
Solving the equations above for G results in G=Coν
- {square root over (2)}Coν
+2. Thus, the eigenvectors of the matrix G can be derived from the covariance matrices at the respective subcarriers.
Reference is now made to FIG. 6 for a further description of the underlying derivation of the method depicted by FIG. 4 described above. FIG. 6 shows a complex domain (real-imaginary). By applying
SVD on Coν
a single SVD vector of h
can be obtained. Similarly for Coν
+1, h
is obtained. The channel vectors h
and h
can be obtained by solving the set of equations for h
and h
. However, due to non-uniqueness of SVD vectors, there is a random phase between the SVD of Coν
and Coν
+1. The random phase is not recoverable. Before solving the linear set of equations, w
is multiplied with a correction phase term φ
. As shown in FIG. 6, an estimate is applied so that above set of equation can be solved with little effect from the random phase. When rotating h
by 45°, h
rotates between 0 to 45 degrees depending on the magnitude of h
and h
. While this rotation is unknown, on average it is 22.5° or π/8. Thus, in one embodiment, φ
is set equal to 22.5° or π/8. There are various other correction phase values that may be used as well. The SVD of Coν
+1 can be forced to have phase difference of π/8 with respect to the SVD of Coν
. In general, for any CSD rotation of α, the phase correction term is sqrt(α).
Generalization to any Number of Antennas at Transmitting Device
The techniques described above can be generalized to a CSD scheme employed with any number of antennas. The following describes such a generalization where the transmitting device (client device)
uses a CSD scheme with four antennas. This generalization is applicable to 2- and 3-antenna CSD schemes as well.
The notation 1, 2, 3, 4 is used to denote the respective adjacent subcarriers instead of i, i+1, i+2, i+3 for sake of simplicity. The received signals are:
[1] r[2]
.gam- ma.)s
[2] r[3]
e.su- p.j2γ)s
[3] r[4]
e.su- p.j3γ)s
[4] where
α, β and γ are the phases associated with CSDs. For r
, r
, r
, and r
, covariance matrices, denoted by Coν
, Coν
, Coν
and Coν
are obtained.
The largest eigenvector associated with Coν
gives a vector representing: g
. The largest eigenvector associated with Coν
gives a vector representing: g
γ. The largest eigenvector associated with Coν
gives a vector representing: g
e.sup- .j2γ. The largest eigenvector associated with the Coν
gives a vector representing: g
e.sup- .j3γ.
Having vectors g
, g
, g
and g
, the resulting linear set of equations can be solved to obtain h
, h
, h
and h
. However, before solving the linear set of equations, vectors g
, g
and g
are multiplied with some correction phase terms. Vectors g
, g
and g
are multiplied with correction phase φ
, φ
and φ
respectively, which in one embodiment, are:
There are various other correction phase values that may be used as well
While the foregoing is described for 4 antennas, when used for 2 and 3 antennas, zero vectors would be computed for non-existing antennas, i.e., for 3 antennas h
would be a zero vector (or close to zero in magnitude).
CSD Estimation Techniques
As explained above, the downlink beamforming weights may be computed based on a predetermined (fixed) CSD amount to represent CSD performed at the client device (without specific knowledge of the CSD
amount used by that client device). The concepts described above in which particular values are used for CSD may be generalized for any CSD value. For example, while IEEE 802.11 specifics CSD values
to be used by client devices, it is possible that a client device or chipset vendors may deviate from the recommended values. Accordingly, a technique is now described to estimate the CSD value used
by a client device for its transmissions and then to use the estimated CSD in the beamforming derivation techniques presented above in connection with FIGS. 4 and 5. Moreover, these CSD estimation
techniques can also be used for other methods that require estimation of CSD before performing their full spatial dimension extraction. These CSD estimation techniques do not require a larger
coherence bandwidth.
The CSD performed at a particular client device can be estimated base on values of the received uplink signals across the plurality of antennas of the AP. In this method, the High Throughput-Long
Training Field (HT-LTF) portion of the received signal (transmitted in accordance with the IEEE 802.11n wireless communication standard and amendments of the IEEE 802.11 wireless communication
standard) is used to estimate CSD. Three consecutive subcarriers of the received HT-LTF signal are denoted:
[1] r[2]
[2] r[3]
[3] where s[1]
, s
, and s
are the HT-LTF signals in the three consecutive subcarriers and they are known at the receiver, r
, r
and r
are vectors representing the received values at the three consecutive subcarriers across the plurality of antennas and θ=e
α. The channel is assumed to be all white Gaussian noise (AWGN), but this is not shown for simplicity.
A set of three equations and three unknowns (h
, h
, θ) is obtained. Solving these equations for θ gives the following: θ=average[(r
)] where average is over the received signal of all antennas at the AP and over all sets of three consecutive subcarriers. Alternatively, other statistical methods can be used to average over set of
receive antennas and all tones. A similar technique can be used to estimate CSD when the transmitting device transmits from three and four-antenna cases.
In summary, the AP receives a training pattern of known values from the client device, and estimates the CSD by computing an average with respect to values of the received signal and the known values
of the training pattern of the plurality of antennas at the AP at the consecutive subcarriers.
Further still, techniques may be employed to estimate the number of antennas of the transmitting client device. In other words, an estimation is made as to the maximum number of spatial streams ever
used by that client device, or for each candidate number of antennas, the CSD estimation techniques described above are performed and the number of antennas that solves the foregoing equations
(perhaps with minimum degrees of freedom) is chosen as the estimate of the number of antennas of that client device.
An example of a procedure to estimate the number of antennas of the transmitting client device is as follows. First, it is assumed that the number of receive antennas at the client device is the same
as maximum number of spatial streams that it can support. This can be obtained from messages each client device sends to the AP at the time of association (contained in, for example, the
High-Throughput Capabilities Element of the IEEE 802.11n protocol). The AP continues using this assumption as the number of receive antennas at the client device. If the AP determines that
beamforming performance to this client device is worse than what it expects, then the AP continues with the following. The AP first assumes that the client has less than four transmit antennas, then
the AP uses the foregoing generalized algorithm for up to four antennas. If the estimated channel for one or more antennas has a low magnitude compared to the rest of the antennas, the AP may
conclude that the low-magnitude antennas are in fact non-existent antennas (the AP averages the magnitude over several observations). The AP then updates its assumption of the number of the antennas
at the client device. The above two steps include almost all practical scenarios (i.e., the client devices which have four or less receiver chains). However, if the AP still determines that the
beamforming performance is worse than what it expects, then the client device may have more than four antennas. In this case, the AP continues without downlink beamforming to this client device (when
the number of spatial streams contained in the uplink signals cannot be accurately determined/estimated).
In summary, the techniques described herein allow for deriving the full downlink spatial information in an implicit manner even when the uplink information is designed to reveal channel information
for a lesser number of spatial streams. This allows the downlink and uplinks to support unequal spatial multiplexing and yet maintain improved capacity. These techniques are useful when the client
device sends uplink transmissions using all of its antennas but with a less number of spatial streams that would otherwise directly reveal the full channel information needed for downlink
transmissions with a greater number of spatial streams. These techniques are applicable to any multiple spatial stream MIMO communication standard.
The above description is intended by way of example only. Various modifications and structural changes may be made therein without departing from the scope of the concepts described herein and within
the scope and range of equivalents of the claims.
Patent applications by Ahmadreza Hedayat, Allen, TX US
Patent applications by Matthew A. Silverman, Shaker Heights, OH US
Patent applications by Mohammad Janani, Plano, TX US
Patent applications by Cisco Technology, Inc.
Patent applications in class Modems (data sets)
Patent applications in all subclasses Modems (data sets)
User Contributions:
Comment about this patent or add new information about this topic: | {"url":"http://www.faqs.org/patents/app/20130128935","timestamp":"2014-04-24T14:39:50Z","content_type":null,"content_length":"69787","record_id":"<urn:uuid:cbc29645-70e4-41b2-8c6d-6b30779ce664>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00286-ip-10-147-4-33.ec2.internal.warc.gz"} |
at University of Wisconsin Oshkosh
Physical Geology
at University of Wisconsin Oshkosh
Implementor(s): Jennifer Wenner, Christie Demosthenous, Kate Kramer
Enrollment: 140-180
Challenges to using math in geoscience
University of Wisconsin Oshkosh is a four-year comprehensive university located in NE Wisconsin and serves about 12,000 students who mostly come from within a 100-150 mile radius; over 50% of our
students are first-generation college students. Wisconsin High Schools have standards that require students to take only two years of mathematics (Algebra and Geometry); thus, a large number of these
students place into remedial mathematics courses (College Algebra). We perceive that
The Math You Need, When You Need It
can provide a way of "leveling the playing field" for all students in introductory geoscience. In addition, it has helped us to reduce the amount of time that we spend teaching to the "lowest common
denominator" by allowing us to have the expectation that students come to lab prepared to do the mathematics needed for that particular exercise.
More about your geoscience course
Physical Geology is the first in a sequence of two general education courses designed to satisfy the lab science general education requirement for BS majors. Of the 140-180 students who enroll in
Physical Geology each semester, many report that they choose it because it is seen as a mostly descriptive science - "rocks for jocks" or that it seemed like the least scary of the lab sciences. When
students are asked to complete basic mathematical problems in the laboratory sections, we often hear the phrase "This is geology, why do I have to do math?". We also have many students who, because
of the perception of geology as a descriptive science, are surprised that they need to know basic mathematics to perform science. On the other hand, there are also many students in Physical Geology
who are perfectly capable of performing higher level mathematics and find the math we ask them to do menial. The range of abilities has presented many challenges to instructors of introductory
geoscience at Oshkosh.
Inclusion of quantitative content pre-TMYN
Before we implemented
The Math You Need,
instructors of Physical Geology addressed the quantitative content covered in the course on an ad hoc basis. Because a variety of instructors teach both the course and the labs, we tried a number of
strategies. We have spent valuable class/lab time reviewing mathematics, assigned pre-lab homework (that often had little support attached to it for the math, e.g., a half page worksheet that
required students to do unit conversions with little introduction to the mathematics) and used office hours or lab time to help students one-on-one. Prior to implementing The Math You Need, we would
spend a significant portion of at least 6 labs going over the basic mathematics.
Which Math You Need Modules will/do you use in your course?
• Density
• Graphing
• Plotting Points
• Best Fit Line
• Rearranging Equations
• Unit Conversions
Strategies for successfully implementing The Math You Need
As stated above, we think of this program as giving the instructor and students a base level of mathematics that they can be expected to perform in our labs, thus we require The Math You Need from
every student.
Because students at UWO do not perform tasks unless they see them as benefiting them or increasing their grade, we have always counted module completion as a portion of their lab grade. We also
reinforce the reasoning behind using these modules by reminding students often about the support it provides and reinforcing connections between the modules and geoscience in class and lab kdflsdghk.
The number of modules used varies from instructor to instructor but we have used Density, Plotting Points, Constructing a Best-Fit Line, Rearranging Equations, and Unit Conversions. Modules are
always attached to a given lab with reference to exercises in that lab. Instructors who use The Math You Need administer the pre-test within the first week of classes and then introduce an initial
module (usually rearranging equations) to be due before the first lab (in the second week of class). Anecdotally, instructors indicate that the strength of The Math You Need lies in being able to
refer back to it and knowing that all of the students are expected to be familiar with the content of each module. Furthermore, students are suggesting that they feel like it helps them with the
course itself.
Reflections and Results (after implementing)
In general the use of TMYN has been a success at UW Oshkosh. Over the three semesters that we have used The Math You Need, we have had anywhere from 84-95% of students in our large lecture courses
fully engage and participate in the online resources. Instructors feel that the success lies in the fact that they can refer to the modules and students are familiar with the tools provided there.
Students have improved scores on quantitative tasks by up to 30 percentage points from the pre-test to the post-module quizzes. There are two aspects of the program that we see as strengths for our
students: 1) the use of this across the entire course allows all students to see what is expected of them in terms of math and 2) it gives the instructor some class time back to use for geologic
problem solving, instead of mathematics remediation.
We have implemented The Math You Need three times over the past two years and have modified it each time. When we started using The Math You Need, we only allowed students to take the post-module
quizzes once and gave them full credit for just taking the quiz. However, in the second and third implementations, we felt that our goal was mastery of the material so that students would come to lab
completely prepared to do the mathematics, thus we allowed students to take the quizzes until they were satisfied with their scores (within a time limit). Students seemed to like this and the
instructors saw an increase in preparedness for the lab. | {"url":"http://serc.carleton.edu/mathyouneed/implementations/47577.html","timestamp":"2014-04-17T22:01:33Z","content_type":null,"content_length":"26816","record_id":"<urn:uuid:d99cc7a1-d08c-4cdc-a62c-8b634322cf8a>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00422-ip-10-147-4-33.ec2.internal.warc.gz"} |
How does one map regression depth to undirected depth of a point?
up vote 1 down vote favorite
The regression depth of a line is the minimum number of points it has to cross to take it from its initial position to vertical. The undirected depth of a point is the minimum number of lines a ray
originating at the point will cross before escaping. If we use projective duality, then regression depth is same as undirected depth. I cannot see how that would be.
My attempt is as follows: By duality, points will map to lines, and lines will map to points. Suppose I am trying to compute the undirected depth of a point. So there is a point such that all rays
emanating from it will meet n/3 points or more in each direction. In the dual plane, the point will coincide with a line. But what will rays correspond to?
EDIT: Quoting from paper by Nina Amenta, Marshall Bern et al, "Regression Depth and Center Points": Geometrically, the regression depth of a hyperplane is the minimum number of points intersected by
the hyperplane as it undergoes any continuous motion taking it from its initial position to vertical. In the dual setting of hyperplane arrangements, the undirected depth of a point in an arrangement
is the minimum number of hyperplanes touched by or parallel to a ray originating at the point. Standard techniques of projective duality transform any statement about regression depth to a
mathematically equivalent statement about undirected depth and vice versa.
I was trying to come up with this duality but havent so far succeeded. In R_2, when we consider lines and points, I was considering a point that has high undirected depth, but am unable to make a
transformation that would show exactly how the undirected depth of the point will map to the regression depth of a line in the dual plane.
1 There is presumably some background arrangement of lines and points going on that makes this question make sense, but you really need to be more precise about this. – Qiaochu Yuan Apr 30 '10 at
I have added the exact definition to be clearer. I am trying to come up with the transformation that would show how regression depth will map to undirected depth and vice versa. – Umar Sheikh Apr
30 '10 at 19:38
add comment
2 Answers
active oldest votes
(For some context: the question here is written in terms of points in a plane, and my answer is in the same terms, but the same duality works in higher dimensions as well. The paper
in question is arxiv:cs.CG/9809037.)
Let P be the plane in which you are measuring regression depth; regression depth is defined in a Euclidean plane, not a projective one, but we view P as being a Euclidean plane
embedded as a subset of its projective completion P*, so that a line is vertical iff it passes through the point at vertical infinity (the intersection point in P* of two vertical
up vote 0 down
vote accepted Now take a dual projective plane Q*, and again view Q* as containing a Euclidean plane Q and a line at infinity; choose Q and Q* in such a way that the line at infinity in Q* is dual
to the point at vertical infinity in P*. (It's easy enough to do this with coordinates, if you prefer them to this sort of conceptual explanation.)
Then rotating a line in P until it is vertical is the same thing as moving the projectively dual point in Q until it reaches infinity, and the number of points crossed by the
rotating line is the same as the number of lines crossed by the moving point.
add comment
The construction probably goes like this.
Let $l$ be the line whose regression depth you are evaluating, and le $l'$ be its final (vertical) position after some continuous motion. Now assuming $l \ne l'$, there is some unique
point $p$ that they intersect in.
up vote 0 Without loss of generality now, we can assume that the continuous motion taking $l$ to $l'$ is a rotation centered at $p$, since it will sweep the same area. Let $S$ be the set of lines
down vote in the double wedge between $l$ and $l'$ with apex at $p$.
The dual of a double wedge is a line segment. Construct the dual of $l$ and $l'$. these are two points that mark the endpoints of the line segment. By duality, this line segment
intersects the duals of all points in the wedge and is the desired ray originating at a point (which is the dual point of $l$)
Your answer is very easy to follow without defining or worrying about projective vs euclidean plane. It also resolves what a ray is. it is a line segment whose dual is a wedge. – Umar
Sheikh May 1 '10 at 0:14
add comment
Not the answer you're looking for? Browse other questions tagged geometry or ask your own question. | {"url":"http://mathoverflow.net/questions/23136/how-does-one-map-regression-depth-to-undirected-depth-of-a-point","timestamp":"2014-04-19T05:14:09Z","content_type":null,"content_length":"59091","record_id":"<urn:uuid:e94b0f40-6a4d-4cdc-b4cb-85ff9168be8d>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00052-ip-10-147-4-33.ec2.internal.warc.gz"} |
Law of Cosines or Cosine Rule (with worked solutions & videos)
Law of Cosines / Cosine Rule
In this lesson, we will learn:
• the Law of Cosines
• how to use the Law of Cosines when given two sides and an included angle
• how to use the Law of Cosines when given three sides
• how to proof the Law of Cosines
• how to solve applications or word problems using the Law of Cosine
Related Topics:
More Lessons on Trigonometry
Law of Cosines
The Law of Cosines relates the lengths of the sides of a triangle with the cosine of one of its angles.
The Law of Cosines, for any triangle ABC is
a ^2 = b ^2 + c ^2 – 2bc cos A
b ^2 = a ^2 + c ^2 – 2ac cos B
c ^2 = a ^2 + b ^2 – 2ab cos C
The Law of Cosines is also sometimes called the Cosine Rule or Cosine Formula.
If we are given two sides and an included angle (SAS) or three sides (SSS) then we can use the Law of Cosines to solve the triangle ie. to find all the unknown sides and angles.
We can use the Law of Sines to solve triangles when we are given two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA).
Law of Cosines: Given two sides and an included-angle
Solve triangle PQR in which p = 6.5 cm, q = 7.4 cm and ∠R = 58°.
Using the Cosine rule,
r^2 = p^2 + q^2– 2pq cos R
r^2 = (6.5)^2 + (7.4)^2– 2(6.5)(7.4) cos58°
= 46.03
r = 6.78 cm
Using the Sine rule,
∠Q = 180° – 58° – 54.39°
= 67.61°
∠P = 54.39°, ∠Q = 67.61° and r = 6.78 cm
Law of Cosines: Given three sides
In triangle ABC, a = 9 cm, b = 10 cm and c = 13 cm. Find the size of the largest angle.
The largest angle is the one facing the longest side, i.e. C.
c^2 = a^2 + b^2– 2ab cos C
= 0.067
∠C = 86.2°
The following video gives two examples of how to use the cosine rule. One given SAS and the other given SSS. Using Law of Cosines to solve triangles given 3 sides (SSS) and 2 sides and the angle in
between (SAS).
The following video shows how to solve an oblique triangle given SSS using the Law of Cosines.
Proof of the Law of Cosines
The following video shows how to prove the law of cosines by using coordinate geometry and the Pythagorean theorem.
Applications using the Law of Cosines
The video shows how to solve two word problems using the Law of Cosines
Example 1: An engineering firm decides to bid on a proposed tunnel through a mountain. Find how long is the tunnel and the bid amount.
Example 2: Find the range of service of a transmission tower.
Use the Law of Cosines to determine the length across the lake.
Use the Law of Cosines to find the distance a plane has traveled after a change in direction.
Use the Law of Cosines to determine the length of a diagonal of a parallelogram.
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Tits Geometry, Arithmetic Groups, and the Proof of a Conjecture of Siegel
Enrico Leuzinger
Enrico Leuzinger
Mathematisches Institut II
Universität Karlsruhe
Englerstr. 2
D-76131 Karlsruhe
Abstract: Let $X = G/K$ be a Riemannian symmetric space of noncompact type and of rank $\geq 2$. An irreducible, non-uniform lattice $\Gamma\subset G$ in the isometry group of $X$ is arithmetic and
gives rise to a locally symmetric space $V=\Gamma\backslash X$. Let $\pi:X\rightarrow V$ be the canonical projection. Reduction theory for arithmetic groups provides a dissection $V=\coprod_{i=1}^k \
pi(X_i)$ with $\pi(X_0)$ compact and such that the restiction of $\pi$ to $X_i$ is injective for each $i$. In this paper we complete reduction theory by focusing on metric properties of the sets
$X_i$. We detect subsets $C_i$ of $X_i$ (${\Bbb Q}$--Weyl chambers) such that $\pi_{\mid C_i}$ is an isometry and such that $C_i$ is a net in $X_i$. This result is then used to prove a conjecture of
C.L. Siegel. We also show that $V$ is quasi-isometric to the Euclidean cone over a finite simplicial complex and study the Tits geometry of $V$.
Full text of the article:
Electronic version published on: 1 Sep 2004. This page was last modified: 1 Sep 2004.
© 2004 Heldermann Verlag
© 2004 ELibM and FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition | {"url":"http://www.kurims.kyoto-u.ac.jp/EMIS/journals/JLT/vol.14_no.2/6.html","timestamp":"2014-04-17T18:37:02Z","content_type":null,"content_length":"4040","record_id":"<urn:uuid:ba9a4e1d-e536-4eb7-bffa-cab7dcf629f9>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00197-ip-10-147-4-33.ec2.internal.warc.gz"} |
Scheduling Algorithms
Results 1 - 10 of 30
- Algorithmica , 1996
"... Given a set of n tasks and m resources, where each task x has a rational weight x:w = x:e=x:p; 0 < x:w < 1, a periodic schedule is one that allocates a resource to a task x for exactly x:e time
units in each interval [x:p k; x:p (k + 1)) for all k 2 N. We de ne a notion of proportionate progress, ca ..."
Cited by 242 (25 self)
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Given a set of n tasks and m resources, where each task x has a rational weight x:w = x:e=x:p; 0 < x:w < 1, a periodic schedule is one that allocates a resource to a task x for exactly x:e time units
in each interval [x:p k; x:p (k + 1)) for all k 2 N. We de ne a notion of proportionate progress, called P-fairness, and use it to design an e cient algorithm which solves the periodic scheduling
problem. Keywords: Euclid's algorithm, fairness, network ow, periodic scheduling, resource allocation.
- In Proceedings of the 9th International Parallel Processing Symposium
"... Given n periodic tasks, each characterized by an execution requirement and a period, and m identical copies of a resource, the periodic scheduling problem is concerned with generating a schedule
for the n tasks on the m resources. We present an algorithm that schedules every feasible instance of t ..."
Cited by 101 (15 self)
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Given n periodic tasks, each characterized by an execution requirement and a period, and m identical copies of a resource, the periodic scheduling problem is concerned with generating a schedule for
the n tasks on the m resources. We present an algorithm that schedules every feasible instance of the periodic scheduling problem, and runs in O(minfm lg n; ng) time per slot scheduled. 1
Introduction Given a set \Gamma of n tasks, where each task x is characterized by two integer parameters x:e and x:p, and m identical copies of a resource, a periodic schedule is one that allocates a
resource to each task x in \Gamma for exactly x:e time units in each interval [k \Delta x:p; (k+1) \Delta x:p) for all k in N, subject to the following constraints: Constraint 1: A resource can only
be allocated to a task for an entire "slot" of time, where for each i in N slot i is the unit interval from time i to time i + 1. Constraint 2: No task may be allocated more than one copy of the
resource ...
, 2001
"... The scheduling of systems of periodic tasks upon multiprocessor platforms is considered. ..."
, 1993
"... It is now more than a quarter of a century since researchers started publishing papers on mapping strategies for distributing computation across the computation resource of multiprocessor
systems. There exists a large body of literature on the subject, but there is no commonly-accepted framework ..."
Cited by 79 (1 self)
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It is now more than a quarter of a century since researchers started publishing papers on mapping strategies for distributing computation across the computation resource of multiprocessor systems.
There exists a large body of literature on the subject, but there is no commonly-accepted framework whereby results in the field can be compared. Nor is it always easy to assess the relevance of a
new result to a particular problem. Furthermore, changes in parallel computing technology have made some of the earlier work of less relevance to current multiprocessor systems. Versions of the
mapping problem are classified, and research in the field is considered in terms of its relevance to the problem of programming currently available hardware in the form of a distributed memory
multiple instruction stream multiple data stream computer: a multicomputer.
- HANDBOOK ON SCHEDULING ALGORITHMS, METHODS, AND MODELS , 2004
"... ..."
, 2000
"... Traditional multiprocessor real-time scheduling partitions a task set and applies uniprocessor scheduling on each processor. For architectures where the penalty of migration is low, such as
uniform-memory access shared-memory multiprocessors, the non-partitioned method becomes a viable alternative. ..."
Cited by 37 (2 self)
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Traditional multiprocessor real-time scheduling partitions a task set and applies uniprocessor scheduling on each processor. For architectures where the penalty of migration is low, such as
uniform-memory access shared-memory multiprocessors, the non-partitioned method becomes a viable alternative. By allowing a task to resume on another processor than the task was preempted on, some
task sets can be scheduled where the partitioned method fails. We address fixed-priority scheduling of periodically arriving tasks on Ñ equally powerful processors having a non-partitioned ready
queue. We propose a new priorityassignment scheme for the non-partitioned method. Using an extensive simulation study, we show that the priorityassignment scheme has equivalent performance to the
best existing partitioning algorithms, and outperforms existing fixed-priority assignment schemes for the non-partitioned method. We also propose a dispatcher for the nonpartitioned method which
reduces the number of preemptions to levels below the best partitioning schemes.
, 1995
"... The issue of temporal fairness in periodic real-time scheduling is considered. It is argued that such fairness is often a desirable characteristic in real-time schedules. A concrete criterion
for temporal fairness -- pfairness -- is described. The weight-monotonic scheduling algorithm, a static prio ..."
Cited by 26 (3 self)
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The issue of temporal fairness in periodic real-time scheduling is considered. It is argued that such fairness is often a desirable characteristic in real-time schedules. A concrete criterion for
temporal fairness -- pfairness -- is described. The weight-monotonic scheduling algorithm, a static priority scheduling algorithm for generating pfair schedules, is presented and proven correct. A
feasibility test is presented which, if satisfied by a system of periodic tasks, ensures that the weight-monotonic scheduling algorithm will schedule the system in a pfair manner.
- ACM COMPUTING SURVEYS , 2011
"... This survey covers hard real-time scheduling algorithms and schedulability analysis techniques for homogeneous multiprocessor systems. It reviews the key results in this field from its origins
in the late 1960s to the latest research published in late 2009. The survey outlines fundamental results ab ..."
Cited by 23 (4 self)
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This survey covers hard real-time scheduling algorithms and schedulability analysis techniques for homogeneous multiprocessor systems. It reviews the key results in this field from its origins in the
late 1960s to the latest research published in late 2009. The survey outlines fundamental results about multiprocessor real-time scheduling that hold independent of the scheduling algorithms
employed. It provides a taxonomy of the different scheduling methods, and considers the various performance metrics that can be used for comparison purposes. A detailed review is provided covering
partitioned, global, and hybrid scheduling algorithms, approaches to resource sharing, and the latest results from empirical investigations. The survey identifies open issues, key research
challenges, and likely productive research directions.
, 2009
"... This paper addresses the problem of priority assignment in multiprocessor real-time systems using global fixed task-priority pre-emptive scheduling. In this paper, we prove that Audsley’s
Optimal Priority Assignment (OPA) algorithm, originally devised for uniprocessor scheduling, is applicable to th ..."
Cited by 19 (9 self)
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This paper addresses the problem of priority assignment in multiprocessor real-time systems using global fixed task-priority pre-emptive scheduling. In this paper, we prove that Audsley’s Optimal
Priority Assignment (OPA) algorithm, originally devised for uniprocessor scheduling, is applicable to the multiprocessor case, provided that three conditions hold with respect to the schedulability
tests used. Our empirical investigations show that the combination of optimal priority assignment policy and a simple compatible schedulability test is highly effective, in terms of the number of
tasksets deemed to be schedulable. We also examine the performance of heuristic priority assignment policies such as Deadline Monotonic, and an extension of the TkC priority assignment policy called
DkC that can be used with any schedulability test. Here we find that Deadline Monotonic priority assignment has relatively poor performance in the multiprocessor case, while DkC priority assignment
is highly effective. 1.
"... This survey covers hard real-time scheduling algorithms and schedulability analysis techniques for homogeneous multiprocessor systems. It reviews the key results in this field from its origins
in the late 1960’s to the latest research published in late 2009. The survey outlines fundamental results a ..."
Cited by 13 (0 self)
Add to MetaCart
This survey covers hard real-time scheduling algorithms and schedulability analysis techniques for homogeneous multiprocessor systems. It reviews the key results in this field from its origins in the
late 1960’s to the latest research published in late 2009. The survey outlines fundamental results about multiprocessor realtime scheduling that hold independent of the scheduling algorithms
employed. It provides a taxonomy of the different scheduling methods, and considers the various performance metrics that can be used for comparison purposes. A detailed review is provided covering
partitioned, global, and hybrid scheduling algorithms, approaches to resource sharing, and the latest results from empirical investigations. The survey identifies open | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=103124","timestamp":"2014-04-16T22:32:48Z","content_type":null,"content_length":"36680","record_id":"<urn:uuid:e9d0f57e-6865-4bcd-8ca7-7cdafe921e81>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00590-ip-10-147-4-33.ec2.internal.warc.gz"} |
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I am not able to understand field between oppositely charged parallel conducting plates using Gauss law.Can anyone explain?
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Yea.. its quite easy... first and fore most.. you need to understand what Gauss law is.. i want you to look at a link that ll post in a while.. explanation by Professor Walter Lewin!
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http://www.youtube.com/watch?v=XaaP1bWFjDA m not able to copy at the particular time unfortunately.. but start watching from exactly 23.51.. if you know what gauss law is.. else watch the whole
video from beginning!
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p.s. after that, proceed to use the pillbox surface and Gauss law to derive the E field on one parallel plate.
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Function composition and raising functions to a power.
December 8th 2011, 09:32 PM #1
Super Member
Dec 2009
Function composition and raising functions to a power.
$f(x)=\frac{1}{1-x}$ (x all real numbers excluding 0 and 1)
$g(x)=1-\frac{1}{x}$ (x all real numbers excluding 0 and 1)
a) Express $fg$ in a similar form and hence describe the relationship between $f$ and $g.$
b) Evaluate $f^{2011}g^{1994}(\frac{1}{2})$
I was able to express $fg$ and $fg=x$, whereby (x all real numbers excluding 0 and 1) but was unable to find the relationship between $f$ and $g$
for part b, I couldn't figure out how to start...
Re: Functions
Function composition is denoted by $f\circ g$, or f o g in plain text. Some sources may skip the circle, but I would expect them to note this convention explicitly. In any case, it is much better
to write a bit more explanations than to let people wonder about notation.
You showed that $(f\circ g)(x) = x$, which means that g is the (right) inverse of f. This is the relationship between the functions. There are many other relationships, e.g., $(g\circ f)(x) = x$,
$f(x) = (g\circ g)(x)$, $g(x)=(f\circ f)(x)$, $f(x)=1-\frac{1}{g(x)}$, $g(x)=f(x+1)+1$, etc., but apparently they are not relevant to the question (except maybe the first).
Note also that both sides of equality should have the same type: either functions or real numbers. For example, $f\circ g$ is a function (the composition of f and g), while x is a number. So,
writing $f\circ g=x$ is not the best. One should write $(f\circ g)(x)=x$ or $f\circ g=\mathrm{id}$ where $\mathrm{id}(x)=x$ is the identity function.
Since $f(g(x))=x$, you can cancel 1994 applications $f(g(\dots))$ in $f^{2011}g^{1994}(1/2)$ and get $f^{2011-1994}(1/2)$. In evaluating this, note also that $f^3(x)=x$ (this can be verified
directly; it also follows from $g=f^2$ and $f\circ g=\mathrm{id}$).
Re: Functions
Function composition is denoted by $f\circ g$, or f o g in plain text. Some sources may skip the circle, but I would expect them to note this convention explicitly. In any case, it is much better
to write a bit more explanations than to let people wonder about notation.
You showed that $(f\circ g)(x) = x$, which means that g is the (right) inverse of f. This is the relationship between the functions. There are many other relationships, e.g., $(g\circ f)(x) = x$,
$f(x) = (g\circ g)(x)$, $g(x)=(f\circ f)(x)$, $f(x)=1-\frac{1}{g(x)}$, $g(x)=f(x+1)+1$, etc., but apparently they are not relevant to the question (except maybe the first).
Note also that both sides of equality should have the same type: either functions or real numbers. For example, $f\circ g$ is a function (the composition of f and g), while x is a number. So,
writing $f\circ g=x$ is not the best. One should write $(f\circ g)(x)=x$ or $f\circ g=\mathrm{id}$ where $\mathrm{id}(x)=x$ is the identity function.
Since $f(g(x))=x$, you can cancel 1994 applications $f(g(\dots))$ in $f^{2011}g^{1994}(1/2)$ and get $f^{2011-1994}(1/2)$. In evaluating this, note also that $f^3(x)=x$ (this can be verified
directly; it also follows from $g=f^2$ and $f\circ g=\mathrm{id}$).
Thank you, one important point I missed was to realise that g is the inverse of f but I have another
doubt, is it true that $f^3=f^{13}=f^{15}=f^{17}= ...$ or for any odd number always
applies? And is it also true that $f^{2}=f^{60}=f^{90}=f^{100}=f^{102}=...$ always?
Re: Functions
You are allowed to cross out $f^3$. Therefore, $f^{13}=f^{10}=f^{7}=f^{4}=f$. You are not allowed to cross $f^2$, so, e.g., you can't conclude that $f^4=f^2$.
Re: Functions
I see, so the fact that $f^3$ can be crossed out is applicable on a general basis and not only applicable to this question... Thank you!
Re: Functions
Of course it's applicable only to this question. The fact that $f^3=\mathrm{id}$ is a very specific property that is true only of some functions. Also, it does not follow from $f^3=\mathrm{id}$
that $f^2=f^{60}$.
Re: Functions
Re: Functions
You can only use those equalities that follow by the laws of mathematics. In post #2, I stated that for this particular $f$ we have
$f(f(f(x)))=x$ (*)
No, this does not follow from (*).
Some of these equalities follow from (*) (e.g., $f^{60}=f^{90}$) and some don't (e.g., $f^{90}=f^{100}$).
No, (*) applies only to this particular $f$. Have you checked (*) for this $f$? In doing so, did you use the definition of $f$?
You can repeatedly replace $f(f(f(x)))$ with $x$ or vice versa. This implies, for example, that $x=f^{3k}(x)$ for all natural numbers $k$. All such equalities have to be proved, not just guessed.
Re: Functions
You can only use those equalities that follow by the laws of mathematics. In post #2, I stated that for this particular $f$ we have
$f(f(f(x)))=x$ (*)
No, this does not follow from (*).
Some of these equalities follow from (*) (e.g., $f^{60}=f^{90}$) and some don't (e.g., $f^{90}=f^{100}$).
No, (*) applies only to this particular $f$. Have you checked (*) for this $f$? In doing so, did you use the definition of $f$?
You can repeatedly replace $f(f(f(x)))$ with $x$ or vice versa. This implies, for example, that $x=f^{3k}(x)$ for all natural numbers $k$. All such equalities have to be proved, not just guessed.
I understand now. So you had tried that f(f(f(x)))=x and that is the reason why we can use f^3=x. Am I right to then say that whether f^2 or f^3 can be cancelled must be tried by showing f^2 or f
^3=x before we can use them?
Re: Functions
Yes. Since $f(x)=\frac{1}{1-x}$, $f(f(x))=\frac{1}{1-\frac{1}{1-x}}=\frac{x-1}{x}=1-\frac{1}{x}=g(x)$. Applying $f$ to both sides we get $f^3(x)=f(g(x))$, and you have already showed that $f(g
(x))=x$. As I said in post #2:
December 9th 2011, 12:22 AM #2
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December 9th 2011, 12:53 AM #3
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Skilled and Semi-Skilled Workers
Date: 09/04/2002 at 04:42:54
From: Riya Bajaj
Subject: Work and time problem
Hi Dr. Math,
This is regarding a problem on "work and time" in the Dr. Math FAQ:
Working Together
Jack can paint a house in 5 days, and Richard can paint the same house
in 7 days. Working together, how long will it take them to finish the
A very easy method was used:
Jack and Richard = 1 house - 5*7 / 5+7
= 35/12 days
I tried to apply the same formula to the following question but was
unable to get the answer.
Four skilled workers do a job in 5 days, and five semi-skilled workers
do the same job in 6 days. How many days will it take for two skilled
and one semi-skilled worker to do that job?
Here is what I did:
4 skilled workers 5 days
1 skilled worker 5/4 days
2 skilled workers 5*2/4 = 5/2 days.
5 semi-skilled 6 days
1 semi-skilled 6/5 days
Now applying the same method used in the first question, here we say:
1 skilled and 1 semi-skilled (5/2 * 6/5)
5/2 + 6/5
= 30/37
But the right answer is 60/7.
Why can't we use the same method for harder word problems? Could you
please suggest a common method or way to solve such problems?
Thank you,
Riya Bajaj
Date: 09/04/2002 at 10:29:01
From: Doctor Ian
Subject: Re: Work and time problem
You can't just divide the time by the number of workers - which is
exactly what makes these problems so hard.
Think of it this way:
four skilled workers can do 1 job in 5 days.
We could just have them repeat what they did, so
four skilled workers can do 4 jobs in 20 days.
Now, if four workers are doing 4 jobs, each of them is essentially
doing a job by himself. So we can just eliminate any number of jobs by
eliminating the same number of workers:
1 skilled worker can do 1 job in 20 days.
> 2 skilled workers 5*2/4 = 5/2 days.
> Then,
> 5 semi-skilled 6 days
> 1 semi-skilled 6/5 days
Doing the same thing as before:
5 semi-skilled workers can do 1 job in 6 days.
5 semi-skilled workers can do 5 jobs in 30 days.
1 semi-skilled worker can do 1 job in 30 days.
Now, what can two skilled workers and one semi-skilled worker do
together? Suppose we give them 60 days. Each of the skilled workers
can do 3 jobs, for a total of 6 jobs. The semi-skilled worker can do 2
jobs. So in 60 days, the three of them could do 8 jobs, which means
that they can do one job in 60/8 days.
>The right answer is 60/7.
I think it's 60/8, or 15/2.
Can we check that another way? Let's start from the beginning:
4 skilled workers can do 1 job in 5 days
5 semi-skilled workers can do 1 job in 6 days
We'd like to get a ratio of 2 skilled workers to 1 semi-skilled
4 skilled workers can do 1 job in 5 days
20 skilled workers can do 5 jobs in 5 days
5 semi-skilled workers can do 1 job in 6 days
10 semi-skilled workers can do 2 jobs in 6 days
And we'd like to consider the same number of days:
4 skilled workers can do 1 job in 5 days
20 skilled workers can do 5 jobs in 5 days
20 skilled workers can do 30 jobs in 30 days
5 semi-skilled workers can do 1 job in 6 days
10 semi-skilled workers can do 2 jobs in 6 days
10 semi-skilled workers can do 10 jobs in 30 days
Now, working together, 20 skilled workers and 10 semi-skilled workers
can do 40 jobs in 30 days. We can divide them into 10 work crews, each
with 2 skilled workers and 1 semi-skilled worker. Each of those crews
can do 4 of the jobs in those 30 days.
So two skilled and one semi-skilled worker can do 4 jobs in 30 days,
or one job in 30/4 days, or 15/2 days, which is what we got before.
> Why can't we use the same method for harder word problems? Could
> you please suggest a common method or way to solve such problems?
I think I've shown that we _can_ use a single method for all the
various kinds of problems. The key is to avoid the temptation to use
formulas without understanding why they work, and instead use methods
where each step makes sense.
I hope this helps. Write back if you'd like to talk more about this,
or anything else.
- Doctor Ian, The Math Forum
Date: 09/04/2002 at 15:22:40
From: Riya Bajaj
Subject: Thank you (Work and time problem)
Hi Dr. Math,
Thank you so much for your response. Things seem to be clearer than | {"url":"http://mathforum.org/library/drmath/view/61163.html","timestamp":"2014-04-17T16:32:12Z","content_type":null,"content_length":"9845","record_id":"<urn:uuid:16e3b1e8-382d-4fba-a1a1-8f2cb744f9c5>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00412-ip-10-147-4-33.ec2.internal.warc.gz"} |
find the dimensions of the rectangle that will enclose the most area, and show steps please. A fence is to be built... - Homework Help - eNotes.com
find the dimensions of the rectangle that will enclose the most area, and show steps please.
A fence is to be built to enclose a rectangular area. The fence along three sides is to be made of material
that costs $5 per foot. The material for the fourth side costs $15 per foot.
(B) If $3,000 is available for the fencing, find the dimensions of the rectangle that will enclose the most area.
We need to find the maximum area that can be enclosed in the rectangular area with a cost of 3000.
Let the sides be x and y.
Let the three sides that will cost 5 dollars per foot are x, y, and y and for the fourth side y will cost 15 dollars per foot.
The cost for the three sides is 5*(x+y+y)= 5(x+2y)
The cost for the forth side is 15*(x).
The total cost is 3000
==> 5(x+2y) + 15x = 3000
==> 5x + 10y + 15x = 3000
==> 20x + 10y= 3000
==> 2x + y = 300
Let the area of the rectangle be A.
==> A= xy
But we know that 2x+y = 300 ==> y= 300-2x
==> A = x(300-2x)
==> A = 300x - 2x^2
Now we need to find the maximum area which is the zero of the derivative A'.
==> A'= 300-4x = 0
==> 4x = 300
==> x = 300/4 = 75.
==> y= 300-2x = 300-150 = 150
Then, the dimensions for a maximum area are x=75 ft and y=150 ft.
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MMP Activities
IHE Network
Upcoming Events
There are no upcoming events at this time.
IHE Resources
UWM & MATC Mathematics Course Equivalencies Comparision Sheet word document
For background information about the IHE Network, click here.
Past Events
21st annual Marden Lecture on Mathematics, Wednesday April 28, 2010 from 4-5 p.m.
This Marden Lecture featured Roger Howe of Yale University. Dr. Howe gave a presentation entitled "Symmetry: More Than Pretty Pictures". In it, he reviewed the phenomena, concepts and history of
symmetry, and concluded with some observations about symmetry's role in scientific development.
View Flyer PDF document
Jerry Murdoch lecture, Friday, April 16, 2010, from 4:30-5:30 pm.
Are you interested in high school mathematics curricula? Jerry Murdoch, the principal author of "Discovering Algebra" and "Discovering Advanced Algebra", recently selected by Milwaukee Public Schools
as the district-wide curriculum, spoke in the UW-Milwaukee Department of Mathematical Sciences (room E495A). This was an excellent opportunity to talk with Jerry about his philosophy of teaching
mathematics, and the curricular and pedagogical decisions that went into the design of the books.
View Flyer PDF document
Kepner Becomes President of NCTM
Kepner, a Professor of Mathematics and Education at University of Wisconsin-Milwaukee, will now serve a two-year term as President and one-year term as Past President as spokesman for the
organization after assuming duties last year as President-Elect. It is an enormous honor both to Dr. Kepner and to Milwaukee to have him serve this capacity. Congratulations Dr. Kepner!
The Mathematical Preparation of Middle School Teachers of Mathematics at the Wisconsin Mathematics Council Annual Meeting; Friday May 2, 2008 -- 8:00-11:00 am
This workshop, built from the successful conference The Mathematical Preparation of Middle School Teachers of Mathematics: a Wisconsin Concern, held in Wisconsin Dells last October, presented two
visions for the mathematical preparation of teachers at the middle grades.
Download Announcement Flyer Word document
WMC Conference Registration Form PDF document
Green Lake Conference Center Food/Lodging Form PDF document
Math Colloquium, Monday, November 26, 2007--1:00 p.m.
Math Colloquium at Marquette University, Monday, November 26, 2007, at 1:00 p.m. in Room 401 of Katharine Reed Cudahy Hall. Marta Magiera of Illinois Institute of Technology, will give a
presentation entitled Metacognition in Solving Complex Problems: A Case Study of Situations and Circumstances that Prompt Metacognitive Behaviors. Ms. Magiera is a candidate for a mathematics
education faculty position at Marquette University.
For additional information on this event, please contact Dr. Jack Moyer, phone (414) 288-5299, or e-mail johnm@mscs.mu.edu
Math Colloquium, Wednesday, November 28, 2007--1:00 p.m.
Math Colloquium at Marquette University, Wednesday, November 28, 2007, at 1:00 p.m. in Room 401 of Katharine Reed Cudahy Hall. Jennifer Kaminski of the Ohio State University will give a
presentation entitled Promoting Transfer of Mathematical Knowledge. Ms. Kaminski is a candidate for a mathematics education faculty position at Marquette University.
For additional information on this event, please contact Dr. Jack Moyer, phone (414) 288-5299, or e-mail johnm@mscs.mu.edu
IHE-DPI Joint Conference, October 5-6, 2007
"Mathematical Preparations for Middle School Teachers of Mathematics: A Wisconsin Concern"
Conference Announcement Word document
Expense Details Word document
Work Groups Word document
Final Agenda Word document
S.Nickerson, "Reconceptualizing Mathematics: Courses for Prospective and Practicing Teachers" Powerpoint Presentation
S.Nickerson, "Reconceptualizing Mathematics Revisited: Prespectives and Views on Building and Delivering a Coherent Program" Powerpoint Presentation
Huinker, McLeod, "The MCEA Math Minor at UWM" Powerpoint Presentation
Summary of MP Powerpoint Presentation
Marden Lecture, April 27, 2007 at 4 pm - University of Wisconsin-Milwaukee
UWM's April Marden Lecture featured David Keyes of Columbia University. Dr. Keyes gave a presentation entitled "Scientific Discovery through Advanced Computing". The Scientific Discovery through
Advanced Computing (SciDAC) initiative is a set of interconnected projects--science, software development, and research directed toward the latter--designed to support simulation, data exploration,
and collaboration in many thrust areas of the U.S. Department of Energy, including: climate modeling, fusion energy, chemistry and materials science, astrophysics, and high energy and particle
physics. The lecture briefly reviewed the sweep of SciDAC and then focuses on some particular advances in the U.S. magnetic fusion energy program.
Download Flyer PDF document
Math Colloquium - February 21, 2007 at 1 pm - Marquette University
Denise Forrest of The Ohio State University-Newark gave a presentation entitled "Re-searching Mathematics Teachers' Verbal Communication." Ms. Forrest is a candidate for a mathematics education
faculty position with Marquette University. The presentation was Wednesday, February 21, 2007, at Marquette University.
Download Flyer Word document
Mathematics Colloquium, February 2, 2007, 4 pm, Marquette University
Chris Hruska of the University of Wisconsin-Milwaukee gave a presentation entitled "Nonpositively Curved Spaces with Isolated Flats." The event was sponsored by Marquette's Dept. of Mathematics,
and Computer Science.
Mathematics Colloquium, January 19, 2007, 1pm, Marquette University
Tetyana Berezovski of Simon Fraser University, Canada, gave a presentation entitled "Towards
Effective Teaching: The Case for Logarithms." Ms. Berezovski is a candidate for a mathematics education faculty position with Marquette University. She has focused her research on examination of
pre-service and in-service teachers’ understanding of logarithms and logarithmic functions and to explore how their understanding influences their choices of the approaches to teaching these
concepts. Research supplies consistent evidence that teachers' conceptions of mathematics strongly impact their instructional practice. In addition, research findings confirm that teachers'
instructional practices, especially in mathematics, reflect the teachers' conceptions of the subject matter.
Download Flyer Word document
Mathematics Colloquium, October 20, 2006, 4pm, University of Wisconsin-Miwlaukee
"NCTM Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Mathematical Analysis and Perspective on U.S. Mathematics Education" by Henry S. Kepner, Jr., University of
Dr. Kepner discussed the recently released Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics from the National Council of Teachers of Mathematics. His talk focused on the
document's relevance in American school mathematics, and examined the recommendations of the Curriculum Focal Points and its intended purpose. He also brought to attention varied commentaries and
editorials appeared in the Wall Street Journal, New York Times, and other national media, and how these relate to the historical context of school mathematics in the US and contrast this context to
approaches in other countries.
Download Flyer PDF document
Mathematics Workshop, August 31, 2006, 5-6:30 pm, University of Wisconsin-Milwaukee
"Origami Revisited: Connecting Your Students to Learning Mathematics" by Greg Oates, University of Auckland, New Zealand
As a secondary mathematics teacher, Greg Oates often used Origami as a diversionary activity for Friday afternoons, or various activities such as school open days, or science fairs. However, recent
scientific and mathematical developments incorporating Origami have stimulated a growing international awareness of Origami, and suggest some justification for greater inclusion of Origami in the
mathematics curriculum. The wider applications of Origami to the curriculum became apparent during the development of the Great Origami Maths and Science Show, a project funded by the Royal Society
of New Zealand (RSNZ), with collaboration between Origami New Zealand and the Mathematics Department at The University of Auckland. The one-hour show is accompanied by a detailed resource aligned to
the New Zealand mathematics curriculum, and is described in advertising to schools as the ultimate maths field trip, with an aim of turning people on to the realms of possibilities within the origami
art-form. It examines some of the mathematical and scientific concepts contained in the extraordinary array of shapes and the folds involved.
This presentation looked at some of the features of the show, the speaker’s conversion to regarding Origami as a serious curricular activity, and provided examples of how Origami may be included in
everyday curricular activities.
Download flyer Word document
Mathematics Colloquium, August 30, 2006, 2-3 pm, University of Wisconsin-Milwaukee
"The Great Origami Maths and Science Show - What’s up in Origami and Mathematics Down-under?" by Greg Oates, University of Auckland, New Zealand
Two years ago origami was regarded largely as a children’s recreational activity in the mathematics department at The University of Auckland. This status has changed remarkably as a result of two
main influences. Firstly, access via the Internet to the materials and research of international origami experts such as Robert Lang and Tom Hull. Secondly and more recently, Jonathan Baxter has
stimulated interest in this field through presentations to students and staff within the department, particularly those involved in the secondary mathematics teacher training programme, and topology
An exciting development of this growing awareness has been a project called the Great Origami Maths and Science Show. This project, funded by the Royal Society of New Zealand (RSNZ), is a
collaboration between Origami New Zealand and the Mathematics Department at The University of Auckland. The one-hour show is accompanied by a detailed resource aligned to the New Zealand mathematics
curriculum. The show, described in advertising to schools as the ‘ultimate maths field trip’, aims to turn people on the realms of possibilities within the origami art-form. It examines some of the
mathematical and scientific concepts contained in the extraordinary array of shapes and the folds involved. Multiple presentations will be made during a national tour of seven major centres through
July, August and early September of 2006, commencing with the Incredible Science Day at The University of Auckland.
To the best of our knowledge, this is the first ever project anywhere in the world to receive government funding on such a scale for origami and maths, science and education-related pursuits. This
presentation will discuss the design and implementation of the project, with video excerpts from the show, and examples from the accompanying resource book. Details of the project, including feedback
from educators and students, tips on how to start a similar project, places to look for funding, pit-falls, and possible improvements on the original concept will be shared.
Download Flyer Word document
Math Colloquium, August 30, 2006; University of Wisconsin-Milwaukee
"Mathematics Tutoring in Auckland", by Greg Oates, University of Auckland, New Zealand
Mathematics courses at The University of Auckland have for many years used collaborative tutorials and peer-tutors in their teaching. This seminar, led by one of the supervisors of this tutorial
programme, will describe and discuss some of the unique aspects of the programme; for example, the innovative Tutoring in Mathematics course that allows students to earn degree-status credits in
mathematics while training them as peer-group tutors, and the recent extension of the collaborative small-class tutorials to cover all first and second-year undergraduate courses in the department.
Evidence will be provided of the effectiveness of the tutorial programme, as gauged from a survey of students’ perceptions. Some of the tutors also subsequently enroll in the University’s Graduate
Diploma in Teaching (Secondary). The effects of the tutoring course on tutors continuing to tutor at higher levels, and on their performances as novice teachers in the Diploma course, will be
Download Flyer Word document
Workshop for Mathematics Teachers, April 8, 2006 8:30 AM - 11 AM, MPS School Support Center
"Exploring the Shape of Space" by Dr. Jeff Weeks
Dr. Weeks presented an interactive talk with Middle and High School Mathematics teachers challenging the idea of the universe as infinite. Through demonstration, Weeks shared activities from his book
(The Shape of Space, Marcel Dekker, 1985; second ed. 2002), and curriculum unit (Exploring the Shape of Space, Key Curriculum Press, 2001). This included small group activities, paper-and-scissors
constructions and computer games to introduce students in grades 5-10 to the mind-stretching possibility of a "multiconnected universe". Interactive 3D graphics let them explore many possible shapes
for space. To conclude the workshop, the group examined how recent satellite data provides tantalizing clues to the true shape of our universe.
Download Flyer Word document
Workshop for Mathematics Teachers, May 20, 2006, 8:30 am - 4 pm, University of Wisconsin-Milwaukee
"Engaging Mathematics: Connecting Your Students to Learning Mathematics"
Guest Speakers:
• Judy Paterson, University of Auckland, New Zealand;
• Gabriella Pinter, UW-Milwaukee;
• Harvey Keynes & Simon Morgan, University of Minnesota
Download Flyer Word Document
The number of participants at the May 20th workshops were as follows:
64 people attended at least one of the sessions (AM or PM)
37 people attended both sessions
7 attended only the AM session
20 attended only the PM session
44 attended the AM session
57 attended the PM session
Math Department Colloquium, May 22, 2006, 3 pm, University of Wisconsin-Milwaukee
Dr. Judy Paterson from the Department of Mathematics at the University of Auckland visited UWM to present "Using Mathematics to open up windows in teachers’ minds: Encouraging teacher talk about
learning and teaching" at a colloquium on May 22, 2006. Approximately 35 attendees from several local institutions of higher education were in attendance to hear and discuss their reactions to Dr.
Paterson’s talk, which related her involvement in a longitudinal study that encourages interaction between mathematics teachers and the greater community through stimulation of mathematical input
during professional development opportunities.
Download Flyer Word Document
Mathematics Colloquium, March 8, 2006, 4 pm, Marquette University
Marquette University’s Department of Mathematics, Statistics, and Computer Science hosted Professor Guershon Harel, of the University of California-San Diego, for a colloquium talk on March 8, 2006.
Professor Harel’s talk, “What is Mathematics? A Pedagogical Answer to a Philosophical Question”, enabled educators from several institutions of higher education to convene and benefit from Harel’s
extensive research in the teaching and learning of mathematical proof. The colloquium coincided with two other events geared to mathematics educators and education majors at Marquette University.
Download Flyer Word Document
Math Colloquium, October 14, 2005, Marquette University
Over forty-five mathematicians and mathematics educators from twenty Milwaukee area high schools and institutions of higher education attended a Colloquium Event featuring Deborah Hughes-Hallett on
Friday, October 14, 2005. The event, co-sponsored the MMP, was a one-day seminar held in the Department of Mathematics, Statistics and Computer Science at Marquette University. During the event, Dr.
Hughes-Hallett taught a demonstration calculus lesson, led a question-and-answer session at a luncheon following the lesson, and conducted a colloquium on the topic of Calculus Teaching from a Reform
View the Colloquium invitation PDF Document
IHE Network Conference, August 25-26, 2005, Carroll College
"Mathematical Knowledge Needed for Teaching in K-12 and Collegiate Mathematics and The Role of Definition in Mathematics Instruction" with guest speakers Hyman Bass and Deborah Ball, University of
This two-day conference had participants sharing in the conjectures and observations of the work of Deborah Ball and Hyman Bass. In their quest to describe the mathematical knowledge needed for
teaching, Ball and Bass have observed that teaching requires extensive mathematical problem solving, which occurs constantly as teachers. Conference sessions dealt with topics such as the role and
significance of definitions in mathematics instruction; calculus students' grasp of mathematical definition; mathematical knowledge needed for teaching, and the relevance of methematical knowledge
for teaching to instructional quality and student learning. Sessions were highly interactive.
Flyer Word Document
Registration Word Document
Directions Word Document
Invitation Letter Word Document
Marden Lecture on Mathematics - "Math in the Movies", March 10, 2005
"Math in the Movies" presented by Dr. Tony DeRose, Senior Scientist and head of Research, Pixar Animation Studios
Film making is undergoing a digital revolution brought on by advances in areas such as computer technology, computational physics and computer graphics. This talk will provide a behind the scenes
look at how fully digital films, such as Pixar's "Finding Nemo" and "The Incredibles", are made, with particular emphasis on the role that mathematics plays in the revolution.
Tony DeRose is currently a Senior Scientist and head of Research at Pixar Animation Studios. He received a Ph.D. in Computer Science from the University of California, Berkeley in 1985. From 1985 to
1995 Dr. DeRose was a Professor of Computer Science and Engineering at the University of Washington. In 1998, he was a major contributor to the Oscar winning short film "Geri's game", and in 1999 he
received the ACM SIGGRAPH Computer Graphics Achievement Award.
The Marden Lectures were established by Morris and Mirian Marden. Dr. Marden was a Distinguished Professor of Mathematics at UWM and was responsible for the inauguration of its graduate program, the
first at the University. The Lectures are designed to bring distinguished mathematicians to UWM to speak to a general audience on a topic of mathematical interest. They have been given annually since
This event was held Thursday, March 10, 2005, 4:00 - 5:00 PM, Bolton Hall 150, at UWM. The lecture was sponsored by the Mirian and Morris Marden Fund and co-sponsored by the Department of
Mathematical Sciences, the College of Letters and Science, and the College of Engineering.
View flyer Word Document
Math Colloquium, December 10, 2004
"The Mathematics Education of Teachers: One Example of an Evolving Partnership Between Mathematicians and Mathematics Educators" presented by Gail Burrill of Michigan State University.
Over forty mathematicians and mathematics educators from ten Milwaukee area institutions of higher education were in attendance at “The Mathematics Education of Teachers: One Example of an Evolving
Partnership between Mathematicians and Mathematics Educators” presented by Gail Burrill of Michigan State University. This colloquium was the first in a series sponsored by the Department of
Mathematical Sciences in conjunction with the School of Education through the Milwaukee Mathematics Partnership (MMP).
Gail Burrill, who is the Former Director of the Mathematical Sciences Education Board at the National Research Council, and Past President of the National Council of Teachers of Mathematics, spent
the day at other MMP events before conversing animatedly about the capstone course for secondary mathematics majors that she co-developed and taught at Michigan State University. Successes,
challenges, sample math problems and other resources used, co-teaching procedures, and surprise findings from teaching the course were all discussed to an interactive crowd of educators from local
colleges and universities including UWM, Marquette, Milwaukee Area Technical College, Alverno College, Carroll College, and others.
UWM is in the process of developing a similar type of course through collaboration between faculty in the School of Education and those in the Department of Mathematical Sciences.
View the Powerpoint presentation.
IHE Kickoff Meeting, August 16 & 17, 2004
The kickoff for the Institutions of Higher Education (IHE) Network was held in conjunction with the annual NPRIME (Networking Project for the Improvement of Mathematics Education) conference August
16 and 17, 2004 at Alverno College . The conference topic, “What is the mathematical knowledge needed for teaching?” provided a springboard for discussion on topics related to the improvement of
teacher education at the partnering institutions. Group sessions related experiences in developing teachers' mathematical knowledge and the process of preparing future mathematics teachers. The
two-day interactive breakout sessions dealt with specific areas of mathematical preparation, such as geometry, probability and statistics, problem solving and proof, math for elementary teachers, and
preparing secondary students' transition to college mathematics.
Download the Agenda. PDF Document | {"url":"http://www4.uwm.edu/Org/mmp/_activities/IHE.html","timestamp":"2014-04-20T19:27:29Z","content_type":null,"content_length":"49670","record_id":"<urn:uuid:51379b37-66d8-4c4b-bd3f-73ea46191a72>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00644-ip-10-147-4-33.ec2.internal.warc.gz"} |
n-localic (infinity,1)-topos
$(\infty,1)$-Topos Theory
Extra stuff, structure and property
structures in a cohesive (∞,1)-topos
An (∞,1)-topos is $n$-localic if
More precisely: if (∞,1)-geometric morphisms into it are fixed by their restriction to the underlying (n,1)-toposes of $(n-1)$-truncated objects.
To the tower of (n,1)-toposes of $(n-1)$-truncated objects
$\cdots \to \tau_{\leq 3-1} \mathcal{X} \to \tau_{\leq 2-1} \mathcal{X} \to \tau_{\leq 1-1} \mathcal{X} \to \tau_{\leq 0-1} \mathcal{X} \to *$
of a given (∞,1)-topos $\mathcal{X}$ corresponds a tower of $n$-localic toposes $\mathcal{X}_n$ such that $\tau_{\leq n -1} \mathcal{X} \simeq \tau_{\leq n-1} \mathcal{X}_n$. We may think of the $n$
-localic $\mathcal{X}_n$ as being $n$th stage in the Postnikov tower decomposition of $\mathcal{X}$.
A 0-localic $(1,1)$-topos is a localic topos from ordinary topos theory.
We write (∞,1)Topos for the (∞,1)-category of (∞,1)-toposes and (∞,1)-geometric morphisms between them.
For $\mathcal{X}$ an (∞,1)-topos we denote by
$\tau_{\leq n-1} \mathcal{X} \hookrightarrow \mathcal{X}$
the (n,1)-topos of $(n-1)$-truncated objects of $\mathcal{X}$.
We write $(n,1)Topos$ for the (n+1,1)-category of (n,1)-toposes and $(n,1)$-geometric morphisms between them.
($n$-localic $(\infty,1)$-topos)
An (∞,1)-topos $\mathcal{X}$ is $n$-localic if for any other $(\infty,1)$-topos $\mathcal{Y}$ the canonical morphism
$(\infty,1)Topos(\mathcal{Y},\mathcal{X}) \to (n,1)Topos(\tau_{\leq n-1} \mathcal{Y}, \tau_{\leq n-1}\mathcal{Y})$
is an equivalence of (∞,1)-categories (of ∞-groupoids).
More generally,
a (k,1)-topos $\mathcal{X}$ is $n$-localic for $0 \leq n \leq k \leq \infty$ if for any other $(k,1)$-topos $\mathcal{Y}$ the canonical morphism
$(k,1)Topos(\mathcal{Y},\mathcal{X}) \to (n,1)Topos(\tau_{\leq n-1} \mathcal{Y}, \tau_{\leq n-1}\mathcal{Y})$
is an equivalence of (∞,1)-categories (of ∞-groupoids).
This is (HTT, def. 6.4.5.8).
This is (HTT, lemma 6.4.5.6).
This is (LurieStructured, lemma 2.3.16).
For $n \in \mathbb{N}$ and $\mathcal{X}$ an $n$-localic $(\infty,1)$-topos, the over-(∞,1)-topos $\mathcal{X}/U$ is $n$-localic precisely if the object $U$ is $n$-truncated.
This is (StrSp, lemma 2.3.14).
For $\mathcal{X}$ an $n$-localic $(\infty,1)$-topos let $U \in \mathcal{X}$ be an object. Then the following are equivalent
1. the restriction of the inverse image $U^* : \mathcal{X} \to \mathcal{X}/U$ (of the etale geometric morphism from the over-(∞,1)-topos) to $(n-1)$-truncated objects is an equivalence of (∞,1)
2. the object $U$ is $n$-connected.
This is (StrSp, lemma 2.3.14).
Every (n,1)-topos $\mathcal{Y}$ is the (n,1)-category of $(n-1)$-truncated objects in an $n$-localic $(\infty,1)$-topos $\mathcal{X}_n$
$\tau_{n-1} X_n \stackrel{\simeq}{\to} \mathcal{Y} \,.$
This is (HTT, prop. 6.4.5.7).
Let $\mathcal{G}$ be a geometry (for structured (∞,1)-toposes).
This is StrSp, lemma 2.6.17
The general noion is the topic of section 6.4.5 of
Remarks on the application of $n$-localic $(\infty,1)$-toposes in higher geometry are in | {"url":"http://nlab.mathforge.org/nlab/show/n-localic+(infinity%2C1)-topos","timestamp":"2014-04-19T07:49:58Z","content_type":null,"content_length":"50979","record_id":"<urn:uuid:fcb40d23-11a0-4538-a685-1c31508572d0>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00505-ip-10-147-4-33.ec2.internal.warc.gz"} |
Taking Percentage Away
June 3rd 2011, 06:27 AM #1
Jun 2011
Taking Percentage Away
Hi There,
I am arguing with my boss over how to minus a percentage.
If you want to work out the VAT on £100 you divide by 1.2 and then minus the £100 which is £16.67 VAT
If i then want to pay someone 12% of that £100 my boss says u times by 0.12 which gives you £12 where as i think you divide by 1.12 and then minus the £100 which gives you £10.72
Can somebody please tell me which way is the right one
Many Thanks
Your method is correct
If i then want to pay someone 12% of that £100 my boss says u times by 0.12 which gives you £12 where as i think you divide by 1.12 and then minus the £100 which gives you £10.72
Your boss is correct
Last edited by e^(i*pi); June 3rd 2011 at 06:43 AM. Reason: quoting for clarity
ok this is where i do not understand vat is taking off 20% of a figure u divide so why multiply when taking 12%? its just a different percentage
well what i am saying is you divide vat so why multiply something that is exactly the same its just a different percentage! VAT you want 20% of the amount and i want 12% of the amount so why not
divide by 1.12 rather than 1.2
You multiply for the 12% because you want to find 12% of £100. If your £100 was including the 12% then you would indeed divide by 1.12
You divide for VAT because the £100 figure already includes VAT. If your £100 was before the addition of VAT then you'd multiply
June 3rd 2011, 06:39 AM #2
June 3rd 2011, 06:42 AM #3
Jun 2011
June 3rd 2011, 06:45 AM #4
June 3rd 2011, 06:50 AM #5
Jun 2011
June 3rd 2011, 01:23 PM #6 | {"url":"http://mathhelpforum.com/math-topics/182302-taking-percentage-away.html","timestamp":"2014-04-16T16:10:20Z","content_type":null,"content_length":"46921","record_id":"<urn:uuid:45d364f2-ab47-449f-87bc-43e144576c7e>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00129-ip-10-147-4-33.ec2.internal.warc.gz"} |
HowStuffWorks "How Fractals Work"
Fractals are a paradox. Amazingly simple, yet infinitely complex. New, but older than dirt. What are fractals? Where did they come from? Why should I care?
Unconventional 20th century mathematician Benoit Mandelbrot created the term fractal from the Latin word fractus (meaning irregular or fragmented) in 1975. These irregular and fragmented shapes are
all around us. At their most basic, fractals are a visual expression of a repeating pattern or formula that starts out simple and gets progressively more complex.
One of the earliest applications of fractals came about well before the term was even used. Lewis Fry Richardson was an English mathematician in the early 20th century studying the length of the
English coastline. He reasoned that the length of a coastline depends on the length of the measurement tool. Measure with a yardstick, you get one number, but measure with a more detailed foot-long
ruler, which takes into account more of the coastline's irregularity, and you get a larger number, and so on.
Carry this to its logical conclusion and you end up with an infinitely long coastline containing a finite space, the same paradox put forward by Helge von Koch in the Koch Snowflake. This fractal
involves taking a triangle and turning the central third of each segment into a triangular bump in a way that makes the fractal symmetric. Each bump is, of course, longer than the original segment,
yet still contains the finite space within. Weird, but rather than converging on a particular number, the perimeter moves towards infinity. Mandelbrot saw this and used this example to explore the
concept of fractal dimension, along the way proving that measuring a coastline is an exercise in approximation [source: NOVA].
If fractals have really been around all this time, why have we only been hearing about them in the past 30 years or so? | {"url":"http://science.howstuffworks.com/math-concepts/fractals.htm","timestamp":"2014-04-21T00:34:22Z","content_type":null,"content_length":"118794","record_id":"<urn:uuid:aeaf201a-06a3-4068-8e96-9b662c8ecac3>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00453-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathFiction: The Mystery of Khufu's Tomb (Mundy Talbot)
Contributed by Vijay Fafat
A rapid-read, reasonably entertaining novel about the real location of the Pharaoh Khufu’s (Cheops) tomb and the fabulous treasury buried therein. An old, Chinese mathematician spends decades
decoding the mathematical information encoded in the construction of the pyramid of Cheops to pinpoint the burial site of Khufu (and as a true, platonic mathematician, “He was really only interested
in the pyramid. The treasure did not attract him; what gave him exquisite delight was seeing the proof that his deductions were correct. He was too old to care for money, or too wise” and “[he was]
much more delighted with his mathematical solution and with our bewilderment than with any thought about the value of the gold”). General intrigue follows with an altruistic ending that the recovered
treasure is to be used for the general good of the entire world.
Some quotables from the story:
(quoted from The Mystery of Khufu's Tomb)
“To Chu Chi Ying mathematics were religion. From the moment the little old man started figuring, and interpreting the figures, he felt himself in touch with the Infinite, and was happy” […]
(quoted from The Mystery of Khufu's Tomb)
"[Mathematics is the] Business of think, not guess! You know tligonometly? You know tliangulation? You know what base is? So. Look, see."
(quoted from The Mystery of Khufu's Tomb)
"Mathlematics no can lie," he answered. "You no savvy. Me savvy."
(quoted from The Mystery of Khufu's Tomb)
"He was obsessed by mathematics; but, according, to him, as I understood his explanations, music and mathematics are the interpretation of law that governs the whole universe, and he who understands
them owns the key to everything.
(quoted from The Mystery of Khufu's Tomb)
“To him, the Pyramid expressed - by means of some abstruse relation between the number of courses of stone and the height and weight of the finished building - not only the number of ingots of gold
and silver that Khufu caused to be buried with him in his tomb, but their exact dimensions, purity or fineness, and the order of their arrangement underground.”
(quoted from The Mystery of Khufu's Tomb)
“Chu Chi Ying's theory was this: There were men in the days when the Pyramid was built who knew Knowledge. Abstract knowledge. And abstract knowledge was their notion of the after-life and what we
call heaven. Therefore, the attainment of abstract knowledge meant eternal life. But--and here was the rub, as I understood it--abstract knowledge could not be understood unless first concretely
expressed in some way. In other words, he who believed he had attained to abstract knowledge had to prove it, and to leave his proof for others to follow if they could. So the Pyramid was an effort
on the part of old King Khufu to express concretely the sum total of the abstract knowledge that had been taught to him by the sages of his day…”
(quoted from The Mystery of Khufu's Tomb)
“He went into a maze of calculations then that would baffle an astronomer who hadn't tables to fall back on. Chu Chi Ying used never a note, set down no figures, hesitated not one second, but reeled
off--in English, mind you--numbers running into billions, pointing with the long nail of his left forefinger to the various details of the Pyramid's construction as he dealt with them mathematically,
one by one. He calculated for an hour. He dragged in the precession of the equinoxes and law of gravity, the speed of light, and the mean distance between the earth and sun, and related all that-in
some inscrutable fashion that seemed plausible while he was doing it--to the inside measurements of the empty granite sarcophagus--so called--that was all they ever found in the Pyramid when Al
Mahmoun's men broke in, A.D. 800. And the long and short of all that was, as he announced triumphantly at the conclusion, that the base of the Pyramid on the side opposed to the Sphinx is the base of
a theoretical triangle, whose apex falls exactly on the opening into Khufu's real tomb!” | {"url":"http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf867","timestamp":"2014-04-21T15:35:40Z","content_type":null,"content_length":"12602","record_id":"<urn:uuid:13b5ab63-fb05-4e18-9a37-0501cb97b0ed>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00304-ip-10-147-4-33.ec2.internal.warc.gz"} |
Extending Jordan loops
up vote 4 down vote favorite
I encountered this issue recently, but do not know of any general results to deal with it, so I would appreciate any pointers.
Let $\mathbb T=\{z\in\mathbb C\mid |z|=1\}$, and let $f:\mathbb T\to\mathbb C$ be continuous and injective, so its image $\mathbb T'$ is a Jordan loop. Under what (general) conditions can we
ensure that there is a homeomorphism between the unit disc and the interior of $\mathbb T'$ whose extension to the boundary is $f$?
Moreover, if there are reasonable conditions that ensure this, and $f$ is $C^\infty$, can we further require some nice regularity (perhaps even $C^\infty$) of the extension as well?
cv.complex-variables real-analysis
Doesn't Schoenflies give what you want? Or Google Schoenflies extension theorem. – Bill Johnson Sep 22 '12 at 16:06
1 en.wikipedia.org/wiki/Schoenflies_problem – Bill Johnson Sep 22 '12 at 16:18
I should have mentioned Schoenflies. I am asking for a sort of converse. Schoenflies theorem ensures that a homeomorphism of the interiors can be extended to the boundary and that, in general,
given a Jordan loop $\mathbb T'$, there is an $f$ with range $\mathbb T'$ that can be extended. Here I am starting with a given $f$ and want to extend it to the interior. – Andres Caicedo Sep 22
'12 at 16:19
1 Since you asked for pointers: There is an extension version of Schoenflies stating that every homeomorphism $f\colon \mathbb{T} \to \mathbb{T}'$ can be extended to a homeomorphism of $\mathbb{C}$.
This is mentioned (without proof) as a consequence of an extension theorem of Carathéodory in Remmert, Classical Topics in Complex Function Theory, page 187: books.google.com/books?id=BHc2b0iCoy8C
&pg=PA187 and the book refers to Pommerenke, Boundary Behavior of Conformal Maps, Springer 1991 for details. – Theo Buehler Sep 23 '12 at 13:07
Thank you, Theo. – Andres Caicedo Sep 23 '12 at 14:48
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1 Answer
active oldest votes
What you're asking is equivalent to asking whether any homeomorphism $g : S^1 \rightarrow S^1$ can be extended to a homeomorphism of the disc. This is easy -- write the disc in polar
coordinates $(t,\theta)$ with $\theta \in S^1$, and define an extension $G(t,\theta) = (t,g(\theta))$.
The question about whether this can be done smoothly if $g$ is smooth is more subtle. Observe that the above also works for $S^k$ with $k > 1$. The smooth version fails in higher
dimensions and is responsible for the existence of exotic spheres. However, for $k=1,2$ there is no problem. For $k=1$, this is a theorem of Smale; see
up vote 7 down Smale, Stephen Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc. 10 1959 621–626.
vote accepted
For $k=2$, it is a much deeper theorem of Hatcher; see
Hatcher, Allen E. A proof of the Smale conjecture, Diff(S3)≃O(4). Ann. of Math. (2) 117 (1983), no. 3, 553–607.
I don't think this is what the question is asking. I think the question refers to the interior of the curve, which is a subset of the same complex plane where the curve is
embedded, and which is guaranteed to exist by the Jordan curve theorem. The question asks for a homeomorphism to this interior, not just to any other space whose boundary is the
curve. – Zsbán Ambrus Sep 22 '12 at 19:03
2 @Zsbán, it is equivalent. – Anton Petrunin Sep 22 '12 at 19:08
2 Hi Andy. Thanks! This does it nicely. – Andres Caicedo Sep 22 '12 at 19:25
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Not the answer you're looking for? Browse other questions tagged cv.complex-variables real-analysis or ask your own question. | {"url":"https://mathoverflow.net/questions/107851/extending-jordan-loops?sort=oldest","timestamp":"2014-04-21T02:49:04Z","content_type":null,"content_length":"60251","record_id":"<urn:uuid:a7e8e502-f39f-4ee0-b80e-83d124fcb579>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00301-ip-10-147-4-33.ec2.internal.warc.gz"} |
Word Problem Database
Addition and Subtraction Word Problems - 3 Digit Numbers
1. Todd read 137 pages of his book last week.
He read 95 pages this week.
He only has 54 pages left to read.
How many pages are in Todd's book?
2. Ethan has a balance scale.
He found that 2 blue blocks will balance 1 red block.
A blue block weighs 258 grams.
How much does a red block weigh?
3. Rachel collected 635 soccer cards.
She collected 241 more cards than Tori.
How many cards did Tori collect?
4. 427 tickets were sold for the play this weekend.
Eric counted 295 chairs in the auditorium.
How many more chairs are needed?
5. Brian ordered 440 turkey sandwiches for the party.
He also ordered 365 ham sandwiches.
There were 187 sandwiches left after the party.
How many sandwiches were eaten? | {"url":"http://www.mathplayground.com/wpdatabase/Addition_Subtraction_3Digit_8.htm","timestamp":"2014-04-20T08:19:36Z","content_type":null,"content_length":"50409","record_id":"<urn:uuid:dbd7b232-335f-4c18-ba3e-8798a1470d60>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00586-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Finding the area of the shaded region.
January 1st 2011, 12:20 PM #1
Jan 2010
Finding the area of the shaded region.
Hi I'm a college student and I'm having a bit of difficulty with my Solid Mensuration homework. Can someone help me out?
I'm trying to find the area of the shaded region in the problems which I've posted pictures of in this thread.
I'm having a hard time because I think the data given is insufficient, but maybe I'm wrong so I'm asking for help
Thank you very much for any help you can give
Last edited by purplesky828; January 1st 2011 at 12:45 PM.
What exactly are you trying to do? Please post the whole question, not just the pictures.
Edit: The OP has since edited his/her post.
Last edited by mr fantastic; January 1st 2011 at 01:07 PM.
I agree with you that technically the data is insufficient.
However, a good guess on #1 would be $(85)(42.5)-\dfrac{(42.5)^2}{2}$.
Do you see why?
Yes I see how it would be a good guess. I was thinking of the same thing but I was unsure because the data wasn't given.
Thank you very much for your help!
Hi I'm a college student and I'm having a bit of difficulty with my Solid Mensuration homework. Can someone help me out?
I'm trying to find the area of the shaded region in the problems which I've posted pictures of in this thread.
I'm having a hard time because I think the data given is insufficient, but maybe I'm wrong so I'm asking for help
Thank you very much for any help you can give
To #2:
1. I assume that the white rectangle has a width of 1 m.
2. I've modified your sketch alittle bit. If the triangles marked in red are isosceles right triangles you can compose them to a complete square.
January 1st 2011, 12:26 PM #2
January 1st 2011, 01:06 PM #3
January 1st 2011, 05:07 PM #4
Jan 2010
January 4th 2011, 12:24 AM #5 | {"url":"http://mathhelpforum.com/geometry/167244-finding-area-shaded-region.html","timestamp":"2014-04-19T22:07:13Z","content_type":null,"content_length":"48855","record_id":"<urn:uuid:bdb7dfd6-9458-4e7f-b136-165d29c41f09>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00526-ip-10-147-4-33.ec2.internal.warc.gz"} |
How To Multiply Repeatig Decimals
May 5th 2006, 04:27 PM #1
How To Multiply Repeatig Decimals
I was wondering, is it possible to multiply 0.333...*0.666... and get the correct result of 0.222... without first turning the repeating decimals into fractions.
I have played with a couple of ideas
Is it possible to multiply the 2 summations together?
I have also looked at 0.333...*0.666... = 3(6/10+6/10^2+6/10^3+....)
But I am wodering where to go from here.
Any ideas would be appreciated.
My conclusion is that turning them into fractions then multipying is the simplist way. I do want to see other methods, practical or not.
Thank you for your time
Ranger SVO
I was wondering, is it possible to multiply 0.333...*0.666... and get the correct result of 0.222... without first turning the repeating decimals into fractions.
I have played with a couple of ideas
Is it possible to multiply the 2 summations together?
I have also looked at 0.333...*0.666... = 3(6/10+6/10^2+6/10^3+....)
But I am wodering where to go from here.
Any ideas would be appreciated.
My conclusion is that turning them into fractions then multipying is the simplist way. I do want to see other methods, practical or not.
Thank you for your time
I do not see any nice method. The proper method is to find their limits i.e. there fractions and multiply them.
Multiplying, infinite series is rather complicated.
If you have,
$S'=b_0+b_1+b_2+...<br />$
I have tryed that and it works, but I think it could get a little complicated
Assume 0.3 and 0.6 is a repeating decimal. Also I should note that 0.19 is 0.1999...
The answer 0.2 is a repeating decimal.
Excuse the mess I did this in a hurry. Also 1.999... = 2 in the work above.
Any critisism is welcome
Last edited by Ranger SVO; May 7th 2006 at 03:49 PM.
Ranger SVO
Also 1.999... = 2 in the work above.
If you are asking whether it is a mistake it is not. I noticed many people thinking that such an equality is a mistake. It is a very common fallacy thinking that a decimal can be expressed in two
different ways. Because you define 1.9999.... to be the value of the convergent of the real number which is 2.
Also, in set theory if you ever studied it. The Cantor's Diagnol Argument starts out as,
...if a number is able to be expressed in two different ways as a decimal then....
It is clearly true.
May 6th 2006, 06:42 PM #2
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Numerical study on particle velocity and sound pressure by circular
ASA 130th Meeting - St. Louis, MO - 1995 Nov 27 .. Dec 01
1pPA10. Numerical study on particle velocity and sound pressure by circular flat transducers.
Tohru Imamura
Natl. Res. Lab. of Metrol., 1-1-4, Umezono, Tsukuba, Ibaraki 305, Japan
During years of investigation on the ultrasonic near field, only the sound pressure, namely, the spatial distribution of velocity potential, has been studied. In this report, particle velocity and
acoustic impedance density of the ultrasonic field by circular flat transducers are derived and computed together with sound pressure. Sound pressure is proportional to the velocity potential of the
ultrasonic field. Its particle velocity is the space differential of the velocity potential, and the acoustic impedance density is the quotient of the sound pressure by the particle velocity. On the
axis of the transmitting circular flat transducer, the phase delay of the sound pressure has peculiar leaps. But, acoustic impedance density has constant leaps from -(pi)/2 to (pi)/2, where the
amplitude is zero. The mean value over a receiving coaxial circular flat transducer is also computed changing the ratio of the radius (a) of the circular flat transducer to the wavelength ((lambda))
of the ultrasonic wave. Mean amplitudes of sound pressure, particle velocity, and acoustic impedance density are tabulated with the normalized distance (z(lambda)/a[sup 2]) in the computing precision
of 0.1%. The mean amplitude of the z component of the particle velocity is always less than 1.0 and seems to be an appropriate response for the ultrasonic system of a pair of circular flat | {"url":"http://www.auditory.org/asamtgs/asa95stl/1pPA/1pPA10.html","timestamp":"2014-04-16T16:30:45Z","content_type":null,"content_length":"2076","record_id":"<urn:uuid:1a9756d0-c994-409e-a106-542e16bc1ef7>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00343-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Here's the question you clicked on:
The fourth term of an arithmetic sequence is 141, and the seventh term is 132. The first term is _____.
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\[a, a+d,a+2d,a+3d,a+4d,...\] you got \(a+3d=141\) and \(a+6d=132\)
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that means \(132-141=(a+6d)-(a+3d)=3d=-9\) and so \(d=-3\)
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okay so then d=4?
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actually \(d=-3\)
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umm please explain!!
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k lets go slow
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please and thanks
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actually before we even start, since \(132<141\) is it clear that the terms are getting smaller?
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in other words, "\(d\)" the "common difference" must be negative right?
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if we call the first term \(a\) then the second term is \(a+d\) for some \(d\) and the third term is \(a+2d\), the fourth term is \(a+3d\) the fifth term is \(a+4d\) etc
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in other words, you keep adding \(d\) to each term to get the next term, with the sophistication that you might be "adding" a negative number
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The fourth term of an arithmetic sequence is 141 tells you that \(a+3d=141\)
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you see that it is the fourth term, so it is \(a+3d\) not \(a+4d\)
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and the seventh term is 132 means \[a+6d=132\]
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from these two pieced of information we can solve for \(d\) and then solve for \(a\)
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a bit of algebra shows that \[a+6d-(a+3d)=3d\] right?
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ohh okay i get it..
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so we see that \[3d=132-141=-9\]
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so far so good?
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and so since it is 7-4= 3 then -9/3 would be -3 right? giving us the difference
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yeah \(-3\) is the difference
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what you said
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ohhh okay!!! i get it!! :D
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you are still not done though right?
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your question asked "The first term is _____"
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hmm well pluggin in the difference and then your equation... a4=141 a3=141+3=144 a2=144+3=147 a1=147+3=150 so then the first term would be 150 right? i just did it backwards
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yeah i guess so i would have said \(a+3\times (-3)=141\)or \[a-9=141\] making \(a=150\) your method means you understand what is going on, which is good but you certainly wouldn't want to use
that if you had say \(a_{75}\) and wanted \(a_1\)
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:D yay thank you!!! :D ohh okay... ill keep in mind that equation!! thank you soo much!
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Linear map eigenvalues
October 28th 2012, 01:40 PM #1
Oct 2012
Linear map eigenvalues
Could someone please explain to me the following?
Suppose you have a vector space, and a linear map T. need to find eigenvalues. when T is a matrix (2 by 2 or 3 by 3 or whichever square matrix with actual VALUES) this is easy, just work out the
characteristic polynomial det(I*x - T)=0 and then the eigenvalues come out.
what does one do when T is a function? say, T(f) = f + f ' where f ' denotes the derivative of f?
so we don't have T as a matrix but as a function...
say, T is applied along the basis (1, x, x^2, x^3, ... , x^n)
do we need to apply T to every element of the basis and then get a matrix out of that to work with? or how??
Re: Linear map eigenvalues
Hey cassius2020.
You will have to either calculate the value for a specific value given your input (like a particular value of x) or do a symbolic calculation using the construction of your function and
derivative if it has an explicit form.
There is an area of mathematics known as operator algebra's and it allows you to calculate a function of an operator so you can use this to get the value of your operator given a function and if
this can be evaluated, then you can get a specific value of your operator under some transformation and apply the usual technique.
Re: Linear map eigenvalues
in the example you gave, you are looking for eigenfunctions. in this case, T is the linear operator 1+D, where D(f) = f'.
solving (1+D)(f) = λf leads to the differential equation:
(1-λ)f + f' = 0. what the solution set is going to be is going to depend a LOT on what kind of function "f" can be (real-valued, complex-valued, polynomial, etc.).
for example, if f is a polynomial, then f must be the 0-polynomial. if f is a real-valued function on the interval [a,b], then f is of the form:
$f(t) = Ke^{-\left(\frac{t}{1-\lambda}\right)}$
in this example, 1+D might have every real eigenvalue except 1 (the spectrum is not a finite set). if we are given more information about f (often as "boundary conditions"), say f is defined on
[0,1], with f(0) = 1, f(1) = 2:
we get K = 1 (from f(0) = 1), and λ = log(2) + 1 (from f(1) = 2).
October 28th 2012, 07:13 PM #2
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Sep 2012
October 28th 2012, 11:24 PM #3
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Heckscher–Ohlin trade theory
From The New Palgrave Dictionary of Economics, Second Edition, 2008
Edited by Steven N. Durlauf and Lawrence E. Blume
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Heckscher–Ohlin trade theory consists of four principal theorems, viz. the Heckscher–Ohlin trade theorem whereby relatively capital-abundant countries export relatively capital-intensive commodities,
the factor-price equalization theorem whereby trade in goods may serve to equalize wage rates between countries, the Stolper–Samuelson theorem whereby an increase in the price of the relatively
labour-intensive commodity unambiguously improves the real wage rate, and the Rybczynski theorem stating that an increase in capital endowment by itself must cause some output to fall if prices are
held constant. The article discusses the nature and fate of these theorems.
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How to cite this article
Jones, Ronald W. "Heckscher–Ohlin trade theory." The New Palgrave Dictionary of Economics. Second Edition. Eds. Steven N. Durlauf and Lawrence E. Blume. Palgrave Macmillan, 2008. The New Palgrave
Dictionary of Economics Online. Palgrave Macmillan. 16 April 2014 <http://www.dictionaryofeconomics.com/article?id=pde2008_H000034> doi:10.1057/9780230226203.0718
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Computational Geometry
Items marked
In the future, this page may include short reviews of each book. If you are familiar with any of these books and would like to contribute a review, please send me email. I would especially like to
hear from anyone who has used any of these books as textbooks, either as students or teachers.
Michel Pocchiola also has several web pages listing computational geometry books, collections, and monographs, and a few books on polytopes.
SpringerVerlag also publishes several relevant monographs and conference proceedings - far too many to list here! - in their Lecture Notes in Computer Science series.
• Keisankikagaku Risankagaku (Computational and Discrete Geometry)
by David Avis and Hiroshi Imai
Asakura, Tokyo, 1994. In Japanese, 150pp.
• Advances in Discrete and Computational Geometry
Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference "Discrete & Computational Geometry: Ten Years Later", July 1996.
Edited by Bernard Chazelle, Jacob E. Goodman, and Richard Pollack,
Contemporary Mathematics series
Americal Mathematical Society, Providence, in preparation.
• Computational Geometry
Edited by Godfried T. Toussaint
North-Holland, Amsterdam, 1985
Out of print.
• Computational Morphology
Edited by Godfried T. Toussaint
North-Holland, Amsterdam, 1988
Out of print.
• Computing in Euclidean Geometry (2nd edition)
Edited by Ding-Zhu Du and Frank Hwang
Lectures Notes Series on Computing 4
World Scientific, Singapore, 1995
• Algorithmic Foundations of Robotics
Edited by Ken Goldberg, Dan Halperin, Jean-Claude Latombe, and Randall Wilson
A K Peters, Ltd., 1995
• Handbook of Discrete and Computational Geometry
Edited by Jacob E. Goodman and Joseph O'Rourke
CRC Press, 1997.
• Discrete and Computational Geometry: Papers from the DIMACS Special Year
Edited by Jacob E. Goodman, Richard Pollack, and William Steiger
DIMACS Series in Discrete Mathematics and Computer Science 6
American Mathematical Society, 1992
• Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift
Edited by Peter Gritzmann and Bernd Sturmfels
DIMACS Series in Discrete Mathematics and Computer Science 4
American Mathematical Society, 1991
• Directions in Geometric Computing
Edited by Ralph Martin
Information Geometers Ltd, 1993
• New Trends in Discrete and Computational Geometry
Edited by János Pach
Algorithms and Combinatorics 10
SpringerVerlag, 1995.
• Fundamentos de Geometria Computacional (em Português)
by Pedro J. de Resende and Jorge Stolfi
Preparado para a IX Escola de Computação (Recife, Brasil, 1994)
Departmento de Informatica da Universidade Federal de Pernambuco, Brasil, 1994
• Handbook for Computational Geometry
Edited by Jorge Urrutia and Jörg-Rudiger Sack
North-Holland, in preparation.
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Fairmount Heights, MD Math Tutor
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Calculate grade needed on a final based on current grade
June 6th 2010, 05:50 PM #1
Junior Member
Jan 2010
Calculate grade needed on a final based on current grade
Hello, I am wondering, because my grade in Math right now is so high and above what you need for an A in my class, that if I could completely bomb the final and still have an A.
Right now, I have a 95.00% exactly. I need to have greater than or equal to an 86.00% in the class to have an A.
If I have:
37.1 / 40.0 points in Tests category which is worth 50% of my grade,
19.1 / 20.0 points in Quiz category which is worth 30% of my grade,
45.0 / 45.0 points in Homework category which is worth 20% of my grade,
And the final will go under the tests category as two independent grades each being worth 5 points. Example: If I get a 4.2 on the final (out of 5) then it will go in as 4.2 two times. Just
mainly for emphasis just because it's a final exam.
Therefore, what do I need to get on this final to keep my A, if nothing else is added to the gradebook?
Hello qcom
Hello, I am wondering, because my grade in Math right now is so high and above what you need for an A in my class, that if I could completely bomb the final and still have an A.
Right now, I have a 95.00% exactly. I need to have greater than or equal to an 86.00% in the class to have an A.
If I have:
37.1 / 40.0 points in Tests category which is worth 50% of my grade,
19.1 / 20.0 points in Quiz category which is worth 30% of my grade,
45.0 / 45.0 points in Homework category which is worth 20% of my grade,
And the final will go under the tests category as two independent grades each being worth 5 points. Example: If I get a 4.2 on the final (out of 5) then it will go in as 4.2 two times. Just
mainly for emphasis just because it's a final exam.
Therefore, what do I need to get on this final to keep my A, if nothing else is added to the gradebook?
At present, your mark is 95.025%.
If your final test is effectively worth 10 marks (being counted double) and you score 0 on it, you will have 37.1 / 50.0 in Tests category. This would bring your final mark down to 85.75% - which
would give you a grade A only if it gets rounded up to the nearest whole number.
To be sure, you would need to score at least 0.15 / 5 (counting as 0.3 out of 10 and giving you 37.4 / 50.0). This would give you a final mark of 86.05%.
But you need to be careful. Some grading systems operate on a 'minimum threshold' basis, whereby you need to score a minimum of a certain minimum mark in each test. So - go for the highest mark
you can!
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More Cache Craziness
Posted by Louis Brandy on 16 March 2009
I've written before about the importance of cpu cache efficient algorithms. This is yet another adventure in that world. First, however, let's start with a simple cache quiz: which code is faster?
/* going vertical /
for (x=0;x<width;x++)
for (y=0;y<height;y++)
a[ywidth+x] = b[ywidth+x] + c[ywidth+x]
/* going horizontal /
for (y=0;y<height;y++)
for (x=0;x<width;x++)
a[ywidth+x] = b[ywidth+x] + c[ywidth+x]
The second one. Right? Every programmer knows that the second is faster, and they know why (I hope). A simple matrix addition is simple enough that we can "reason" our way into which is theoretically
faster without actually trying it. I recently ran a much more complex example of a similar problem.
The problem: For face recognition, it is often necessary to compute a huge face similarity matrix. Face X compared with face Y returns a score Z. We place Z at (X,Y). It's a fairly straightforward
output to a huge comparison between two sets of faces. The naive method would look very similar to the matrix adds at the top. You start in the upper left corner, do the entire first row, one by one,
and then move onto the second row, etc.
The hypothesis: If we break the computation into sub-blocks, it will more effectively use cache and thus be faster. As an example, for a 1000x1000 comparison, the naive method would do all 1000
comparisons in the first row, then 1000 more for the second, and so on. The block method would break the computation into, for example, 100x100 blocks. By only using these sub-blocks, it allows you
to use the same input faces more frequently and thus, hopefully, more cache efficiently.
So let's try it
I took the 20 required minutes, randomly picked a block size (I think it was on the order of 50x50), and lo and behold, it was substantially faster than the naive method. Alright. Now, how do I go
about finding a good block size? Unlike our matrix addition example, this is a tough answer to reason your way towards, so I decided to play around to build up some intuition for the problem.
Does block area matter?
Yes, it matters. 2x2 blocks are just too small. There's too much overhead introduced by breaking the problem into that many blocks (especially with threading enabled). There needs to be enough "meat"
in the blocks to make it worth your while.
Are square blocks best?
It seemed like a logical question. It took almost no time for me to figure out that absolutely not, square blocks were not better. The very first test I ran was something along the lines of 50x50
blocks versus 25x100 blocks. The 25x100 blocks crushed the 50x50 blocks. Interesting.
How about the direction of travel?
Which was faster 10x40 or 40x10? Turns out 10x40 was substantially faster than 40x10. This is getting confusing. I spent another hour playing with numbers and eventually came to the conclusion that
putting the smallest dimension in the horizontal direction made it faster. Inside the inner function, if a block size comes in that doesn't follow this rule, I switched them (so if you asked for a
block size of 40x10, you got 10x40).
Time to stop screwing around
I was about 2 hours into this little adventure when I realized there was absolutely no way I was going to meander to the right answer. This wasn't a simple problem. The only solution was my old
friend science. Let's actually run the experiment and see what happens. I picked 16 different lengths ranging from 2 to 1000, and ran every permutation of block sizes and graphed the results. Here's
what it looked like:
This graph is just so cool. Let me explain what you are looking at. It's a 2d plot in a heat-graph format. The darker the color, the faster the code ran. The position in the grid represents the block
size. As we travel right along the graph, we are increasing the longer dimension and as we travel upward along the graph, we are increasing the shorter dimension. The diagonal from bottom-left to
top-right, then, is the square block sizes. The graph is also symmetric because of the rule I put in to ensure the longer dimension is always height (explained above). This graph has three
interesting features:
1. The extremely "hot" (slow) region hugging the bottom and the left axes. This region is when one (or both) dimensions is extremely small. With extremely small windows, there's too much overhead
(especially with threading) and it begins to dominate the actual computation.
2. The bizarre rectangle in the upper-right. More on this in a second.
3. The sweet-spot dark regions that essentially correspond to extremely long and fairly skinny rectangles.
Making Sense of it All
That rectangle in the upper right perplexed me and so I showed it around. Someone at the office had a conjecture that sounded plausible (versus my previous theory of plain old black magic). For small
widths, the entire horizontal input is able to be cached. As you grow the width, eventually you reach a point where the horizontal input no longer fits, and by the time you start on the next row of
the block, the first input is largely gone. In other words, my little graph shows quite clearly the moment that our block blows up our cache.
That particular theory meant, then, that the point where the block was "too" wide depended heavily on the machine and the environment the process was running. Since I didn't want to tune too much to
a single machine, I thought it wise to avoid going anywhere near that line.
My final block-size selection algorithm looks something like this: Use as tall a rectangle as is reasonable, make it as skinny as possible such that you ensure there is enough area to drown out
threading overhead.
Measure. Measure. Measure.
At the end of the day, you can pretty much "explain" any cache behavior, once you've seen it. You can find a plausible reason why certain things make the code run faster. Maybe in hindsight your
explanation will even makes sense. Its become obvious to me, however, that for any non-trivial problem, you positively need to rigoriously experiment.
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PHIL P551 28113 Philosophy and the Foundations of Math
Philosophy | Philosophy and the Foundations of Math
P551 | 28113 | McCarty
A survey of recent and contemporary philosophy of mathematics in the
analytical tradition, based on readings from the
anthology "Philosophy of Mathematics," edited by Paul Benacerraf and
Hilary Putnam, and on the monograph “Thinking about Mathematics” by
Stewart Shapiro. Close and detailed consideration will be given to
logicism, formalism, and intuitionism, as well as to recent
philosophical issues concerning our knowledge of mathematical
truths, and the precise nature of sets.
Required texts: Benacerraf, P. and H. Putnam (eds) "Philosophy of
Mathematics: Selected Readings." Second Edition. Cambridge
University Press. 1984. Paperback.
Shapiro, Stewart. “Thinking about Mathematics.” Oxford University
Press. 2000. Paperback.
Assignments and Grading: A student's grade in the course will be
determined by his or her performance on one in-class midterm
examination, one take-home final examination, frequent short papers,
quizzes, presentations and classroom participation.
Prerequisites: Students should have completed a solid and
mathematically sophisticated course in mathematical logic at the
graduate level. | {"url":"http://www.indiana.edu/~deanfac/blfal10/phil/phil_p551_28113.html","timestamp":"2014-04-20T23:45:25Z","content_type":null,"content_length":"1704","record_id":"<urn:uuid:6b95b582-8c0a-4c84-a74a-91806bace6bb>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00507-ip-10-147-4-33.ec2.internal.warc.gz"} |
Trace Elements in West Virginia Coals
Elemental Distribution (Histogram) Example
The statistical distribution (histogram) of each trace element is included to convey the actual distribution of trace element values in West Virginia coals. These charts whole coal basis for all coal
samples. The average trace element value in ppm is given as well as the number of samples determined. Some trace elements display a somewhat normal distribution, while other element distributions are
highly skewed. Arsenic, for example, averages 17.13 ppm in 848 samples, but is less than 5 ppm in 363 samples, and less than 10 ppm in 510 samples with a small number of samples containing very high
(>100 ppm) arsenic concentrations. | {"url":"http://www.wvgs.wvnet.edu/www/datastat/te/hisexampl.htm","timestamp":"2014-04-18T13:27:19Z","content_type":null,"content_length":"2064","record_id":"<urn:uuid:a99ff430-1d47-4477-8564-b8b1e892c230>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00215-ip-10-147-4-33.ec2.internal.warc.gz"} |
Some bounds on the computational power of piecewise constant derivative systems
, 1997
"... We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential
equation and can be considered as computational machines working on a continuous space with a continuous tim ..."
Cited by 26 (6 self)
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We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential
equation and can be considered as computational machines working on a continuous space with a continuous time. We prove that the languages recognized by rational PCD systems in dimension d = 2k + 3
(respectively: d = 2k + 4), k 0, in finite continuous time are precisely the languages of the ! k th (resp. ! k + 1 th ) level of the hyper-arithmetical hierarchy. Hence the reachability problem for
rational PCD systems of dimension d = 2k + 3 (resp. d = 2k + 4), k 1, is hyper-arithmetical and is \Sigma ! k-complete (resp. \Sigma ! k +1 -complete).
- Formal Modeling and Analysis of Timed Systems, 4th Int. Conf. FORMATS 2006, volume 4202 of LNCS , 2006
"... Abstract. Well-known hierarchies discriminate between the computational power of discrete time and space dynamical systems. A contrario the situation is more confused for dynamical systems when
time and space are continuous. A possible way to discriminate between these models is to state whether the ..."
Cited by 3 (1 self)
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Abstract. Well-known hierarchies discriminate between the computational power of discrete time and space dynamical systems. A contrario the situation is more confused for dynamical systems when time
and space are continuous. A possible way to discriminate between these models is to state whether they can simulate Turing machine. For instance, it is known that continuous systems described by an
ordinary differential equation (ODE) have this power. However, since the involved ODE is defined by overlapping local ODEs inside an infinite number of regions, this result has no significant
application for differentiable models whose ODE is defined by an explicit representation. In this work, we considerably strengthen this result by showing that Time Differentiable Petri Nets (TDPN)
can simulate Turing machines. Indeed the ODE ruling this model is expressed by an explicit linear expression enlarged with the “minimum ” operator. More precisely, we present two simulations of a two
counter machine by a TDPN in order to fulfill opposite requirements: robustness and boundedness. These simulations are performed by nets whose dimension of associated ODEs is constant. At last, we
prove that marking coverability, submarking reachability and the existence of a steady-state are undecidable for TDPNs. 1
, 2004
"... In the Cellular Automata (CA) literature, discrete lines inside (discrete) space-time diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special
purposes. In this article, we present a parallel analog model of computation corresponding to this ideali ..."
Cited by 3 (1 self)
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In the Cellular Automata (CA) literature, discrete lines inside (discrete) space-time diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special
purposes. In this article, we present a parallel analog model of computation corresponding to this idealization: dimensionless signals are moving on a continuous space in continuous time generating
Euclidean lines on (continuous) space-time diagrams. Like CA, this model is parallel, synchronous, uniform in space and time, and uses local updating. The main difference is that space and time are
continuous and not discrete (i.e. R instead of Z). In this article, the model is restricted to Q in order to remain inside Turing-computation theory. We prove that our model can carry out any
Turing-computation through two-counter automata simulation and provide some undecidability results.
- Journal of Computer and System Sciences , 1995
"... In this paper we show how to construct for every set P of integers in the arithmetical hierarchy a dynamical system H with piecewiseconstant derivatives (PCD) such that deciding membership in P
can be reduced to solving the reachability problem between two rational points for H. The ability of s ..."
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In this paper we show how to construct for every set P of integers in the arithmetical hierarchy a dynamical system H with piecewiseconstant derivatives (PCD) such that deciding membership in P can
be reduced to solving the reachability problem between two rational points for H. The ability of such apparently-simple dynamical systems, whose definition involves only rational parameters, to # A
preliminary version of the paper appeared in P.S. Thiagarajan (Ed.), "Proc. FST/TCS'95", 471-483, LNCS 1026, Springer, 1995. This research was supported in part by the European Community projects
HYBRID EC-US-043 and INTAS-94-697 as well as by Research Grants #93-012-884, 97-01-00692 and 96-15-96048 of Russian Foundation for Basic Research. Verimag is a joint laboratory of cnrs, ujf and inpg.
Some of the results were obtained while the first author was a visiting professor at ensimag, inpg, Grenoble. 1 "solve" highly unsolvable problems is closely related to Zeno's paradox, namely the
ability to pack infinitely many discrete steps in a bounded interval of time. 1
, 2010
"... beyond the ..."
, 2010
"... (will be inserted by the editor) Abstract geometrical computation 3: black holes for classical and analog computating ..."
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(will be inserted by the editor) Abstract geometrical computation 3: black holes for classical and analog computating | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1156668","timestamp":"2014-04-17T13:43:54Z","content_type":null,"content_length":"25601","record_id":"<urn:uuid:f0848269-87e3-48a0-8d7a-49307958adf2>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00152-ip-10-147-4-33.ec2.internal.warc.gz"} |
Digital Nightmares
Digital Nightmares
the bad news: this is how i spent my weekend...
the good news: I think i got it
i ♥ parametric variations in closed solid models...
• 6 Comments
Sep 25, 06 10:46 am
for the ashes of a weekend...
Sep 25, 06 10:51 am
parametric variations ♥ you too!
This is one of those examples about what happens when I overthink assignments (which coincidentally happens to me quite a bit). SolidWorks is actually pretty easy, and this is one of the many,
many exercises I did to get to know the properties and how to control them. I spent soooo much time thinking that there was an obscure objective to the assignment, but all along I was doing it
right. I am now finally working on my final model of assignment 1 due tomorrow in class.
Sep 25, 06 9:09 pm
mmmm parametric models... yummy.
vado retro
Sep 25, 06 11:51 pm | {"url":"http://archinect.com/gsd-quilian/digital-nightmares","timestamp":"2014-04-20T19:01:10Z","content_type":null,"content_length":"85679","record_id":"<urn:uuid:97673628-7a44-49fe-be0f-61764ec7b6fc>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00080-ip-10-147-4-33.ec2.internal.warc.gz"} |
East Elmhurst, New York, NY
New York, NY 10028
STAT/MATH/Actuarial Science/CFA/MBA/Fin. - Ivy League Exp & Prof
Holding First Class Honours Degrees in
ematical Sciences and Psychology from Newcastle University, and an MSc. in Applied Statistics from Cambridge University, England, I became a full-time Private
ematics Coach over ten years ago, tutoring students to undergraduate...
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Bensalem Geometry Tutor
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What is the automorphism group of this geometry?
up vote 3 down vote favorite
Define the following incidence structure of rank three. The points are the elements of $\mathbb{Z}_7=$ {$0,\ldots,6$}. The lines of type 1 are the triples $(x,x+1,x+3)$ modulo $7$. The lines of type
2 are the triples $(x,x+1,x+5)$ modulo 7. Define the incidence relation as follows. A point is incident to a line of type 1 (resp.2) if it is contained in the line. A line of type 1 is incident to a
line of type 2 if they have two points in common.
It is not difficult to check that this incidence structure is a geometry (in the sense of Buekenhout). Somehow it looks like two superposed Fano planes.
Here is my question: what is the full automorphism group of this geometry, and what is the type preserving automorphism group of this geometry?
gr.group-theory incidence-geometry
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1 Answer
active oldest votes
Your geometry has the property that each of its rank 2 restrictions is a Fano plane. In particular, the type-preserving automorphism group (let's call it $G$) is a subgroup of the
automorphism group of the Fano plane, which is $PSL(3,2)$. The group $G$ has the property that the pointwise stabilizer of any two points is trivial (indeed, if $g \in G$ fixes for
instance $0$ and $1$, then it has to fix $3$ and $5$, and from that you deduce that it has to fix everything).
On the other hand, $G$ contains the Singer cycle $x \mapsto x+1 \pmod{7}$, and it contains, for instance, the element $(1 2 4)(3 6 5)$, so it has order divisible by $21$. It follows
up vote 8 down that $G$ is isomorphic to the Frobenius group of order $21$, which is a maximal subgroup of $PSL(3,2)$.
vote accepted
Note that in the full automorphism group of the geometry, you can interchange all types, since you can for instance interchange lines of type 1 and type 2 by the permutation $(24)(35)
$, and by symmetry you can interchange every two types and hence the induced action on the types is $\operatorname{Sym}(3)$. Hence the full automorphism group of the geometry is an
extension of $G$ by $\operatorname{Sym}(3)$ (which is a group of order $126$).
$S_{7}$ has no subgroup of order $126$, since a Sylow $7$-subgroup normalizer has order $42$ (and any group of order $126$ has a normal Sylow $7$-subgroup). Then how is this
automorphism group defined? – DavidLHarden Apr 11 '11 at 17:31
It is no longer a subgroup of $S_7$, because an automorphism interchanging points with lines (of type $1$ or type $2$) is not an element of $S_7$. – Tom De Medts Apr 11 '11 at 18:00
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Not the answer you're looking for? Browse other questions tagged gr.group-theory incidence-geometry or ask your own question. | {"url":"http://mathoverflow.net/questions/61204/what-is-the-automorphism-group-of-this-geometry","timestamp":"2014-04-19T17:50:40Z","content_type":null,"content_length":"54071","record_id":"<urn:uuid:b4557e73-91d5-4c0a-924c-e52256e331ba>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00371-ip-10-147-4-33.ec2.internal.warc.gz"} |
McMaster University - Academic Integrity
Plagiarism - Engineering/Science Example
In [1] Mokhberi et al. write:
One of the difficulties associated with Hall measurement is in the interpretation of the measured quantities, the Hall coefficient, and the conductivity. From these two quantities, one can calculate
mobility and active dose, where based on a simple first-order approach, the relations are
However, these relations neglect the statistical variations in the velocities of free carriers. In general, this is taken into account by introducing the Hall factor (or Hall ratio) r
One could use this article as a source to discuss the concept of a Hall factor. The following are examples of how one might use this information along with a discussion of whether it is plagiarism:
1. Exact copying
□ In this example, a copy of the entire text is not called for. The text is not so unique that a student should feel that he/she needs to reproduce it in its entirety.
□ If a student did decide that he/she wishes to copy the entire text, it would have to be set out as a direct quote using indentation as above.
□ Without explicitly citing the text as a direct quote, exact copying is, by definition, plagiarism.
2. Slight rewording
□ In general, slight rewording is not enough to avoid plagiarism. For example, if instead of “…. From these two quantities, one can calculate mobility…” one said, “…. From these quantities, we
can calculate mobility…” this would still rank as plagiarism.
3. Citing the equations
□ In general, citing equations without giving a reference specific to the equation is plagiarism.
□ In some cases the equations are so standard and so well-known in the field that a specific citation is not required. This is the case, for example, with the Hall equations used in the
paragraph above. The authors do not give a specific reference because the equations would be known to any graduate student well-versed in the field of electrical measurements of
□ When in doubt, it is always better to give a citation. Even if one is reproducing a standard and well-known equation it is better to give a citation by saying (see for example [n]) where "n"
is the number of a reference in your list of references pointing to a particular textbook or article containing the equation.
4. Using your own words
□ This would be the appropriate method of giving this information.
Students sometimes struggle with how to “reword” a particular paragraph. In fact, for a scientific paper this is the wrong way to think about the task. One should rather think of the task as
made up of two steps: (1) understand the concept completely; and (2) explain the concept from scratch to someone else who desires to understand it.
□ The best way to do this in general is to understand the concept from more than one source then attempt an explanation without looking at any of those sources. It may help to actually try to
verbally explain the concept to a friend and allow that friend to ask questions regarding this concept.
□ Even without the benefit of multiple sources, it is always possible to use your own words. Again, thinking of the task as an attempt to educate others about a concept, we have to realize that
we make a large number of choices when we explain something: we choose which steps of a derivation to include, which steps to skip over, which assumptions we state, which objections we
address, the specific order in which the different parts of the explanation are to be presented, the specific vocabulary, etc.
□ As an example, consider the above paragraphs on Hall measurements, and compare with an alternate explanation of the same concept:
"The results of a Hall measurement are two numbers: conductivity, and the Hall coefficient, RH. We use these to compute mobility, c, and carrier concentration, p or n, using:
where r is an empirically-determined coefficient known as the Hall factor or Hall ratio. The Hall ratio is a correction required because the velocities of free electrons and holes are known
to have statistical fluctuations that will affect the Hall measurement."
In writing the above, I made a number of choices different from the original authors. For example, I gave the symbols for each quantity when I first mentioned it instead of assuming that the
reader would be familiar with it. I went straight to the second set of equations without discussing the simplified version first. I did not approach the derivation as a "difficulty associated
with Hall measurement" but rather presented it as a well-known procedure for measurement. You may agree with some of these choices or you may agree with none. The point is that you will have
your own choices to make.
Plagiarism in engineering and science writing is a problem precisely because it means that you did not make your own choices regarding what merits presentation and how. And this may mean that
you did not understand the material to begin with.
[1] A. Mokhberi, P. B. Griffin, J. D. Plummer, E. Paton, S. McCoy, and K. Elliott, "A Comparative Study of Dopant Activation in Boron, BF2, Arsenic, and Phosphorus Implanted Silicon," IEEE Trans.
Electron Dev., vol. 49, pp. 1183-1191, 2002. | {"url":"http://www.mcmaster.ca/academicintegrity/students/typeofad/plagiarism/engsci.html","timestamp":"2014-04-18T05:30:08Z","content_type":null,"content_length":"22535","record_id":"<urn:uuid:06e67e9a-2c12-40fe-89a6-6ce6af1ef8ec>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00050-ip-10-147-4-33.ec2.internal.warc.gz"} |
How do metrics behave under joining along a manifold embedded in the boundary?
up vote 5 down vote favorite
How do metrics behave under joining along a manifold embedded in the boundary?
This is, more-or-less, part of Problem 4.66 in Kirby's List:
Problem 4.66 How do metrics (e.g. Riemannian, Lorentz, constant curvature) behave under standard topological constructions such as connected sum, plumbing, handle addition? Same question for $\
eta$-invariants, moduli spaces, etc.
So, in theory it is an open problem. However, Kirby states of the problem:
Much has probably been done on this open ended problem, and the editor has not attempted to update it.
So, it would seem that this question may indeed have an answer in the literature. My question is: What is that answer and where does one find it?
add comment
Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.
Browse other questions tagged differential-topology smooth-manifolds mg.metric-geometry gt.geometric-topology reference-request or ask your own question. | {"url":"http://mathoverflow.net/questions/58393/how-do-metrics-behave-under-joining-along-a-manifold-embedded-in-the-boundary","timestamp":"2014-04-20T06:32:03Z","content_type":null,"content_length":"48113","record_id":"<urn:uuid:b823df4d-59e8-4616-a34e-589f852d2fa1>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00499-ip-10-147-4-33.ec2.internal.warc.gz"} |
NLVM Number & Operations Manipulatives
Bar Chart – Create a bar chart showing quantities or percentages by labeling columns and clicking on values.
Base Blocks – Illustrate addition and subtraction in a variety of bases.
Base Blocks Addition – Use base ten blocks to model grouping in addition.
Base Blocks Decimals – Add and subtract decimal values using base blocks.
Base Blocks Subtraction – Use base ten blocks to model separation of groups in subtraction.
Chip Abacus – Learn about carrying and digits using chips.
Circle 99 – A puzzle involving adding positive and negative integers to sum to ninety nine.
Color Chips - Addition – Use color chips to illustrate addition of integers.
Color Patterns – Arrange colors to complete a pattern.
Diffy – Solve an interesting puzzle involving the differences of given numbers.
Fraction Bars – Learn about fractions using fraction bars.
Fractions - Naming – Write the fraction corresponding to the highlighted portion of a shape.
Fractions - Parts of a Whole – Relates parts of a whole unit to written description and fraction.
Fractions - Visualizing – Illustrate a fraction by dividing a shape and highlighting the appropriate parts.
Hundreds Chart – Practice counting and visualize number patterns using a hundreds chart.
Mastermind – Use inference and logic to play a game and guess a hidden pattern of pegs.
Money – Learn about money by counting and making change.
Number Line Arithmetic – Illustrates arithmetic operations using a number line.
Number Line Bars – Use bars to show addition, subtraction, multiplication, and division on a number line.
Number Line Bars - Fractions – Divide fractions using number line bars.
Number Line Bounce – Number line addition and subtraction game.
Number Patterns – Discover the pattern and complete a sequence of numbers.
Percentages – Discover relationships between fractions, percents, and decimals.
Pie Chart – Explore percentages and fractions using pie charts.
Place Value Number Line – Explore place value by placing dots on number lines.
Rectangle Division – Visualize and practice dividing numbers by using an area representation.
Rectangle Multiplication – Visualize the multiplication of two numbers as an area.
Rectangle Multiplication of Integers – Visualize and practice multiplying integers using an area representation.
Sieve of Eratosthenes – Relate number patterns with visual patterns.
Spinners – Work with spinners to learn about numbers and probabilities.
Abacus – An electronic abacus that can be used to do arithmetic.
Bar Chart – Create a bar chart showing quantities or percentages by labeling columns and clicking on values.
Base Blocks – Illustrate addition and subtraction in a variety of bases.
Base Blocks Addition – Use base ten blocks to model grouping in addition.
Base Blocks Decimals – Add and subtract decimal values using base blocks.
Base Blocks Subtraction – Use base ten blocks to model separation of groups in subtraction.
Chip Abacus – Learn about carrying and digits using chips.
Circle 0 – A puzzle involving adding positive and negative integers to sum to zero.
Circle 21 – A puzzle involving adding positive and negative integers to sum to twenty one.
Circle 3 – A puzzle involving adding positive real numbers to sum to three.
Circle 99 – A puzzle involving adding positive and negative integers to sum to ninety nine.
Color Chips - Addition – Use color chips to illustrate addition of integers.
Color Chips - Subtraction – Use color chips to illustrate subtraction of integers.
Color Patterns – Arrange colors to complete a pattern.
Diffy – Solve an interesting puzzle involving the differences of given numbers.
Factor Tree – Factor numbers using a tree diagram.
Fraction Bars – Learn about fractions using fraction bars.
Fraction Pieces – Work with parts and wholes to learn about fractions.
Fractions - Adding – Illustrates what it means to find a common denominator and combine.
Fractions - Comparing – Judge the size of fractions and plot them on a number line.
Fractions - Equivalent – Illustrates relationships between equivalent fractions.
Fractions - Naming – Write the fraction corresponding to the highlighted portion of a shape.
Fractions - Parts of a Whole – Relates parts of a whole unit to written description and fraction.
Fractions - Rectangle Multiplication – Visualize and practice multiplying fractions using an area representation.
Fractions - Visualizing – Illustrate a fraction by dividing a shape and highlighting the appropriate parts.
Grapher – A tool for graphing and exploring functions.
Hundreds Chart – Practice counting and visualize number patterns using a hundreds chart.
Mastermind – Use inference and logic to play a game and guess a hidden pattern of pegs.
Money – Learn about money by counting and making change.
Number Line Arithmetic – Illustrates arithmetic operations using a number line.
Number Line Bars – Use bars to show addition, subtraction, multiplication, and division on a number line.
Number Line Bars - Fractions – Divide fractions using number line bars.
Number Line Bounce – Number line addition and subtraction game.
Number Patterns – Discover the pattern and complete a sequence of numbers.
Number Puzzles – Solve puzzles involving arranging numbers on a diagram so that they add up to a given value.
Peg Puzzle – Win this game by moving the pegs on the left past the pegs on the right.
Percent Grids – Represent, name, and explore percentages using hundreds grids.
Percentages – Discover relationships between fractions, percents, and decimals.
Pie Chart – Explore percentages and fractions using pie charts.
Place Value Number Line – Explore place value by placing dots on number lines.
Rectangle Division – Visualize and practice dividing numbers by using an area representation.
Rectangle Multiplication – Visualize the multiplication of two numbers as an area.
Rectangle Multiplication of Integers – Visualize and practice multiplying integers using an area representation.
Sieve of Eratosthenes – Relate number patterns with visual patterns.
Tangrams – Use all seven Chinese puzzle pieces to make shapes and solve problems.
Venn Diagrams – Investigate common features of sets.
Abacus – An electronic abacus that can be used to do arithmetic.
Base Blocks – Illustrate addition and subtraction in a variety of bases.
Base Blocks Addition – Use base ten blocks to model grouping in addition.
Base Blocks Decimals – Add and subtract decimal values using base blocks.
Base Blocks Subtraction – Use base ten blocks to model separation of groups in subtraction.
Chip Abacus – Learn about carrying and digits using chips.
Circle 0 – A puzzle involving adding positive and negative integers to sum to zero.
Circle 21 – A puzzle involving adding positive and negative integers to sum to twenty one.
Circle 3 – A puzzle involving adding positive real numbers to sum to three.
Circle 99 – A puzzle involving adding positive and negative integers to sum to ninety nine.
Color Chips - Subtraction – Use color chips to illustrate subtraction of integers.
Conway's Game of Life – Discover the rules that determine change in these simulations.
Diffy – Solve an interesting puzzle involving the differences of given numbers.
Dueling Calculators – Visualize a dramatic simulation of the effect of propagating rounding errors.
Factor Tree – Factor numbers using a tree diagram.
Fibonacci Sequence – Explore the Fibonacci sequence and the golden ratio.
Fraction Pieces – Work with parts and wholes to learn about fractions.
Fractions - Adding – Illustrates what it means to find a common denominator and combine.
Fractions - Comparing – Judge the size of fractions and plot them on a number line.
Fractions - Equivalent – Illustrates relationships between equivalent fractions.
Fractions - Rectangle Multiplication – Visualize and practice multiplying fractions using an area representation.
Function Machine – Explore the concept of functions by putting values into this machine and observing its output.
Golden Rectangle – Illustrates iterations of the Golden Section.
Grapher – A tool for graphing and exploring functions.
Mastermind – Use inference and logic to play a game and guess a hidden pattern of pegs.
Money – Learn about money by counting and making change.
Number Line Bars - Fractions – Divide fractions using number line bars.
Number Line Bounce – Number line addition and subtraction game.
Number Puzzles – Solve puzzles involving arranging numbers on a diagram so that they add up to a given value.
Pascal's Triangle – Explore patterns created by selecting elements of Pascal's triangle.
Peg Puzzle – Win this game by moving the pegs on the left past the pegs on the right.
Percent Grids – Represent, name, and explore percentages using hundreds grids.
Percentages – Discover relationships between fractions, percents, and decimals.
Sieve of Eratosthenes – Relate number patterns with visual patterns.
Spinners – Work with spinners to learn about numbers and probabilities.
Tangrams – Use all seven Chinese puzzle pieces to make shapes and solve problems.
Tight Weave – Visualize the creation of the Sierpinski Carpet, an iterative geometric pattern that resembles a woven mat.
Turtle Geometry – Explore numbers, shapes, and logic by programming a turtle to move.
Venn Diagrams – Investigate common features of sets.
Abacus – An electronic abacus that can be used to do arithmetic.
Circle 0 – A puzzle involving adding positive and negative integers to sum to zero.
Circle 21 – A puzzle involving adding positive and negative integers to sum to twenty one.
Circle 3 – A puzzle involving adding positive real numbers to sum to three.
Circle 99 – A puzzle involving adding positive and negative integers to sum to ninety nine.
Conway's Game of Life – Discover the rules that determine change in these simulations.
Counting All Pairs – Create a path that sets up a one-to-one correspondence between the counting numbers and infinite sets of ordered pairs of integers.
Diffy – Solve an interesting puzzle involving the differences of given numbers.
Dueling Calculators – Visualize a dramatic simulation of the effect of propagating rounding errors.
Fibonacci Sequence – Explore the Fibonacci sequence and the golden ratio.
Fractions - Adding – Illustrates what it means to find a common denominator and combine.
Fractions - Equivalent – Illustrates relationships between equivalent fractions.
Function Machine – Explore the concept of functions by putting values into this machine and observing its output.
Golden Rectangle – Illustrates iterations of the Golden Section.
Grapher – A tool for graphing and exploring functions.
Mastermind – Use inference and logic to play a game and guess a hidden pattern of pegs.
Number Puzzles – Solve puzzles involving arranging numbers on a diagram so that they add up to a given value.
Pascal's Triangle – Explore patterns created by selecting elements of Pascal's triangle.
Peg Puzzle – Win this game by moving the pegs on the left past the pegs on the right.
Percentages – Discover relationships between fractions, percents, and decimals.
Rational Numbers Triangle – Explore a triangular array that contains every positive rational number exactly once.
Sieve of Eratosthenes – Relate number patterns with visual patterns.
Tight Weave – Visualize the creation of the Sierpinski Carpet, an iterative geometric pattern that resembles a woven mat.
Turtle Geometry – Explore numbers, shapes, and logic by programming a turtle to move.
Venn Diagrams – Investigate common features of sets. | {"url":"http://nlvm.usu.edu/en/nav/topic_t_1.html","timestamp":"2014-04-16T10:18:26Z","content_type":null,"content_length":"54184","record_id":"<urn:uuid:55de5943-8465-40e6-9282-900710490e1d>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00493-ip-10-147-4-33.ec2.internal.warc.gz"} |
egrees of
┃ Illustrating degrees of freedom ┃
┃ in terms of sample size and dimensionality ┃
┃ ┃
┃ Dr. Chong Ho (Alex) Yu (2009) ┃
"Degree of freedom" (df) is an "intimate stranger" to statistics students. Every quantitative-based research paper requires reporting of degrees of freedom associated with the test results such
as "F(df1, df2)," yet very few people understand why it is essential to do so. Although the concept "degree of freedom" is taught in introductory statistics classes, many students learn the
literal definition of this term rather than its deeper meaning. Failure to understand "degrees of freedom" has two side effects. First, students and inexperienced researchers tend to
mis-interpret a "perfect-fitted" model or an "over-fitted" model as a good model. Second, they have a false sense of security that df is adequate while n is large. This reflects the problem that
most failed to comprehend that df is a function of both the number of observations and the number of variables in one's model. Frustration by this problem among statistical instructors is
manifested by the fact that the issue "how df should be taught" has been recurring in several statistical-related discussion groups (e.g. edstat-l, sci.stat.edu, sci.stat.math).
Many elementary statistics textbooks introduce this concept in terms of the numbers that are "free to vary" (Howell, 1992; Jaccard & Becker, 1990). Some statistics textbooks just give the df of
various distributions (e.g. Moore & McCabe, 1989; Agresti & Finlay, 1986). Johnson (1992) simply said that degree of freedom is the "index number" for identifying which distribution is used. Some
definitions given by statistical instructors can be as obscured as "a mathematical property of a distribution related to the number of values in a sample that can be freely specified once you
know something about the sample." (cited in Flatto, 1996) The preceding explanations cannot clearly show the purpose of df. Even advanced statistics textbooks do not discuss the degrees of
freedom in detail (e.g. Hays, 1981; Maxwell and Delany, 1986; Winner, 1985). It is not uncommon that many advanced statistics students and experienced researchers have a vague idea of the degrees
of freedom concept.
There are other approaches taken to present the concept of degree of freedom. Most of them are mathematical in essence (see Appendix A). While these mathematical explanations carry some merits,
they may still be difficult to statistical students, especially in social sciences, who generally do not have a strong mathematical background. In the following section, it is recommended that df
can be explained in terms of sample size and dimensionality. Both can represent the number of pieces of useful information.
Df in terms of sample size
Toothaker (1986) explained df as the number of independent components minus the number of parameters estimated. This approach is based upon the definition provided by Walker (1940): the number of
observations minus the number of necessary relations, which is obtainable from these observations (df = n - r). Although Good (1973) criticized that Walker's approach is not obvious in the
meaning of necessary relations, the number of necessary relationships is indeed intuitive when there are just a few variables. The definition of "necessary relationship" is beyond the scope of
this article. To avoid confusion, in this article, it is simply defined as the relationship between a dependent variable (Y) and each independent variable (X) in the research.
Please keep in mind that this illustration is simplified for conceptual clarity. Although Walker regards the preceding equation as a universal rule, don't think that df = n - r can really be
applied to all situations.
No degree of freedom and effective sample size
Figure 1 shows that there is one relationship under investigation (r = 1) when there are two variables. In the scatterplot where there is only one datum point. The analyst cannot do any
estimation of the regression line because the line can go in any direction, as shown in Figure 1.In other words, there isn't any useful information.
Figure 1. No degree of freedom with one datum point.
When the degree of freedom is zero (df = n - r = 1 - 1 = 0), there is no way to affirm or reject the model! In this sense, the data have no "freedom" to vary and you don't have any "freedom" to
conduct research with this data set. Put it bluntly, one subject is basically useless, and obviously, df defines the effective sample size (Eisenhauer, 2008).
Perfect fitting
In order to plot a regression line, you must have at least two data points as indicated in the following scattergram.
Figure 2. Perfect fit with two data points.
In this case, there is one degree of freedom for estimation (n - 1 = 1, where n = 2). When there are two data points only, one can always join them to be a straight regression line and get a
perfect correlation (r = 1.00). Since the slope goes through all data points and there is no residual, it is considered a "perfect" fit. The word "perfect-fit" can be misleading. Naive students
may regard this as a good sign. Indeed, the opposite is true. When you marry a perfect man/woman, it may be too good to be true! The so-called "perfect-fit" results from the lack of useful
information. Since the data do not have much "freedom" to vary and no alternate models could be explored, the researcher has no "freedom" to further the study. Again, the effective sample size is
defined by df = n -1.
This point is extremely important because very few researchers are aware that perfect fitting is a sign of serious problems. For instance, when Mendel conducted research on heredity, the
conclusion was derived from almost "perfect" data. Later R. A. Fisher questioned that the data are too good to be true. After re-analyzing the data, Fisher found that the "perfectly-fitted" data
are actually erroneous (Press & Tanur, 2001).
In addition, when there are too many variables in a regression model i.e. the number of parameters to be estimated is larger than the number of observations, this model is said to lacking degrees
of freedom and thus is over-fit. To simplify the illustration, a scenario with three observations and two variables are presented.
Figure 3. Over-fit with three data points.
Conceptually speaking, there should be four or more variables, and three or fewer observations to make a model over-fitting. Nevertheless, when only three subjects are used to estimate the
strength of association between two variables, the situation is bad enough. Since there are just a few observations, the residuals are small and it gives an illustration that the model and the
data fit each other very well. When the sample size is larger and data points scatter around the plot, the residuals are higher, of course. In this case, the model tends to be have a lesser
degree of fit. Nevertheless, a less fitted model resulted from more degrees of freedom carry more merits.
Useful information
Finally, you should see that the degree of freedom is the number of pieces of useful information.
┃ Sample size │ Degree(s) of freedom │ Amount of information ┃
┃ 1 │ 0 │ no information ┃
┃ 2 │ 1 │ not enough information ┃
┃ 3 │ 2 │ still not enough information ┃
df in terms of dimensions and parameters
Now degrees of freedom are illustrated in terms of dimensionality and parameters. According to I. J. Good, degrees of freedom can be expressed as
D(K) - D(H),
D(K) = the dimensionality of a broader hypothesis,
such as a full model in regression
D(H) = the dimensionality of the null hypothesis,
such as a restricted or null model
In the following, vectors (variables) in hyperspace are used for illustration (Saville & Wood, 1991; Wickens, 1995). It is important to point out that the illustration is only a metaphor to make
comprehension easier. Vectors do not behave literally as shown.
Figure 4. Vectors in hyperspace.
For the time being, let's ignore the intercept. What is(are) the degree(s) of freedom when there is one variable (vector) in a regression model? First, we need to find out the number of parameter
(s) in a one-predictor model. Since only one predictor is present, there is only one beta weight to be estimated. The answer is straight-forward. There is one parameter to be estimated.
How about a null model? In a null model, the number of parameters is set to zero. The expected Y score is equal to the mean of Y and there is no beta weight to be estimated.
Based upon df = D(K) - D(H), when there is only one predictor, the degree of freedom is just one (1 - 0 = 1). It means that there is only one piece of useful information for estimation. In this
case, the model is not well-supported.
As you notice, a 2-predictor model (df = 2 - 0 = 2) is better-supported than the 1-predictor model (df = 1 - 0 = 1). When the number of orthogonal vectors increases, we have more peices of
independent information to predict Y and the model tends to be more stable.
In short, the degree of freedom can be defined in the context of dimensionality, which conveys the amount of useful information. However, increasing the number of variables is not always
The section regarding df as n - r mentions the problem of "overfitting," in which there are too few observations for too many variables. When you add more variables into the model, the R2
(variance explained) will definitely increase. However, adding more variables into a model without enough observations to support the model is another way to create the problems of "overfitting."
Simply, the more variables you have, the more observations you need.
However, it is important to note that some regression methods, such as ridge regression, linear smoothers and smoothing splines, are not based on least-squares, and thus df defined in terms of
dimensionality is not applicable to these modeling.
Putting both together
The above illustrations (Part I and Part II) compartmentalize df in terms of sample size and df in terms of dimensionality (variables). Observations (n) and parameters (k), in the context of df,
must be taken into consideration together.
For instance, in regression, the working definition of degrees of freedom involves the information of both observations and dimensionality: df = n - k - 1 whereas n = sample size and k = the
number of variables. Take the 3-observation and 2-variable case as an example. In this case, df = 3 - 2- 1 = 0!
View the flash version of this tutorial
Agresti, A., & Finlay, B. (1986). Statistical methods for the social sciences. San Francisco, CA: Dellen.
Cramer, H. (1946). Mathematical methods of statistics. Princeton, NJ: Princeton University Press.
Eisenhauer, J. G. (2008). Degrees of Freedom. Teaching Statistics, 30(3), 75–78.
Flatto, J. (1996, May 3). Degrees of freedom question. Computer Software System-SPSS Newsgroup (comp.soft-sys.spss).
Galfo, A. J. (1985). Teaching degrees of freedom as a concept in inferential statistics: An elementary approach. School Science and Mathematics. 85(3), 240-247.
Good, I. J. (1973). What are degrees of freedom? American Statisticians, 27, 227-228.
Hays, W. L. (1981), Statistics. New York: Holt, Rinehart and Winston.
Howell, D. C. (1992). Statistical methods for psychology. (3rd ed.). Belmont, CA: Duxberry.
Jaccard, J. & Becker, M.A. (1990). Statistics for the behavioral sciences. (2nd ed.). Belmont, CA: Wadsworth.
Johnson, R. A. & Wichern, D. W. (1998). Applied multivariate statistical analysis. Englewood Cliffs, NJ: Prentice Hall.
Maxwell, S., & Delany, H. (1990). Designing experiments and analyzing data. Belmont, CA: Wadworth.
Moore, D. S. & McCabe, G. P. (1989). Introduction to the practice of statistics. New York: W. H. Freeman and Company.
Popper, K. R. (1959). Logic of scientific discovery. London : Hutchinson.
Popper, K. R. (1974). Replies to my critics. In P. A. Schilpp (Eds.), The philosophy of Karl Popper (pp.963-1197). La Salle: Open Court.
Press, S. J., & Tanur, J. M. (2001). The subjectivity of scientists and the Bayesian approach. New York: John Wiley & Sons.
Rawlings, J.O., (1988). Applied regression analysis: A research tool. Pacific Grove, CA: Wadsworth and Brooks/Cole.
Saville, D. & Wood, G. R. (1991). Statistical methods: The geometric approach. New York: Springer-Verlag.
Toothaker, L. E., & Miller, L. (1996). Introductory statistics for the behavioral sciences. (2nd ed.). Pacific Grove, CA: Brooks/Cole.
Walker, H. W. (1940). Degrees of Freedom. Journal of Educational Psychology, 31, 253-269.
Wickens, T. (1995). The geometry of multivariate statistics. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
Winer, B. J., Brown, D. R., & Michels, K. M. (1991). Statistical principles in experimental design. (3rd ed.). New York: McGraw-Hill.
Different approaches of illustrating degrees of freedom
1. Cramer (1946) defined degrees of freedom as the rank of a quadratic form. Muirhead (1994) also adopted a geometrical approach to explain this concept. Degrees of freedom typically refer to
Chi-square distributions (and to F distributions, but they're just ratios of chi-squares). Chi-square distributed random variables are sums of squares (or quadratic forms), and can be represented
as the squared lengths of vectors. The dimension of the subspace in which the vector is free to roam is exactly the degrees of freedom.
All commonly occurring situations involving Chi-square distributions are similar. The most common of these are in analysis of variance (or regression) settings. F-ratios here are ratios of
independent Chi-square random variables, and inherit their degrees of freedom from the subspaces in which the corresponding vectors must lie.
2. Galfo (1985) viewed degrees of freedom as the representation of the quality in the given statistic, which is computed using the sample X values. Since in the computation of m, the X values can
take on any of the values present in the population, the number of X values, n, selected for the given sample is the df for m. The n for the computation of m also expresses the "rung of the
ladder" of quality of the m computed; i.e. if n = 1, the df, or restriction, placed on the computation is at the lowest quality level.
3. Rawlings (1988) associated degrees of freedom with each sum of squares (in multiple regression) as the number of dimensions in which that vector is "free to move." Y is free to fall anywhere
in n-dimensional space and, hence, has n degrees of freedom. Y-hat, on the other hand, must fall in the X-space, and hence, has degrees of freedom equal to the dimension of the X-space -- [p', or
the number of independent variable's in the model]. The residual vector e can fall anywhere in the subspace of the n-dimensional space that is orthogonal to the X-space. This subspace has
dimensionality (n-p') and hence, e has (n-p') degrees of freedom.
4. Chen Xi (Personal communication) asserted that the best way to describe the concept of the degree of freedom is in control theory: the degree of freedom is a number indicating constraints.
With the same number of the more constraints, the whole system is determined. For example, a particle moving in a three-dimensional space has 9 degrees of freedom: 3 for positions, 3 for
velocities, and 3 for accelerations. If it is a free falling and 4 degrees of the freedom is removed, there are 2 velocities and 2 accelerations in x-y plane. There are infinite ways to add
constraints, but each of the constraints will limit the moving in a certain way. The order of the state equation for a controllable and observable system is in fact the degree of the freedom.
5. Selig (personal communication) stated that degrees of freedom are lost for each parameter in a model that is estimated in the process of estimating another parameter. For example, one degree
of freedom is lost when we estimate the population mean using the sample mean; two degrees of freedom are lost when we estimate the standard error of estimate (in regression) using Y-hat (one
degree of freedom for the Y-intercept and one degree of freedom for the slope of the regression line).
6. Lambert (personal communication) regarded degrees of freedom as the number of measurements exceeding the amount absolutely necessary to measure the "object" in question. For example, to
measure the diameter of a steel rod would require a minimum of one measurement. If ten measurements are taken instead, the set of ten measurements has nine degrees of freedom. In Lambert's view,
once the concept is explained in this way, it is not difficult to extend it to explain applications to statistical estimators. i.e. if n measurements are made on m unknown quantities then the
degrees of freedom are n-m.
┃ ┃
┃ ┃
┃ Married Man: There is only one subject and my degree of freedom ┃
┃ is zero. So I shall increase my "sample size." ┃
┃ ┃
┃ ┃ | {"url":"http://www.creative-wisdom.com/computer/sas/df.html","timestamp":"2014-04-23T18:09:47Z","content_type":null,"content_length":"27979","record_id":"<urn:uuid:5f6faba5-0091-4491-a82a-a3a5cf005002>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00342-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
You haven't written a testimonial for Owlfred. | {"url":"http://openstudy.com/updates/50504c17e4b0a6140ad4df54","timestamp":"2014-04-20T00:59:37Z","content_type":null,"content_length":"52808","record_id":"<urn:uuid:85f2c4e2-57f6-413a-8433-f208d56a16fe>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00141-ip-10-147-4-33.ec2.internal.warc.gz"} |
Version 4 (modified by blamario, 4 years ago)
Package monad-coroutine
This library, implemented by the Control.Monad.Coroutine module, provides a limited coroutine functionality in Haskell. The centerpiece of the approach is the monad transformer Coroutine, which
transforms an arbitrary monadic computation into a suspendable and resumable one. The basic definition is simple:
newtype Coroutine s m r = Coroutine {resume :: m (Either (s (Coroutine s m r)) r)}
instance (Functor s, Monad m) => Monad (Coroutine s m) where
return = Coroutine . return . Right
t >>= f = Coroutine (resume t >>= either (return . Left . fmap (>>= f)) (resume . f))
Suspension Functors
The Coroutine transformer type is parameterized by a functor. The functor in question wraps the resumption of a suspended coroutine, and it can carry other information as well. Module
Control.Monad.Coroutine.SuspensionFunctors exports some useful functors, one of which is Yield:
data Yield x y = Yield x y
instance Functor (Yield x) where
fmap f (Yield x y) = Yield x (f y)
A coroutine parameterized by this functor is a generator which yields a value every time it suspends. For example, the following function generates the program's command-line arguments:
genArgs :: Coroutine (Yield String) IO ()
genArgs = getArgs >>= mapM_ yield
The Await functor is dual to Yield; a coroutine that suspends using this functor is a consumer coroutine that on every suspension expects to be given a value before it resumes. The following example
is a consumer coroutine that prints every received value to standard output:
printer :: Show x => Coroutine (Await x) IO ()
printer = await >>= print >> printer
Running a coroutine | {"url":"http://trac.haskell.org/SCC/wiki/monad-coroutine?version=4","timestamp":"2014-04-21T02:15:02Z","content_type":null,"content_length":"7438","record_id":"<urn:uuid:679a7cd4-583f-4257-8a94-2087b82ab1a7>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00281-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Supersymmetry Weinberg Vol. 3
Thank you for your help, but its only half the truth (I've figured it out myself this morning).
You're right, its the F-Term, but its only a tricky way writing it. If you take the [tex]\Phi[/tex] you suggested, and construct a polynom out of different [tex]\Phi_i[/tex]'s and then only take
terms of the Order [tex]\theta_L^2[/tex] you get my Term. To see this, take only the [tex]\phi[/tex]-Terms of the polynom but either two of the [tex]\psi[/tex]'s, or one [tex]F[/tex]. So all terms
are of the order [tex]\theta_L^2[/tex]. If you now count the possibilities of replacing one of the possibilities [tex]\phi[/tex]'s with a [tex]\psi[/tex] (two times, so you get two [tex]\theta[/
tex]), you get same factor as if you just take the whole polynom in [tex]\phi[/tex] and derive in respect to [tex]\phi[/tex]. Thats the whole trick in there.
You are right, I'm jsut learning SUSY, but Weinbergs is the best book I could find. In my opinion, all the other books are too brief or just incomplete, including Argyres or Wess and Bagger. And
you're right, Weinberg is a hard text, but at least he gives enough motivation to the things he does. I'm only missing some comments here and there. So if you know a script (other than Argyres. I got
this one.) somewhere, based on Weinbergs Book with some extra remarks and comments, that would be great.
And yeah, this is a students presentation. 5 weeks to go.
And again, thank you for your help. | {"url":"http://www.physicsforums.com/showpost.php?p=1566009&postcount=3","timestamp":"2014-04-18T13:57:01Z","content_type":null,"content_length":"8516","record_id":"<urn:uuid:cecc59f4-2f2c-458d-b5cb-8d9ba79b49ac>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00314-ip-10-147-4-33.ec2.internal.warc.gz"} |
This is the part II of the Matrix Alignment project:
Concepts include: class extensions, refactoring, Big Oh.
The original Global Alignment project provided students with a basic understanding of the methods behind DNA Matrix Alignment. This portion of the project aims to show code improvements by extension
of the class Global Alignment to use an improved algorithm, explained here. This model of Matrix alignment is closer to the BLAST algorithm itself.
Global Alignment created a double matrix of SeqA x SeqB size. In the diagram below, most of the cells are pointing towards the center.
The thin matrix shown here is the basic d x k array derived from the improved matrix. At first, I was a little confused as to how to implement the traceBack() method, since the Cell locations in the
d x k matrix no longer directly represent their respective symbols at those locations in the Sequences. After some consideration, it became clear that the information stored in the dxk matrix could
translate back to the original matrix by using the threshold and the location on the vertical axis of the dxk matrix. This is shown below.
In other news, I’ve added the progress bar to the iTunes Project. First, the song is selected:
And then, is read. A similar progress bar appears for the writing portion of the program. | {"url":"http://www.cs.duke.edu/csed/mobile/bio/ImprovedGlobalAlignment.html","timestamp":"2014-04-19T04:22:13Z","content_type":null,"content_length":"2513","record_id":"<urn:uuid:9b07cb2f-382f-487c-a291-de54e4b7f0ad>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00495-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is the value of csc (-2pi/3)?
Hi Kishor.
Here are some trig identities that will help get you started:
csc x = 1 / (sin x)
sin -x = - sin x
So csc (-2pi/3) = 1 / (-sin (2pi/3)).
Now you should be able to use your table of exact values to determine the rest.
Hope this helps,
Stephen La Rocque. | {"url":"http://mathcentral.uregina.ca/QQ/database/QQ.02.06/kishor1.html","timestamp":"2014-04-17T06:41:09Z","content_type":null,"content_length":"5442","record_id":"<urn:uuid:23fa913d-03f6-4e25-9c00-57a77d79c278>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00414-ip-10-147-4-33.ec2.internal.warc.gz"} |
arithmetic progression
November 24th 2009, 03:37 PM #1
Nov 2009
arithmetic progression
I need to find first term and common difference for an arithmetic progression where Un denotes the nth term and the sum of the first n terms is denoted Sn.
I am told U5 + U16 = 44 and that S18=3*S10
I cant see a way to find either of them... any help please.
$U_5 + U_{16} = 44$ and that $S_{18}=3*S_{10}$
Okay, 1st fact to remember
An A.P. is of the form: $a+(k-1)d$, where $k\geq1$
where $U_k=a+(k-1)d$ and $S_k=\frac{k(a+a+(k-1)d)}{2}$
so $U_5=a+4d$ and $U_{16}=a+15d$
and $U_5+U_{16}=44$
$\frac{18(a+a+17d)}{2}=3(\frac{10(a+a+9d)}{2})....( 2)$
Solve equations (1) and (2) simultaneously to obtain your answer.
November 24th 2009, 06:09 PM #2
November 24th 2009, 07:27 PM #3
Nov 2009 | {"url":"http://mathhelpforum.com/algebra/116568-arithmetic-progression.html","timestamp":"2014-04-19T19:12:44Z","content_type":null,"content_length":"36196","record_id":"<urn:uuid:b7746d92-85bf-4a95-af9b-ba408f9a8d00>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00433-ip-10-147-4-33.ec2.internal.warc.gz"} |
Markov chain monte carlo
May 19th 2009, 06:36 PM #1
Mar 2009
Gold Coast, Queensland, Australia
Markov chain monte carlo
who knows the answer to this multiple choice question?
Regarding Markov Chain Monte Carlo (MCMC) techniques, it is incorrect to state that:
a) an ergodic Markov chain can have multiple equilibrium distributions.
b) if the Markov chain satisfies detailed balance, the required distribution will be invariant to it,
c) if the Markov chain is reversible, the required distribution will be invariant to it,
d) MCMC simulates a Markov chain such that states from the Markov chain converge to some desired probability distribution.
I'm assuming ergodic means irreducible, positive recurrent and aperiodic. In which case, a) is incorrect because an ergodic Markov chain has a unique equilibrium distribution.
b) is correct
c) satisfying detailed balance is equivalent to reversible.
d) That's the purpose of MCMC
May 21st 2009, 06:40 AM #2
Junior Member
Nov 2008 | {"url":"http://mathhelpforum.com/advanced-statistics/89730-markov-chain-monte-carlo.html","timestamp":"2014-04-21T11:17:35Z","content_type":null,"content_length":"31976","record_id":"<urn:uuid:288b6da1-665a-449c-8409-3967fa67507e>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00015-ip-10-147-4-33.ec2.internal.warc.gz"} |
Need help with an inequality
October 1st 2012, 09:11 PM #1
Oct 2012
Need help with an inequality
Hi, could anybody please help with this inequality? I've tried it several times, but my answers are always absurd.
Find a constant A for which abs(log(cos(x))+(x^2)/2) =< Ax^4 for -pi/3 <= x <= pi/3
Any help at all would be greatly appreciated.
Re: Need help with an inequality
Hey Halcyon.
One suggestion that comes to mind is to use the triangle inequality which says that |a+b| <= |a| + |b| and if this is <= A*x^4 then you're done.
Now for cos(x) when you have pi/3, log(cos(x)) = log(1/2) and -pi/3 log(cos(x)) = log(1/2) and |log(cos(x))| will take on every value between 0 and ln(2). So |log(cos(x))| <= |log(1/2)| = |ln(2)|
and |x^2/2|. So now find an A such that it fits this identity.
Also another hint: log(x) < x^2 for x > 1 and recall that |log(1/x)| = log(x) if x > 0. To show that log(x) < x^2 show that the initial value satisfies this (i.e. log(1) = 0 < 1) and that the
derivative is always less for log(x) than for x^2 (which is easy since 1/x < 2x implies 1 < 2x^2 which is true when x>=1).
October 1st 2012, 11:02 PM #2
MHF Contributor
Sep 2012 | {"url":"http://mathhelpforum.com/advanced-algebra/204478-need-help-inequality.html","timestamp":"2014-04-21T16:06:21Z","content_type":null,"content_length":"31456","record_id":"<urn:uuid:2f77dae4-e995-45ed-a56f-278266e441de>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00372-ip-10-147-4-33.ec2.internal.warc.gz"} |
Consider An MOS Device With 20 Nm Thick Gate Oxide ... | Chegg.com
Image text transcribed for accessibility: Consider an MOS device with 20 nm thick gate oxide and uniform p-type substrate doping of 1017 cm-3. The gate work function is that of n+ Si. What is the
flatband voltage? What is the threshold voltage for strong inversion? Sketch the high frequency C-V curve. Label where the flatband voltage and threshold voltage are. Calculate the maximum and the
minimum capacitance (per area) values. Draw the energy band diagram at strong inversion.
Electrical Engineering | {"url":"http://www.chegg.com/homework-help/questions-and-answers/consider-mos-device-20-nm-thick-gate-oxide-uniform-p-type-substrate-doping-1017-cm-3-gate--q984631","timestamp":"2014-04-20T06:11:12Z","content_type":null,"content_length":"20714","record_id":"<urn:uuid:ccd9a114-0c40-4427-903a-4f4e34a429a7>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00302-ip-10-147-4-33.ec2.internal.warc.gz"} |
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How Do You Simplify an Algebraic Fraction by Canceling Common Factors?
Numerators and denominators are the key ingredients that make fractions, so if you want to work with fractions, you have to know what numerators and denominators are. Lucky for you, this tutorial
will teach you some great tricks for remembering what numerators and denominators are all about. | {"url":"http://www.virtualnerd.com/pre-algebra/factors-fractions-exponents/simplify-algebraic-fraction-common-factors.php","timestamp":"2014-04-19T01:48:09Z","content_type":null,"content_length":"26725","record_id":"<urn:uuid:399d46d2-5255-42c9-8bff-fb7f3d618a59>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00649-ip-10-147-4-33.ec2.internal.warc.gz"} |
Stationary Points
December 2nd 2008, 04:06 AM #1
Oct 2008
Stationary Points
Locate 1) the position and nature of the stationary points and 2) the points of inflection of the following curve
y = x^4 - 6x^2 +15
Find $\frac{dy}{dx}$ of the $y$, then put that expression equal to zero, which will give you the x co-ordinates of the points at which the gradient is 0 i.e. stationary points etc.
To classify the inflexion point, find the gradient at the points either side of the place where the gradient is equal, if both results provide gradients with the same sign i.e. +ve or -ve, then
it is inflection.
$\frac{dy}{dx} = 4x^3 - 12x$
i got that far , thanks.
i just got stuck after that , when i simplified i was ended up with x + / - roots and i wasnt sure if both were roots, or only one was ...etc
Could you post your working so that I can see what you mean?
i got dy/dx = 4x^3 - 12x
which i simplified to 4x ( x^2 - 3) = 0
so then x values of stationary points are x= o and x = + or - root 3
and this is where i get confused
December 2nd 2008, 04:12 AM #2
Oct 2008
December 2nd 2008, 04:19 AM #3
Oct 2008
December 2nd 2008, 04:23 AM #4
Oct 2008
December 3rd 2008, 12:18 PM #5
Oct 2008 | {"url":"http://mathhelpforum.com/calculus/62816-stationary-points.html","timestamp":"2014-04-18T12:31:03Z","content_type":null,"content_length":"37954","record_id":"<urn:uuid:a2b27d9a-ca5c-4161-a7b3-69361d55cbf0>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00369-ip-10-147-4-33.ec2.internal.warc.gz"} |
Taking Stock of the "Higgs" (Jan. 2013)
Taking Stock of the “Higgs” (Jan. 2013)
Matt Strassler [January 17, 2013]
Particle physicists have recently discovered a previously unknown, possibly elementary, and certainly important particle. And if you’ve been following along in even the slightest degree, you know the
new particle resembles to some degree a type of long-awaited Higgs particle. (For laypersons, here’s my FAQ about the Higgs field and particle, and here’s an article about why the Higgs particle
matters so much.) The discovery of such a particle, a ripple in the mass-providing Higgs field, apparently confirms that the Higgs field, without which we and all ordinary matter would explode,
really does exist in nature.
The discovery was made back in July. With six months having passed, it seems a good moment to step back and take stock, as some of us did at the Higgs Symposium last week in Edinburgh, Scotland, UK.
The purpose of this article is to bring non-experts up to date on what we’ve learned over the past six months since the discovery, and to elucidate the questions that lie ahead… using the talks given
at the Symposium as source material.
In this article, I want to take a look at what we know so far about the new particle.
• How Higgs-like is this new particle?
• How likely is it that it is the only Higgs particle?
• How likely is it that it is an elementary particle, as the electron is thought to be, as opposed to made from more elementary particles, like a proton is known to be?
• And how likely, if it is elementary and the only type of Higgs particle in nature, is it to be a Higgs particle of the simplest possible type: the so-called Standard Model Higgs particle?
Several of the speakers at the Higgs Symposium addressed these issues in one way or another, and I’ll be reviewing what they had to say in a non-technical fashion (except that now and then I’ll
mention a few techy details — which you can easily skip over.).
1. Is the Newly Discovered Particle a Higgs Particle of Some Type?
2. If the New Particle is a Higgs Particle, Might it be Composite?
3. Might the New Particle be a Higgs of the Simplest Possible Type (the Standard Model Higgs)?
4. Might there be more than one type of Higgs particle? Has our recent discovery been just the first of several that are still to come?
5. Supersymmetry: What Does The New Particle Teach Us About This Possibility?
6. Even If It’s The Only Higgs Particle, How Might It Surprise Us?
7. Suppose the Standard Model is Right; What Then? [Coming soon: two perspectives, one from Shaposhnikov who offers an argument as to why the Higgs has a mass of close to 125 GeV/c², and one from me,
about how we ought to view this situation, if it turns out to be true.]
17 responses to “Taking Stock of the “Higgs” (Jan. 2013)”
1. Hi Matt, big follower, thank you very much for taking time to explain this things. I’m interested in the argument that composite higgs suggests composite quarks and leptons. I’m trying to find a
more technical account, but failed so far besides slides for presentations which are kind of hard to follow. Do you have any references that are technical, like any papers or so. Thanks
□ Did you look at the papers that are referenced in Rattazzi’s talk at the Symposium?
☆ p.s. I was thinking I should explain this, at least briefly. It would take just a few sentences to get the basic point across… the problem is that it needs a good picture to go with it,
and I haven’t invented one yet…
☆ I tried to look at some of the papers there referenced, and found some discussion of the partial compositeness cenario, but could not find how one compares the fundamental fermions vs
composite fermions supposing a composite higgs. At some point there is a reference to a paper in preparation, I’m not sure if will discuss this point at lenght. In any case I was
interested because I’ve seen an argument for a composite higgs in which the fermions must be elementary.
○ I don’t know any such argument — where did you hear it? You can make some of the lighter fermions elementary (or at least vastly smaller in size than the Higgs) but it is really quite
difficult to do this with the top quark. And with what has recently been learned about conformal field theory, it seems that making the lighter fermions elementary without creating
large flavor-changing neutral currents is even harder than people used to think. You can look at Luty and Okui, for instance; the type of model they write down (inspired by two
decades of work before them, cf. walking techicolor) has a dynamical requirement (one I also wrote about), which Riccardo Rattazzi, Slava Rychkov, Alessandro Vichi. [arXiv:1009.5985
[hep-th]] showed is impossible to satisfy.
■ The paper by Luty and Okui, together with the last reference is exactly what I was looking for. Need a bit time to diggest it though, but I’ll try to navigate by myself. Somewhat
embarassed to missed the Rattazzi, Rychkov and Vichi paper, they indeed were in the slides. As for the argument, I should say I find it a “numerological” one by the gravity folks
based on a intriguing coincidence. Basically they argue that every fermion from the SM obeys the inequality for naked singularities and not by the Kerr-Newman black holes, and
then postulate a connection between elementary particles and naked singularities. It’s in the conclusions from [arXiv:0905.1077[gr-qc]] by George Matsas et al, if you would like
to take a look. Just to be clear, I was not convinced, just intrigued. Anyway, thanks for the references
2. When you say “A Higgs particle must be a boson, and must not be impacted directly by the strong nuclear force or electromagnetic force (i.e. it must not have electric charge). This is true of the
new particle; we know this because it can decay to two photons (which are also bosons with no electric charge and with no direct effects from the strong nuclear force.)”… do you have to also
assume that the Higgs is not composite? Given that a neutral pion can decay to two photons, yet is affected by the strong force?
□ Since you’re the second expert to ask this question, I’ll change the wording so that you won’t complain. The neutral pion is made from particles that carry BOTH electromagnetic AND strong
nuclear forces, so yes, WHEN INVOLVED IN SUFFICIENTLY HIGH ENERGY PROCESSES, it can be affected by BOTH strong nuclear forces AND electromagnetic forces. Similarly the Higgs is color-neutral
as well as electrically neutral — but if it is composite, then WHEN INVOLVED IN SUFFICENTLY HIGH ENERGY PROCESSES (which would be well above LHC scales, given what Rattazzi had to say) it
might be affected in a similar way.
☆ Matt; there’s a problem with trying to write for non-experts without introducing other demons, but here I think one may be digging a hole in trying to draw a conclusion that actually
requires rather a lot of deep argument. The Higgs (even a colour neutral elementary Higgs, which is what this beast may well be) couples to quarks (top antitop in particular) which is
what drives the two gamma decay. It is the strength of the H to quarks coupling that is what matters. The production and decay rates at LHC are consistent with that coupling being
“standard Higgsish” (i.e. not strong); simply seeing it decay to two gammas cannot allow one to draw the conclusion that the beast is not strong. Now I dont know how to go through all
that and retain the interest of a general reader (!) but I also am uneasy at drawing inferences without doing so. Its one of the many hazards of popularization (I like your article very
much by the way, and this is one of the issues that I would like to find a neat way of explaining myself)
○ Hmm… I’m honestly really confused. I feel like we’re talking at cross-purposes, but can’t quite figure out why.
My only and entire point about Higgs –> two photons is that it proves the Higgs isn’t colored or charged [i.e., carries no electric monopole or color monopole charge.] Now I know
you’re not disagreeing with that.
I think you’re trying to say that Higgs –> two photons does not prove the Higgs isn’t composite; and that if it is composite, what it is made from may carry charge or color. And I of
course agree with that. I certainly didn’t say the Higgs isn’t composite and internally-strongly coupled; in fact I have a whole section about that: http://profmattstrassler.com/
articles-and-posts/the-higgs-particle/taking-stock-of-the-higgs-jan-2013/2-perhaps-composite/ And in such a scenario, the Higgs is very much like a neutral pion, with similar
So when you say “digging a hole in trying to draw a conclusion” — I am puzzled about what conclusion you’re talking about, or how I am digging the hole. I don’t think the conclusions
I’m drawing differ from yours. Please clarify… thanks!
3. What is a top-prime? Is it some excited state of the third-generation top quark, or is it a fourth generation quark, or what? I thought there were strong arguments that there were only three
generations of quarks, unless the subsequent generations were REALLY massive?
□ It’s different from a fourth-generation quark. This is not the appropriate article for me to go into all sorts of details — remember this is an article for non-experts.
4. If I were an expert, I would probably know what a top-prime is :-) Honestly not trying to nitpick, just trying to understand… take care and thanks
□ Sorry, I was rushing the last answer and should have done a better job.
The main difference between a top-prime (and as I said that is a provisional name — over the years lots of things have been called “top-prime” for different reason) and a fourth-generation
quark is that a fourth generation quark gets its mass from the Higgs field (just the way quarks of the other three generations do) but a top-prime does not. Therefore (a) although the
arguments that there are only three generations of quarks are indeed correct (and in fact do NOT allow for very massive generations), it happens that (b) the top-prime is not affected by the
arguments in (a).
5. バーバリーメンズ男物腕時計 http://www.hloap.com/
6. なかなか買えないのでセール行ったりですが 34歳、一番いい時ではないでしょうか私は20代から愛用してい … | {"url":"http://profmattstrassler.com/articles-and-posts/the-higgs-particle/taking-stock-of-the-higgs-jan-2013/","timestamp":"2014-04-17T19:37:30Z","content_type":null,"content_length":"119725","record_id":"<urn:uuid:a01d08f8-b18d-4fba-8ec3-c90877ed3f3e>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00059-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Prove that lim_{x to x_0} f(x) = L if and only if lim_{x to 0} f(x+x_0) = L. Assume
February 28th 2010, 06:42 PM #1
Feb 2010
Prove that lim_{x to x_0} f(x) = L if and only if lim_{x to 0} f(x+x_0) = L. Assume
second to last question.
Prove that lim_{x to x_0} f(x) = L if and only if lim_{x to 0} f(x+x_0) = L. Assume that L is finite.
That's pretty straight forward. In terms of the definitions of those limits you want to prove that
"Given $\epsilon> 0$ there exist $\delta> 0$ such that if $|x-x_0|< \delta$ then $|f(x)- L|< \epsilon$"
if and only if
"Given $\epsilon> 0$ there exist $\delta> 0$ such that if $|y|< \delta$ then $|f(x_0+ y)- L|< \epsilon$.
(I've changed "x" to "y" in the second part so as not to confuse the two uses of "x".)
Comparing those two, they will be the same if $x_0+ y= x$. In other words, let $y= x- x_0$.
In other words, let y= x- x_0.....do i have to show anything more or do the two definitions take of it? Sorry i'm really bad at proofs..
Do exactly as HallsOfIvy said. The basic idea is that in the limit $\lim_{x\to x-0}f(x)=0$ if we let $z=x-x_0$ then we get $\lim_{z\to 0}f(z)=0$. Formalize this.
March 1st 2010, 03:17 AM #2
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Apr 2005
March 1st 2010, 08:53 AM #3
Feb 2010
March 1st 2010, 12:46 PM #4 | {"url":"http://mathhelpforum.com/differential-geometry/131309-prove-lim_-x-x_0-f-x-l-if-only-if-lim_-x-0-f-x-x_0-l-assume.html","timestamp":"2014-04-18T07:24:30Z","content_type":null,"content_length":"43399","record_id":"<urn:uuid:e9cfc21e-ec95-47c5-8cb0-5fe1beddf256>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00136-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Associated prime ideals of skew polynomial rings
V. K. Bhat
School of Applied Physics and Mathematics, SMVD University, P/o Kakryal, Katra, J and K, India - 182301, e-mail: vijaykumarbhat2000@yahoo.com
Abstract: In this paper, it has been proved that for a Noetherian ring $R$ and an automorphism $\sigma$ of $R$, an associated prime ideal of $R[x,\sigma]$ or $R[x,x^{-1}, \sigma]$ is the extension of
its contraction to $R$ and this contraction is the intersection of the orbit under $\sigma$ of some associated prime ideal of $R$. The same statement is true for minimal prime ideals also. It has
also been proved that for a Noetherian $\mathbb{Q}$-algebra ($\mathbb{Q}$ the field of rational numbers) and a derivation $\delta$ of $R$, an associated prime ideal of $R[x,\delta]$ is the extension
of its contraction to $R$ and this contraction is an associated prime ideal of $R$.
Keywords: automorphism, associated prime, minimal prime, derivation, skew polynomial ring
Classification (MSC2000): 16-XX; 16S36, 16P40, 16P50, 16U20
Full text of the article:
Electronic version published on: 26 Feb 2008. This page was last modified: 28 Jan 2013.
© 2008 Heldermann Verlag
© 2008–2013 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition | {"url":"http://www.kurims.kyoto-u.ac.jp/EMIS/journals/BAG/vol.49/no.1/23.html","timestamp":"2014-04-20T16:35:53Z","content_type":null,"content_length":"4356","record_id":"<urn:uuid:5437380e-0456-4e8a-90df-d4fd8d70d90e>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00617-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Numpy-discussion] Cython numerical syntax revisited
Francesc Alted faltet@pytables....
Thu Mar 5 10:30:12 CST 2009
A Thursday 05 March 2009, David Cournapeau escrigué:
> I don't understand your argument about Row vs Column matters: which
> one is best depends on your linear algebra equations. You give an
> example where Fortran is better, but I can find example where C order
> will be more appropriate. Most of the time, for anything non trivial,
> which one is best depends on the dimensions of the problem (Kalman
> filtering in high dimensional spaces for example), because some parts
> of the equations are better handled in a row-order fashion, and some
> other parts in a column order fashion.
Yeah. Yet another (simple) example coming from linear algebra: a matrix
multiplied by a vector. Given a (matrix):
a = [[0,1,2],
and b (vector):
b = [[1],
the most intuitive way to do the multiplication is to take the 1st row
of a and do a dot product against b, repeating the process for 2nd and
3rd rows of a. C order coincides with this rule, and it is optimal
from the point of view of memory access, while Fortran order is not.
Francesc Alted
More information about the Numpy-discussion mailing list | {"url":"http://mail.scipy.org/pipermail/numpy-discussion/2009-March/040875.html","timestamp":"2014-04-17T14:03:48Z","content_type":null,"content_length":"3783","record_id":"<urn:uuid:a01c34c8-bd82-401c-9d06-af8988194852>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00221-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - wind vs relative wind (A Thermdynamics and aerodynamics issue)
So an airfoil cannot achieve greater lift force than drag force
I never stated that. In the case of high end gliders, such as a Nimbus 4T:
the glide ratio of the entire glider is 60:1, which corresponds to a lift to drag ratio of 4:1.
If the Nimbus weight was 1500lbs, and acheived 60:1 glide ratio at 60.0083327547 mph, the decent rate would be 1 mph, the forward speed would be 60mph, and the power consumed would be only 4hp (4 hp
= 1500 lbs x 1 mph / 375 conversion factor). The power input is 4 hp from gravity, and the glider in turn transfers the energy gained from gravity to the air at the rate of 4 hp.
What I stated was that an airfoil can't produce thrust from an apparent headwind. Continuing with the glider example, imagine there's a an updraft of 1 mph. The glider can then maintain level flight
at 60 mph forward. The updraft of 1 mph is the apparent crosswind, which the wings divert into thrust sufficient to propel the glider forward at 60mph. At this speed the thrust from the wings is
equally opposed by the drag, which is related to the apparent headwind, so the glider ceases to accelerate once at 60 mph. The wings cannot generate any thrust from the apparent headwind of 60mph. If
the updraft ceases, and the glider maintains horizontal flight, it slows down (until it can no longer maintain horizontal flight) because the wings can't generate thrust from the apparent headwind,
only from an apparent crosswind (the updraft in this case).
The text at some sailcraft web sites make the implication that the faster the glider moves forward, the more thrust (forwards force) the wings can produce, which is false. In the case of a sailcraft,
the total force will increase, but not the component of this force in the direction of travel (thrust). Getting back to the glider, the amount of thrust the wings can produce is related to the
apparent crosswind (in this example the updraft) only; the apparent headwind is an overhead, and actually reduces the thrust produced as forward speed increases due to drag. What these web sites
don't make clear is that the role that the apparent headwind plays is that it and the apparent crosswind are diverted with sufficient "upwind" component against the "true wind" to slow down the true
wind (in this case the updraft) and extract energy from it. | {"url":"http://www.physicsforums.com/showpost.php?p=2274391&postcount=10","timestamp":"2014-04-19T09:35:48Z","content_type":null,"content_length":"10133","record_id":"<urn:uuid:f838cf66-318d-45bf-95a7-0380a2d796d5>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00215-ip-10-147-4-33.ec2.internal.warc.gz"} |
Conics: Parabolas: Introduction
Conics: Parabolas: Introduction (page 1 of 4)
Sections: Introduction, Finding information from the equation, Finding the equation from information, Word problems & Calculators
In algebra, dealing with parabolas usually means graphing quadratics or finding the max/min points (that is, the vertices) of parabolas for quadratic word problems. In the context of conics, however,
there are some additional considerations.
To form a parabola according to ancient Greek definitions, you would start with a line and a point off to one side. The line is called the "directrix"; the point is called the "focus". The parabola
is the curve formed from all the points (x, y) that are equidistant from the directrix and the focus. The line perpendicular to the directrix and passing through the focus (that is, the line that
splits the parabola up the middle) is called the "axis of symmetry". The point on this axis which is exactly midway between the focus and the directrix is the "vertex"; the vertex is the point where
the parabola changes direction.
│ "regular", or vertical, parabola (in blue), with the focus (in green) "inside" the parabola, │ │ "sideways", or horizontal, parabola (in blue), with the focus (in green) "inside" the parabola, │
│ the directrix (in purple) below the graph, the axis of symmetry (in red) passing through the │ │ the directrix (in purple) to the left of the graph, the axis of symmetry (in red) passing │
│ focus and perpendicular to the directrix, and the vertex (in orange) on the graph │ │ through the focus and perpendicular to the directrix, and the vertex (in orange) on the graph │
The name "parabola" is derived from a New Latin term that means something similar to "compare" or "balance", and refers to the fact that the distance from the parabola to the focus is always equal to
(that is, is always in balance with) the distance from the parabola to the directrix. In practical terms, you'll probably only need to know that the vertex is exactly midway between the directrix and
the focus.
In previous contexts, your parabolas have either been "right side up" or "upside down" graph, depending on whether the leading coefficient was positive or negative, respectively. In the context of
conics, however, you will also be working with "sideways" parabolas, parabolas whose axes of symmetry parallel the x-axis and which open to the right or to the left.
A basic property of parabolas "in real life" is that any light or sound ray entering the parabola parallel to the axis of symmetry and hitting the inner surface of the parabolic "bowl" will be
reflected back to the focus. "Parabolic dishes", such as "bionic ears" and radio telescopes, take advantage of this property to concentrate a signal onto a receiver. The focus of a parabola is always
inside the parabola; the vertex is always on the parabola; the directrix is always outside the parabola.
The "general" form of a parabola's equation is the one you're used to, y = ax^2 + bx + c — unless the quadratic is "sideways", in which case the equation will look something like x = ay^2 + by + c.
The important difference in the two equations is in which variable is squared: for regular (vertical) parabolas, the x part is squared; for sideways (horizontal) parabolas, the y part is squared.
The "vertex" form of a parabola with its vertex at (h, k) is:
regular: y = a(x – h)^2 + k
sideways: x = a(y – k)^2 + h Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved
The conics form of the parabola equation (the one you'll find in advanced or older texts) is:
regular: 4p(y – k) = (x – h)^2
sideways: 4p(x – h) = (y – k)^2
Why "(h, k)" for the vertex? Why "p" instead of "a" in the old-time conics formula? Dunno. The important thing to notice, though, is that the h always stays with the x, that the k always stays with
the y, and that the p is always on the unsquared variable part. The relationship between the "vertex" form of the equation and the "conics" form of the equation is nothing more than a rearrangement:
y = a(x – h)^2 + k
y – k = a(x – h)^2 ^
(1/a)(y – k) = (x – h)^2
4p(y – k) = (x – h)^2
In other words, the value of 4p is actually the same as the value of 1/a; they're just two ways of saying the exact same thing. But this new variable p is one you'll need to be able to work with when
you're doing parabolas in the context of conics: it represents the distance between the vertex and the focus, and also the same (that is, equal) distance between the vertex and the directrix. And 2p
is then clearly the distance between the focus and the directrix.
• State the vertex and focus of the parabola having the equation (y – 3)^2 = 8(x – 5).
Comparing this equation with the conics form, and remembering that the h always goes with the x and the k always goes with the y, I can see that the center is at (h, k) = (5, 3). The coefficient
of the unsquared part is 4p; in this case, that gives me 4p = 8, so p = 2. Since the y part is squared and p is positive, then this is a sideways parabola that opens to the right. The focus is
inside the parabola, so it has to be two units to the right of the vertex:
vertex: (5, 3); focus: (7, 3)
• State the vertex and directrix of the parabola having the equation (x + 3)^2 = –20(y – 1).
The temptation is to say that the vertex is at (3, 1), but that would be wrong. The conics form of the equation has subtraction inside the parentheses, so the (x + 3)^2 is really (x – (–3))^2,
and the vertex is at (–3, 1). The coefficient of the unsquared part is –20, and this is also the value of 4p, so p = –5. Since the x part is squared and p is negative, then this is a regular
parabola that opens downward. This means that the directrix, being on the outside of the parabola, is five units above the vertex.
vertex: (–3, 1); directrix: y = 6
Top | 1 | 2 | 3 | 4 | Return to Index Next >>
Cite this article as: Stapel, Elizabeth. "Conics: Parabolas: Introduction." Purplemath. Available from
http://www.purplemath.com/modules/parabola.htm. Accessed | {"url":"http://www.purplemath.com/modules/parabola.htm","timestamp":"2014-04-20T01:56:51Z","content_type":null,"content_length":"35625","record_id":"<urn:uuid:6ed0ea70-2e47-4511-b6da-d770d9782de2>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00485-ip-10-147-4-33.ec2.internal.warc.gz"} |
Trends in AL run scoring (using R)
July 14, 2012
By Martin Monkman
I have started to explore the functionality of
R, the statistical and graphics programming language
. And with what better data to play than that of Major League Baseball?
There have already been some good examples of using R to analyze baseball data. The most comprehensive is the on-going series at
The Prince of Slides
(Brian Mills, aka Millsy), cross-posted at the
site. I am nowhere near that level, but explaining what I've done is a valuable exercise for me -- as
Joseph Joubert
said (no doubt in French) "To teach is to learn twice over."
So after some reading (I have found Paul Teetor's
R Cookbook
particularly helpful) and working through some examples I found on the web, I decided to plot some time series data, calculate a trend line, and then plot the points and trend line.
I started with the American League data, from its origins in 1901 through to the All Star break of 2012. For this, I relied on this handy table at Baseball Reference.
Step 1: load the data into the R workspace.
This required a bit of finessing in software outside R. Any text editor such as Notepad or TextPad would do the trick. What I did was paste it into the text editor, tidied up the things listed
below, and then saved the file with a .csv extension.Read more »
for the author, please follow the link and comment on his blog:
Bayes Ball
daily e-mail updates
news and
on topics such as: visualization (
), programming (
Web Scraping
) statistics (
time series
) and more...
If you got this far, why not
subscribe for updates
from the site? Choose your flavor:
, or | {"url":"http://www.r-bloggers.com/trends-in-al-run-scoring-using-r/","timestamp":"2014-04-20T18:56:12Z","content_type":null,"content_length":"36757","record_id":"<urn:uuid:8d2c062b-187b-4fd4-8f95-ead3fea73f7e>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00351-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: BULLETIN OF THE
Volume 83, Number 4, July 1977
Communicated by J. A. Wolf, January 26, 1977
Let G be a reductive group defined over Q. Index the parabolic subgroups
defined over Q, which are standard with respect to a minimal (O)P, by a partially
ordered set 4. Let 0 and 1 denote the least and greatest elements of 9 respec-
tively, so that (l)P is G itself. Given u E 9, we let (")N be the unipotent radical
of (u)p, (u)M a fixed Levi component, and (U)A the split component of the cen-
ter of (u)M. Following [1, p. 328], we define a map (u)H from (")M(A) to
(u)a =
Hom(X((U)M)Q, R) by
ex,(u)H(m)> = Ix(m)l, x E X((U)M)Q, m E ()M(A).
If K is a maximal compact subgroup of G(A), defined as in [1, p. 328], we ex-
tend the definition of (")H to G(A) by setting
(u)H(nmk) = ()H(m), n e (u)N(A), m E ()M(A), k e K.
Identify (°)a with its dual space via a fixed positive definite form ( ,) on (°)a
which is invariant under the restricted Weyl group Q2. This embeds any (u)a into | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/266/3013865.html","timestamp":"2014-04-16T04:27:38Z","content_type":null,"content_length":"8108","record_id":"<urn:uuid:4038416b-9083-4a18-b09e-77f7c6181d08>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00453-ip-10-147-4-33.ec2.internal.warc.gz"} |
Common Core Standards : CCSS.Math.Content.HSF-BF.B.3
Common Core Standards: Math
3. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx) and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment
with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Much like replacing sugar with salt can transform a blueberry pie to a foul-tasting disaster, replacing x with x + k or kx can transform a graph. Students should know that adding a constant k to a
function will change the graph of the function depending not only on the value of the constant, but on where it is inserted as well.
If y = f(x) is changed to y = f(x) + k, the curve will shift vertically (up for k > 0, down if k < 0). Adding k to x such that y = f(x + k) will shift the curve horizontally (left for k > 0, right
for k < 0).
Multiplying f(x) by a constant k stretches (k > 1) or squishes (0 < k < 1) the graph vertically. If k < 0, the graph is also flipped over the x-axis. Multiplying x by k stretches (k > 0) or squishes
(k < 0) the graph horizontally.
Students should also know that by definition, a function is even if f(-x) = f(x). If students are confused as to how this happens, give them the function f(x) = x^2. It's even because f(-x) = (-x)^2
= (-1)^2 × (x)^2 = 1 × x^2 = x^2 = f(x). Make sure they know not all functions with even numbers are even functions! It's an unfortunate and too common mistake. Even functions are symmetrical across
the y-axis.
An odd function is a misnomer because plenty of odd functions aren't strange in the slightest. A function is odd if f(-x) = -f(x). One such function is f(x) = x^3, because f(-x) = (-x)^3 = (-1)^3 × x
^3 = -1 × x^3 = -x^3 = -f(x). Convinced? Odd functions are symmetrical about the origin, not across any axis.
The best part is that students can have fun squishing and moving and flipping curves to their hearts' content without any nauseating repercussions. That's more than we can say for that pukeberry pie. | {"url":"http://www.shmoop.com/common-core-standards/ccss-hs-f-bf-3.html","timestamp":"2014-04-19T14:48:34Z","content_type":null,"content_length":"47108","record_id":"<urn:uuid:23d271b6-d6cc-4a90-bdc7-b16db160c578>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00530-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: February 2003 [00295]
[Date Index] [Thread Index] [Author Index]
RE: ParametricPlot - a feature or a bug?
• To: mathgroup at smc.vnet.net
• Subject: [mg39459] RE: [mg39447] ParametricPlot - a feature or a bug?
• From: "David Park" <djmp at earthlink.net>
• Date: Mon, 17 Feb 2003 04:33:56 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com
Not a bug. ParametricPlot has a default for how many plot points and
subdivisions it will use.
Options[ParametricPlot, {PlotPoints, PlotDivision}]
{PlotDivision -> 30., PlotPoints -> 25}
Let's look at the most illustrative case. With...
plot1 =
ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 500 Pi},
AspectRatio -> Automatic];
you obtain an inaccurate plot because you are only obtaining 3 or 4 points
each time you traverse the circle. If more PlotPoints are used a better plot
is obtained.
ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 500 Pi},
PlotPoints -> 1000,
AspectRatio -> Automatic];
To understand better what is happening to your plot, let's extract the
points that Mathematica used and plot them.
pts = (First[plot1] /. Line[a_] :> a)[[1,1]];
We have only 901 points for 250 circuits of the circle.
If just the points are plotted a fairly good circle is obtained.
{Point /@ pts}],
AspectRatio -> Automatic];
If only the first 15 points are used to draw a line, then you can see how an
annular region will slowly be filled in.
I believe the same phenomenon explains all of your cases. If Mathematica
attempted to continue subdividing the plot until a smooth curve was
obtained, it could easily fall into an infinite recursion, for example when
the curve had a cusp.
It is up to the user to specify an appropriate domain and number of plot
points. Generally, better looking curves are obtained if regions are not
David Park
djmp at earthlink.net
From: Vladimir Bondarenko [mailto:vvb at mail.strace.net]
To: mathgroup at smc.vnet.net
While trying to plot complex parametric plots with large values
of the parameter I run into a problem which boils down to the
following simple observation.
a) ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 1 Pi}, AspectRatio -> Automatic];
A perfect circumference.
b) ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 200 Pi}, AspectRatio ->
Instead of a circumference, not a very wide annulus.
c) ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 500 Pi}, AspectRatio ->
An annulus which width is equal to the radius of the inner
d) ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 1000 Pi}, AspectRatio ->
A black ring with tiny white spots.
e) ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 10^19 Pi}, AspectRatio ->
A funny net.
f) ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 10^20 Pi}, AspectRatio ->
A segment.
g) ParametricPlot[{Cos[t], Sin[t]}, {t, 0, 10^26 Pi}, AspectRatio ->
Only the axes are shown. There is no graph itself.
Is (at least a part of the shown output) a feature or a bug?
(By the way, before answering why do not try to solve the same
problems with a couple of other systems? ;-)
Best wishes,
Vladimir Bondarenko
Mathematical and Production Director
Symbolic Testing Group
Web : No other my site is permitted to me to quote here
http://www.CAS-testing.org/ GEMM Project (95% ready)
Email: vvb at mail.strace.net
Voice: (380)-652-447325 Mon-Fri 6 a.m. - 3 p.m. GMT
ICQ : 173050619
Mail : 76 Zalesskaya Str, Simferopol, Crimea, Ukraine | {"url":"http://forums.wolfram.com/mathgroup/archive/2003/Feb/msg00295.html","timestamp":"2014-04-19T04:59:35Z","content_type":null,"content_length":"37519","record_id":"<urn:uuid:85b0c331-364c-4e25-ba11-00fbe8ed970f>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00498-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calculating geodetic ellipse
In a geographic program I have the following problem:
I have an ellipse given as ellipse(longitude,latitude,semiMajor,semiMinor,angle)
The semiMajor/Minor are in
Now I need to plot this shape acurately onto a map .In order to do this I need to create a whole lot of points on the perimeter of this ellipse and plot each one.
I have a function (see below) which does this .
The problem is that this function does not take into acount the curvature of the earth.
I have another funtion (opensource from http://www.gavaghan.org/blog/free-source-code/geodesy-library-vincentys-formula/)
which takes a starting point ,bearing and distance in meters and evaluates the destination coordinates reached on the earth.
I have tried in the code below to use these two functions together in order to produce an ellipse correct for the earths curvature however for some reason I get a peanut shape instead of an ellipse!
Could someone maybe see where I've gone wrong?
protected void buildEllipse(double centerX,double centerY,
double semiMajor,double semiMinor,double angleOfEllipse)
//the amount of points on the perimeter to calculate
double angleStepSize = 0.5;
//the array of normal points (using classes from the JTS library)
Coordinate[] coords1 = new Coordinate[(int)(360/angleStepSize)];
//the array of geodetically corrected points
Coordinate[] coords2 = new Coordinate[(int)(360/angleStepSize)];
double beta = -angleOfEllipse * (Math.PI / 180);
double sinbeta = Math.sin(beta);
double cosbeta = Math.cos(beta);
int i = 0;
for (double angleOffset = 0; angleOffset < 360; angleOffset += angleStepSize)
double alpha = angleOffset * (Math.PI / 180) ;
double sinalpha = Math.sin(alpha);
double cosalpha = Math.cos(alpha);
double xDiff =
(semiMajor * cosalpha * cosbeta - semiMinor * sinalpha * sinbeta);
double x = centerX + xDiff;
double yDiff =
(semiMajor * cosalpha * sinbeta + semiMinor * sinalpha * cosbeta);
double y = centerY + yDiff;
coords1[i] = new Coordinate(x,y);
//here I find the distance in meters from the center of the ellipse
//to the new x y point found (I hope?)
double hyp = Math.sqrt((xDiff*xDiff)+(yDiff*yDiff));
double bearing = angleOfEllipse + angleOffset;
//This function finds the destination coordinates on the surface
//of the earth (I'm sure it works, used it elswhere)
GlobalCoordinates dest = calculateEndingCoords(centerX,centerY,hyp,bearing);
coords2[i++] = new Coordinate(dest.getLongitude(),dest.getLatitude());
//At this stage coords1 draws an ellipse,
//coords2 draws a peanut! | {"url":"https://community.oracle.com/message/6968743","timestamp":"2014-04-23T13:42:01Z","content_type":null,"content_length":"127038","record_id":"<urn:uuid:280f1e42-cee5-4b87-af9f-cc88b9319e9c>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00088-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hardy, G(odfrey) H(arold) (1877
Hardy, G(odfrey) H(arold) (1877–1947)
One of the most prominent English mathematicians of the 20th century; his legendary collaboration with John Littlewood lasted 35 years and produced nearly 100 papers. Hardy was a precocious child,
whose tricks include factorizing hymn numbers during sermons. In 1919 he became Savilian Professor of Geometry at Oxford but returned to Cambridge in 1931 as professor of pure mathematics. His work
was mainly in analysis and number theory.
Hardy had only one other passion in his life – the game of cricket. His daily routine would begin with reading The Times and studying the cricket scores over breakfast. Then he would do mathematical
research from 9 o'clock till 1 o'clock. After a light lunch, he would walk down to the university cricket ground to watch a game. In the late afternoon he would walk slowly back to his rooms in
College, and take dinner followed by a glass of wine. Hardy was known for his eccentricities. He couldn't stand having his photo taken and only five snapshots of him are known to exist. He also hated
mirrors and his first action on entering any hotel room was to cover any mirror with a towel.
Hardy's book A Mathematician's Apology (1940)^1 is one of the most vivid descriptions of how a mathematician thinks and the pleasure of mathematics. But the book is more, as C. P. Snow writes:
A Mathematician's Apology is ... a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits: yes, the crystalline clarity and candor are still there: yes, it is the
testament of a creative artist. But it is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again. I know nothing like it
in the language: partly because most people with the literary gift to express such a lament don't come to feel it: it is very rare for a writer to realise, with the finality of truth, that he is
absolutely finished.
1. Hardy, G. H. A Mathematician's Apology. London: Cambridge University Press, 1941.
Related category | {"url":"http://www.daviddarling.info/encyclopedia/H/Hardy.html","timestamp":"2014-04-18T10:40:19Z","content_type":null,"content_length":"8478","record_id":"<urn:uuid:b8745444-61e7-44b4-9fc7-3331eaec8ac7>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00031-ip-10-147-4-33.ec2.internal.warc.gz"} |
Angular momentum
Classical mechanics
$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \vec{v})$
Newton's Second Law
History of ...
In physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its
velocity vector. The angular momentum of a system of particles is the sum of that of the particles within it.
Angular momentum is an important concept in both physics and engineering, with numerous applications. Angular momentum is important in physics because it is a conserved quantity: a system's angular
momentum stays constant unless an external torque acts on it. Rotational symmetry of space is related to the conservation of angular momentum as an example of Noether's theorem. The conservation of
angular momentum explains many phenomena in nature.
Angular momentum in classical mechanics
Angular momentum of a particle about a given origin is defined as:
$\mathbf{L}$ is the angular momentum of the particle,
$\mathbf{r}$ is the position vector of the particle relative to the origin,
$\mathbf{p}$ is the linear momentum of the particle, and
$\times\,$ is the vector cross product.
As seen from the definition, the derived SI units of angular momentum are newton metre seconds (N·m·s or kg·m^2s^-1). Because of the cross product, L is a pseudovector perpendicular to both the
radial vector r and the momentum vector p and it is assigned a sign by the right-hand rule.
Angular momentum of a collection of particles
If a system consists of several particles, the total angular momentum about an origin can be obtained by adding (or integrating) all the angular momenta of the constituent particles. Angular momentum
can also be calculated by multiplying the square of the displacement r, the mass of the particle and the angular velocity.
Angular momentum in the centre of mass frame
It is very often convenient to consider the angular momentum of a collection of particles about their centre of mass, since this simplifies the mathematics considerably. The angular momentum of a
collection of particles is the sum of the angular momentum of each particle:
$\mathbf{L}=\sum_i \mathbf{R}_i\times m_i \mathbf{V}_i$
where R[i] is the distance of particle i from the reference point, m[i] is its mass, and V[i] is its velocity. The centre of mass is defined by:
$\mathbf{R}=\frac{1}{M}\sum_i m_i \mathbf{R}_i$
where the total mass of all particles is given by
$M=\sum_i m_i\,$
It follows that the velocity of the centre of mass is
$\mathbf{V}=\frac{1}{M}\sum_i m_i \mathbf{V}_i\,$
If we define $\mathbf{r}_i$ as the displacement of particle i from the centre of mass, and $\mathbf{v}_i$ as the velocity of particle i with respect to the centre of mass, then we have
$\mathbf{R}_i=\mathbf{R}+\mathbf{r}_i\,$ and $\mathbf{V}_i=\mathbf{V}+\mathbf{v}_i\,$
and also
$\sum_i m_i \mathbf{r}_i=0\,$ and $\sum_i m_i \mathbf{v}_i=0\,$
so that the total angular momentum is
$\mathbf{L}=\sum_i (\mathbf{R}+\mathbf{r}_i)\times m_i (\mathbf{V}+\mathbf{v}_i) = \left(\mathbf{R}\times M\mathbf{V}\right) + \left(\sum_i \mathbf{r}_i\times m_i \mathbf{v}_i\right)$
The first term is just the angular momentum of the centre of mass. It is the same angular momentum one would obtain if there were just one particle of mass M moving at velocity V located at the
center of mass. The second term is the angular momentum that is the result of the particles moving relative to their centre of mass. This second term can be even further simplified if the particles
form a rigid body, in which case a spin appears. An analogous result is obtained for a continuous distribution of matter.
Fixed axis of rotation
For many applications where one is only concerned about rotation around one axis, it is sufficient to discard the pseudovector nature of angular momentum, and treat it like a scalar where it is
positive when it corresponds to a counter-clockwise rotations, and negative clockwise. To do this, just take the definition of the cross product and discard the unit vector, so that angular momentum
$L = |\mathbf{r}||\mathbf{p}|\sin \theta_{r,p}$
where θ[r,p] is the angle between r and p measured from r to p; an important distinction because without it, the sign of the cross product would be meaningless. From the above, it is possible to
reformulate the definition to either of the following:
$L = \pm|\mathbf{p}||\mathbf{r}_{\perp}|$
where $\mathbf{r}_{\perp}$ is called the lever arm distance to p.
The easiest way to conceptualize this is to consider the lever arm distance to be the distance from the origin to the line that p travels along. With this definition, it is necessary to consider the
direction of p (pointed clockwise or counter-clockwise) to figure out the sign of L. Equivalently:
$L = \pm|\mathbf{r}||\mathbf{p}_{\perp}|$
where $\mathbf{p}_{\perp}$ is the component of p that is perpendicular to r. As above, the sign is decided based on the sense of rotation.
For an object with a fixed mass that is rotating about a fixed symmetry axis, the angular momentum is expressed as the product of the moment of inertia of the object and its angular velocity vector:
$\mathbf{L}= I \mathbf{\omega}$
$I\,$ is the moment of inertia of the object (in general, a tensor quantity)
$\mathbf{\omega}$ is the angular velocity.
As the kinetic energy K of a massive rotating body is given by
$\mathbf{K}= I \mathbf{\omega^2}/2$
it is proportional to the square of the angular momentum.
Conservation of angular momentum
In a closed system angular momentum is constant. This conservation law mathematically follows from continuous directional symmetry of space (no direction in space is any different from any other
direction). See Noether's theorem.
The time derivative of angular momentum is called torque:
$\tau = \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \mathbf{r} \times \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = \mathbf{r} \times \mathbf{F}$
So requiring the system to be "closed" here is mathematically equivalent to zero external torque acting on the system:
$\mathbf{L}_{\mathrm{system}} = \mathrm{constant} \leftrightarrow \sum \tau_{\mathrm{ext}} = 0$
where τ[ext] is any torque applied to the system of particles.
In orbits, the angular momentum is distributed between the spin of the planet itself and the angular momentum of its orbit:
$\mathbf{L}_{\mathrm{total}} = \mathbf{L}_{\mathrm{spin}} + \mathbf{L}_{\mathrm{orbit}}$;
If a planet is found to rotate slower than expected, then astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared between the planet and its
satellite in order to be conserved.
The conservation of angular momentum is used extensively in analyzing what is called central force motion. If the net force on some body is directed always toward some fixed point, the centre, then
there is no torque on the body with respect to the center, and so the angular momentum of the body about the centre is constant. Constant angular momentum is extremely useful when dealing with the
orbits of planets and satellites, and also when analyzing the Bohr model of the atom.
The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation. By bringing part of mass of her body
closer to the axis she decreases her body's moment of inertia. Because angular momentum is constant in the absence of external torques, the angular velocity (rotational speed) of the skater has to
The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron stars and black holes) when they are formed out of much larger and slower rotating stars (indeed,
decreasing the size of object 10^4 times results in increase of its angular velocity by the factor 10^8).
The conservation of angular momentum in Earth-Moon system results in the transfer of angular momentum from Earth to Moon (due to tidal torque the Moon exerts on the Earth). This in turn results in
the slowing down of the rotation rate of Earth (at about 42 nsec/day), and in gradual increase of the radius of Moon's orbit (at ~4.5 cm/year rate).
Angular momentum in relativistic mechanics
In modern (late 20th century) theoretical physics, angular momentum is described using a different formalism. Under this formalism, angular momentum is the 2-form Noether charge associated with
rotational invariance (As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant). For a system of point particles
without any intrinsic angular momentum, it turns out to be
$\sum_i \bold{r}_i\wedge \bold{p}_i$
(Here, the wedge product is used.).
Angular momentum in quantum mechanics
In quantum mechanics, angular momentum is quantized -- that is, it cannot vary continuously, but only in " quantum leaps" between certain allowed values. The angular momentum of a subatomic particle,
due to its motion through space, is always a whole-number multiple of $\hbar$ ("h-bar," known as Dirac's constant), defined as Planck's constant divided by 2π. Furthermore, experiments show that most
subatomic particles have a permanent, built-in angular momentum, which is not due to their motion through space. This spin angular momentum comes in units of $\hbar/2$. For example, an electron
standing at rest has an angular momentum of $\hbar/2$.
Basic definition
The classical definition of angular momentum as $\mathbf{L}=\mathbf{r}\times\mathbf{p}$ depends on six numbers: r[x], r[y], r[z], p[x], p[y], and p[z]. Translating this into quantum-mechanical terms,
the Heisenberg uncertainty principle tells us that it is not possible for all six of these numbers to be measured simultaneously with arbitrary precision. Therefore, there are limits to what can be
known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one
Mathematically, angular momentum in quantum mechanics is defined like momentum - not as a quantity but as an operator on the wave function:
where r and p are the position and momentum operators respectively. In particular, for a single particle with no electric charge and no spin, the angular momentum operator can be written in the
position basis as
where $abla$ is the vector differential operator " Del" (also called " Nabla"). This orbital angular momentum operator is the most commonly encountered form of the angular momentum operator, though
not the only one. It satisfies the following canonical commutation relations:
$[L_l, L_m ] = i \hbar \sum_{n=1}^3 \varepsilon_{lmn} L_n$,
where ε[lmn] is the (antisymmetric) Levi-Civita symbol. From this follows
$\left[L_i, L^2 \right] = 0$
$L_x = -i\hbar (y {\partial\over \partial z} - z {\partial\over \partial y})$
$L_y = -i\hbar (z {\partial\over \partial x} - x {\partial\over \partial z})$
$L_z = -i\hbar (x {\partial\over \partial y} - y {\partial\over \partial x})$
it follows, for example,
\begin{align} \left[L_x,L_y\right] & = -\hbar^2 \left( (y {\partial \over \partial z} - z {\partial\over \partial y})(z {\partial\over \partial x} - x {\partial\over \partial z}) - (z {\partial\
over \partial x} - x {\partial\over \partial z})(y {\partial \over \partial z} - z {\partial\over \partial y})\right) \\ & = -\hbar^2 \left( y {\partial\over \partial x} - x {\partial\over \
partial y}\right) = i \hbar L_z. \\ \end{align}
Addition of quantized angular momenta
Given a quantized total angular momentum $\overrightarrow{j}$ which is the sum of two individual quantized angular momenta $\overrightarrow{l_1}$ and $\overrightarrow{l_2}$,
$\overrightarrow{j} = \overrightarrow{l_1} + \overrightarrow{l_2}$
the quantum number j associated with its magnitude can range from | l[1] − l[2] | to l[1] + l[2] in integer steps where l[1] and l[2] are quantum numbers corresponding to the magnitudes of the
individual angular momenta.
Angular momentum as a generator of rotations
If φ is the angle around a specific axis, for example the azimuthal angle around the z axis, then the angular momentum along this axis is the generator of rotations around this axis:
$L_z = -i\hbar {\partial\over \partial \phi}.$
The eigenfunctions of L[z] are therefore $e^{i m_l \phi}$, and since φ has a period of 2π, m[l] must be an integer.
For a particle with a spin S, this takes into account only the angular dependence of the location of the particle, for example its orbit in an atom. It is therefore known as orbital angular momentum.
However, when one rotates the system, one also changes the spin. Therefore the total angular momentum, which is the full generator of rotations, is J[i] = L[i] + S[i] Being an angular momentum, J
satisfies the same commutation relations as L, as will explained below. namely
$[J_\ell, J_m ] = i \hbar \sum_n \varepsilon_{lmn} J_n$
from which follows
$\left[J_\ell, J^2 \right] = 0.$
Acting with J on the wavefunction ψ of a particle generates a rotation: $e^{i \phi J_z} \psi$ is the wavefunction ψ rotated around the z axis by an angle φ. For an infinitesmal rotation by an angle d
φ, the rotated wavefunction is ψ + idφJ[z]ψ. This is similarly true for rotations around any axis.
In a charged particle the momentum gets a contribution from the electromagnetic field, and the angular momenta L and J change accordingly.
If the Hamiltonian is invariant under rotations, as in spherically symmetric problems, then according to Noether's theorem, it commutes with the total angular momentum. So the total angular momentum
is a conserved quantity
$\left[J_l, H \right] = 0$
Since angular momentum is the generator of rotations, its commutation relations follow the commutation relations of the generators of the three-dimensional rotation group SO(3). This is why J always
satisfies these commutation relations. In d dimensions, the angular momentum will satisfy the same commutation relations as the generators of the d-dimensional rotation group SO(d).
SO(3) has the same Lie algebra (i.e. the same commutation relations) as SU(2). Generators of SU(2) can have half-integer eigenvalues, and so can mj. Indeed for fermions the spin S and total angular
momentum J are half-integer. In fact this is the most general case: j and mj are either integers or half-integers.
Technically, this is because the universal cover of SO(3) is isomorphic SU(2), and the representations of the latter are fully known. J[i] span the Lie algebra and J^2 is the Casimir invariant, and
it can be shown that if the eigenvalues of J[z] and J^2 are m[j] and j(j+1) then m[j] and j are both integer multiples of one-half. j is non-negative and m[j] takes values between -j and j.
Relation to spherical harmonics
Angular momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. Then, the angular momentum in space representation is:
$L^2 = -\frac{\hbar^2}{\sin\theta}\frac{\partial}{\partial \theta}\left( \sin\theta \frac{\partial}{\partial \theta}\right) - \frac{\hbar^2}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2}$
When solving to find eigenstates of this operator, we obtain the following
$L^2 | l, m \rang = {\hbar}^2 l(l+1) | l, m \rang$
$L_z | l, m \rang = \hbar m | l, m \rang$
$\lang \theta , \phi | l, m \rang = Y_{l,m}(\theta,\phi)$
are the spherical harmonics.
Angular momentum in electrodynamics
When describing the motion of a charged particle in the presence of an electromagnetic field, the "kinetic momentum" p is not gauge invariant. As a consequence, the canonical angular momentum $\
mathbf{L} = \mathbf{r} \times \mathbf{p}$ is not gauge invariant either. Instead, the momentum that is physical, the so-called canonical momentum, is
$\mathbf{p} -\frac {e \mathbf{A} }{c}$
where e is the electric charge, c the speed of light and A the vector potential. Thus, for example, the Hamiltonian of a charged particle of mass m in an electromagnetic field is then
$H =\frac{1}{2m} \left( \mathbf{p} -\frac {e \mathbf{A} }{c}\right)^2 + e\phi$
where φ is the scalar potential. This is the Hamiltonian that gives the Lorentz force law. The gauge-invariant angular momentum, or "kinetic angular momentum" is given by
$K= \mathbf{r} \times \left( \mathbf{p} -\frac {e \mathbf{A} }{c}\right)$
The interplay with quantum mechanics is discussed further in the article on canonical commutation relations. | {"url":"http://www.pustakalaya.org/wiki/wp/a/Angular_momentum.htm","timestamp":"2014-04-18T06:35:37Z","content_type":null,"content_length":"37629","record_id":"<urn:uuid:33a61fe6-2a06-48c0-9dff-e94ce3c28189>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00120-ip-10-147-4-33.ec2.internal.warc.gz"} |
CHAPTER 6 Small-Signal RF Amplifier Desi
Design Using S Parameters 141 6. Once Y in is known, set B S equal to the negative of the imaginary part of Y in , or: B S = -B in 7. Calculate the gain of the stage using Equation 6-12. From this
point forward, it is only necessary to provide input and output networks that will present the calculated Y S and Y L to the transistor. Example 6-3 illustrates the procedure. D E S I G N U S I N G S
PA RA M ET E R S As we discussed in Chapter 5, transistors can also be com- pletely characterized by their scattering or S parameters. With these parameters, it is possible to calculate potential
instabilities (tendency toward oscillation), maximum available gain, input and output impedances, and transducer gain. It is also possi- ble to calculate optimum source and load impedances either for
simultaneous conjugate matching or simply to help you choose specific source and load impedances for a specified transducer gain. EXAMPLE 6-3 Consider a transistor with the following Y parameters at
200 MHz: y i = 2.25 + j7.2 y o = 0.4 + j1.9 y f = 40 - j20 y r = 0.05 - j0.7 Set B L equal to -b o of the transistor, B L = -j1.9 mmhos The load admittance is now defined. Y L = 4.24 - j1.9 mmhos
Calculate the input admittance of the transistor using Equation 6-13 and Y L . Y in = y i - y r y f y o + Y L | {"url":"http://my.safaribooksonline.com/book/-/9780750685184/chapter-6-small-signal-rf-amplifier-design/design_using_s_parameters","timestamp":"2014-04-17T11:14:21Z","content_type":null,"content_length":"50146","record_id":"<urn:uuid:cd13972c-50aa-4d0d-913b-88b30eb5ef35>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00415-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Back in second grade, I was dissatisfied with the algorithm we were being taught for doing subtraction. So I “invented” my own
– 185
5 is bigger than 3, so we subtract them in the opposite order (5–3=2) and take the tens-complement (8) of the result. As usual, we borrow from the 6 (which becomes a 5) and we repeat: 8–5=3 and take
the tens-complement (7). Finally 2–1=1, so the answer is 178.
While only slightly different from the conventional algorithm, I felt this one to be an improvement because I never had to know how to subtract from numbers larger than 10 (e.g. who cares what 13–5
I haven’t thought much about this little juvenile act of rebellion until a couple of months ago, when I was going over my daughter’s 3^rd grade math homework with her. She was doing similar
subtraction problems. But, in keeping with the times, she was charged with explaining her methods for arriving at the answer.
Imagine my surprise when she explained her method to me. It was exactly the same “unconventional” algorithm that I had used when I was her age. It was not what the teacher had taught; she had figured
it out on her own.
[Her method was the same, but her accuracy was not the greatest. So I taught her the other trick that I learned in that era: check your work by doing arithmetic modulo 9: 178=1+7+8= 7 mod 9, 185=
1+8+5=5 mod 9. So 178+185=5+7=12=1+2 = 3 mod 9, which agrees with 363=3+6+3=3 mod 9.]
Now, I don’t know what this has to do with Larry Summers’ remarks on the dearth of women in the Hard Sciences (at least in this country). My personal experience echoes that of the AIP Study. An
alarmingly large majority of the women who arrived at Harvard the year I did, intending to major in Physics, had decided by sophomore year to do something else. As a consequence, it was unsurprising
that, by the time I started graduate school, there was only one woman in an entering class of 28. Sean Carroll takes on the thankless task of confronting Summers hypotheses with the data. I’m afraid
I can’t muster the energy.
I’m much too busy trying to nurture that spark of creativity in my daughter, hoping that, a decade from now, she doesn’t face the stark choice that my classmates at Harvard/Radcliffe faced a
generation ago.
Posted by distler at February 23, 2005 4:28 PM
Re: Reinvention
I am surprised you don’t mention another Harvard man’s approach to a similar problem.
Posted by: Robert on February 24, 2005 1:19 AM | Permalink | Reply to this
Re: Reinvention
Like father, like daughter :)
Posted by: Srijith on February 24, 2005 3:05 AM | Permalink | PGP Sig | Reply to this
Re: Reinvention
Fan that spark into a fire! Especially the abiity and creativity to explore alternative methods.
And thanks for the “unconventional” method. I would be interested to hear (since they now have to explain their methods) what her teacher thinks of her methods. I was rather discouraged early in my
education by “educators” who accept any deviations from the text book.
Posted by: Eric on February 24, 2005 8:43 PM | Permalink | Reply to this
As I’ve written before, I’m generally pretty pleased with her teachers’ efforts to challenge the students to explore mathematics.
One of the things both pleased and surprised me was when I explained to her the “arithmetic modulo 9” trick for checking her answers. We did a few examples, and she was quickly convinced that it
worked. But then she turned to me and said, “OK … but why does it work?”
I actually had to go through the proof for her, before she was satisfied. (Or was it bored? Hard to tell with a 9 year old.)
Posted by: Jacques Distler on February 25, 2005 5:09 PM | Permalink | PGP Sig | Reply to this
Re: Reinvention
Hi Jacques,
When I was about 12 or 13 I remember I read a book called the “Trachtenberg Speed System of Applied Mathematics”. This is a number of clever and powerful tricks to multiply and divide huge numbers in
your head, as well as many other things. Remarkably, it was developed by Jakow Trachtenberg when he was a prisoner in a concentration camp.
Focusing his mind this way must have helped him survive.
I can’t remember how to do it now but the book is available on Amazon.
Posted by: Steve M on February 25, 2005 1:25 PM | Permalink | Reply to this | {"url":"http://golem.ph.utexas.edu/~distler/blog/archives/000516.html","timestamp":"2014-04-19T06:52:23Z","content_type":null,"content_length":"17787","record_id":"<urn:uuid:b01b8ba7-393b-470d-84a3-2b2c979b472c>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00052-ip-10-147-4-33.ec2.internal.warc.gz"} |
Odds and ends: the 2010 Joint Mathematics Meeting and Euler's Gem
Posted by: Dave Richeson | January 5, 2010
Odds and ends: the 2010 Joint Mathematics Meeting and Euler’s Gem
I’ll be heading to the 2010 Joint Mathematics Meeting in San Francisco next week. In case any of you are interested in meeting up, here are a few of the items on my (busy) schedule. Please introduce
yourself; it would be nice to put faces with names.
• I’m giving a talk on some work with my collaborator, Jim Wiseman, entitled “Symbolic Dynamics for Nonhyperbolic Systems.” It is in the AMS special session on Dynamical Systems, Friday, January 15
at 5:15. I have no idea how I’ll be able to say anything in my 10 minute time slot, but I’ll do my best.
• I’m on the MAA Committee for Minicourses and will be monitoring two minicourses (one of the perks of being on the committee!):
□ Using GeoGebra to create activities and applets for visualization and exploration, by Michael K. May
□ The hitchhiker’s guide to mathematics, by Dan Kalman and Bruce F. Torrence
• I’m having a book signing for my book Euler’s Gem at the Princeton University Press booth in the exhibit hall, Friday, January 15 at 10:30. Please come by!
If you are going to JMM 2010 and are giving a talk, post it in the comments below. Also, if you’re on Twitter, the hashtag for the meeting is #jointmath. I don’t have a smart phone, so I’m not sure
how much I’ll be able to tweet. But I’ll try to contribute some.
Speaking of Euler’s Gem,… in case you are interested…
I’ll hope to see you at your book signing.
By: Sue VanHattum on January 6, 2010
at 2:00 pm
I asked my son Dan to get me a copy at your book signing but he may be too busy with interviews.
By: David Freeman on January 6, 2010
at 4:17 pm
Great, Sue, I’m looking forward to meeting you.
David, I hope your son can make it. Please wish him luck with the interviews. It will be an exhausting week for him.
By: Dave Richeson on January 6, 2010
at 11:28 pm
Posted in Math | Tags: AMS, Euler's Gem, Joint Mathematics Meeting, MAA, Princeton University Press | {"url":"http://divisbyzero.com/2010/01/05/odds-and-ends-the-2010-joint-mathematics-meeting-and-eulers-gem/","timestamp":"2014-04-17T09:55:35Z","content_type":null,"content_length":"64768","record_id":"<urn:uuid:d3cde9a3-643a-408b-bfe8-a5ea2d3c7537>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00422-ip-10-147-4-33.ec2.internal.warc.gz"} |
Alwyn Scott Turner
Chicago | Los Angeles | Miami | New York | San Francisco | Santa Fe
Amsterdam | Berlin | Brussels | London | Paris | São Paulo | Toronto | China | India | Worldwide
Alwyn Scott Turner
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Leonard, Texas
Lives in
Gearhart, Oregon
Works in
United States of America
Representing galleries
Contact me directly by eMail: alwyn@pacifier.com
photography, traditional, realism, The American People Archive. Alwyn Scott Turner. documentary photography
The American People
Alwyn Scott Turner
Historical Portfolios of Documentary Photographs During The Golden Age Of The United States of America
DOCUMENTARY PHOTOGRAPHS - THE GOLDEN AGE OF AMERICA - AN ARCHIVE OF HISTORICAL PORTFOLIOS OF DOCUMENTARY IMAGES MADE DURING THE LAST HALF OF THE 20th CENTURY AND INTO THE 21st CENTURY: 1950 -2014:
Harmonic Composition and Perspective based on The Golden Theme in the historical Documentary Photography Portfolios - each Portfolio accompanied by harmonic sequences in The Musical Composition
based on natural harmonic music titled: MusicScience: The Golden Theme - Variations and Improvisations by Alwyn Scott Turner.Abstract of Priority dated May 1, 2012, Published on ArtSlant.com as
MusicScience: THE GOLDEN THEME - TradeMark & Copyright by Alwyn Scott Turner. All Rights Reserved Pending Patents.
Selections from the Documentary Photography Portfolios at ArtSlant:
Master Selections from the Documentary Photography Portfolios - The American People Archive.
MusicScience: Abstracts of Priority/Copyrights
Background Composition for the Portfolio: "The Golden Theme". Composed, performed and copyright by Alwyn Scott Turner.
These images in the Portfolio are dedicated to the historical documentary photographs of artists whose work personified their realism, ideas, integrity and ethics. Mathew Brady, Edward Curtis, Lewis
Hine, Robert & Frances Flaherty, Walker Evans, W. Eugene Smith, Enrico Natali and many other influential documentary photographers true an uncontrived and authenic view of reality in their
historical perspective.
The Oregon Coast Portfolio 1984-2013/ Master Selections from the North Oregon Coast Portfolios in The American People Archive, including series of images from Seaside, Gearhart, Cannon Beach,
Manzanita, Oceonside, and elsewhere - including a Porfolio of Documentary Landscapes from 'Spuce Run'.
Documentary Portraits Portfolio
An Era of Famous Americans/ A Selection of documentary portraits and video images of personalities in various fields of endeavor during The Golden Age of America. Including: Rev.Martin Luther King,
Jr. (Civil Rights Leader), Robert (Bobbie) Kennedy (Pres.Candidate), Pres.Richard Nixon, Pres.Lyndon Johnson, Chief Justice Earl Warren (U.S.Supreme Court), Nelson Rockefeller (Pres.Candidate),
Hubert Humphrey (Pres.Candidate), Eugene MacCarthy (Pres.Candidate, George Wallace (Pres.Candidate), Gus Hall (American Communist Party), Walter Reuther (UAW Labor Leader), Henry Ford (Ford Mtr
Co.), Ralph Nader (Pres.Candidate), , John Szarkowski (MoMA Photography Curator), Samuel J. Wagstaff, Jr.,(Photography Curator & Collector), Ken Burns (FilmMaker), Alfred Eisenstadt (TimeLife
Photojournalist), Gorden Parks (photographer), Yousuf Karsh (Studio Portrait Photographer), Cornell Capa (Photo Curator), Ralph Ginsberg (Avant-Garde Publisher), Timothy Leary (LSD Proponent), Elia
Kazan (Film Director), Henry Fonda (Film Actor), Isaac Perlman (Violinist), Doc Watson (Bluegrass Musician), Willie Nelson, (Singer Musician), Pete Fountain (New Orleans Jazz Clarinetist), Herbie
Mann (Jazz Flutist), Tiny Tim (Ukulele Actor), , MoonDog (NYC Poet), Eddie Rickenbacker (WW1 Ace & Author), Monte Montana (Rodeo Trick Rider), Larry Mehan (5-Time Rodeo Champion), Peter MacDonald
(Navajo Indian Leader), Chief Grey Squirrel (Pueblo de Taos), Gordie Howe (Hockey Player), Jerry Cooney (Boxer), Tex Cobb (Boxer), Sylvester Stallone (Film Actor & Fight Manager), Mordecai Persky
(Editor & Publisher), Frank Ditto (Black Activist), and More. Background Composition for the Portfolio: MusicScience: The Golden Theme: Let's Cross Over The River/ composed and performed on Piano by
Alwyn Scott Turner in memory of Martin Luther King, Jr.
Martin Luther King, Jr Portraits
BackStory: This is the last formal portrait made of Martin Luther King, Jr. It was made during his last public speaking performance in Detroit, Michigan in late March, on the evening prior to his
airplane departure for Memphis, Tennessee - He was assassinated a few days later. He had agreed to do an interview/portrait with me after his final speech in Detroit, and after making this
documentary portrait - his schedule was running late to do the hour-long interview - he agreed to write an article to be published with his historical portrait. On the day he was assassinated, I
received the article titled: 'The Other America" - by Martin Luther King, Jr. - Two weeks after his assassination, I published this documentary portrait with his article in The Detroit Magazine -
which I owned. In this last speech written to be published by Martin Luther King, Jr. - titled 'The Other America' - arrived by mail on the day of his assassination, and was subsequently published
in The Detroit Magazine - Dr. King states in his opening paragraph:
"I still believe that freedom is the bonus you receive for telling the truth - ye shall know the truth and the truth shall set you free. I do not see how we will ever solve the turbulent problem of
race confronting our nation until there is an honest confrontation with it and a willing search for the truth and a willingness to admit the truth when we discover it."
The Detroit People Portfolio 1969/ Documentary Portraits. Self-published in 1970, Photographs of The Detroit People by Alwyn Scott Turner was distrubuted through the bookstores at The Museum of
Modern Art and The George Eastman House - with the assistance of John Szarkowki and Minor White. The book of 142 full-frame black & white documentary photographs was credited by Collier's
Encyclopedia as being the publication that stated the self-publishing industry, and influential in reviving the photography publishing market, subsequently leading to a computer era of
self-publishing. Szarkowski at MoMA purchased several prints for their permanent collection, and offered to exhibit The Detroit Portfolio. However, the larger documentary project on The American
People was in an early stage of compilation, and with Szarkowski's assistance, funded by Guggenheim and National Endowment for the Arts (NEA) fellowship grants.
Documentary Kodachromes Portfolio 1969-79/ Color Images Verite: The American People. A selection of color prints from Kodachrome sllides selected from various Portfolios of documentary photographs
in The American People Archive. Dedicated to Minor White and the curatorial staff at George Eastman House in Rochester, NY., whose assistance made this Portfolio possible.
The Samuel J. Wagstaff, Jr. Memorial Portfolio/Opening Night Reception at the Detroit Institute of Arts. Spring, 1969. Photographs of the opening night reception at Sam Wagstaff's first curated
exhibition at the DIA. Portraits of attending artists and local celebrities. Including photographs of Wagstaff and photographs of Alwyn Scott Turner by Wagstaff. The newspaper review photographic
outlay of the event by Alwyn Scott Turner, Contributing Editor to the Sunday Magazine section of The Detroit Free and The Detroit News, which published my story essay on Wagsstaff - which brought
him to national attention and led to his awakened interest and subsequent knowledge in collecting photography as works of art, and his historic role as an archive collector of historical aesthetic
photographs....and by the power of his personal insight and wealth, established the vintage photographic print market.
The Washington DC Portfolio. November, 1969/ Moratorium Day Rally. The largest national massive Rally of citizens in American history, as a half million people in the nation's Capitol gathered to
demonstrate and celebrate their support for President Nixon's pledge to the American people that he would end The Vietnam War. Consequently, what was scheduled as a violent protest against the war
became a victory celebration for advocates of peace and freedom of speech, and the right of civil protest to challenge tyrantic totalitarian authority. This historical Portfolio is dedicated to the
memory of the peace policies of President John F. Kennedy, assasinated because of his efforts to prevent proliferation of nuclear weapons into the Middle East.
The Avant-Garde Magazine #13 Portfolio. 1970/ A Monumental Portfolio of Photographs by Alwyn Scott Turner. The first prominant magazine of the era to publish an entire issue of 70 documentary
photographs printed full frame, uncropped, uncontrived and unposed images, sequenced and justaposed - images not influenced by editorial assignment, making a clear distinction between the historical
integrity of documentary content as opposed to that of supervised photojournalism - and with this issue of Avant-Garde, as a sequel to the self-publilshed book of images on The Detroit People
Portfolio, was a primary factor in establishing a public interest in documentary portraits as an art form for publication. Dedicated to Ralph Ginsberg, Editor and Publisher of Avant-Garde Magazine,
for his insight and initiative in publishing an entire issue of documentary photographs when publishing photographic books was not being considered in the marketplace.
Documentary Portraits Portfolio/The American Communist Party Leaders. Portraits of Gus Hall and leaders of The American Communist Party - Leaders photographed at their headquarters in New York City.
Dedicated to James Angleton, former head of the CIA whose assistance made this Portfolio possible.
The New York Portfolio 1969-84/ Dedicated to John Szarkowski, former Director of Photography at the Museum of Modern Art in New York City. A photographer/curator who was instrumental in photography
being accepted as a fine art, and how encouraged and promoted opportunity for artist when the commercial market did not exist.
The New England Portfolio 1969/Dedicated to Francis Flarety - whose documentary portraits of ordinary people set a precedent for reality in film.
The Tampa Portfolio 1970. The Missing Portfolio consisting of 144 B&W negatives that are being retained by AOL Time/Life, which was requested by Life photo Editor Alice Rose George with the promise
of returning them to me after their perusal. These negatives have never been returned by Time/Life. The documentary images were made at the Florida State Mental Hospital where indigent elderly
citizens were being commited to the mental institution. A few of the images were part of a documentary expose that was published in the Tampa Times/Tribune Sunday Magazine when I was Editor of the
The Cheyenne Portfolio 1970-71. Dedicated to John Denver, Singer and Musician, who personified the American spirit during The Golden Age of America.
The New Orleans Portfolio 1971-74. The French Quarter Portfolio was presented by the New Orleans Museum of Art as a One-Man Exhibition in 1974, and the exhibit of 33 archival prints were purchased
for their permanent collection. The Images were again exhibited recently as a part of a four-man exhibition with Van Der Zee, Alvarez Bravo and Doisneau - titled: Inner Cities.
The Rappahannock Portfolio 1974-78/ Dedicated to Linda Beck-Turner.
The Philadelphia Portfolio 1978-84/ Dedicated to Linda Beck-Turner - and the many people whose silent consent to the historical documentary portraits in this Portfolio - to the people that live in
this great cultural city...that they refer to as Philly.
The Oregon Coast Portfolio 1984-2012/The Oregon Coast Portfolio/ Master Selections from the North Oregon Coast Portfolios in The American People Archive, including series of images from Seaside,
Gearhart, Cannon Beach, Manzanita, Oceonside, Astoria and elsewhere - plus a Porfolio of Documentary and Abstract Landscapes. Dedicated to Sandra Claire Foushee, Master Poet and Editor of Poetry &
Prose Annual, and 2010 winner of The William Stafford Annual Award for Poetry for her entry Surfing On Intuition. Known for her clarity of style, sense harmony and intimate reality of substance, and
the enlightened nature of her higher consciousness - and to other poets whose influence focused the light of consciousness on the vitality of life - rather than the darkness that prevades the
contemporary conntrol of poetry and prose.
Documentary Portraits/The American People - a comprehensive portfolio of ordinary people, unposed and uncontrived, made with their silent consent. Accomplished without painting the camera black or
secretly photographing from a distance, or with a telephoto lens. John Szarkowski suggested calling this innovative new approach to documentary photographs as 'Frontal Documentary Portraits' - and
acquired a selection of these images for their permanent collection at MoMA.
The American People Archive was funded by grants and fellowships from the Guggenheim Foundation and the National Endowment of the Arts. A selection of vintage prints are in the permanent collections
of the Museum of Modern Art in New York and the New Orleans Museum of Art. Portfolios consist of sequenced images arranged by the Artist.
Original Prints: All Photographs are Original One-of-Kind Original 16x20 Archival 'Silver Prints' Master Exhibiton prints made by the Artist. The Black & White 'Silver' prints were made from vintage
35mm TriX Film, and the color prints from vintage 35mm Kodachrome color slides. All prints full-frame and unmanipulated. All original prints are signed, sealed and autographed by Alwyn Scott Turner.
None of the original images were made as a product of commercial or editorial direction.
MusicScience is the original scientific discovery of the Natural Harmonic Matrix - of The Numerical Code of The Universe' - It is a 'Harmonic Universe' - and Music is no longer a 'Theory' - but a
Harmonic Science (MusicScience) that is the foundation for musical compositions based on the natural variations of the harmonic code that governs all energy and matter - It is 'Harmonic Universe'
based on a 'Numerical Code' and 'The Golden Theme' - the 'divine' ratio of life functions as 'The Code' - a Numerical System - the scientific knowledge that unifies all Sciences and expands the
creative base for 'New Game Changing Technology' that will rapidly alter the present primeval political, economic and cultural paradigm of humanity - including all forms of energy and matter.
Hopefully this monumental discovery of 'The Harmonic Code' - an integral function of 'The Numerical Code of The Universe' - will create a human bond of enlightened consciousness in which each human
being has the freedom to evolve according to their gifted nature, and to have the privacy to research and create new new scientific inventions and original art forms that benefit all mankind -
rather than scientific discoveries used as weapons of war or means to terrify people. It is my hope that MusicScience will be used to bring illumination to the minds of human beings, and that it
will not be used to destroy what nature has created - for we must freely share our scientific discoveries with all of mankind...and work harmoniously together for the survival of all life on planet
Earth. Each human being by the nature of their unique individuality - makes an original contribution to life in the Harmonic Universe...
The search for higher Consciousness is a science and an art - and a grace that unites all humanity - each person is endowed with natural Rights to enjoy their personal evolution - with a sense of
security that exists only when Freedom, Peace and Properity prevails - the Pursuit of Happiness is essential for the natural course of evolution toward a higher sense of Cosmic Consciousness - which
is our instinctive path to wisdom - for everyone...
Harmonic Fusion and The Nuclear Code.
& THE PRIME ROOT CODE FOR THE ATOMIC AND MOLECULAR MATRIX:
THE GOLDEN THEME: A Mathematical System and Numerical Method of Natural Variations derived from the Cosmic Code of the Harmonic Universe
Announcing the Discovery of The Numerical Code of the Universe, the Harmonic Code, the Decimal Code (including the harmonic decimal system for all 'irrational numbers), The Prime Root Code, the
Binary Matrix of Prime Roots for the Periodic Table of Atoms and Molecules, Parallel Matrix of the Gravitational Code; Harmonic Calculus for the MagicSineWave Vectors of the Six Harmonic Triads -
Quarks, Phonons & Photons; Triadal Code (Progression of 18 Triads) of the Diatonic Scale and Harmonic Rhythm of the Chromatic Spectrum; Prime Roots for the Velocity of Light, The Prime Roots of the
Energy Constant (Planck's) , Prime Root frequencies of Hydrogen & all Atomic and Molecular combinations, including Harmonic Fusion; Harmonic Encryption of Genetic Sequences of the DNA Molecule;
Calculus Logarithms of 3-dimensional Angular Momentum of Triadal Planes on the Nuclear Axis in nautical terms, etc.
All these discoveries will lead to New Technologies of Scientific Invention - including Harmonic Fusion of the Elements, which will replace all forms of enegy - the primitive period of 'fossil' fuel
as the main energy source on is over. And to all the Musicans of the world - it is no longer Music 'Theory' - it is now - MusicScience.
MusicScience is The Universal Harmonic Number System that Unifies Time, Space & Motion in the Micro/Macro Cosmos; the Harmonic Wave Vectors of Conscious Thought and the Mechanics of Levity of the
Astral Body, and the Method of entering the tate of Cosmic Consciousness and the Illumination of the Mind. - In essence - all these original scientific discoveries are elements embodied in the
Musical Composition: MusicScience - THE GOLDEN THEME' - Variations On The Golden Theme - Copyright by Alwyn Scott Turner - with all rights pending patents.
1.1.13 Abstract of Priority: MusicScience - The Harmonic Code of The Universe.
‘THE GOLDEN THEME’ - A Musical Arrangements - Copyright January 1, 2001 by Alwyn Scott Turner - Trademark and All Rights Reserved Pending Patents. A futher disclosure on the 'Harmonic Code' can be
viewed at the following sites: MusicScience.org - and Abstracts of Priority: 'The Numerical Code Of The Universe' - posted at Academia.edu and ResearchGate.com.
(MusicScience is deliberately not being listed in search engines by Google - in order to suppress this great discovery that will lead to new technologies that will antiquate current power structures
- and those that conspire to prevent 'game changing' scientific discoveries.)
The Harmonic Calculus of Atoms & Molecules based on the Prime Root Frequencies of Each Atom in the Atomic Matrix - including the frequencies of those yet to be discovered - based on their Electonic
Orbits.- and the Harmonic Combinations of each atom in the Triadal System of all Molecules.
See: Abstract Notes published at Academia.edu & ResearchGate.com & MusicScience.org
The Quantum Parallel Matrix Of Gravity
(The Gravity Matrix Abstract of Priority - has been published to a limited distribution.)
OF HARMONIC 'BASE PAIRS' IN THE DNA MOLECULE
There is a Natural Numerical Order in the Decimal Integer Sequences that govern the Harmonic Quantum frequencies and wavelengths of the Chromatic Spectrum. The Harmonic Universe is based on
frequencies of vibration that originate from the natural order of Number Systems that that arise from The Periodic Matrix of Prime Roots.
The Quantum Gravity is a Parallel Matrix based on the Odd and Even numbers - that correspond to the Prime Roots Matrix that governs electronic frequencies. The Harmonic Calculus of the Functions
derive from The Decimal Code - the Decimal Expansion of Prime Roots including 'irrational' sequences which in fact are not numerically irrational - but which functions in orderly numerical order in
a multiple system based on prime roots, that create perfect harmony within all forms of vibration, motion and energy, and forms of matter. The natural harmonic order numerical integer sequences in
decimal expansion is a domain system of 'Base Pair' Prime Roots in the calculus of all Universal functions - especially, the harmonic architecture of 'base pairs' in the structure of Genetic
sequences of the DNA Molecule.
(.000000000000000000000000065470944 )
65470944 = ( 31416 x 521 x 4 )
The Real Numerical Constant for Planck's (h) Energy Interval
And the Quantum Prime Root for the Frequency of the Photon
The 'Decimal Code' for the Photon Prime Root:
Prime Root #521
This is the Harmonic Vibration in Erg Seconds of the Fundamental Quantum Energy Numerical Consant that Spans 40 Octaves (40 x 13 = 520) in the Chromatic Spectrum - the Decimal Code in the Expansion
of Prime Root 521 - Governs the Numerical Systems of the Photon, Electron, Neutron and all Atomic Frequencies in the Hydrogen Atoms and basic to all frequencies in The Periodic Table of Elements
Quantum Units & The Speed of Light
931416 x .2 = 186283.2 Miles Per Second
Times 5280 Feet = 983575296 Feet Per Second
THE PHOTON: Quantum Prime Root: #197
(197x197) x (48x48) x (11)
The Universal Constant fo the Frequency of Light
The 'Second' and the 'Foot' being natural units of measure to Gravity, and the only units of measure in the Harmonic Calculus of functions in all the decimal systems in MusicScience and 'The
Numerical Code of the Universe'.
All natural numerical calculations and formulas are in these natural units, and it is only in these precise units of measusre that the decimal integer sequences of the Numerical Systems relative to
Time and Space.
2.21.12 Abstract of Priority: ‘THE GOLDEN THEME’ & THE SIX QUANTUM TRIADS - MusicScience & -The Cosmic Code Of The Harmonic Universe - ‘TRIADAL MATRIX OF QUANTUM INTERVALS’ and Musical Arrangements
- Copyright January 1, 2001 by Alwyn Scott Turner - Trademark and All Rights Reserved Pending Patents. A futher edification on the Method and System in the Abstract of Priority can be viewed at:
MusicScience.org - which is being blacklisted from review and currently not being indexed by Google. _________________________________________________________________________
'The Golden Theme - 3:4:5 Matrix'
The Harmonic Code:
Quarks & The Six Quantum Triads
Binary Matrix of 2 Sets of 3 Major & 3 Minor Triads Ascending & Descending in the Octave:
Major Set Minor Set
435 : 534 = Root Triad = 345:543
453 : 453 = 1st Inversion = 354:453
543 : 345 = 2nd Inversion = 534:435
Binary Matrix of 2 Sets of 3 Major & 3 Minor Chords Ascending & Descending Across the Octave:
435 : 534 = Root Triadal = 345 : 543
354 : 453 = 1st Inversion = 453 : 354
543 : 345 = 2nd Inversion = 534 : 435
There are only SIX (6) QUANTUM TRIADS in 'The Nuclear Code'.
‘THE GOLDEN THEME’
MusicScience & 'The Cosmic Code' Of The Harmonic Universe
The Natural Harmonic Order of (18) Quantum Triads Ascending & Descending Inside the Octave and Diatonic Scales of the Chromatic Spectrum - Expanding Across the Octaves in a Universal Quantum System
of Frequencies & Wavelengths based on the Decimal Expansion and Partition of the 12th Root of 2.
The Triadal Progression Of 'The Six Quantum Triads'
The full Octave Chord (13 Keys) begins the Triadal progression with the first three Triads in C' Scale, and proceeds through the natural set of Triads in each of the 7 Diatonic Scales. In serial
order, each Scale has a set of Quantum Triads, Ascending & Descending in a Binary Progression arranged in their natural decimal sequence across the Diatonic Scale - the series 18 Triads are played
in the same decimal order in each of the 12 Chromatic Keys. (The Sequence Order of 12 Chromatic Keys in which the Triads are arranged in their primary numerical order are rearranged into a new
hierarachy of functions by changing the decimal harmonic series by The Decimal Code expansion and place series of Integers derived from Prime Root 19.
Music: The Triadal Progression
Harmonic Matrix of (18) Quantum Triads of The (7) Diatonic Scales
Root - 1st Inversion - 2nd Inversion of the Ascending Triadal Progression
(1) 1.1 C’: Major >( 435 = CEGC ) + [ 354 = EGCE ] + ( 543 = GCEG )
(2) 1.2 C’: Major >( 453 = CEAC ) + [ 534 = EACE ] + ( 345 = ACEA )
(3) 1.3 C’: Major >( 543 = CFAC ) + [ 435 = FACF ] + ( 354 = ACFA )
(4) 2.1 D’: minor >( 345 = DFAD ) + [ 453 = FADF ] + ( 534 = ADFA )
(5) 2.2 D’: Major >( 543 = DGBD ) + [ 435 = GBDG ] + ( 354 = BDGB )
(6) 3.1 E’: minor >( 345 = EGBE ) + [ 453 = GBEG ] + ( 534 = BEGB )
(7) 3.2 E’: minor >( 354 = EGCE ) + [ 543 = GCEG ] + ( 435 = CEGC )
(8) 3.3 E’: minor >( 534 = EACE ) + [ 345 = ACEA ] + ( 453 = CEAC )
(9) 4.1 F’: Major >( 435 = FACF ) + [ 354 = ACFA ] + ( 543 = CFAC )
(10) 4.2. F’: Major >( 453 = FADF ) + [ 534 = ADFA ] + ( 345 = DFAD )
(11) 5.1 G’: Major >( 435 = GBDG ) + [ 354 = BDGB ] + ( 543 = DGBD )
(12) 5.2 G’: Major >( 453 = GBEG ) + [ 534 = BEGB ] + ( 345 = EGBE )
(13) 5.3 G’: Major >( 543 = GCEG ) + [ 435 = CEGC ] + ( 354 = EGCE )
(14) 6.1 A’: minor >( 345 = ACEA ) + [ 453 = CEAC ] + ( 534 = EACE )
(15) 6.2 A’: minor >( 354 = ACFA ) + [ 543 = CFAC ] + (435 = FACF )
(16) 6.3 A’: minor >( 534 = ADFA ) + [ 345 = DFAD ] + ( 453 = FADF )
(17) 7.1 B’: minor >( 354 = BDGB ) + [ 543 = DGBD ] + ( 435 = GBDG )
(18) 7.2 B’: minor >( 534 = BDGB ) + [ 345 = EGBE ] + ( 453 = GBEG )
'THE GOLDEN THEME' - And The Natural Order of the Harmonic Triadal Progression of Quantum Intervals in the Diatonic Scales: Beginning in C’ Scale - All 'Keys' in the Chromatic Spectrum follow this
Harmonic Numerical Sequence which Ascends from the Root & Descends from The Octave.
The Triadal Root, 1st & 2nd Inversions in Harmonic Progression Across the Octave - through the 7 Diatonic Scales and the 12 Keys of the Chromatic Scale:
The 18 Triads Ascending from the Root in each of the 12 Keys (12 x 18) = 216 Triads. Plus the 18 Triads Descending from the Octave in the same System: (2 x 216) = 432 Triads.
& THE SQUARE ROOT OF TWO
The Decimal Code for the Harmonic Octave and the Diatonic Scale.
The scientific origin of the Chromatic Scale was discovered by Sir Issac Newton in the 16th Century - by measuring the frets of the rainbow, he discovered the true harmonic rainbow structure of the
Chromatic Scale by discovering the decimal sequential order based on the 12th Root of 2 = 1.059 463 094 - which defines the 12 intervals of the Chromatic Scale and the 13 intervals in the musical
Octave. Newton defined the Diatonic Scale - but had problems with computing the decimal interger sequences and their expansions - deterred by the insoluable complexity of cracking the harmonic
Decimal Code - Newton turned his genius and his discovery of The Calculus on the physics of Music - onto the abstract principles of Universal Gravitation - however, he was also unable to compute the
numerical harmony of the Decimal Code to the harmonic functions of gravity...which in fact, Galileo numerically cracked. Thus, after Newton the diatonic scales were expanded by Bach, and the science
of Music remained a 'Theory' until now - ith the discovery of 'The Triadal Progression' and the 'Harmonic Code' and so has the numerical system discovered by Galileo - which has is now exspanded
into The Gravity Matrix from which the Prime Root Code is derived- all essential elements in MusicScience.
2.17.12: Abstract of Priority. MusicScience: THE QUANTUM INTERVALS & THE TRIADAL MATRIX - 'Harmoni Fusion & The Harmonic Code Of The Universe' - 'The Golden Theme', 'Variations On The Golden Theme',
and 'Improvisations On The Golden Theme': The Harmonic Method - Copyright & Tradmark with All Rights Reserved Pending Patents by Alwyn Scott Turner.
System of Quantum Intervals Essential to Nuclear Harmonic Fusion.
In MusicScience - An original Method based on the Natural System of Major and minor Sets of Quantum Triads - that Ascend and Descend within the Chromatic Octave, and the Triadal Array Across the set
of Octaves.
The Quantum Intervals Ascend & Descend - Inside The Octave & Across The Octaves:
C' Scale = THREE (3) Major Quantum Triads in a Natural Series - (435:453:543).
E' Scale = THREE (3) minor Quantum Triads in a Natural Series - (345:354:534).
The Root, 1st Inversion & 2nd Inversion of the Triadal Chord within the Octave, is repeated across the set of Octaves; that is: The Triad consists of a triplet: CEG chord, in which C is the Major
Root, E is the minor 1st Inversion, and G is the Major 2nd Inversion. Thus, C' is the Tonic Key for the Root; E' is the Tonic Key of the 1st Inversion; and G' is the Tonic Key of the 2nd Inversion.
The Root, 1st Inversion, 2nd Inversion, and Octave consist of (4) full Octave Chords each consisting of 13 notes = 52 Notes. The same Triadal pattern of Root, Inversions and Octave - Arrayed Acrosss
the Octaves, consists of (5) Octaves (Including the spaces between the Chordal set of Octaves) - for a sum total of 61 Intervals in the Octave Span.
The Frequencies and Wavelengths of the Hydrogen Atom are based on the Binary Base Pair of Prime Roots: 61 and 47....(61x61) x (47x47) x (4) = 32878756 - which is the first 8 integer sequences of the
16 digit Code. Also, (61 + 47 = 108; 60 + 48 = 108; 59 + 49 = 108; etc) is the Integer Series for Quantum Calculus on Harmonic Variations of the System Functions. Thus, it will be revealed, that the
'Periodic Table of Prime Roots' - are also naturally arranged into a precise System of Base Pairs.
(The Original Formula for the natural order of Base Pairs in the Binary Matrix of Prime Roots, and the Harmonic System of the DNA Molecule - has been Abstracted and Published for Priority, and will
be forthcoming here...)
The Unit Circle of The Octave consists of 13 Chromatic Keys, and 7 Diatonic Scales: C' Scale Ascends 12 Chromatic Keys from the Root = E' Scale Descends 12 Chromatic Keys from the Octave - A 'Full
Chromatic Chord' (CEGC) consists of (8) Keys in the Octave. A Binary System of Triadal Chords Ascend & Descend from the Root & Octave - that change the Binary Base Pair of Chromatic Keys - into two
(2) different Arrays. In MusicScience, 'The Golden Theme' Progression of Ascending Base Pairs = 4th from the Root is a 5th Descending from the Octave - and 5th Ascending from the Root is a 4th
Descending from the Octave.
The Octave is the Unit Circle. Thus, Binary Triplets in the Circle of 4th & 5ths Ascending & Descending from the Root & Octave - form a harmonic Chromatic quantum series of Binary Base Pairs of
Prime Roots (The Periodic Table of Prime Roots), Binary Base Pairs of Chromatic Keys, & Binary Base Pairs of Triple Triads:
Binary Pairs From Root: (1) C:G; (2) D:A; (3) E:B; (4) F#:Db; (5) Ab:Eb; Bb:F.
Binary Pairs from Octave: (1) C:F; (2) Bb:Eb; (3) Ab:Db; (4)F#:B; (5) E;A; (6) D:G.
(A 5th Degree Ascending is a 4th Degree Descending. There are 7 Degrees (Scales) in the Diatonic and 13 Keys in the Chromatic Octave.)
All (7) of The Diatonic Scales in the Ascending Triadal Progression total (18) Triads - The Descending order in Mirror Image Reverse (MIR) replicates the Series, as a Binary System of Base Pair
Triads Ascending & Descending from the Prime Root & the Octave.
SIX QUANTUM INTERVALS: 'THE TRIADAL MATRIX':
Triadal Matrix of Quantum Intervals of the same Tonic Root, 1st Inversion, and 2nd Inversion Inside the Octave Same as Across the Octaves.
Root: C' Scale Triad: 435 = Major
1st Inv: E' Scale Triad: 354 = minor
2nd Inv: G' Scale Triad: 543 = Major
The Triadal Triple Keys of C'Scale = 4:3:5 (CEG) Within the Octave Ascending from the Root - are Reversed in Mirror Image Descending the Diatonic From the C' Octave in the Triadal Triple Keys of E'
Scale in the Order 5:3:4 (CGE).
The Quantum Method of the Harmonic Spectrum is a Binary System of Triple Triads - Ascending & Descending between the Root and the Octave:
C' Scale is E' Scale in Reverse, and the 3 Major Triads in C' Scale are the Reverse Mirror Image of the 3 minor Triads in E' Scale Descending from the Octave.
Harmonic Rhythm. The Integer Series of the Major Triadal Chord (C'435) Reverses to (E'534) - changing the Rhythmic sequence (4:3:5 Major set of intervals) - into a new sequence (5:3:4 minor set of
rhythmic beats) in the measure. However, The Reversal of Polarity in numerical order does not change the Harmonic set of Keys in the Major Chord - but only Reverses the Series in the sequence of
their order.
The Harmonic & Rythym have the same numerical sets of Quantum Intervals - in MusicScience terms, the 12 beat measure is divided into three Major sets: (4:3:5 = 4 beats, 3 beats, 5 beats); the 12
Keys of the Harmonic measure is divided into the same Triadal sets: (4:3:5 = 4 notes, 3 notes, 4 notes) - the accent patterns means that the Harmonic & Rhythmic system are played together in unity,
and that changes in one system occur at the same time that the harmonic order changes, and in the same degree.
1. 435 Triad (CEG) Ascends the Octave; Reverse Mirror Image 534 Triad (CGE) Descends From the C' Octave as E' Scale.
( 1st Major Triad in C’ Scale: 435 Ascends the Diatonic from the Prime Root; Descends from C' Octave as the minor Triad: 534 in E’ Scale. )
2. 354:453 = 1st Inversion
( 1st Inversion in minor E’ Scale: E'354 >Ascends from the 2nd Key of E' (the Middle Note) In the CEG Triad; Descends from the E' Octave as Major Triad: 453 in C’ Scale. )
3. 543:345 = 2nd Inversion
( 2nd Inversion in G’ Scale: 543 >Ascending from the 3rd Key in the C' Triad; Descends from the G' Octave as minor Triad: 345 in A’ Scale.
C' = Root: Major; 1st Inversion: minor; 2nd Inversion: Major.
C' Scale = (3) Major Triads: 435:453:543 Ascends the Diatonic from Root.
E' Scale = (3) minor Triads: 534:354:345 Descends tje Diatonic from Octave.
Within the Octave, the Tonic Progression of Root, 1st Inversion, and 2nd Inversion is - Major, minor, and Major.
Across the Octaves, the Tonic Progression of Octave chords are also Major, minor, and Major. What occurs within the Octave determines what Occurs across the Octaves.
C' Scale Triads Ascend from the Root - E' Scale Triads descend in Reverse from the Octave. C' Scale is E' Scale in Reverse. E' Scale is C' Scale in Reverse. G' Scale is A' Scale in Reverse. The
other scales are: D' Ascends & Descends as D' (Palindromic); F' Scale is B' Scale in Reverse; A' Scale is G' Scale in Reverse; B' Scale is F' Scale in Reverse. C' Scale Ascends from the Root to the
Octave - and Descends from the Octave to the Root as E' Scale. All the Scales follow the same mirror-imge patterns within the Diatonic system.
The Chromatic Octave consists of 13 keys from Root to Octave: The Diatonic Scale consists of 12 keys Ascscending from the Root, and 12 Keys Descending from the Octave. Each measure consists of 12
Beats of Quantum Intervals of Rhythm, and 12 Quantum Harmonic Intervals. Both Harmonic & Rhythmic Systems Function in counterpoint to the same 435 Triad - by the division of the harmonic notes into
(3) sets of keys, and corresponding to dividing the time factor of the rhytmic beat into the same (3) sets of Quantum Intervals: The Triad of 4 notes, 3 notes and 5 notes - in the same series of a
12 beat measure as: 4 beats, 3 beats, and 5 beats - the natural melody of HarmonicRhythm.
The same Triadal Set of Quantum Intervals: The 1st Major Triad 435 (CEG) in C’ Scale has 3 sets of Quantum Intervals that apply to both armonic and rhytmnic systems: C' Scale Triad: 435 consists: C
= 4 notes, E = 3 notes, G = 5 notes - within the same measure. The Rhythm is divided into the same set of three sets of Quantum Intervals: 4 beats, 3 beats & 5 beats = Root, 1st Inversion, 2nd
Inversion. Both rhythm and harmony have the same intervals derived from the same Quantum Triad. Consequently, the changes in the Triadal Progression governs both harmony and rhythm. (Instead of
playing a 12 beat measure either as 3 sets of 4 beats, or 4 sets of three beats - the 435 Triad is played as a set of 4 beats, 3 beats and 5 beats. The changing within the Harmonic Calculus proceeds
throughout the Triadal Progression of the Chromatic Spectrum and Diatonic Scale.
All things that vibrate in the Harmonic Universe are based on the harmonics of the Prime Roots - 'The Code' - The true order and function of the Prime Roots has finally been discovered - for the
first time in history, science has access to the master keys for all quantum frequencies and wavelengths of the Energy Spectrum and Cosmic Functions. The Periodic Table of Prime Roots - is the key
to all functions of time and space, all frequencies and wavelengths, and all forms of matter. Harmonic Calculus of Prime Roots is a 'The Code' that will effect a harmonic Unification of Science -
and create new initiatives in a giant leap into Harmonic Fusion technology and the natural functions of energy - and extrapolation of the genetic code. It is time that everyone joined the Family of
Man to use technical advances of knowledge for peaceful means - instead of destructive secret technological advances used for weapons of war and terrorism against innocent people. America must throw
off the yoke of Zionist tyranny and return to the natural state of being Free...where Freedom of Privacy, Thought and Speech make creative thinking and vital knowledge possible, and share great
discoveries...and by individual achievements, creat peace and prosperity among nations, and inspire Mankind in the search for Cosmic Cosciousness - and the quest for greatness of spirit and wisdom -
that such knowledge commands...
HarmonicCalculus defines the ElectroMagnetic Wave Mechanics integral to the domain of functions on the Gravitational Axis - alternating harmonic patterns of the Triadal Progression which govern the
changing vectors of Angular Momentum and the harmonic sequence of alternating polarities of energy in three-dimensional coordinates. The integrals of Harmonic Calculus are functions that derive from
the Prime Root expansion of decimal partials that are harmonic and rhythmic sequences based on a number system - aparallel quadratic matrix of 'base pair' combinations.
The Triadal Series of quantum intervals expanded from the Periodic Matrix of Prime Roots govern all frequencies and wavelengths - including atomic electron orbitals based on the Octave and Triad
which governs the precise prime root frequencies of all atomic and molecular combinations based on the harmonics of the Octave . The Method of HarmonicEncryption is original - extraction of the
place series and the array of decimal numbers (including a method for cracking the Decimal Code of all irrational numbers) in the electronic system. HarmonicEncryption is a method of rendering
precise wave vectors based on The Prime Root Code frequencies - that systematically alternate sequential changes of polarity in the gravity field by within a numerical system based on The Prime Root
Code of quadratic harmonic frequencies - the natural order that extends from the frequencies of the Photon to molecular chemical combinations- a state of harmonic fusion in which the periodic
Triadal wave mechanics follow predictable numerical patterns. For instance, by the combination of 'base pair' frequencies in the Periodic Matrix of Prime Roots - based on the harmonic order of
decimal partials in the expansion of Prime Root 19 - the Harmonic Calculus for a MagicSineWave is generated that produces Harmonic Fusion in the molecular array of chemical combinations generated by
the 18th Atom in the Perioditic Table of Atomic Orbitals.
Prime Root 19 - Harmonic Decimal Expansion
{ 1.10.5.12.6.3.11.15.17:18.9.14.7.13.16.8.4.2 }
HarmonicCalculus: Base Pair Wave Vectors
for 3-Dimentional Changes in Polarity on the Triadal Axis in Nautical Terms:
(1. 1) = Sway & Roll : Heave & Pitch : Surge & Yaw / (18. 2) = Sway & Roll : Surge & Yaw : Heave & Pitch
(2. 10) = Sway & Roll : Surge & Yaw : Heave & Pitch / (17.4) = Heave & Pitch : Sway & Roll : Surge & Yaw
(3. 5) = Surge & Yaw : Sway & Roll : Heave & Pitch / (16.8) = Surge & Yaw : Heave & Pitch : Sway & Roll
(4. 12) = Sway & Roll : Surge & Yaw : Heave & Pitch / (15.16) = Surge & Yaw : Heave & Pitch : Sway & Roll
(5. 6) = Heave & Pitch : Sway & Roll : Surge & Yaw / (14.13) = Surge & Yaw : Sway & Roll : Heave & Pitch
(6. 3) = Surge & Yaw : Sway & Roll : Heave & Pitch / (13.7) = Heave & Pitch : Surge & Yaw : Sway & Roll
(7. 11) = Sway & Roll : Heave & Pitch : Surge & Yaw / (12.14) = Heave & Pitch : Sway & Roll : Surge & Yaw
(8. 15) = Heave & Pitch : Surge & Yaw : Sway & Roll / (11.9) = Sway & Roll : Heave & Pitch : Surge & Yaw
(9. 17) = Heave & Pitch : Surge & Yaw : Sway & Roll / (10.18) = Surge & Yaw : Heave & Pitch : Sway & Roll
12 QUANTUM INTERVALS & 7 DIATONIC DEGREES OF THE OCTAVE
MusicScience: The 7 Keys of the Diatonic Scales Ascend from the Root & Descend in Reverse Mirror Image Order descending down the Octave to the Root: From C' Root to C' Octave there are 8 Keys in the
Full Chromatic Chord (4 notes: CEGC) of the 13 Quantum Intervals inside the Octave.
It takes some analysis to understand - although the 435:CEGC Major Triad Ascends, Reverses in Mirror Image, and Descends from the Octave as a 534 E'Scale Triad - it retains the same notes in the
C'Scale - but changes the order to a minor series! Thus, keeping the same Major Harmonic Division of Quantum Intervals as it descends from the Octave. The minor Triadal numerical structure
descending from the Octave retains the same set of major harmonic keys in from the reverse order: The Major and Minor Scales Reverse - the but same notes are retained so that the sound of the full
chord remains Major. It is an important insight into the deeper structure of the Harmonic Code.
1. 2212221: 1222122 C' Scale Ascends from Root, E' Scale Descends from Octave
2. 2122212: 2122212 D' Scale Ascends from Root, D' Scale Descends from Octave
3. 1222122: 2212221 E' Scale Ascends from Root, C' Scale Descends from Octave
4. 2221221: 1221222 F' Scale Ascends from Root, B' Scale Descends from Octave
5. 2212212: 2122122 G' Scale Ascends from Root, A' Scale Descends from Octave
6. 2122122: 2212212 A' Scale Ascends from Root, G' Scale Descends from Octave
7. 1221222: 2221221 B' Scale Ascends from Root, F' Scale Descends from Octave
The Harmonic Octave consists of 13 Quantum Intervals.
The Chromatic Scale consists of 12 Quantum Intervals.
The Diatonic Scale consists of 7 Degrees Ascending from the Root and 7 Degrees Descending from the Octave. The ascending order of scales and triads reverse descending from the Octave, the harmonic
and rhythmic (HarmonicRhythm) structure within the measure function with the same Triadal Progression - that is, the 12 key measure is divided into three harmonic sections, and the 12 beat rhythm is
equally divided into three sections - The Triad within the scale, determines the triple counterpoint of both harmonic and rhythmic sections: Thus, the triad of the measure consists of Root, 1st
Inversion and 2nd Inversion within the Octave, and the same keys as Roots expand across the Octaves. The rhythm section has the same beat structure as the harmony. Simply, the same triple division
of the Triad - governs both elements. It should be noted that within the Octave - the first Triad in C' is Major, whereas across the Octave, the Root is Major, the 1st Inversion is minor and the 2nd
Inversion is Major. Expansion across the Octaves by other scales and their Triads follow the same procedure - in essence, what occurs within the Octave governs what occurs across the Octaves.
1. The 1st Degree Ascends from the Root - and Descends as the 3rd Degree from the Octave.
2. The 2nd Degree Ascending from the Root - is palindromic Descending from the Octave.
3. The 3rd Degree Ascending from the Root - and Descends as the 1st Degree from the Octave.
4. The 4th Degree Ascending from the Root - and Descends as the 7th Degree from the Octave.
5. The 5th Degree Ascending from the Root - and Descends as the 6th Degree from the Octave.
6. The 6th Degree Ascending from the Root - and Descends as the 5th Degree from the Octave.
7. The 7th Degree ascending from the root - and Descends as the 4th Degree from the Octave.
The Keys D,F & B each have two triads and are related in the Triadal Progression by the fact that D' Scale has two Triads: 345 and 543 - one minor & one Major in palindromic number; F' Scale has two
Major Triads: 435 & 453 - missing the 3rd 543 Major Triad; B' Scale has two minor Triads: 354 & 534 - missisng the 1st 345 minor Triad. There are 7 Keys in the Diatonic and there are 6 Quantum
Triads - the natural order of the numerical expansion within the harmonic structure is expanded and arranged into a functional system within the Chromatic Spectrum. It is interesting to note that
the 1st minor Triad in D' Scale is one interval from the C' Tonic...while the 2nd Triad in D' Scale, a Major 543 is Six Intervals from the Root. Also, the 345 Triad is a Mirror Image to the 543
Traid - also, a 345 Triad Ascending is a 543 Triad in the Descending order. This nuance of this is imperative in the playing the accent patterns of arranging and orchestrating 'Variations and
Improvisations On The Golden Theme'.
2.19.12: Abstract of Priority. MusicScience: QuantumGravity & The Periodic Table of Atomic Elements; 'The Harmonic Code Of The Universe' , 'The Golden Theme', 'Variations On The Golden Theme', and
'Improvisations On The Golden Theme'; The Method & All Systems - Copyright & Trademark with All Rights Reserved Pending Patents by Alwyn Scott Turner.
PRIME ROOTS OF ALL ATOMS & ELECTON ORBITS IN
THE 'PERIODIC TABLE OF ELEMENTS & MOLECULES'
The Harmonic Structure of the Periodic Table of Elements & Triadal Combinations of Molecules - The Quantum Chemistry of All Atomic Orbits based on The Octave & Triadal Matrix of The Prime Root Code
Each Atom is part of an Harmonic Matrix aligned with other atoms arranged in Triads & Octaves in accordance to the Prime Root Matrix - the number system that govens the molecular combinations of
Atoms. All the atomic mass in the Harmonic Universe... -
The Numerical Order of Electrons in Atomic Orbit is determined by the Prime Root of each Atom - based on the Octave and Triadal Combinations - all Atoms combine as Molecules in accordance to the
their Numerical Sequence in the The Prime Root Matrix. In effect, each Atom and Molecule has a 'harmonic quadrature set of Prime Roots' frequencies based on their Electronic configuration.
The periodicity of Electronic Orbits of all Atom, and their harmonic molecular combinations - are governed by the natural sequential combinations of The Prime Root Matrix which determine their
combined frequencies - just as The Prime Root Matrix is the Numerical Code that governs all combinations of 'base pairs' at the root all harmonic vibration including the DNA Molecule and its
harmonic 'base pair' integer sequences.
The Quantum Number System for the Harmonic Orbit of Electrons:
The Fundamental Binary Arrangement for each side of the Atomic Nucleus:
1. 2 x (1x1) = 1st Orbit - Electrons: 2 = 2 (DS= 2:2)
2. 2 x (2x2) = 2nd Orbit - Electrons: 8 = 10 (8:1)
3. 2 x (3x3) = 3rd Orbit - Electrons: 18 = 28 (9:1)
4. 2 x (4x4) = 4th Orbit - Electrons: 32 = 60 (5:6)
5. 2 x (5x5) = 5th Orbit - Electrons: 50 = 110. (5:2)
6. 2 x (6x6) = 6th Orbit - Electrons: 72 = 182 (9:2)
7. 2 x (7x7) = 7th Orbit - Electrons: 98 = 280 (8:1)
8. 2 x (8x8) = 8th Orbit - Electrons: 128 = 408 (2:3)
9. 2 x (9x9) = 9th Orbit - Electrons: 162 = 570 (9:3)
10. 2x (10x10) = 10th Orbit - Electrons: 200 = 770 (2:5)
11. 2 x (11x11) = 11th Orbit - Electrons: 242 = 1012 (8:4)
12. 2 x (12x12) = 12th Orbit - Electrons: 288 = 1300 (9:4)
13. Etc. Etc.
The 'Digit Sum Serial Number = 289.559.829 for the Decimal Code Matrix.
This actua 'Base Pair' arrangement of electrons in orbit is a System of parallel positive and negative electronic orbits around the nucleus is a binary system that doubles the arrangement of
electrons into triadal sets of Octaves for all the Atoms, including predetermining all undiscovered atoms - and their natural harmonic (octave and triadal) combinations based on the 'Prime Root
Matrix' frequencies and the inverse system of 'Base Pairs'.
2.28.12: Abstract of Priority. MusicScience: GyroTorus.com; QuantumTorus.com; HarmonicTorus.com; HarmonicQuantum.com; QuantumGravity & TheHarmonicCodeOfTheUniverse.com; TheMagicMatrix & The
Chromatic Scale - The Method & All Systems - Copyright & Trademark with All Rights Reserved Pending Patents by Alwyn Scott Turner.
The Harmonic Matrix: Based on the Decimal Expansion of Prime Root 13
'Base Pairs' of The Quantum Chromatic Spectrum:
1.& 12. = ( 1 : 8 )
2. & 11. = ( 10 : 6 )
3. & 10. = ( 9 : 11 )
4. & 9. = ( 12 : 5 )
5. & 8. = ( 3 : 7 )
6. & 7. = ( 4 : 2 )
'The Golden Theme' Series of Quantum Intervals in The Octave:
The Chromatic Quantum Spectrum of Base Pairs by 4th & 5th Degrees
1. & 12. = ( 1 : 6 )
2. & 11. = ( 8 : 11 )
3. & 10. = ( 3 : 4 )
4. & 9. = (10 : 9 )
5. & 8. = ( 5 : 2 )
6. & 7. = ( 12 : 7 )
The Harmonic DiHedral Decimal Expansion of Chromatic 'Base Pairs'
1. & 12. = ( 1 : 7 )
2. & 11 = ( 3 : 2 )
3. & 10 = ( 5 : 8 )
4. & 9. = ( 12 : 8 )
5. & 8. = ( 10 : 11 )
6. & 7. = ( 9 : 4 )
Abstract of Priority: MusicScience - THE HARMONIC CODE OF THE UNIVERSE.COM; - The Calculus: Harmonic Fusion and The Nuclear Code. (TheNuclearCode.com) - Variations On The Golden Theme: Published on
ARTslant.com. Summer Solstice, June 21, 2011 by MusicScience, Trademark & Copyright with All Rights Reserved Pending Patents by Alwyn Scott Turner.
Notice of Abstract Priority: March 21, 2012 - Vernal Equinox. MusicScience: The PRIME ROOT MATRIX and the PRIME ROOT MATRIX & THE PRIME ROOTS OF THE PERIODIC TABLE OF ELEMENTS & MOLECULES -
Published on this date establishing priority to the copyright, trademark and all rights reserved and pending patents.
The original discovery and invention of the natural order of the Prime Root Matrix - the core harmonic matrix number system of the Universe based on the periodic order of prime root integers in
their natural sequence that governs all numerical systems that function as mathematical codes in all natural geometrical forms and vibrational frequencies, micro and macro scale, in the unification
of all time, energy and matter. Everything that exists in the vast cosmos of time and motion in the Harmonic Universe vibrates and functions in accordance to the Natural periodic order of the
In effect, a numerical system and method for harmonic molecular fusion and genetic engineering.
MusicScience and TheGoldenTheme embodies these mathematical codes published in musical composition, original discoveries based on the Prime Root Periodic Table and the physics of The Harmonic Code
of the Universe.
This unique discovery of the Prime Root Periodic Table ends a search for periodic order of the prime roots for all frequencies since the beginning of science, and the gravity of this knowledge will
change the world by revealing the method by which Harmonic Fusion of natural elements in perpetual motion, unlimited access to natural energy to replace fossil fuels, and the alchemy of elements -
and the scientific molecular engineering of the genetic code - enabling humans to essentially live forever - to live in peace with Nature, in good health and prosperity, and illuminate the natural
path toward the illumination of cosmic consciousness and wisdom for all mankind.
From this historical moment into the future - we are on the edge of miraculous times.
Dedicated to Pythagoras, Galileo, Sir Issac Newton, and to Bertrand Russell - Noble Prize winner for literature, philosopher, mathematician and peace activist. Imprisoned for his anti-war writing,
the author of unilateral renunciation of atomic weapons. A mind of great clarity and knowledge, of ideas composed with intellectual wit and vivid metaphor, author of the Principia Mathematica
(symbolic logic), Analysis of Mind, Analysis of Matter, and History of Western Philosophy. Dedicated Anti-war spokesman (AND one of the first well-known advocates that the assassination of President
John F. Kennedy was the result of a political conspiracy) - which repressed his humanitarian viewpoint and quest for Freedom....the clarity and truth of his noble views prevail with dignity.
Recent Exhibits
Alwyn Scott Turner
participated in these exhibits:
Apr, 2010 Between the Bricks and the Blood: Transgressive Typologies
Steven Kasher Gallery
Jan, 2004 The American People Archive/New Orleans Portfolio/The French Quarter
The New Orleans Museum of Art
Exhibited with these artists
Alwyn Scott Turner
has Exhibited with these artists:
Hyers and Mebane Bernd and Hilla Becher
the IRT Corporation the Minneapolis Police Department
Mike Disfarmer Emory Douglas
Walker Evans Robert Mapplethorpe
Tetsu Okuhara Alexandra Penney
the Associated Press Stephen Shames
Andy Warhol
Exhibited at these venues
Alwyn Scott Turner
has Exhibited at these venues:
Steven Kasher Gallery The New Orleans Museum of Art
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