content stringlengths 86 994k | meta stringlengths 288 619 |
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Issaquah Algebra 2 Tutor
Find a Issaquah Algebra 2 Tutor
...I feel that this is a very important asset, as I will be able to relate to the student on a more personal level. I have a strong understanding of chemistry and neuroscience, as my undergraduate
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23 Subjects: including algebra 2, English, reading, chemistry
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11 Subjects: including algebra 2, physics, geometry, Chinese
...Organic Chemistry was covered in my high school. When I attended Washington State University, I was placed directly into O-Chem; where I earned A's in lecture and lab. Immediately following the
class, I was selected by the WSU Chem department to be an O-chem peer-tutor (a tutorial instructor). I hold a BS in Chem E (GPA 3.7) with minors in Mathematics and Material Science Engineering.
62 Subjects: including algebra 2, English, chemistry, reading
...It is my belief that students are capable of understanding their weaker subjects as long as it is explained in a manner most fitted for the student: whether that be in drawing, words, or
otherwise. Schedule: My current schedule (as of March 2014), is fairly competitive and any scheduled sessions should be made a week in advanced. Cancellations should be made 6 hours in advanced.
17 Subjects: including algebra 2, chemistry, calculus, physics
...I mean, who moves from Hawaii to eastern Washington? Well, I did, because despite how beautiful and easygoing life in the islands is, there just wasn't the right opportunities for me to pursue
the higher education I wanted. So I moved to Pullman where I studied at Washington State University and received my B.S in biotechnology and continued on for a master's in science.
14 Subjects: including algebra 2, writing, geometry, biology | {"url":"http://www.purplemath.com/Issaquah_Algebra_2_tutors.php","timestamp":"2014-04-18T13:59:36Z","content_type":null,"content_length":"24122","record_id":"<urn:uuid:db447889-d523-43e1-a456-60f3e7778cd2>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00423-ip-10-147-4-33.ec2.internal.warc.gz"} |
2349 -- Arctic Network
Arctic Network
Time Limit: 2000MS Memory Limit: 65536K
Total Submissions: 9086 Accepted: 3036
The Department of National Defence (DND) wishes to connect several northern outposts by a wireless network. Two different communication technologies are to be used in establishing the network: every
outpost will have a radio transceiver and some outposts will in addition have a satellite channel.
Any two outposts with a satellite channel can communicate via the satellite, regardless of their location. Otherwise, two outposts can communicate by radio only if the distance between them does not
exceed D, which depends of the power of the transceivers. Higher power yields higher D but costs more. Due to purchasing and maintenance considerations, the transceivers at the outposts must be
identical; that is, the value of D is the same for every pair of outposts.
Your job is to determine the minimum D required for the transceivers. There must be at least one communication path (direct or indirect) between every pair of outposts.
The first line of input contains N, the number of test cases. The first line of each test case contains 1 <= S <= 100, the number of satellite channels, and S < P <= 500, the number of outposts. P
lines follow, giving the (x,y) coordinates of each outpost in km (coordinates are integers between 0 and 10,000).
For each case, output should consist of a single line giving the minimum D required to connect the network. Output should be specified to 2 decimal points.
Sample Input
Sample Output | {"url":"http://poj.org/problem?id=2349","timestamp":"2014-04-18T10:36:05Z","content_type":null,"content_length":"6931","record_id":"<urn:uuid:9795c2b6-63b9-4b57-9bdf-d68d4e48dedc>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00033-ip-10-147-4-33.ec2.internal.warc.gz"} |
Bridgewater, MA Statistics Tutor
Find a Bridgewater, MA Statistics Tutor
...I first encountered Statistics in High School where I took the AP test and scored a 4 out of 5. In college I had many statistics based classes and really increased my knowledge. Since freshman
year of college I have majored in Economics and have done well in all of the courses I have taken, starting with macro and going until graduate level courses on econometrics and other fields.
18 Subjects: including statistics, chemistry, economics, biology
I have spent the last five years tutoring mathematics for the Academic Achievement Center at Bridgewater State University. My tutoring experience includes teaching topics in college algebra,
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I am a retired university math lecturer looking for students, who need experienced tutor. Relying on more than 30 years experience in teaching and tutoring, I strongly believe that my profile is a
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14 Subjects: including statistics, calculus, ACT Math, algebra 1 | {"url":"http://www.purplemath.com/Bridgewater_MA_statistics_tutors.php","timestamp":"2014-04-19T17:10:58Z","content_type":null,"content_length":"24462","record_id":"<urn:uuid:91e228b4-b7df-43f8-885d-f3c076f24f8d>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00489-ip-10-147-4-33.ec2.internal.warc.gz"} |
Step size adaptation in reproducing kernel Hilbert space
- In ICML , 2006
"... We apply Stochastic Meta-Descent (SMD), a stochastic gradient optimization method with gain vector adaptation, to the training of Conditional Random Fields (CRFs). On several large data sets,
the resulting optimizer converges to the same quality of solution over an order of magnitude faster than lim ..."
Cited by 95 (4 self)
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We apply Stochastic Meta-Descent (SMD), a stochastic gradient optimization method with gain vector adaptation, to the training of Conditional Random Fields (CRFs). On several large data sets, the
resulting optimizer converges to the same quality of solution over an order of magnitude faster than limited-memory BFGS, the leading method reported to date. We report results for both exact and
inexact inference techniques. 1.
"... We develop gain adaptation methods that improve convergence of the kernel Hebbian algorithm (KHA) for iterative kernel PCA (Kim et al., 2005). KHA has a scalar gain parameter which is either
held constant or decreased according to a predetermined annealing schedule, leading to slow convergence. We a ..."
Cited by 7 (0 self)
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We develop gain adaptation methods that improve convergence of the kernel Hebbian algorithm (KHA) for iterative kernel PCA (Kim et al., 2005). KHA has a scalar gain parameter which is either held
constant or decreased according to a predetermined annealing schedule, leading to slow convergence. We accelerate it by incorporating the reciprocal of the current estimated eigenvalues as part of a
gain vector. An additional normalization term then allows us to eliminate a tuning parameter in the annealing schedule. Finally we derive and apply stochastic meta-descent (SMD) gain vector
adaptation (Schraudolph, 1999, 2002) in reproducing kernel Hilbert space to further speed up convergence. Experimental results on kernel PCA and spectral clustering of USPS digits, motion capture and
image denoising, and image super-resolution tasks confirm that our methods converge substantially faster than conventional KHA. To demonstrate scalability, we perform kernel PCA on the entire MNIST
data set.
"... Abstract. In many applicative contexts in which textual documents are labelled with thematic categories, a distinction is made between the primary categories of a document, which represent the
topics that are central to it, and and its secondary categories, which represent topics that the document o ..."
Cited by 2 (1 self)
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Abstract. In many applicative contexts in which textual documents are labelled with thematic categories, a distinction is made between the primary categories of a document, which represent the topics
that are central to it, and and its secondary categories, which represent topics that the document only touches upon. We contend that this distinction, so far neglected in text categorization
research, is important and deserves to be explicitly tackled. The contribution of this paper is three-fold. First, we propose an evaluation measure for this preferential text categorization task,
whereby different kinds of misclassifications involving either primary or secondary categories have a different impact on effectiveness. Second, we establish several baseline results for this task on
a well-known benchmark for patent classification in which the distinction between primary and secondary categories is present; these results are obtained by reformulating the preferential text
categorization task in terms of well established classification problems, such as single and/or multi-label multiclass classification; state-of-the-art learning technology such as SVMs and
kernel-based methods are used. Third, we improve on these results by using a recently proposed class of algorithms explicitly devised for learning from training data expressed in
- Advances in Neural Information Processing Systems , 2007
"... We introduce two methods to improve convergence of the Kernel Hebbian Algorithm (KHA) for iterative kernel PCA. KHA has a scalar gain parameter which is either held constant or decreased as 1/t,
leading to slow convergence. Our KHA/et algorithm accelerates KHA by incorporating the reciprocal of the ..."
Cited by 1 (1 self)
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We introduce two methods to improve convergence of the Kernel Hebbian Algorithm (KHA) for iterative kernel PCA. KHA has a scalar gain parameter which is either held constant or decreased as 1/t,
leading to slow convergence. Our KHA/et algorithm accelerates KHA by incorporating the reciprocal of the current estimated eigenvalues as a gain vector. We then derive and apply Stochastic
Meta-Descent (SMD) to KHA/et; this further speeds convergence by performing gain adaptation in RKHS. Experimental results for kernel PCA and spectral clustering of USPS digits as well as motion
capture and image de-noising problems confirm that our methods converge substantially faster than conventional KHA. 1
"... Recent approaches to independent component analysis have used kernel independence measures to obtain very good performance in ICA, particularly in areas where classical methods experience
difficulty (for instance, sources with near-zero kurtosis). In this chapter, we compare two efficient extension ..."
Cited by 1 (1 self)
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Recent approaches to independent component analysis have used kernel independence measures to obtain very good performance in ICA, particularly in areas where classical methods experience difficulty
(for instance, sources with near-zero kurtosis). In this chapter, we compare two efficient extensions of these methods for large-scale problems: random subsampling of entries in the Gram matrices
used in defining the independence measures, and incomplete Cholesky decomposition of these matrices. We derive closed-form, efficiently computable approximations for the gradients of these measures,
and compare their performance on ICA using both artificial and music data. We show that kernel ICA can scale up to larger problems than yet attempted, and that incomplete Cholesky decomposition
performs better than random sampling.
"... A successful class of image denoising methods is based on Bayesian approaches working in wavelet representations. The performance of these methods improves when relations among the local
frequency coefficients are explicitly included. However, in these techniques, analytical estimates can be obtaine ..."
Cited by 1 (0 self)
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A successful class of image denoising methods is based on Bayesian approaches working in wavelet representations. The performance of these methods improves when relations among the local frequency
coefficients are explicitly included. However, in these techniques, analytical estimates can be obtained only for particular combinations of analytical models of signal and noise, thus precluding its
straightforward extension to deal with other arbitrary noise sources. In this paper, we propose an alternative non-explicit way to take into account the relations among natural image wavelet
coefficients for denoising: we use support vector regression (SVR) in the wavelet domain to enforce these relations in the estimated signal. Since relations among the coefficients are specific to the
signal, the regularization property of SVR is exploited to remove the noise, which does not share this feature. The specific signal relations are encoded in an anisotropic kernel obtained from mutual
information measures computed on a representative image database. In the proposed scheme, training considers minimizing the Kullback-Leibler divergence (KLD) between the estimated and actual
probability functions (or histograms) of signal and noise in order to enforce similarity up to the higher (computationally estimable) order. Due to its non-parametric
"... In many applicative contexts in which textual documents are labelled with thematic categories, a distinction is made between the primary categories of a document, which represent the topics that
are central to it, and its secondary categories, which represent topics that the document only touches up ..."
Add to MetaCart
In many applicative contexts in which textual documents are labelled with thematic categories, a distinction is made between the primary categories of a document, which represent the topics that are
central to it, and its secondary categories, which represent topics that the document only touches upon. We contend that this distinction, so far neglected in text categorization research, is
important and deserves to be explicitly tackled. The contribution of this paper is threefold. First, we propose an evaluation measure for this preferential text categorization task, whereby different
kinds of misclassifications involving either primary or secondary categories have a different impact on effectiveness. Second, we establish several baseline results for this task on a well-known
benchmark for patent classification in which the distinction between primary and secondary categories is present; these results are obtained by reformulating the preferential text categorization task
in terms of well established classification problems, such as single and/or multi-label multiclass classification; state-of-the-art learning technology such as SVMs and kernelbased methods are used.
Third, we improve on these results by using a recently proposed class of algorithms explicitly devised for learning from training data expressed in preferential
"... A successful class of image denoising methods is based on Bayesian approaches working in wavelet representations. The performance of these methods improves when relations among the local
frequency coefficients are explicitly included. However, in these techniques, analytical estimates can be obtaine ..."
Add to MetaCart
A successful class of image denoising methods is based on Bayesian approaches working in wavelet representations. The performance of these methods improves when relations among the local frequency
coefficients are explicitly included. However, in these techniques, analytical estimates can be obtained only for particular combinations of analytical models of signal and noise, thus precluding its
straightforward extension to deal with other arbitrary noise sources. In this paper, we propose an alternative non-explicit way to take into account the relations among natural image wavelet
coefficients for denoising: we use support vector regression (SVR) in the wavelet domain to enforce these relations in the estimated signal. Since relations among the coefficients are specific to the
signal, the regularization property of SVR is exploited to remove the noise, which does not share this feature. The specific signal relations are encoded in an anisotropic kernel obtained from mutual
information measures computed on a representative image database. In the proposed scheme, training considers minimizing the Kullback-Leibler divergence (KLD) between the estimated and actual
probability functions (or histograms) of signal and noise in order to enforce similarity up to the higher (computationally estimable) order. Due to its non-parametric
"... Periodic step-size adaptation in second-order gradient ..." | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=4106479","timestamp":"2014-04-19T23:47:20Z","content_type":null,"content_length":"37485","record_id":"<urn:uuid:e009e95b-3ba7-4bf5-904f-f688727a0ae5>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00548-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Surface Measure for Probabilistic Structural Computations
Jeanette P. Schmidt, Cheng C. Chen, Jonathan L. Cooper, and Russ B. Altman
Computing three-dimensional structures from sparse experimental constraints requires methods for combining heterogeneous sources of information, such as distances, angles, and measures of total
volume, shape, and surface. For some types of information, such as distances between atoms, numerous methods are available for computing structures that satisfy the provided constraints. It is more
difficult, however, to use information about the degree to which an atom is on the surface or buried as a useful constraint during structure computations. Surface measures have been used as accept/
reject criteria for previously computed structures, but this is not an efficient strategy. In this paper, we investigate the efficacy of applying a surface measure in the computation of molecular
structure, using a method of probabilistic least square computations which facilitates the introduction of multiple, noisy, heterogeneous data sources. For this purpose, we introduce a simple purely
geometrical measure of surface proximity called max/real conic view (MCV). MCV is efficiently computable and differentiable, and is hence well suited to driving a structural optimization method
based, in part, on surface data. As an initial validation, we show that MCV correlates well with known measures for total exposed surface area. We use this measure in our experiments to show that
information about surface proximity (derived from theory or experiment, for example) can be added to a set of distance measurements to increase significantly the quality of the computed structure. In
particular, when 30 to 50 percent of all possible shortrange distances are provided, the addition of surface information improves the quality of the computed structure (as measured by RMS fit) by as
much as 80 percent. Our results demonstrate that knowledge of which atoms are on the surface and which are buried can be used as a powerful constraint in estimating molecular structure.
This page is copyrighted by AAAI. All rights reserved. Your use of this site constitutes acceptance of all of AAAI's terms and conditions and privacy policy. | {"url":"http://aaai.org/Library/ISMB/1998/ismb98-018.php","timestamp":"2014-04-17T01:26:53Z","content_type":null,"content_length":"3904","record_id":"<urn:uuid:886aab6c-fba0-46da-b241-a6a606727bee>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00366-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: December 2004 [00267]
[Date Index] [Thread Index] [Author Index]
How to express ODE?
• To: mathgroup at smc.vnet.net
• Subject: [mg52817] How to express ODE?
• From: Adam Getchell <agetchell at physics.ucdavis.edu>
• Date: Mon, 13 Dec 2004 04:24:15 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com
Still (unfortunately) not getting another ODE to work, even using Dr.
Bob's example. I want to express this equation:
m = 10^16;
mp = 1.2211*10^19;
G = mp^(-2);
V[\[Phi]_] := (1/2)*m^2*\[Phi][t]
\[Rho][\[Phi]_] :=
Derivative[1][\[Phi]][t]^2/2 +
H := Sqrt[((8*Pi*G)/3)*
Inflaton :=
Derivative[2][\[Phi]][t] +
t] + D[V[\[Phi]], \[Phi]]
That is, phi double-dot of t + 3 H phi dot of t + V' of phi, where phi
is a function of t; V is a function of phi; H is a function of rho; rho
is a function of phi dot and V.
However, solving Inflaton for phi as a function of t:
f = \[Phi] /. First[NDSolve[
{Inflaton == 0,
t] == 0}, \[Phi],
{t, 0, 100*m}]]
Generates NDSolve::overdet and ReplaceAll::reps.
I *think* part of my problem is that dV/d(phi) is not getting properly
evaluated (it returns 0 when I look at it explicitly).
I thought that Derivative[1][\[Phi]][x] would give the above, but not
the way I'm doing it. | {"url":"http://forums.wolfram.com/mathgroup/archive/2004/Dec/msg00267.html","timestamp":"2014-04-19T17:21:31Z","content_type":null,"content_length":"35061","record_id":"<urn:uuid:4a274641-3c6d-493f-a6a1-7c9e296e74d7>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00613-ip-10-147-4-33.ec2.internal.warc.gz"} |
Making it stick.
Warren Harrison writes in the recent IEEE Software magazine about the dangers of end user programming. I will admit the dangers he describes are real.
However I am surprised, shocked really, at his prescription.
Rather than assuming the blame for himself and other computer professionals, rather than taking these dangers as impetus to improve the state of the industry, rather than recognizing the natural and
unstoppable demand for end user programming, Harrison would simply ban end user programming. "Dabblers" he calls them. "It’s simply unfathomable..." he writes, that these dabblers can program
Unfathomable indeed.
Warren, the roots of the problem do not lay with the "dabblers". The root cause is the poor state of tools and languages professionals like you and I have given them. Recognize that *we* are the ones
who can and must do better. End user programming is to be encouraged. We have the ability and responsibility to make them safe and productive, and we will have failed completely if we do not.
Yes, static languages can be Nice (read these comments), nicer than the popular static, rather, rigid languages.
A fun addition to Croquet would be something like SmoothTeddy. Maybe this is in there already. I can't wait for the download.
Looking at the Teddy pages... Teddy is now available on Sony's PS2 in Japan. Hope it makes it to the US. My kids have an EyeToy camera for the PS2. How great would an integration of Teddy and the
EyeToy be?
This blog hasn't been taken down in months, but I've made major changes to the underlying code base...
And none of the weird jumping through hoops that .NET requires for this sort of thing either. If you want zero downtime, you need a dynamic language. If you like restarting every time you need to
make a change - sure, go grab one of those mainstream systems.
...James Robertson, wherein he also quotes Keith Mantell...
With Smalltalk (at least with Visualworks) this was a breeze: added the code for a variable, accepted it and, hey presto, any inspector open on an instance got a new variable field set to nil.
Marc Stiegler in the e-lang mail list...
It is possible to visualize a capability-based world in which the human being remembers one and only one pass phrase, the phrase that unlocks all his capabilities on his personal machine. This
pass phrase would never be sent over the wire, never be shared with anybody. It would be used solely to to enable the human to authenticate himself to his computer. From there on, ACLs need not
apply. Hallelujah.
Alan Karp's site password is an interesting approximation of this idea.
Ralph Johnson writes more about Kyma...
Of course, part of the reason that they are able to come out with new products regularly in spite of the small size of their company is because they use Smalltalk...
The music is generated by an array of digital signal processors. The UI is written in Smalltalk and generates code that runs on the DSPs. So, the actual code that produces the music is produced
by Smalltalk, but isn't Smalltalk.
I think we're in a wonderful age of heresy, where it's getting easier to buck the establishment to get things done.
Webmin is a web-based interface for system administration for Unix [including Linux]. Using any browser that supports tables and forms (and Java for the File Manager module), you can setup user
accounts, Apache, DNS, file sharing and so on.
Webmin consists of a simple web server, and a number of CGI programs which directly update system files like /etc/inetd.conf and /etc/passwd. The web server and all CGI programs are written in
Perl version 5, and use no non-standard Perl modules.
What is Usermin?
Usermin is a web interface that can be used by any user on a Unix system to easily perform tasks like reading mail, setting up SSH or configuring mail forwarding. It can be thought of as a
simplified version of Webmin designed for use by normal users rather than system administrators.
Like Webmin, Usermin consists of a simple web server, and a number of CGI programs which directly update user config files like ~/.cshrc and ~/.forward. The web server and all CGI programs are
written in Perl version 5, and use only the non-standard Authen::PAM perl module.
An interesting Sony patent related to the cell processors Ted mentioned in his blog...
A hardware sandbox structure is provided for security against the corruption of data among the programs being processed by the processing units. The uniform software cells contain both data and
applications and are structured for processing by any of the processors of the network. Each software cell is uniquely identified on the network.
Now it is Java's turn to play with young programmers' minds sliding downhill on the nature of unnaturally boxed objects.
How can a student think clearly with dull tools when they could otherwise move on toward thinking about problem domains instead of unboxing memory allocations.
From "Structure and Interpretation of Classical Mechanics"...
The advantage of Scheme over other languages for the exposition of classical mechanics is that the manipulation of procedures that implement mathematical functions is easier and more natural in
Scheme than in other computer languages. Indeed, many theorems of mechanics are directly representable as Scheme programs.
The version of Scheme that we use in this book is MIT Scheme, augmented with a large library of software called Scmutils that extends the Scheme operators to be generic over a variety of
mathematical objects, including symbolic expressions. The Scmutils library also provides support for the numerical methods we use in this book, such as quadrature, integration of systems of
differential equations, and multivariate minimization.
The Scheme system, augmented with the Scmutils library, is free software. We provide this system, complete with documentation and source code, in a form that can be used with the GNU/Linux
operating system, on the Internet at http://www-mitpress.mit.edu/sicm.
This book presents classical mechanics from an unusual perspective. It focuses on understanding motion rather than deriving equations of motion. It weaves recent discoveries in nonlinear dynamics
throughout the presentation, rather than presenting them as an afterthought. It uses functional mathematical notation that allows precise understanding of fundamental properties of classical
mechanics. It uses computation to constrain notation, to capture and formalize methods, for simulation, and for symbolic analysis.
Jim points out that more flexible languages like Smalltalk and Lisp have issues with "identity" and "equality". He also makes the point though that the more useful notion of "equality" is typically
employed in Smalltalk via = (and by the way in Scheme via equal? versus eq?).
In the common rigid languages, unfortunately, the typical programmer uses identity everywhere, which is just an artifact of those languages' rigid foundation in squeezing an instruction or two out of
the code here and there. "Hey, let's use an identity test for equality because it's *fast* and it's meaningful a lot of the time." Only now, it's not.
I just wish in Smalltalk "identity" wasn't the easily confused == and in Scheme I wish eq? was a more distinguished symbol like identical? or even something better. Maybe in Smalltalk, #isIdentical:
anObject. But it's decades too late.
On the Fourth of July in Nebraska...
My family and I spent the 4th in Grand Island a few years ago in the middle of a cross-country trip. One of the better shows I've seen, IIRC.
"Monad is as programmatic as Perl or Ruby."
Looks fairly cumbersome to extend, and I'm not sure how supportive the interpreter is compared to, say, a typical Python interpreter, let alone a Lisp REPL or Smalltalk workspace.
Am I just whining? I hope not. Monad is a great big leap beyond the DOS prompt, I'll grant you that. But hopefully they are learning something from the industry folks who've been down this path
For example, I hope they're spending a week to bring in an expert to sit down at a Symbolics machine for an in-depth tour of the Genera command processor; likewise for a tour of a Smalltalk workspace
and tools. Hey, one week for both: 2 days with each technology and then 1 day to solicit feedback on Monad.
Would that be worth the effort? Certainly. They will be affecting developers and administrators for a good long while.
"Why else would we be in power, if we were not a better judge of international realities?" -- Britt Blaser, writing with irony at Escapable Logic
Tyler Close (of Waterken) writes in the e-lang email list...
The status quo of WWW security is unfortunately mired in the ACL model. However, if you ignore the various security add-ons of the WWW, and focus solely on the underlying model, you find an
amazing symmetry with capability-based security. In fact, if you push REST design principles to their logical conclusions, you arrive at some of the core principles of capability-based security.
Alan Kay...
Until real software engineering is developed, the next best practice is to develop with a dynamic system that has extreme late binding in all aspects. The first system to really do this in an
important way was LISP, and many of its great ideas were used in the invention of Squeak’s ancestor Smalltalk—the first dynamic completely object-oriented development and operating environment—in
the early 70s at Xerox PARC.
...this time there *was* an earth-shattering ka-boom. (WAV)
Just in from blowing things up with the kids and neighbors. Ah, freedom.
The burgers were not bad either.
Read things like this testimony to Smalltalk by a formerly rigid language aficianado.
Me being a statically typed language guy at the time thought the numbers were a bunch of BS. I actually learned Smalltalk to prove to myself the numbers were wrong. After learning Smalltalk and
using it in my spare time for over a year I came to the conclusion that I don't like Java, or C++.
Then read things like this effort to bring computer textbook customization into the 21st century.
And wind it up with a search on that site for a Smalltalk text to include in a custom textbook. But...
...a search on the site leads to zero Smalltalk textbooks. (Plenty of Java, C#, and Extreme Programming texts that *refer* to Smalltalk, though.)
The hosting suggestion I rec'd is looking good... linode.com will host a User Mode Linux for you. You simply get root access to your own machine.
(In response to this search for another ISP that supports Smalltalk.)
Kyma is written in VisualWorks Smalltalk and the lengthy article was a review talking about how all of the major electronic and soundtrack composers and sound designers use it and swear by it. | {"url":"http://patricklogan.blogspot.com/2004_07_04_archive.html","timestamp":"2014-04-20T03:54:07Z","content_type":null,"content_length":"199021","record_id":"<urn:uuid:d59dda5c-1e7e-49ca-b5ed-76a3b5cba5ed>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00421-ip-10-147-4-33.ec2.internal.warc.gz"} |
Angle of Deviation - 2 Mirrors
1. The problem statement, all variables and given/known data
Hi, my teacher asked me to conduct an experiment on the question "How does the angle of deviation depend upon the angle between two mirrors?" - We weren't given much else on the topic. I know how to
find the angle between two mirrors, but finding the angle of deviation of the light ray is what I'm lost at.
2. Relevant equations
Law of Reflection?
3. The attempt at a solution
I came up with my own solutions to try and find a method to calculate angle of deviation. Anyway, here is the diagram which I created to hopefully shed some light on the situation
As you can see, θi is the incoming light ray (just a name for it...) and θf is the final, or outgoing ray. Now, as seen, θf (the outgoing ray) has been reflected straight back in the same direction
as the original ray. Do I just say the angle of deviation is 360, or does it have something to do with the angles made at the mirrors (m1 and M2). θ1+θ2 = 180, so is that the angle of deviation?
Now in the image below (which I again made, myself), if you only adjust one of the mirrors (I changed M2), you get a different story. Now, as seen the outgoing light ray continues on in the same
direction as the incoming light ray. Is the Angle of Deviation Zero degrees here? Or do I add up the angles between M1 and M2? Which gives me 180? But how can the Angle of Deviation be 180 when the
light ray simply keeps going on in the same direction?
I'm lost as to how to find the Angle of Deviation of a light ray after reflection of 2 mirrors...
Please help. Thanks !! | {"url":"http://www.physicsforums.com/showthread.php?t=403292","timestamp":"2014-04-16T22:04:44Z","content_type":null,"content_length":"38133","record_id":"<urn:uuid:9b8345bd-de7b-4415-aa2c-4a03f8f8c3f4>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00464-ip-10-147-4-33.ec2.internal.warc.gz"} |
Prove that TU = T0
November 5th 2009, 10:04 AM #1
Prove that TU = T0
I would like help in:
Assume T and U are in L(V,V ) and that V = N(T)+N(U). Prove that TU = T0 . (T0 is the zero linear transformation)
Thank you
Let's see if I succeed in decoding the above: T,U are linear operators and N(T), N(U) are the corresponding null spaces, or kernel, of these operators.
So, if $V = N(T)+N(U)$ then $TU=T_0$.
Assuming I guessed correctly your symbols, the claim is false: as example take $T,\,U\in L(\mathbb{R}^2,\mathbb{R}^2)$ , $T\left(\begin{array}{c}x\\y\end{array}\right)=\lef t(\begin{array}{c}x-y\
\x-y\end{array}\right)\,,\,\,U\left(\begin{array}{c}x \\y\end{array}\right)=\left(\begin{array}{c}0\\y\e nd{array}\right)$
It's easy to check that $\left(\begin{array}{c}a\\b\end{array}\right)=\left (\begin{array}{c}b\\b\end{array}\right)+\left(\beg in{array}{c}a-b\\0\end{array}\right)\in\,N(T)+N(U)$ , but $TUeq T_0$
, as you can easily check.
If by the above symbols you meant something else then the above is worthless (perhaps the above's worthless EVEN if I guessed correctly your symbols).
Last edited by tonio; November 5th 2009 at 11:20 AM.
Yes, you guessed write. And this is the question that I have to prove that TU=T_0.
I wonder if Prof. quotation is to prove TU not equal T_0. maybe typo
Thank you very much
November 5th 2009, 10:44 AM #2
Oct 2009
November 5th 2009, 11:11 AM #3 | {"url":"http://mathhelpforum.com/advanced-algebra/112614-prove-tu-t0.html","timestamp":"2014-04-19T20:50:03Z","content_type":null,"content_length":"44243","record_id":"<urn:uuid:b2de48f7-5dec-4624-9ea7-c512bff8e8b8>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00251-ip-10-147-4-33.ec2.internal.warc.gz"} |
[FOM] 275:Kruskal Theorem/Impredicativity
Harvey Friedman friedman at math.ohio-state.edu
Sun Apr 2 12:16:40 EDT 2006
1. Kruskal's theorem.
2. Minimal bad sequences.
3. Labeled Kruskal's theorem.
4. Open questions.
This is a summary of lots of aspects of KT and LKT, which clarifies some
predicativity issues in one place. Everything is pretty much already under
reasonable control in the literature, [1], [2], [3]. However, an old result
of mine cited in section 3, is probably NOT well known, and probably NOT in
the literature (?).
Also see
[4] S. Simpson, Nonprovability of certain combinatorial properties of finite
trees, 1985, 87-115, in: Harrington, Morley, Scedrov, Simpson (eds.), Harvey
Friedman's Research on the Foundations of Mathematics, Studies in Logic and
the Foundations of Mathematics, North-Holland, 1985.
[5] H. Friedman, Internal finite tree embeddings, Reflections on the
Foundations of Mathematics: Essays in honor of Solomon Feferman, ed. Sieg,
Sommer, Talcott, Lecture Notes in Logic, volume 15, 62-93, 2002, ASL.
[6] H. Friedman, N. Robertson and P. Seymour), The Metamathematics of the
Graph Minor Theorem, Logic and Combinatorics, ed. S. Simpson,
AMS Contemporary Mathematics Series, vol. 65, 1987, 229-261.
1. KRUSKAL'S THEOREM.
Let let us review the situation with the most commonly quoted form of
Kruskal's Theorem.
[1] Joseph B. Kruskal. Well-quasi-ordering, the Tree Theorem, and Vazsonyi's
conjecture. Transactions of the American Mathematical Society , 95:210--225,
May 1960.
KRUSKAL THEOREM. Let T1,T2,... be finite trees. There exist i < j such that
Ti is inf preserving embeddable into Tj.
There is a well known sharper form of this, also in [1]. A structured tree
is a tree (always in the sense of a poset, not a graph), where the children
of every parent are given a linear ordering.
KRUSKAL THEOREM (with structure). Let T1,T2,... be finite structured trees.
There exist i < j such that Ti is inf and structure preserving embeddable
into Tj.
There is also an important weakening with bounded valence.
KRUSKAL THEROEM (bounded valence). Fix k >= 1. Let T1,T2,... be finite trees
of valence at most k. There exist i < j such that Ti is inf preserving
embeddable into Tj.
KRUSKAL THEOREM (bounded valence, with structure). Fix k >= 1. Let T1,T2,...
be finite structured trees of valence at most k. There exist i < j such that
Ti is inf and structure preserving embeddable into Tj.
THEOREM 1.1. KT, KT (with structure), KT (bounded valence), KT (bounded
valence, with structure), are all provably equivalent to
"theta_capitalomega_0(0) is well ordered" over RCA_0. For each fixed k,
these theorems for valence k are provably equivalent over RCA_0 to the well
ordering of some common specific proper initial segment of
COROLLARY 1.2. KT, KT (with structure), KT (bounded valence), KT (bounded
valence, with structure), are all provably equivalent, over RCA_0, to: Pi11
reflection over ACA_0 + Pi12-BI. For each fixed k, these theorems are
provable in ACA_0 + Pi12-BI.
For a treatment of ACA_0 + Pi12-BI see
[2] M. Rathjen, A. Weiermann, Proof-theoretic investigations on Kruskal's
Theorem, Annals of Pure and Applied Logic, 60, 1993.
In light of Theorem 1.2, we say that there is no predicative proof of KT,
under the usual classical interpretation of predicativity by
2. MINIMAL BAD SEQUENCES.
The FIRST, and by far the EASIEST, and most WELL KNOWN proof of KT, is
impredicative on ANY proposed analysis of predicativity. It uses a blatantly
impredicative definition. In fact, the usual proof uses a technique used
throughout this area of combinatorices called
the minimal bad sequence lemma.
MINIMAL BAD SEQUENCE LEMMA. Let Q be a quasi order with domain x1x2,...,
where the x's are distinct. Assume Q is not wqo. Then Q has a minimal bad
sequence. I.e., there exists y1,y2,..., drawn from the x's, such that
1. For no i < j is yi <=Q yj.
2. Suppose z1,z2,..., is a different sequence drawn from the x's, with
property i. There exists i such that yi occurs earlier among the x's, and
for all j < i, yi = zi.
THEOREM 2.1. The minimal bad sequence lemma is provably equivalent to
Pi11-CA0 over RCA0.
[3] A. Marcone, On the logical strength of Nash-William's theorem on
transfinite sequences, 1996, 327-351, in: W. Hodges, M. Hyland, C.
Steinhorn, J. Truss (eds.), Logic: From Foundations to Applications, Keele
1993, Oxford Science Publications, Oxford University Press, 1996.
ALTHOUGH I have seen (and worked with) proofs of Kruskal's theorem WITHOUT
using the minimal bad sequence lemma, I have NEVER seen a proof of Kruskal's
theorem WIHTOUT proving the labeled Kruskal Theorem, perhaps with bounded
We know that we CAN prove Kruskal's theorem WITHOUT resorting to either the
minimal bad sequence lemma AND WITHOUT resorting to any form of labeled
Kruskal's theorem. We know this by PROOF THEORY.
KRUSKAL'S original proof resorts to BOTH the minimal bad sequence lemma AND
labeled Kruskal's theorem.
But I have not seen it done directly. I BELIEVE that it could be done
directly, in an intelligible way, BUT not without additional work, going FAR
BEYOND what J.B. Kruskal did.
We have already seen that if we use the minimal bad sequence lemma, then we
have already done something that is IMPREDICATIVE under any interpretation
of predicativity put forward.
NOW we see what happens with the labeled Kruskal theorem.
3. LABELED KRUSKAL'S THEOREM.
We now discuss the Labeled Kruskal Theorem. It was proved by J.B. Kruskal in
the course of proving the normally quoted Kruskal Theorem.
NOTE: The Labeled Kruskal Theorem is NOT my EKT = Extended Kruskal Theorem.
This is a theorem from J.B. Kruskal, proved as he proved the (ordinary)
Kruskal Theorem.
In the course of proving KT, Kruskal, in [1], proved a sharper form of KT -
with labels from a wqo.
LABELED KRUSKAL THEOREM. Let T1,T2,... be finite trees with the vertices
labeled from a wqo. There exist i < j such that Ti is inf preserving and
label dominating embeddable into Tj.
LABELED KRUSKAL THEOREM (bounded valence). Fix k >= 1. Let T1,T2,... be
finite trees of valence at most k, with the vertices labeled from a wqo.
There exist i < j such that Ti is inf preserving and label dominating
embeddable into Tj.
THEOREM 3.1. RCA_0 + LKT proves theta_capitalomega_0(1) is well ordered. In
fact, we only need LKT with two discrete labels for this claim.
Thus LKT is stronger than KT, proof theoretically.
THEOREM 3.2. For each fixed k, LKT (bounded valence) is provable in ACA_0 +
Note that the labeled Kruskal theorem, with or without bounded valence, is a
Pi12 sentence. HOW does it fit into predicativity?
In the 1970's, we proved the following (unpublished).
THEOREM 3.3. Consider the statement "if X is a well ordering then kappa^X is
a well ordering". Here kappa^X is constructed according to Feferman's
original predicative proof theory, where Gamma_0 is the limit of kappa^0,
kappa^kappa^0,... . This statement is provably equivalent to ATR_0 over
See, things match nicely in terms of quantifier complexity, since the
statement in quotes is Pi12 and ATR_0 is also Pi12.
THEOREM 3.4. LKT, even for valence 2, implies ATR_0.
So clearly LKT, even for valence 2, is not provable predicatively, under any
interpretation put forward for predicativity.
4. Open Questions.
In [[2], the analysis did not go beyond KT and LKT for bounded valence. And
for LKT, and LKT (bounded valence), there was no provable equivalence with
standard formal systems. Extend the analysis to KT with various explicit
labels (e.g., finitely many labels, or labels from omega), and to LKT, LKT
(bounded valence).
I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 275th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-249 can be found at
http://www.cs.nyu.edu/pipermail/fom/2005-June/008999.html in the FOM
archives, 6/15/05, 9:18PM. NOTE: The title of #269 has been corrected from
the original.
250. Extreme Cardinals/Pi01 7/31/05 8:34PM
251. Embedding Axioms 8/1/05 10:40AM
252. Pi01 Revisited 10/25/05 10:35PM
253. Pi01 Progress 10/26/05 6:32AM
254. Pi01 Progress/more 11/10/05 4:37AM
255. Controlling Pi01 11/12 5:10PM
256. NAME:finite inclusion theory 11/21/05 2:34AM
257. FIT/more 11/22/05 5:34AM
258. Pi01/Simplification/Restatement 11/27/05 2:12AM
259. Pi01 pointer 11/30/05 10:36AM
260. Pi01/simplification 12/3/05 3:11PM
261. Pi01/nicer 12/5/05 2:26AM
262. Correction/Restatement 12/9/05 10:13AM
263. Pi01/digraphs 1 1/13/06 1:11AM
264. Pi01/digraphs 2 1/27/06 11:34AM
265. Pi01/digraphs 2/more 1/28/06 2:46PM
266. Pi01/digraphs/unifying 2/4/06 5:27AM
267. Pi01/digraphs/progress 2/8/06 2:44AM
268. Finite to Infinite 1 2/22/06 9:01AM
269. Pi01,Pi00/digraphs 2/25/06 3:09AM
270. Finite to Infinite/Restatement 2/25/06 8:25PM
271. Clarification of Smith Article 3/22/06 5:58PM
272. Sigma01/optimal 3/24/06 1:45PM
273: Sigma01/optimal/size 3/28/06 12:57PM
274: Subcubic Graph Numbers 4/1/06 11:23AM
Harvey Friedman
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Hello and welcome to my webpage!
I am Nishanth Gudapati, a mathematics PhD student at the Albert Einstein Institute (AEI) at Golm within the program of the International Max-Planck Research School (IMPRS).
Research Summary
The general area of my research is geometric analysis, in particular the mathematical study of hyperbolic equations arising in Einstein's equations of general relativity. My PhD project is on global
regularity of wave maps. Wave maps are maps from a Lorentzian manifold to a Riemannian manifold which are critical points of a Lagrangian which is a natural geometrical generalization of the free
wave Lagrangian. I work on a conjecture that associates the formation of blow-up of a wave map to the existence of a nontrivial solution of the corresponding static equation i.e the harmonic maps
equation - thereby establishing a blow up criterion for the Cauchy problem of wave maps. Such a result has been established for flat spacetime by Struwe for the equivariant and spherically symmetric
case and, Tataru and Sterbenz for the general case. In my PhD project I attempt to extend this result to wave maps on dynamical curved backgrounds. My PhD advisors are Lars Andersson and Gerhard
In addition, even though I don't claim by any means to be an expert, I keenly inform myself about the basic ideas and developments in the following topics: regularity of linear and nonlinear wave
equations on various backgrounds, formation of black holes, trapped surfaces, stability of black holes, general relativistic interpretation of gravitational collapse, Penrose inequalities,
Hamiltonians in general relativity and (a bit far fetched topic of ) propagation of electromagnetic waves through different nanophotonic structures. | {"url":"http://www.aei.mpg.de/~nishanth/","timestamp":"2014-04-16T13:17:46Z","content_type":null,"content_length":"2540","record_id":"<urn:uuid:4fa497ce-4501-4c6d-8352-968a9a7e8ee7>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00659-ip-10-147-4-33.ec2.internal.warc.gz"} |
A formulae-as-type notion of control, POPL
, 1998
"... This paper proposes a simple computational interpretation of Parigot's -calculus. The -calculus is an extension of the typed -calculus which corresponds via the CurryHoward correspondence to
classical logic. Whereas other work has given computational interpretations by translating the -calculus int ..."
Cited by 9 (0 self)
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This paper proposes a simple computational interpretation of Parigot's -calculus. The -calculus is an extension of the typed -calculus which corresponds via the CurryHoward correspondence to
classical logic. Whereas other work has given computational interpretations by translating the -calculus into other calculi, I wish to propose here that the -calculus itself has a simple
computational interpretation: it is a typed - calculus which is able to save and restore the runtime environment. This interpretation is best given as a single-step semantics which, in particular,
leads to a relatively simple, but powerful, operational theory.
"... This paper proposes a simple computational interpretation of Parigot's -calculus. The -calculus is an extension of the typed -calculus which corresponds via the CurryHoward correspondence to
classical logic. Whereas other work has given computational interpretations by translating the -calculus int ..."
Add to MetaCart
This paper proposes a simple computational interpretation of Parigot's -calculus. The -calculus is an extension of the typed -calculus which corresponds via the CurryHoward correspondence to
classical logic. Whereas other work has given computational interpretations by translating the -calculus into other calculi, I wish to propose here that the -calculus itself has a simple
computational interpretation: it is a typed - calculus which is able to save and restore the runtime environment. This interpretation is best given as a single-step semantics which, in particular,
leads to a relatively simple, but powerful, operational theory. This is an expanded version of a paper presented at the 23rd International Symposium on Mathematical Foundations of Computer Science.
August 24-- 28, 1998. Brno, Czech Republic. c fl G M B September 2, 1998 i 1 Introduction It is well-known that the typed -calculus can be viewed as a term assignment for natural deduction proofs in
intuitionistic logic ...
, 905
"... Arithmetical proofs of strong normalization results for symmetric λ-calculi ..."
"... Arithmetical proofs of strong normalization results for ..."
, 2009
"... présentée en vue de l’obtention du grade de Docteur de l’Université de Savoie Spécialité: Mathématiques et Informatique par ..."
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présentée en vue de l’obtention du grade de Docteur de l’Université de Savoie Spécialité: Mathématiques et Informatique par | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=2613916","timestamp":"2014-04-17T01:04:19Z","content_type":null,"content_length":"21104","record_id":"<urn:uuid:5a9712b4-d1c5-4645-81c5-c0216d4a7241>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00161-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mathematics Education
Volume I: Mathematics, mathematics education, and the curriculum
Part 1: Mathematics and Mathematics Education
(a) Histories of Mathematics
1. Luis Radford, ‘On Psychology, Historical Epistemology and the Teaching of Mathematics: Towards a Socio-Cultural History of Mathematics’, For the Learning of Mathematics, 1997, 17, 1, 26–33.
2. G. G. Joseph, ‘Different Ways of Knowing: Contrasting Styles of Argument in Indian and Greek Mathematical Traditions’, in P. Ernest (ed.), Mathematics, Education and Philosophy: An International
Perspective (Falmer Press, 1994), pp. 194–204.
3. N. Kleiner and N. Movshovitz-Hadar, ‘The Role of Paradoxes in the Evolution of Mathematics’, The American Mathematical Monthly, 1994, 101, 10, 963–74.
(b) Conceptions of Mathematics from an Educational Standpoint
4. Tommy Dreyfus and Theodore Eisenberg, ‘On the Aesthetics of Mathematical Thought’, For the Learning of Mathematics, 1986, 6, 1.
5. Efraim Fischbein, ‘Intuition and Proof’, For the Learning of Mathematics, 1982, 3, 2.
6. S. MacLane, ‘Mathematical Models: A Sketch for the Philosophy of Mathematics’, American Mathematical Monthly, 1981, 88, 7.
7. S. Restivo, ‘The Social Life of Mathematics’, in S. Restivo, J. P. Van Bendegem, and R. Fischer (eds.), Math Worlds: Philosophical and Social Studies of Mathematics and Mathematics Education
(State University of New York Press, 1993).
(c) Culture, Mathematics, and Mathematics Education
8. Ubiratan D’Ambrosio, ‘Ethnomathematics and its Place in the History and Pedagogy of Mathematics’, For the Learning of Mathematics, 1985, 5, 1, 44–8.
9. B. Barton, ‘Making Sense of Ethnomathematics: Ethnomathematics is Making Sense’, ESM, 1996, 31, 210–33.
10. A. J. Bishop, ‘Western Mathematics: The Secret Weapon of Cultural Imperialism’, Race & Class, 1990, 32, 2, 51–65.
(d) Society, Technology, and Mathematics Education
11. Paulus Gerdes, ‘Conditions and Strategies for Emancipatory Mathematics Education in Underdeveloped Countries’, For the Learning of Mathematics, 1985, 5, 1.
12. C. Keitel, ‘Numeracy and Scientific and Technological Literacy’, in E. W. Jenkins (ed.), Innovations in Science and Technology Education, Vol. 6 (UNESCO, 1997), pp. 165–85.
13. W. Blum and M. Niss, ‘Applied Mathematical Problem Solving, Modelling, Applications, and Links to Other Subjects: State, Trends, and Issues in Mathematics Education’, Educational Studies in
Mathematics, 1991, 22, 37–68.
Part 2: Education and the Mathematics Curriculum
(e) Goals of Mathematics Education
14. R. B. Davis, ‘The Culture of Mathematics and the Culture of Schools’, Journal of Mathematical Behavior, 1989, 8, 143–60.
15. T. A. Romberg and J. J. Kaput, ‘Mathematics Worth Teaching, Mathematics Worth Understanding’, in E. Fennema and T. A. Romberg (eds.), Mathematics Classrooms that Promote Understanding (Lawrence
Erlbaum Associates, 1999), pp. 3–19.
16. P. Davis and R. Hersh, ‘The Ideal Mathematician’, The Mathematical Experience (Penguin, 1980), pp. 34–44.
(f) Mathematics Curricula in Schools
17. D. Robitaille and M. Dirks, ‘Models for the Mathematics Curriculum’, For the Learning of Mathematics, 1982, 2, 3, 3–21.
18. J. Confrey, ‘Conceptual Change Analysis: Implications for Mathematics and Curriculum’, Curriculum Inquiry, 1981, 11, 3, 243–57.
19. L. Streefland, ‘The Design of a Mathematics Course: A Theoretical Reflection’, Educational Studies in Mathematics, 1993, 25, 109–35.
(g) Mathematics Curricula at Tertiary and Vocational Levels
20. M. Harris, ‘Looking for the Maths in Work’, in Harris (ed.), Schools, Mathematics and Work (Falmer Press, 1991), pp. 132–44.
21. D. O. Tall, ‘Comments on the Difficulty and Validity of Various Approaches to the Calculus’, For the Learning of Mathematics, 1981, 2, 2, 16–21.
22. B. Cornu, ‘Limits’, in D. O.Tall (ed.), Advanced Mathematical Thinking (Kluwer, 1992), pp. 153–66.
(h) Assessment, Evaluation, and the Mathematics Curriculum
23. B. Cooper, ‘Authentic Testing in Mathematics? The Boundary Between Everyday and Mathematical Knowledge in National Curriculum Testing in English Schools’, Assessment in Education, 1994, 1,
24. J. Kilpatrick, ‘The Chain and the Arrow: From the History of Mathematics Assessment’, in M. Niss (ed.), Investigations in Assessment in Mathematics Education (Kluwer, 1992), pp. 31–46.
Volume II: Mathematics teachers and teaching
Part 1: Mathematics Teachers
(a) Mathematics Teachers’ Knowledge
25. D. L. Ball, S. T. Lubienski, and D. S. Mewborn, ‘Research on Teaching Mathematics: The Unsolved Problem of Teachers’ Mathematical Knowledge’, in V. Richardson (ed.), Handbook of Research on
Teaching (American Educational Research Association, 2001), pp. 433–56.
26. H. Freudenthal, ‘Should a Mathematics Teacher Know Something about the History of Mathematics?’, For the Learning of Mathematics, 1981, 2, 1, 30–3.
(b) Mathematics Teachers’ Beliefs, Attitudes, and Values
27. A. G. Thompson, ‘The Relationship of Teacher’s Conceptions of Mathematics Teaching to Instructional Practice’, Educational Studies in Mathematics, 1984, 15, 105–27.
28. C. Chin, Y.-C. Leu, and F.-L. Lin, ‘Pedagogical Values, Mathematics Teaching, and Teacher Education: Case Studies of Two Experienced Teachers’, in F.-L. Lin and Thomas J. Cooney (eds.), Making
Sense of Mathematics Teacher Education (Kluwer Academic, 2001). pp. 247–69.
(c) Pre-Service Mathematics Teacher Education
29. T. J. Cooney, B. E. Shealy, and B. Arvold, ‘Conceptualizing Belief Structures of Preservice Secondary Mathematics Teachers’, Journal for Research in Mathematics Education, 1998, 29, 306–33.
30. J. Hiebert, A. K. Morris, and G. Glass, ‘Learning to Learn to Teach: An "Experiment" Model for Teaching and Teacher Preparation in Mathematics’, Journal of Mathematics Teacher Education, 2003, 6,
(d) Mathematics Teachers’ Professional Development
31. D. Clarke, ‘Ten Key Principles from Research for the Professional Development of Mathematics Teachers’, in D. B. Aichele and A. F. Coxford (eds.), Professional Development for Teachers of
Mathematics (National Council of Teachers of Mathematics, 1994), pp. 37–48.
32. C. Laborde, ‘The Use of New Technologies as a Vehicle for Restructuring Teachers’ Mathematics’, in F.-L. Lin and T. J. Cooney, Making Sense of Mathematics Teacher Education (Kluwer Academic
Publishers, 2001), pp. 87–109.
Part 2: Teaching Mathematics
(e) Issues in Teaching Mathematical Topics
33. J. Boaler, ‘Learning from Teaching: Exploring the Relationship Between Reform Curriculum and Equity’, Journal for Research in Mathematics Education, 2002, 13, 4, 239–58.
34. Z. Markovits, B.-S. Eylon, and M. Bruckheimer, ‘Functions Today and Yesterday’, For the Learning of Mathematics, 1986, 6, 2, 18–24.
35. J. Gregg, ‘The Tensions and Contradictions of the School Mathematics Tradition’, Journal for Research in Mathematics Education, 1995, 26, 5, 442–66.
(f) Pedagogical Theories and Practices
36. P. Cobb et al., ‘Characteristics of Classroom Mathematics Traditions: An Interactional Analysis’, American Educational Research Journal, 1992, 28, 573–604.
37. S. Crespo, ‘Learning to Pose Mathematical Problems: Exploring Changes in Pre-Service Teachers’ Practices’, Educational Studies in Mathematics, 2003, 52, 243–70.
(g) Classroom Cultures and Interactions
38. H. Bauersfeld, ‘Hidden Dimensions in the Reality of a Mathematics Classroom’, Educational Studies in Mathematics, 1980, 11, 23–41.
39. I. M. Christiansen, ‘When Negotiation of Meaning is also Negotiation of Task: Analysis of the Communication in an Applied Mathematics High School Course’, Educational Studies in Mathematics,
1997, 34, 1, 1–25.
(h) Teaching and Assessing
40. D. J. Clarke, ‘The Interactive Monitoring of Children’s Learning of Mathematics’, For the Learning of Mathematics, 1987, 7, 1, 2–6.
41. A. G. Thompson and D. J. Briers, ‘Assessing Students’ Learning to Inform Teaching: The Message in the NCTM Evaluation Standards’, Arithmetic Teacher, 1989, 37, 4, 22–6.
42. P. J. Black and D. Wiliam, ‘Inside the Black Box: Raising Standards through Classroom Assessment’, Phi Delta Kappan, 1998, 80, 2, 139–48.
(i) Teachers as Researchers
43. S. Lerman, ‘The Role of Research in the Practice of Mathematics Education’, For the Learning of Mathematics, 1990, 10, 2, 25–8.
44. B. Jaworski, ‘Mathematics Teacher Research: Process, Practice and the Development of Teaching’, Journal of Mathematics Teacher Education, 1998, 1, 3–31.
45. B. Clarke, D. Clarke, and P. Sullivan, ‘The Mathematics Teacher and Curriculum Development’, in A. Bishop et al. (eds.), International Handbook of Mathematics Education (Kluwer Academic
Publishers, 1996), pp. 1207–34).
Volume III: Mathematics Learners and Learning
Part 1: Mathematics Learners
(a) School Learners
46. S. H. Erlwanger, ‘Benny’s Conception of Rules and Answers in IPI Mathematics’, Journal of Children’s Mathematical Behavior, 1973, 1, 2, 7–26.
47. N. Gorgorio, N. Planas, and X. Vilella, ‘The Cultural Conflict in the Mathematics Classroom: Overcoming its "Invisibility"’, in A. Ahmed, H. Williams, and J. M. Kraemer (eds.), Cultural Diversity
in Mathematics (Education) (Horwood Publishing, 2000), pp. 179–85.
(b) Adult Learners
48. D. Coben, ‘Mathematics or Common Sense? Researching "Invisible" Mathematics Through Adults’ Mathematics Life Histories’, in D. Coben, J. O’Donaghue, and G. E. FitzSimons (eds.), Perspectives on
Adults Learning Mathematics: Research and Practice (Kluwer Academic Publishers, 2000), pp. 53–66.
49. C. Hoyles, R. Noss, and S. Pozzi, ‘Proportional Reasoning in Nursing Practice’, Journal for Research in Mathematics Education, 2001, 32, 1, 4–27.
(c) Disadvantaged and Marginalized Learners
50. T. N. Carraher, D. W. Carraher, and A. D. Schliemann, ‘Mathematics in the Streets and in Schools’, British Journal of Developmental Psychology, 1985, 3, 21–9.
51. S. Zeleke, ‘Learning Disabilities in Mathematics: A Review of the Issues and Children’s Performance across Mathematical Texts’, Focus on Learning Problems in Mathematics, 2004, 26, 4, 1–14.
(d) Gifted Learners
52. K. Tirri, ‘How Finland Meets the Needs of Gifted and Talented Pupils’, High Ability Studies, 1997, 8, 2, 213–22.
53. D. Buerk, ‘An Experience with Some Able Women who Avoid Mathematics’, For the Learning of Mathematics, 1982, 3, 2, 19–23.
(e) Gender Issues
54. M. Walshaw, ‘A Foucauldian Gaze on Gender Research: What Do You Do When Confronted with the Tunnel at the End of the Light?’, Journal for Research in Mathematics Education, 2001, 32, 5, 471–92.
55. G. C. Leder and H. J. Forgasz, ‘Single-Sex Classes in a Co-Educational High School: Highlighting Parents’ Perspectives’, Mathematics Education Research Journal, 1997, 9, 3, 274–91.
(f) Cultural Issues
56. A. Chronaki, ‘Researching the School Mathematics Culture of "Others"’, in P. Valero and R. Zevenbergen (eds.), Researching the Socio-Political Dimensions of Mathematics Education: Issues of Power
in Theory and Methodology (Kluwer Academic Publishers), pp. 145–65.
57. G. De Abreu, ‘Understanding How Children Experience the Relationship Between Home and School Mathematics’, Mind, Culture and Activity, 1995, 2, 119–42.
Part 2: Learning Mathematics
(g) Issues in Learning Mathematical Topics
58. C. Kieran, ‘Concepts Associated with the Equality Symbol’, Educational Studies in Mathematics, 1981, 12, 317–26.
59. N. Movshovitz-Hadar, ‘The False Coin Problem, Mathematical Induction and Knowledge Fragility’, Journal of Mathematical Behaviour, 1993, 12, 253–68.
60. G. Vergnaud, ‘Multiplicative Conceptual Field: What and Why?’, in G. Harel and J. Confrey (eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics (SUNY Press, 1994),
pp. 41–59.
61. J. Adler, ‘A Language of Teaching Dilemmas: Unlocking the Complex Multilingual Secondary Mathematics Classroom’, For the Learning of Mathematics, 1998, 18, 1, 24–33.
(h) Theories of Learning Mathematics
62. S. Pirie and T. Kieren, ‘A Recursive Theory of Mathematical Understanding’, For the Learning of Mathematics, 1989, 9, 3, 7–11.
63. A. Sfard, ‘On Two Metaphors for Learning and the Dangers of Choosing Just One’, Educational Researcher, 1998, 27, 2, 4–13.
64. R. Skemp, ‘Relational and Instrumental Understanding’, Mathematics Teaching, 1976, 77, 20–6.
65. L. P. Steffe and T. E. Kieren, ‘Radical Constructivism and Mathematics Education’, Journal for Research in Mathematics Education, 1994, 25, 711–33.
(i) Language, Visualization, and Mathematics Learning
66. N. Presmeg, ‘Visualisation in High School Mathematics’, For the Learning of Mathematics, 1986, 6, 3, 42–6.
67. M. Setati et al., ‘Incomplete Journeys: Code-Switching and Other Language Practices in Mathematics, Science and English Language Classrooms in South Africa’, Language and Education, 2002, 16, 2,
(j) Beliefs and Affective Aspects of Learning Mathematics
68. M. Lampert, ‘When the Problem is Not the Question and the Solution is not the Answer: Mathematical Knowing and Teaching’, American Educational Research Journal, 2001, 27, 29–63.
69. G. A. Goldin, ‘Affect, Meta-Affect, and Mathematical Belief Structures’, in G. C. Leder, E. Pehkonen, and G. Törner (eds.), Beliefs: A Hidden Variable in Mathematics Education (Kluwer, 2002), pp.
70. D. B. McLeod, ‘Affective Issues in Mathematical Problem Solving: Some Theoretical Considerations’, Journal for Research in Mathematics Education, 1988, 19, 2, 134–41.
71. D. Moreira, ‘Facing Exclusion: The Student as Person’, in P. Gates and T. Cotton (eds.), First International Mathematics Education and Society Conference 6th–11th September (Nottingham
University, Center for the Study of Mathematics Education, 1988), pp. 253–61.
Volume IV: The Contexts of Mathematics Education
Part 1: Societal and Cultural Contexts
(a) Parental and Community Aspects
72. R. Merttens, ‘Teaching Not Learning: Listening to Parents and Empowering Students’, For the Learning of Mathematics, 1995, 15, 3, 2–9.
73. M. Civil, ‘Culture and Mathematics: A Community Approach’, Journal of Intercultural Studies, 2002, 23, 2, 133–48.
(b) Numeracies and Mathematics Education
74. R. Noss, ‘New Numeracies for a Technological Culture’, For the Learning of Mathematics, 1998, 18, 2, 2–12.
75. R. Zevenbergen, ‘Technologizing Numeracy: Intergenerational Differences in Working Mathematically in New Times’, Educational Studies in Mathematics, 2004, 56, 97–117.
(c) Technologies and Mathematics Education
76. S. Schuck and G. Foley, ‘Viewing Mathematics in New Ways: Can Electronic Learning Communities Assist?’, Mathematics Teacher Education and Development, 1999, 1, 1, 23–37.
77. W. Dörfler, ‘Computer Use and Views of the Mind’, in C. Keitel and K. Ruthven (eds.), Learning from Computers: Mathematics Education and Technology (Springer-Verlag, 1993), pp. 159–86.
(d) International Comparisons of Mathematics Achievement
78. C. Keitel and J. Kilpatrick, ‘The Rationality and Irrationality of International Comparative Studies’, in G. Kaiser, E. Luna, and I. Huntley (eds.), International Comparisons in Mathematics
Education (Falmer Press, 1999), pp. 241–56.
79. F. K. S. Leung, ‘The Mathematics Classroom in Beijing, Hong Kong and London’, Educational Studies in Mathematics, 1995, 29, 4, 297–325.
80. D. Zhang, S. Li, and R. Tang, ‘The "Two Basics": Mathematics Teaching and Learning in Mainland China’, in L. Fan et al., How Chinese Learn Mathematics (World Scientific, 2004), pp. 189–201.
Part 2: Research and Theoretical Contexts
(e) Developments in Research Approaches
81. H. Ginsburg, ‘The Clinical Interview in Psychological Research on Mathematical Thinking: Aims, Rationales, Techniques’, For the Learning of Mathematics, 1981, 1, 3, 4–11.
82. M. A. Eisenhart, ‘The Ethnographic Research Tradition and Mathematics Education Research’, Journal for Research in Mathematics Education, 1988, 19, 2, 99–114.
(f) Histories of Mathematics Education
83. G. M. A. Stanic, ‘The Growing Crisis in Mathematics Education in the Early Twentieth Century’, Journal for Research in Mathematics Education, 1986, 17, 3, 190–205.
84. A. G. Howson, ‘Seventy-Five Years of the International Commission on Mathematics Instruction’, Educational Studies in Mathematics, 1984, 15, 4, 75–93.
(g) Philosophies of Mathematics Education
85. P. Ernest, ‘The Dialogical Nature of Mathematics’, in Ernest (ed.), Mathematics, Education and Philosophy (Falmer Press, 1994), pp. 33–48.
86. E. Wittmann, ‘Mathematics Education as a "Design Science"’, Educational Studies in Mathematics, 1995, 29, 4, 355–74.
(h) Theories in Mathematics Education
87. P. Cobb, ‘Experiential, Cognitive and Anthropological Perspectives in Mathematics Education’, For the Learning of Mathematics, 1989, 9, 2, 32–42.
88. E. Fennema, H. Walberg, and C. Marrett, ‘Explaining Sex-Related Differences in Mathematics: Theoretical Models’, Educational Studies in Mathematics, 1985, 16, 3, 303–4.
89. G. Leder, ‘Sex-Related Differences in Mathematics: An Overview’, Educational Studies in Mathematics, 1985, 16, 3, 304–9.
90. E. Fennema and P. L. Peterson, ‘Autonomous Learning Behavior: A Possible Explanation of Sex-Related Differences in Mathematics’, Educational Studies in Mathematics, 1985, 16, 3, 309–11.
91. J. Eccles, ‘Model of Students’ Mathematics Enrollment Decisions’, Educational Studies in Mathematics, 1985, 16, 3, 311–14.
92. D. R. Maines, ‘Preliminary Notes on a Theory of Informal Barriers for Women in Mathematics’, Educational Studies in Mathematics, 1985, 16, 3, 314–20.
93. A. Sfard, ‘Reification as the Birth of Metaphor’, For the Learning of Mathematics, 1994, 14, 1, 44–55.
(i) International Cooperation in Mathematics Education Research
94. J. Cai, ‘Why do US and Chinese Students Think Differently in Mathematical Problem-Solving? Exploring the Impact of Early Algebra Learning and Teachers’ Beliefs’, Journal of Mathematical Behavior,
2004, 23, 2, 133–65.
95. B. Nebres, ‘International Benchmarking as a Way to Improve School Mathematics Achievement in the Era of Globalization’, in G. Kaiser, E. Luna, and I. Huntley (eds.), International Comparisons in
Mathematics Education (Falmer Press, 1999), pp. 200–12.
(j) Globalization, Post-Colonialism, and Critical Perspectives
96. B. Atweh and P. Clarkson, ‘Internationalisation and Globalization of Mathematics Education: Towards an Agenda for Research/Action’, in B. Atweh, H. Forgasz, and B. Nebres (eds.), Sociocultural
Research on Mathematics Education (Erlbaum, 2001), pp. 77–94.
97. R. Vithal and O. Skovsmose, ‘The End of Innocence: A Critique of "Ethnomathematics"’, Educational Studies in Mathematics, 1997, 34, 2, 131–57. | {"url":"http://www.psypress.com/books/details/9780415438742/","timestamp":"2014-04-18T19:09:44Z","content_type":null,"content_length":"54187","record_id":"<urn:uuid:86d22a79-9166-4274-a2c8-f6bf69a7c071>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00634-ip-10-147-4-33.ec2.internal.warc.gz"} |
Relativizable and non-relativizable theorems in the polynomial theory of algorithms
Results 1 - 10 of 25
- Journal of Computer and System Sciences , 1997
"... We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic
quantum class BQP. --- BQP is low for PP, i.e., PP BQP = PP. --- There exists a relativized world where P = ..."
Cited by 98 (3 self)
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We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum
class BQP. --- BQP is low for PP, i.e., PP BQP = PP. --- There exists a relativized world where P = BQP and the polynomial-time hierarchy is infinite. --- There exists a relativized world where BQP
does not have complete sets. --- There exists a relativized world where P = BQP but P 6= UP " coUP and one-way functions exist. This gives a relativized answer to an open question of Simon.
- IN PROCEEDINGS 10TH STRUCTURE IN COMPLEXITY THEORY , 1995
"... We investigate function classes h#Pi f which are defined as the closure of #P under the operation f and a set of known closure properties of #P, e.g. summation over an exponential range. First,
we examine operations f under which #P is closed (i.e., h#Pi f = #P) in every relativization. We obtain t ..."
Cited by 32 (9 self)
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We investigate function classes h#Pi f which are defined as the closure of #P under the operation f and a set of known closure properties of #P, e.g. summation over an exponential range. First, we
examine operations f under which #P is closed (i.e., h#Pi f = #P) in every relativization. We obtain the following complete characterization of these operations: #P is closed under f in every
relativization if and only if f is a finite sum of binomial coefficients over constants. Second, we characterize operations f with respect to their power in the counting context in the unrelativized
case. For closure properties f of #P, we have h#Pi f = #P. The other end of the range is marked by operations f for which h#Pi f corresponds to the counting hierarchy. We call these operations
counting hard and give general criteria for hardness. For many operations f we show that h#Pi f corresponds to some subclass C of the counting hierarchy. This will then imply that #P is closed under
f if and only if ...
- THEORETICAL COMPUTER SCIENCE , 1998
"... Going back to the seminal paper [FSS84] by Furst, Saxe, and Sipser, analogues between polynomial time classes and constant depth circuit classes have been considered in a number of papers.
Oracles separating polynomial time classes have been obtained by diagonalization making essential use of lower ..."
Cited by 12 (2 self)
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Going back to the seminal paper [FSS84] by Furst, Saxe, and Sipser, analogues between polynomial time classes and constant depth circuit classes have been considered in a number of papers. Oracles
separating polynomial time classes have been obtained by diagonalization making essential use of lower bounds for circuit classes. In this note we show how separating oracles can be obtained
uniformly from circuit lower bounds without the need of carrying out a particular diagonalization. Our technical tool is the leaf language approach to the definition of complexity classes.
- IN THE PROCEEDINGS OF THE 26TH INTERNATIONAL COLLOQIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING, LECTURE , 2001
"... ..."
- INFORMATION AND COMPUTATION , 2001
"... A polynomial time computable function h : whose range is a set L is called a proof system for L. In this setting, an h-proof for x 2 L is just a string w with h(w) = x. Cook and Reckhow de ned
this concept in [11] and in order to compare the relative strength of dierent proof systems for ..."
Cited by 10 (1 self)
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A polynomial time computable function h : whose range is a set L is called a proof system for L. In this setting, an h-proof for x 2 L is just a string w with h(w) = x. Cook and Reckhow de ned this
concept in [11] and in order to compare the relative strength of dierent proof systems for the set TAUT of tautologies in propositional logic, they considered the notion of p-simulation. Intuitively,
a proof system h psimulates h if any h-proof w can be translated in polynomial time into an h for h(w). Krajcek and Pudlak [18] considered the related notion of simulation between proof systems where
it is only required that for any h-proof w there exists an h whose size is polynomially bounded in the size of w.
, 1997
"... We consider a special kind of non-deterministic Turing machines. Cluster machines are distinguished by a neighbourhood relationship between accepting paths. Based on a formalization using
equivalence relations some subtle properties of these machines are proven. Moreover, by abstraction we gain the ..."
Cited by 9 (1 self)
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We consider a special kind of non-deterministic Turing machines. Cluster machines are distinguished by a neighbourhood relationship between accepting paths. Based on a formalization using equivalence
relations some subtle properties of these machines are proven. Moreover, by abstraction we gain the machine-independend concept of cluster sets which is the starting point to establish cluster
operators. Cluster operators map complexity classes of sets into complexity classes of functions where for the domain classes only cluster sets are allowed. For the counting operator c#\Delta and the
optimization operators cmax\Delta and cmin\Delta the structural relationships between images resulting from these operators on the polynomial-time hierarchy are investigated. Furthermore, we compare
cluster operators with the corresponding common operators #\Delta, max\Delta and min\Delta [Tod90b, HW97].
, 1998
"... For some fixed alphabet A with jAj 2, a language L ` A is in the class L 1=2 of the Straubing-Therien hierarchy if and only if it can be expressed as a finite union of languages A a 1 A a 2 A \
Delta \Delta \Delta A anA , where a i 2 A and n 0. The class L 1 is defined as the boo ..."
Cited by 9 (3 self)
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For some fixed alphabet A with jAj 2, a language L ` A is in the class L 1=2 of the Straubing-Therien hierarchy if and only if it can be expressed as a finite union of languages A a 1 A a 2 A \Delta
\Delta \Delta A anA , where a i 2 A and n 0. The class L 1 is defined as the boolean closure of L 1=2 . It is known that the classes L 1=2 and L 1 are decidable. We give a membership criterion for
the single classes of the boolean hierarchy over L 1=2 . From this criterion we can conclude that this boolean hierarchy is proper and that its classes are decidable. In finite model theory the
latter implies the decidability of the classes of the boolean hierarchy over the class \Sigma 1 of the FO[!]-logic. Moreover we prove a "forbidden-pattern" characterization of L 1 of the type: L 2 L
1 if and only if a certain pattern does not appear in the transition graph of a deterministic finite automaton accepting L. We discuss complexity theoretical consequences of our results. C...
- In Proceedings 25th Symposium on Mathematical Foundations of Computer Science , 2000
"... . The boolean hierarchy of k-partitions over NP for k 2 was introduced as a generalization of the well-known boolean hierarchy of sets. The classes of this hierarchy are exactly those classes of
NPpartitions which are generated by nite labeled lattices. We extend the boolean hierarchy of NP-partiti ..."
Cited by 7 (3 self)
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. The boolean hierarchy of k-partitions over NP for k 2 was introduced as a generalization of the well-known boolean hierarchy of sets. The classes of this hierarchy are exactly those classes of
NPpartitions which are generated by nite labeled lattices. We extend the boolean hierarchy of NP-partitions by considering partition classes which are generated by nite labeled posets. Since we
cannot prove it absolutely, we collect evidence for this extended boolean hierarchy to be strict. We give an exhaustive answer to the question of which relativizable inclusions between partition
classes can occur depending on the relation between their dening posets. The study of the extended boolean hierarchy is closely related to the issue of whether one can reduce the number of solutions
of NP problems. For nite cardinality types, assuming the extended boolean hierarchy of k-partitions over NP is strict, we give a complete characterization when such solution reductions are possible.
1 Introduct...
, 2000
"... We study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP
machines [OH93], we nonetheless show that it often is possible to reduce the number of solutions of NP functi ..."
Cited by 7 (4 self)
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We study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines
[OH93], we nonetheless show that it often is possible to reduce the number of solutions of NP functions. For finite cardinality types, we give a sufficient condition for such solution reduction. We
also give absolute and conditional necessary conditions for solution reduction, and in particular we show that in many cases solution reduction is impossible unless the polynomial hierarchy
- IN READERS OF THE NINTH EUROPEAN SUMMER SCHOOL IN LOGIC, LANGUAGE AND INFORMATION, CHAPTER CL7. CNRS AIX-EN-PROVENCE AND THE EUROPEAN ASSOCIATION FOR LOGIC, LANGUAGE AND INFORMATION , 1996
"... We address the question of the power of several logics with Lindstrom quantifiers over finite ordered structures. We will see that in the first-order case this nicely fits into the framework of
Barrington, Immerman, and Straubing's examination of constant depth circuit classes. In the second-order c ..."
Cited by 7 (2 self)
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We address the question of the power of several logics with Lindstrom quantifiers over finite ordered structures. We will see that in the first-order case this nicely fits into the framework of
Barrington, Immerman, and Straubing's examination of constant depth circuit classes. In the second-order case we get a strong relationship to succinct encodings of languages via circuits. Some of
these logics can be characterized as closures of succinct encodings under appropriate reducibilities, others by certain hierarchies of circuit classes. We will see that in a special case second-order
Lindstrom quantifiers can equivalently be expressed in first-order logic, while in the general case this equivalence seems unlikely. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1648309","timestamp":"2014-04-23T22:40:02Z","content_type":null,"content_length":"38128","record_id":"<urn:uuid:e58432b8-66a6-4caa-8844-dc84ddd8b331>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00300-ip-10-147-4-33.ec2.internal.warc.gz"} |
Brazilian Journal of Physics
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Print version ISSN 0103-9733
Braz. J. Phys. vol.38 no.3b São Paulo Sept. 2008
Different symmetry realizations in relativistic coupled bose systems at finite temperature and densities
R. L. S. Farias; R. O. Ramos; R. Vartuli
Departamento de Física Teórica, Universidade do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil
We revisited the calculation of the effective potential for self-interacting scalar field with U (1) charge at one loop approximation. We show that high charge densities can induce important changes
in the phase structure of the theory. A class of very interesting phenomena appear when we introduce finite density effects, e.g. symmetry nonrestoration, inverse symmetry breaking and anticipation
of the high temperature symmetry restoration. The extension of these calculations in the context of multi-scalar field theory is outlined, with the objectives of studying the effects of a finite
charge on the symmetry breaking phase transition, and to learn how these effects change the number of phases allowed by the system symmetries.
Keywords: Relativistic heavy-ion collisions; Inverse symmetry breaking and symmetry nonrestoration
Much work has been done exploring how field theories behave at finite temperature and densities [1-4]. The study of symmetry breaking (SB) and symmetry restoration (SR) mechanisms have proved to be
extremely useful in the analysis of phenomena related to phase transitions in almost all branches of physics [5]. For a large number of physical systems we have a good idea, both qualitatively as
quantitatively of how symmetries change as the temperature is changed. Usually, we expect that the larger is the temperature the larger is the symmetry manifested by the system and vice-versa. This
behavior is expected in high energy systems like particle physics models (e.g. in the electroweak phase transition) and in lower energy systems like ones found in condensed matter (e.g. Bose-Einstein
condensation in atomic gases).
An almost general rule that arises from studies of how the symmetry changes with temperature is that a symmetry which is broken at zero temperature should get restored as the temperature increases.
Examples range from the traditional ferromagnet to the more up to date chiral symmetry breaking/restoration in Quantum chromodynamics (QCD), with the transition pattern being the simplest one of
going from the broken phase to the symmetric one as temperature goes from below to above some critical value and vice-versa. One very interesting counter example was shown by Weinberg in the context
of multiscalar field theories at finite temperatures [6]. He has shown the possibility of appearing two very interesting phenomena: one that occurs when a symmetry that is not broken at low
temperatures, can get broken at high temperatures, a phenomenon called inverse symmetry breaking (ISB), and the possibility of another case, that can happen when a symmetry that is broken at lower
temperatures, may never get restored at all as we go to higher temperatures, a phenomenon called symmetry nonrestoration (SNR).
These phenomena of ISB and SNR can be found in the context of high energy physics due to possibility of their implementation in realistic particle physics models, like in the context of high
temperature phase transitions in the early Universe [7, 8] and in applications covering problems which involve CP violation and baryogenesis, topological defect formation, inflation, etc [9]. Beside
these interesting applications, there are real physical systems which do exhibit phenomena similar to ISB/SNR, like liquid crystals, spin glass materials and many other systems and materials [10].
One of us have recently analyzed how ISB/SNR manifest themselves in nonrelativistic theories which may be used in condensed matter physics [8, 11] and possible applications of ISB/SNR phenomena to a
coupled two-species dilute Bose gas system [12, 13].
In this paper we are looking for the phenomena of symmetry nonrestoration and inverse symmetry breaking in the context of multi-scalar field theories considering the effects of both finite
temperature and density. In other words we are interested in the different symmetries realizations and the effects of a finite charge on a multi-scalar field theory. After reviewing the results at
finite temperature for a coupled two-scalar field model, we give the results obtained for a scalar self-interacting theory with a fixed U (1) charge [14] at finite temperature. The extension of these
results and calculations for a multi-scalar field theory at both finite temperature and densities are then explained. Our aim here is try to understand how a nonzero charge affects the phase
structure of a multi-scalar field theory, which should be of relevance in processes that may occur in heavy-ion collisions and in the early universe.
2. MULTI-SCALAR FIELD THEORY: LOOKING FOR ISB AND SNR
Let us initially briefly review the symmetry broken (SB) / symmetry restoration (SR) for a self-interacting real scalar field theory at finite temperature. Then, we contrast these results with those
that may originate as a consequence of having more than one scalar field with cross-couplings showing how phenomena ISB or SNR can emerge.
2.1. Symmetry breaking/restoration in O (N) scalar models
We start with the case of a relativistic case self-interacting scalar field with a O(N) symmetry. The Lagrangian density is given by
where Φis a scalar field with N components and the potential V (Φ) is given by
This potential is bounded if λ > 0 and depending on the signal of trie mass term we have the system either in the unbroken phase, (m^2 > 0), or broken phase, (m^2 < 0). By considering initially the
broken phase at T = 0, we have for instance that
for which there is a nonvanishing vacuum expectation value. At finite temperature the potential acquires thermal corrections and, for example at high temperatures [2, 6], it (the effective potential)
acquires the following form
From the above expression we obtain that there is a critical temperature T[c] above which the vacuum expectation value for the effective potential vanishes and the symmetry is restored, with T[c]
given by
It is very interesting to mention that, in the perturbative regime, the dominant contribution giving T[c] comes from a one-loop correction, and it is not expected to change too much due to higher
order contributions.
2.2. O (N) x O (N) relativistic models
Having shown the simple one field case in the previous subsection, let us see now how it is possible to appear different symmetry breaking/restoring patterns, like ISB and/or SNR. These phenomena
were first shown possible in finite temperature quantum field theory by Weinberg in 1974 [6]. Weinberg proposed the following multi-scalar field model (in either the unbroken or broken phase
initially for both the (Φ and Ψ directions, depending on the sign in front of the mass terms),
The potential that appear in this Lagrangian density is bounded from below for λ[Φ]> 0, λ[ψ] > 0 and λ[Φ]λ[ψ] > 9λ^2. One important aspect here is that we can change the signal of the coupling
constant λ and the potential still remain bounded.
The thermal masses and obtained from Eq. (6) can be readily obtained [7],
where Φ and ψ , respectively. We then obtain the thermal masses and as given, in the high temperature approximation, by
For N = 2 for example and using (9) and (10), we obtain the critical temperature (i = Φ,ψ)
Assuming λ< 0, or in other words, |λ|> 2λ[Φ]/3, remembering that the boundness condition assures that |λ|>2λψ/3. We note that by taking < 0 and using the equations (10) and (11), in the Ψsector we
have SR, but in the ( sector we have SNR. In the opposite case, where > 0, in the Ψsector the system remains in the unbroken phase, while in theΦsector we have ISB.
As this simple example illustrates, we note that temperature effects in multiscalar field models can change the symmetry aspects in unexpected ways, e.g., in the O(N) x O(N) example, it shows the
possibilities of phenomena like inverse symmetry breaking (ISB) and symmetry nonrestoration (SNR). One very interesting question that appears is: Can we trust perturbative methods at high
temperatures? Though we know that high temperature field theories require nonperturbative treatments for consistence [7], it has been shown extensively in the literature that these phenomena also
appear in nonperturbative approaches, thus they are not artifacts of perturbation theory. One discussion in a fully nonperturbative context of the phenomena of SNR and ISB was done in Ref. [15] (see
also references therein for other nonperturbative approaches).
We can say that for a relativistic O(N) x O(N) theory, ISB/SNR phenomena is possible. Many interesting application of these phenomena have been proposed in the literature, like in cosmology, in the
context of formation of monopoles/domain walls and also in condensed matter physics, Ref. [10], where several applications were discussed.
Having briefly discussed the effects of temperature in symmetry restoring/breaking of symmetries, let us now see how density effects can also affect the symmetry properties of a theory. Let us
briefly see the derivation of the effective potential for a self-interacting scalar field theory at finite temperature and density (with a fixed U (1) charge). For simplicity we restrict the
calculation at the one-loop order [1,2, 14, 16]. We start with the grand partition function,
where β is the inverse of the temperature, Ω is the volume of the system, J[i] are the external sources, µ is the chemical potential, is the Hamiltonian density and ' is given by
Using standard manipulations [14] like Legendre transformations we obtain the one-loop effective potential,
where is a charge constraint
Rewriting the trace of Eq.(12) as a functional integral (the index i runs from 1 and 2) we get
The functional integral above is over two real fields (Φ1,Φ2) and their conjugate momenta (π[1], π[2]). The integral over t runs from 0 to β since we are working at finite temperature in the
Matsubara imaginary time formalism [16]. The Hamiltonian density for the model is given by
with [0] is a quadratic potential. This Hamiltonian is invariant under a global SO(2) symmetry which is generated by the integral over all space of the charge density The π[i] integrals are easily
done and we can rewrite Z [J[i]] as follows,
where N is a constant and is given by
The last two terms of Eq. (19) represent the effects of the conserved charge. We note that µ^2 serves as a negative mass squared term, or in other words, we can expect spontaneous symmetry breaking
when µ is greater than the mass of Φ. Performing the same calculations that were made in details in sections II and III of Ref. [14], we obtain the one-loop effective potential
and the zero point energy is given by
where the excitation energies are given by
The renormalization can be defined by introduction of counterterms as usual. Below we will quote directly the expression for the renormalized V[eff].
The effective potential is one adequate quantity to extract information of the phase structure (in terms of the temperature and chemical potential) and to determine the symmetry changing phase
transitions. In the high temperature limit we neglect the contribution V[zeropoint], which can be justified as follows. The zero point energy term has no temperature dependence explicitly, but at
high T this term will become temperature dependent due to mass renormalization, but this temperature dependence in this limit remains λ suppressed relative to the contribution of the V(h[ermal]. So
we will neglect V[zeropoint] in our analysis of the effects of finite charge in the phase structure.
Up to now we have not specified the form of the potential. For simplicity, we will use from now on the λΦ^4 potential,
Working with perturbation theory and requiring it to converge, requires that we restrict the the values allowed for the field Φ, such that, using one naive criterium, that
The information about the phase structure is in the Φand µ dependence of the effective potential. The effective potential is then minimized with respect to the expectation value of the scalar field
and the chemical potential. The high temperature effective potential for a 50(2) symmetry at one-loop order is given by
where is the charge density.
The phase structure depends on the minima of the effective potential. Minimizing the effective potential with respect the expectation value of the scalar field we obtain
From Eq. (28), we obtain two minima: from the unbroken symmetry, Φ= 0, while for the broken symmetry,
From the above equation we obtain the critical temperature T[c] at which the vacuum expectation value vanishes,
The relation between n and p follows from minimizing the effective potential with respect to µ,
Again, we can neglect the zero point contribution. In the high temperature limit the sum of the two contributions in Eq. (31) becomes
One of our motivations for this work is the application of this formalism in cosmology, as the universe expands at constant entropy. Using the effective potential we can evaluate the entropy density
So, as in a expanding universe the charge per comoving volume remains constant, in our analysis we also consider constant volume. Keeping n/s constant, we must require [14]
Now we can write n as a function of p, n = (µ), and then by using Eqs. (29) and (32) we obtain that
The analysis of Eq. (34) can be simplified if we work in two limits: at low density and at high density . The high density limit is particularly more interesting since in this regime n and p are
large and their effects on the phase structure of the theory is extreme. From Eq. (34) we get
Using Eq. (30) we obtain
We see that symmetry breaking occurs much earlier at a much higher temperature, than it would in absence of n. Next, we show some numerical results for the high density limit that follows from the
analysis of the temperature and density dependent effective potential.
3.1. Numerical Results
We here concentrate on the behavior of the λΦ^4 theory with a charge n = ηT^3. In Fig. (1) we see that for a small charge the spontaneously broken symmetry gets restored, as expected, at high
temperatures (here we consider η = 0.01). But when the charge increases (increasing the value of η ) the symmetry seems never to get restored, thus given a symmetry non-restoration phenomenon
analogous as the one seen in the twofield case at finite temperature only, shown in the previous section.
In Fig. (2) we show the behavior of the chemical potential and the expectation value of the scalar field Φas a function of the temperature as the number density varies, with n = ηT^3. We note that
this case of unbroken symmetry, we can have a broken phase developing as we increase the density. We can then conclude that a high charge density can induce a high temperature symmetry breaking (or
ISB). The symmetry broken solution is not necessarily valid near T[c] [14], but holds only asymptotically at temperatures when
4. MULTI-SCALAR FIELD THEORY AT FINITE T AND µ
By extending our calculations for a coupled two complex scalar fields (Φ and Ψ, we can consider the Lagrangian density,
We can write our expressions in terms of fields ( Φ[i] andΨi = 1,2) using the transformations
In the Grand Canonical formalism, we introduce the chemical potentials µ[Φ] and µ[ψ] for charge conservation used for each field separately. The resulting action becomes
The next step consist in expanding the action to determine a expression for the effective potential. From Eq. (39) and following similar computation of the effective potential as for the one-field
case, the one-loop effective potential is determined by the functional partition function to one-loop order and given by
where is the matrix operator for the quadratic terms in the fluctuations,
Minimizing the effective potential in relation to (Φ andΨ, the phase structure for this theory can be determined. According to the results of Sec. II.B, the two-field case, for a convenient choice of
couplings, can result naturally in the phenomena of ISB/SNR. From the discussion and analysis of the effects of a finite charge (density) these symmetry change effects can appear even in the
one-field case. Thus, we expect that these density effects included in Eq. (40) will strength the emergence of such phenomena in the two-field, or multi-field models. An extensive analysis of the
resulting phase diagram obtained from Eq. (40) will be shown elsewhere [17].
5. CONCLUSIONS
In this work we have shown that a high charge density can induce strong changes in the phase structure of the theory. Besides high temperature symmetry restoration it can also exhibit symmetry
nonrestoration and inverse symmetry breaking. We can look for this phenomena in different branches of physics, like condensed matter, cosmology and even in applications motivated by the up to date
heavy-ion collision experiments. Since phenomena like ISB/SNR can appear in a theory with one self-interacting complex scalar field at finite density, it is an interesting matter to explore the same
phase structure behavior in models with higher field content. We are currently looking for this kind of phenomena in a multi-scalar field theory at finite temperatures and densities [17].
As a future application we intend to perform the dynamics of the multi-scalar field model in order to probe, dynamically, the emergence of such interesting phase behaviors like ISB/SNR. One step
toward this comparison was done in the classical level for the self-interacting scalar field theory in Ref. [18].
The authors would like to thank FAPERJ, CNPq and CAPES for the financial support.
[1] C. W. Bernard, Phys. Rev. D. 9, 3312 (1974). [ Links ]
[2] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974). [ Links ]
[3] J. Bernstein and S. Dodelson, Phys. Rev. Lett. 66, 683 (1991). [ Links ]
[4] R. L. S. Farias, G. Krein, and R. O. Ramos, arXiv:0809.1449 (in press Phys. Rev. D). [ Links ]
[5] A. Linde, Rep. Prog. Phys. 42, 389 (1979). [ Links ]
[6] S. Weinberg, Phys. Rev. D 9, 3357 (1974). [ Links ]
[7] R. O. Ramos and M. B. Pinto, Phys. Rev. D 61, 125016 (2000). [ Links ]
[8] M. B. Pinto, R. O. Ramos, and J. E. Parreira Phys. Rev. D 71, 123519 (2005). [ Links ]
[9] R. N. Mohapatra and G. Senjanovic Phys. Rev. Lett. 42, 1651 (1979); [ Links ] S. Dodelson and L. M. Widrow, Phys. Rev. Lett. 64, 340 (1990); [ Links ] B. Bajc and G. Senjanovic, Nucl. Phys. Proc.
Suppl. A 52, 246 (1997). [ Links ]
[10] N. Schupper and N. M. Shnerb, Phys. Rev. E 72, 046107 (2005). [ Links ]
[11] R. O. Ramos and M. B. Pinto, J. Phys. A 39, 6687 (2006). [ Links ]
[12] M. B. Pinto, R. O. Ramos, and F. F. de Souza Cruz, Phys. Rev. A 74, 033618 (2006). [ Links ]
[13] R. O. Ramos and M. B. Pinto, J. Phys. A 39, 6649 (2006). [ Links ]
[14] K. M. Benson, J. Bernstein, and S. Dodelson, Phys. Rev. D 44, 2480 (1991). [ Links ]
[15] M. B. Pinto and R. O. Ramos, Phys. Rev. D 61, 125016 (2000). [ Links ]
[16] J. I. Kapusta, Phys. Rev. D 24, 426 (1981); [ Links ] H. E. Haber and H. Weldon, Phys. Rev. Lett. 46, 1497 (1981); [ Links ] Phys. Rev. D 25, 502 (1982); [ Links ] J. I. Kapusta, Finite
Temperature Field Theory (Cambridge University Press, Cambridge, England, 1989). [ Links ]
[17] R. L. S. Farias, R. O. Ramos, and R. Vartuli, in preparation. [ Links ]
[18] R. L. S. Farias, R. O. Ramos, and L. A. da Silva, Langevin Simulations with Colored Noise and Non-Markovian Dissipation, in press Braz. J. Phys., 38, 3B, (2008). [ Links ]
(Received on 17 April, 2008) | {"url":"http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332008000400015&lng=en&nrm=iso&tlng=en","timestamp":"2014-04-16T19:56:23Z","content_type":null,"content_length":"63133","record_id":"<urn:uuid:84766654-5894-4880-952a-bf45e14fad75>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00013-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is the homotopy theory of categories?
up vote 11 down vote favorite
I've heard that Grothendieck, in his letter "Pursuing Stacks," wanted to find alternative models for the classical homotopy category of CW complexes and continuous maps (up to homotopy), and one of
his proposed ideas was a "homotopy theory of categories." What does this mean, precisely?
I know that any category corresponds to a simplicial set (its nerve), and an equivalence of categories introduces a homotopy equivalence (in the category of simplicial sets) of the associated nerves.
I also know that there is a characterization of (the nerves of) categories among simplicial sets in terms of a unique filler extension condition. If this extension condition is weakened, so that one
gets the notion of a quasicategory or $\infty$-category, one can obtain a model structure where the quasicategories are the fibrant objects.
But if we want to just work with ordinary categories, is there a natural model structure on simplicial sets in which they are the fibrant objects? And if so, is this Quillen equivalent to the
(Quillen/Serre) model structure on topological spaces?
homotopy-theory ct.category-theory simplicial-stuff
3 Pursuing Stacks' is erroneously called a letter. The letter to Quillen' that is included in the first few pages is not the main point. It is clear that AG thought of the manuscript as a working
diary on that project. Again he did not want to find find alternative models as such (see David's excellent reply below), but rather to search for the higher dimensional analogues of Covering
Space theory, and to look for a good model of n-categories that would do the job. – Tim Porter Apr 15 '11 at 6:42
Thanks Tim ;) – David Roberts Apr 15 '11 at 7:40
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2 Answers
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I am not knowledgeable enough to have much to say I have not writen in my answer to a previous question of yours, and I think that David Roberts's answer (or, rather immodestly, my previous
one) provides what you were looking for as regards your first question. Just a few additional small points:
Pursuing Stacks is not a letter. See Tim Porter's comment.
As regards Grothendieck's opinion of Thomason's model structure, I do not know. Actually, I am unsure he knew of Thomason's model structure when writing Pursuing Stacks [EDIT: see Tim
Porter's comment below]. What he knew for sure was that the localization of $Cat$ with respect to classical weak equivalences (functors between small categories the nerve of which are
simplicial weak equivalences) is equivalent to the classical homotopy category. The first proof is due to Quillen and Illusie "wrote the details" (his words) in his thesis. (And there is a
quite simpler proof, by the way.) Model structures crop up in Pursuing Stacks at some point, but I am pretty sure the idea is not developed in the beginning, which is much more concerned
with mere models for homotopy types. Here is a citation from Chapter 75: "the notion of asphericity structure — which, together with the closely related notion of contractibility structure,
tentatively dealt with before, and the various "test notions" (e.g. test categories and test functors) seems to me the main payoff so far of our effort to come to a grasp of a general
formalism of "homotopy models"." (Beware: these asphericity structures are not what Maltsiniotis called "asphericity structures" in his own work.)
Another fact Grothendieck knew was, of course, Quillen's Theorem A. It seems he did not write a detailed proof of the relative version, but he gave a sketch of a toposic proof of it,
though, and took it as an axiom for what he called basic localizer.
As for your second question, I do not know, but it seems to me that Grothendieck was not that interested in simplicial sets and thus did not work extensively with them. In a 1991 letter to
up vote 9 Thomason, he wrote: " D’autre part, pour moi le "paradis originel" pour l’algèbre topologique n’est nullement la sempiternelle catégorie ∆∧ semi-simpliciale, si utile soit-elle, et encore
down vote moins celle des espaces topologiques (qui l’une et l’autre s’envoient dans la 2-catégorie des topos, qui en est comme une enveloppe commune), mais bien la catégorie Cat des petites
accepted catégories, vue avec un œil de géomètre par l’ensemble d’intuition, étonnamment riche, provenant des topos. En effet, les topos ayant comme catégories des faisceaux d’ensembles les C∧ ,
avec C dans Cat, sont de loin les plus simples des topos connus, et c’est pour l’avoir senti que j’insiste tant sur l’exemple de ces topos ("catégoriques") dans SGA 4 IV". (See here.)
To conclude, let me mention that, if one takes Grothendieck's viewpoint of homotopical algebra, there should exist not only a homotopy theory of categories, but a homotopy theory of
$n$-categories. In this respect, there should be a "relative Theorem A" for every $n$, which should allow one to define a workable notion of "basic $n$-localizer". (Actually, this is
already done for $n=2$: see this paper by Bullejos and Cegarra for Theorem A.) And then one should work out a theory of test $n$-categories, whose $(n-1)-Cat$-valued presheaves should be
models for homotopy types, and so on. To sum up, what Grothendieck wanted to do amounts to giving new foundations for homotopical algebra, and this is still a work in progress.
David Roberts gives the two most useful available references in his answer. If you want to read Grothendieck's words (and in English), just wait for the upcoming annotated version of
Pursuing Stacks.
EDIT (2013/10/29): Rereading this answer, I realize that I should add something of which I was not aware at the time of my writing, still regarding Grothendieck's knowledge of Thomason's
model category structure (see also Tim Porter's comment and David Roberts's answer). An annotated version of section 69 of Pursuing Stacks is available at http://www.math.jussieu.fr/
~maltsin/groth/ps/ps-69.pdf. On page 4, Grothendieck writes that "it appears very doubtful still that (Cat) is a “model category” in Quillen’s sense, in any reasonable way (with W of course
as the set of “weak equivalences”". Thus, he was not aware of the existence of Thomason's structure then. See also note 6 on that same page: Grothendieck has learnt of the existence of
Thomason's model structure between the writing of Sections 69 and 87.
Jonathan: you say: 'As regards Grothendieck's opinion of Thomason's model structure, I do not know.' The point was made in a letter to me in the middle of the time when he was writing
1 PS. (The letter was to have been included in Maltsinitios' retyping of the PS correspondence but that was stopped by Grothendieck. He does mention Thomason's work (in fact, he learnt of
that at least partially from me and was surprised by the result.) I think he makes some comments about the cofibrations not being that intuitive or geometric. – Tim Porter Apr 15 '11 at
Dear Jonathan: thanks! I hadn't heard of Thomason's model structure, and was only really aware of the concrete example of simplicial sets as being a good model for classical homotopy
theory. (I'll have to look up Quillen's Theorem A and B, as well.) – Akhil Mathew Apr 16 '11 at 1:08
Dear Tim: Thanks a lot for your comment. I have edited my answer accordingly. – Jonathan Chiche Apr 17 '11 at 8:26
Dear Akhil: You're welcome. I had trouble inserting the diagram to state Theorem A, but the relative case is one of the axioms for basic localizers, so you'll find it in Maltsiniotis's
and Cisinski's books. – Jonathan Chiche Apr 17 '11 at 8:28
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This is just to answer your first question. The second one I don't know about.
The homotopy theory of categories is not quite as you envisage it. Really Grothendieck is thinking of the Thomason model structure on $Cat$ (the category of small categories), which is
Quillen equivalent to the Quillen model structure on $sSet$ via the nerve functor. Then Grothedieck considered pairs $(Cat,W)$ where $W$ is a class of functors which acted as weak
equivalences. This he called a basic localizer (nLab). Grothendieck conjectured, and Cisinski proved, that the class of weak equivalences in the Thomason model structure was the smallest
basic localizer.
From there Grothendieck moved to considering pairs $(C,W)$ for any category $C$ and class $W$ of arrows such that $C[W^{-1}]$ was equivalent to the homotopy category of CW-complexes, or even
the homotopy category of some basic localizer, and in particular he was interested in when $C = Pre(S) = Cat(S^{op},Set)$, presheaves on some small category $S$. In particular, we know that
up vote $S=\Delta$ can be used to recover the homotopy theory of CW-complexes. The question was to characterise those $S$ such that $(Pre(S),W')$, where $W'$ was inherited from a basic localizer
11 down (consult Cisinski's or Maltsiniotis' work for details), can be used to model the same homotopy types as $Cat$. Such categories $S$ were called [weak/strict] test categories.
D. Cisinski, Les préfaisceaux comme modèles des types d’homotopie, Astérisque 308 (2006)
G. Maltsiniotis, La théorie de l’homotopie de Grothendieck, Astérisque, 301 (2005)
are central resources in this area.
All the relevant papers can be found on Maltsiniotis' homepage: math.jussieu.fr/~maltsin/ps.html – Theo Buehler Apr 15 '11 at 5:47
Thanks, Theo. I knew they were available electronically somewhere - just not from Cisinski's home page! – David Roberts Apr 15 '11 at 6:32
1 Point of Info: Grothendieck actually did not like Thomason's structure as the cofibrations worried him. – Tim Porter Apr 15 '11 at 6:44
1 Ah, that would explain the emphasis on presheaf models, where the cofibrations are taken to be monomorphisms. – David Roberts Apr 15 '11 at 6:48
1 I think the emphasis on presheaf models is due to the difficulty of the general case! This is explained in the introduction of Maltsiniotis's book. – Jonathan Chiche Apr 15 '11 at 8:19
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Not the answer you're looking for? Browse other questions tagged homotopy-theory ct.category-theory simplicial-stuff or ask your own question. | {"url":"http://mathoverflow.net/questions/61781/what-is-the-homotopy-theory-of-categories","timestamp":"2014-04-16T22:33:44Z","content_type":null,"content_length":"75521","record_id":"<urn:uuid:72993ee4-6618-4c8d-a7df-4b10f48109fc>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00243-ip-10-147-4-33.ec2.internal.warc.gz"} |
Angles of Polygons and Regular Tessellations Exploration
From EscherMath
Objective: Calculate the interior angles of polygons and classify the regular tessellations of the plane.
Interior Angles of Polygons
1. Check that the sum of the angles in a triangle is 180° as follows: Cut out a triangle. Tear off the corners and put them together so that their vertices are touching. What do you see?
2. Draw some quadrilaterals. For each one, show how to cut it into two triangles. Since the angle sum of each triangle is 180°, explain how you know the angle sum of each quadrilateral. What is the
angle sum of a quadrilateral?
3. Any polygon can be cut into triangles by connecting its vertices with additional lines. How many triangles make up a 4-gon? How many triangles make up a 5-gon? How many triangles make up a 6-gon?
How many triangles make up an $ n $-gon?
│ │ │
│ Cutting polygons into triangles. │ A bad way to cut into triangles. │
4. Using the information from question 3 argue that:
The sum of the interior angles of an $ n $-gon is $ (n-2)\times 180^\circ $
5. Why does the "bad way to cut into triangles" fail to find the sum of the interior angles?
Regular Polygons
A regular polygon is a polygon with all sides the same length and all angles having the same angle measure.
6. Explain the following formula:
Each angle of a regular $ n $-gon is $ \frac{(n-2)180^\circ}{n} $.
Would this formula work for just any $ n $-gon? Why or why not?
7. Complete the following table:
│ Number of Sides │3 │4 │5│6│7│8│9│10│11│12│15│20│50│100│
│Corner angle = $ \frac{(n-2)180^\circ}{n} $ │60°│90°│ │ │ │ │ │ │ │ │ │ │ │ │
8. If regular polygons are going to fit around a vertex, then their angle measures have to divide evenly into 360°. Explain. Which of the angle measures in the table divide evenly into 360˚?
9. The table doesn't list every possible number of sides. How do you know that there are no other regular polygons with angles that divide evenly into 360˚, besides the ones mentioned on the list?
10. Which regular $ n $-gons are the only ones that can tessellate the plane using just one type of tile?
11. Three equilateral triangles and two squares can fit together, since 60+60+60+90+90 = 360°. What other combinations of corner angles in the table can be combined to make 360°?
Handin: A sheet with answers to all questions. | {"url":"http://euler.slu.edu/escher/index.php/Angles_of_Polygons_and_Regular_Tessellations_Exploration","timestamp":"2014-04-18T10:35:13Z","content_type":null,"content_length":"20379","record_id":"<urn:uuid:aacf73da-4778-4e35-be34-fcecf752422a>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00079-ip-10-147-4-33.ec2.internal.warc.gz"} |
Meeting Details
For more information about this meeting, contact Robert Vaughan.
Title: The Montgomery-Hooley Theorem and its generalizations
Seminar: Algebra and Number Theory Seminar
Speaker: Robert Vaughan, Penn State University
This theorem is often called the Barban-Davenport-Halberstam theorem but they only proved weak versions. It gives the asympotics for the mean square of the error term in the prime number theorem for
arithmetic progressions. I will give an overview of this theorem and related theorems and generalizations. There are still some interesting open questions.
Room Reservation Information
Room Number: MB106
Date: 03 / 05 / 2009
Time: 11:15am - 12:05pm | {"url":"http://www.math.psu.edu/calendars/meeting.php?id=4404","timestamp":"2014-04-17T15:44:12Z","content_type":null,"content_length":"3398","record_id":"<urn:uuid:4ba00ef8-53fd-4a4e-a3b2-003f6de579a3>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00343-ip-10-147-4-33.ec2.internal.warc.gz"} |
Kinetic energy gained by a rocket
Ok, in the previous posts I mistakenly took "u" to be the initial velocity, since I'm use to a certain convention where "u" is the initial velocity, but in this case it is the exhaust velocity, sorry
Well, I think I'm beginning to understand a bit what's going on here.
The gain in kinetic energy due to the rocket emitting a small amount of matter
from added source, is given by:
(I still don't completely understand why this is the equation and why it is only determined by the exhaust velocity and not other velocity components)
Now you can calculate the
kinetic energy gained by the rocket as follows (this
kinetic energy gained is due to the rocket emitting small amounts of matter
extra supplied kinetic energy from other sources.)
Consider the problem consisting of three different components:
1.) The exhaust cloud
2.) The fuel inside the rocket (this is the part of the total mass of the rocket which will be decreasing)
3.) The shell of the rocket (without fuel) (this is the part of the total mass of the rocket which will stay the same)
Each of these components have a mass and a velocity.
1.) For the shell of the rocket:
Initial mass: [tex]m_{r}[/tex]
Initial velocity: [tex]v_{0}[/tex]
Final mass: [tex]m_{r}[/tex]
Final velocity: [tex]v_{0}+\Delta v[/tex]
2.) For the fuel inside the rocket:
Initial mass: [tex]m_{f}[/tex]
Initial velocity: [tex]v_{0}[/tex]
Final mass: [tex]m_{f}-\Delta m[/tex]
Final velocity: [tex]v_{0}+\Delta v[/tex]
3.) For the exhaust cloud:
Initial mass: 0
Final mass: [tex]\Delta m[/tex]
Final velocity: [tex]v_{0}+\Delta v-u[/tex]
It can be seen that the initial and final velocity of the fuel and the rocket is the same, since they are moving together and the final velocity of the exhaust cloud is again an inertial frame, thus
it has a component due to moving with the rocket ( [tex]v_{0}+\Delta v[/tex] ) minus the component at which it is emitting from the rocket, since it is emitted in the opposite direction to which the
rocket is moving, to accelerate the rocket ( [tex]u[/tex] )
The initial kinetic energy is thus:
which is just due to the shell of the rocket and the fuel, since the exhaust cloud does not exist yet.
The final kinetic energy is thus:
[tex]T_{F}=\frac{1}{2}m_{r}\left(v_{0}+\Delta v\right)^{2}+\frac{1}{2}\left(m_{f}-\Delta m\right)\left(v_{0}+\Delta v\right)^{2}+\frac{1}{2}\Delta m\left(v_{0}+\Delta v-u\right)^{2}[/tex]
[tex]T_{F}=\frac{1}{2}m_{r}\left(v^{2}_{0}+2v_{0}\Delta v+\Delta v^{2}\right)+\frac{1}{2}\left(m_{f}-\Delta m\right)\left(v^{2}_{0}+2v_{0}\Delta v+\Delta v^{2}\right)+\frac{1}{2}\Delta m\left(v^{2}_
{0}+2v_{0}\Delta v-2v_{0}u+\Delta v^{2}-2\Delta vu+u^{2}\right)[/tex]
[tex]T_{F}=\frac{1}{2}m_{r}v^{2}_{0}+m_{r}v_{0}\Delta v+\frac{1}{2}m_{r}\Delta v^{2}+\frac{1}{2}m_{f}v^{2}_{0}+m_{f}v_{0}\Delta v+\frac{1}{2}m_{f}\Delta v^{2}-\frac{1}{2}\Delta mv^{2}_{0}-\Delta mv_
{0}\Delta v-\frac{1}{2}\Delta m\Delta v^{2}[/tex]
[tex]+\frac{1}{2}\Delta mv^{2}_{0}+\Delta mv_{0}\Delta v-\Delta mv_{0}u+\frac{1}{2}\Delta m\Delta v^{2}-\Delta m\Delta vu+\frac{1}{2}\Delta mu^{2}[/tex]
[tex]T_{F}=\frac{1}{2}m_{r}v^{2}_{0}+m_{r}v_{0}\Delta v+\frac{1}{2}m_{r}\Delta v^{2}+\frac{1}{2}m_{f}v^{2}_{0}+m_{f}v_{0}\Delta v+\frac{1}{2}m_{f}\Delta v^{2}-\Delta mv_{0}u-\Delta m\Delta vu+\frac
{1}{2}\Delta mu^{2}[/tex]
Thus the gain in kinetic energy is equal to the final kinetic energy minus the initial kinetic energy:
[tex]T_{F}-T_{0}=m_{r}v_{0}\Delta v+\frac{1}{2}m_{r}\Delta v^{2}+m_{f}v_{0}\Delta v+\frac{1}{2}m_{f}\Delta v^{2}-\Delta mv_{0}u-\Delta m\Delta vu+\frac{1}{2}\Delta mu^{2}[/tex]
Now I know you can group terms together to reduce it more, but I didn't do it, because I don't really know which terms to group together to get the best result, because if I'm right you have subtract
[tex]\frac{1}{2}M_0u^2(1-e^{-v/u})[/tex] from the above answer to get the energy that needs to be supplied by other sources, right?
Plus if the rocket is "held fixed on a test bed" does it imply that it is stationary and have a constant velocity of 0 or that it has a velocity, but can't accelerate? | {"url":"http://www.physicsforums.com/showthread.php?t=315634","timestamp":"2014-04-20T08:40:46Z","content_type":null,"content_length":"85960","record_id":"<urn:uuid:7eadd705-dafd-4962-aa6f-1cc9ee7f299d>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00396-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sources of Magnetic Fields
Having established the magnetic field of the simplest cases, straight wires, we must go through some calculus before analyzing more complex situations. In this section we shall generate an expression
for the small contribution of a segment of a wire to the magnetic field at a given point, and then show how to integrate over the whole wire to generate an expression for the total magnetic field at
that point.
Contribution to the Magnetic Field by a Small Segment of Wire
Consider a randomly shaped wire, with a current I running through it, as shown below.
Figure %: An odd-shaped wire. We find the magnetic field at point P by summing the contributions to the field of each element dl
We want to find the magnetic field at a given point near the wire. First, we find the individual contributions of very small lengths of the wire,
. The concept behind this method is that a very small piece of wire, no matter how the whole wire curves and twists, can be considered a straight line. So we sum over an infinite number of straight
lines (i.e. integrate) to find the total field of the wire. If the distance between our small segment
and the point is
, and the unit vector in this radial direction is denoted by
, then the contribution by the segment
is given by:
The derivation of this equation requires the introduction of the concept of vector potential. As this is beyond the scope of this text, we simply state the equation without justification.
Application of the Magnetic Field Equation
This equation is quite complicated, and is difficult to understand on a theoretical level. Thus, to show its applicability, we will use the equation to calculate something we already know: the field
from a straight wire. We begin by drawing a diagram showing a straight wire, including an element dl , in relation to a point a distance x from the wire:
Figure %: An element dl on a long wire, contributing to the magnetic field at P , a distance x from the wire
From the figure, we see that the distance between
. In addition, the angle between
is given by
sinθ =
. Thus we have the necessary values to plug into our equation:
dB =
Now that we have an expression for the contribution of a small piece, we may sum over the whole wire to find the total magnetic field. We integrate our expression with respect to
, with limits of integration from
- ∞
are constants, we may remove them from the integral, simplifying the calculus. This integral is still quite complicated, and we must use a table of integration to solve it. It turns out that the
integral is equal to
. We evaluate this expression using our limits:
B =
When we plug infinity into our expression we find that
, implying that plugging in a value of infinity yields the value
1/x ^2
. When we plug in our negative infinity, we get
-1/x ^2
in a similar manner. Thus:
B =
This is the equation we saw earlier for the field of a straight wire, implying that our calculus equation derived earlier is correct. The math that accompanies this kind of calculation is difficult,
and rarely used, but it is essential for deriving the formulae we will encounter in the
next section | {"url":"http://www.sparknotes.com/physics/magneticforcesandfields/sourcesofmagneticfields/section2.rhtml","timestamp":"2014-04-19T22:07:47Z","content_type":null,"content_length":"66547","record_id":"<urn:uuid:a301e99c-cd97-4242-b2ad-1a77912ece95>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00602-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Solved] Converting military time to hours/minutes in Excel
I have tried using a simple subtraction equation to determine the difference in hours using military time, however when you get to half hour increments, for example, 1700 - 1230, You see the minutes
as 70, which is the correct subtraction. I need to know how to convert the 100 units to 60 showing 5 hours and 30 minutes, not 4 hours and 70 minutes. I thought if inserting an if statement to add
160 units, but can't remember the condition "ends with" 70. Thank you!.
It is probably happening because you do not have the cells formatted as TIME.In cell A1 enter 17:00 << NOTE the colon - it is REQUIREDIn cell B1 enter 12:30 << NOTE the colon - it is REQUIREDIn cell
C1 enter the formula =A1-B1 you should now have an answer of 4:30MIKEhttp://www.skeptic.com/
In cell A1 enter 17:00 << NOTE the colon - it is REQUIREDIn cell B1 enter 12:30 << NOTE the colon - it is REQUIRED
In cell C1 enter the formula =A1-B1 you should now have an answer of 4:30
Don't start a new thread for the same question. I have deleted your other post as you have a perfectly good answer hereIf you need clarification, post in this thread.Stuart
Thank you for the response. I did try format the cells to time, but when I put in the data, it comes back as 00:00 PM. I'm not sure what I am doing wrong, but I did find a formula that worked:=(60 *
(DOLLARDE(B3/100, 60) - DOLLARDE(B2/100, 60) + 24 * (B3+0 < B3+0) ))Again, thank you Mike!
Are you going past Midnight with your calculations?Did you enter the time with or without the AM/PM?Here is a great source for Date & Time info:http://www.cpearson.com/excel/datet...MIKEhttp://
Here is a great source for Date & Time info: | {"url":"http://www.computing.net/answers/office/converting-military-time-to-hoursminutes-in-excel/17957.html","timestamp":"2014-04-16T04:35:52Z","content_type":null,"content_length":"42827","record_id":"<urn:uuid:9ea550c7-e80a-46c2-bba1-156f1bcfe12d>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00496-ip-10-147-4-33.ec2.internal.warc.gz"} |
Critical values &
degrees of freedom
of the
^2 distribution
The critical value of a statistical test is the numerical value at which an observed result is statistically significant at some pre-determined degree of probability. For example, for a ^2 test with
one df = 1, the table shows that if ^2 > 3.841, the observed result would be expected to occur by chance less than one time in twenty, that it, is has a probability p < 0.05. If the significance
level of the test had been set at p = 0.05, the observed results is said to be statistically significant. Where the observed result exceeds the critical value for p = 0.01, the result is sometimes
said to be highly significant. Strictly, the significance level is set a priori (before the experiment is carried out), and any result is either significant or non-signficant at that level.
The simplest kind of statistical test has only two possible outcomes ("either / or") and is also called a single-classification test. In such a test, the number of events in the first category
automatically determines the number in the second. Because of this, the experiment is said to have only a single degree of freedom, or df = 1. For a more complicated experiment with C possible
categories of outcome for N events, the number of events that can fall into any of the first C-1 categories can vary freely. The number of events in the last category must make the total add up to N,
and is thus constrained. The experimental outcome has lost one degree of freedom, and in general df = (# of possible outcomes - 1). | {"url":"http://www.mun.ca/biology/scarr/MGA2-06-Table1.html","timestamp":"2014-04-19T14:49:28Z","content_type":null,"content_length":"6738","record_id":"<urn:uuid:d848d0d9-aaca-4025-bb2b-fd4f6bf51184>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00512-ip-10-147-4-33.ec2.internal.warc.gz"} |
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[Numpy-discussion] numpy slices limited to 32 bit values?
Glenn Tarbox, PhD glenn@tarbox....
Thu May 14 01:22:41 CDT 2009
On Wed, May 13, 2009 at 11:04 PM, Charles R Harris <
charlesr.harris@gmail.com> wrote:
> On Wed, May 13, 2009 at 10:50 PM, Glenn Tarbox, PhD <glenn@tarbox.org>wrote:
>> I'm using the latest version of Sage (3.4.2) which is python 2.5 and numpy
>> something or other (I will do more digging presently)
>> I'm able to map large files and access all the elements unless I'm using
>> slices
>> so, for example:
>> fp = np.memmap("/mnt/hdd/data/mmap/numpy1e10.mmap", dtype='float64',
>> mode='r+', shape=(10000000000,))
>> which is 1e10 doubles if you don't wanna count the zeros
>> gives full access to a 75 GB memory image
>> But when I do:
>> fp[:] = 1.0
>> np.sum(fp)
>> I get 1410065408.0 as the result
> As doubles, that is more than 2**33 bytes, so I expect there is something
> else going on. How much physical memory/swap memory do you have? This could
> also be a python problem since python does the memmap.
I've been working on some other things lately and that number seemed related
to 2^32... now that I look more closely, I don't know where that number
comes from.
To your question, I have 32GB of RAM and virtually nothing else running...
Top tells me I'm getting between 96% and 98% for this process which seems
about right.
Here's the thing. When I create the mmap file, I get the right number of
bytes. I can, from what I can tell, update individual values within the
array (I'm gonna bang on it a bit more with some other scripts)
Its only when using slicing that things get strange (he says having not
really done a more thorough test)
Of course, I was assuming this is a 32 bit thing... but you're right...
where did that result come from???
The other clue here is that when I create my own slice (as described above)
it returns instantly... numpy doesn't throw an error but it doesn't do
anything with the slice either.
Since I'm IO bound anyways, maybe i'll just write a loop and see if I can't
set all the values. The machine could use a little exercise anyways.
> Chuck
> _______________________________________________
> Numpy-discussion mailing list
> Numpy-discussion@scipy.org
> http://mail.scipy.org/mailman/listinfo/numpy-discussion
Glenn H. Tarbox, PhD || 206-274-6919
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Physics Forums - View Single Post - Synge's Theorem
from Frankel:
Synge's Theorem
Let [tex]M^{2n}[/tex] be an even-dimensional, orientable manifold with positive sectional curvatures, [tex]K(\mathbf{X}\wedge\mathbf{Y}) > 0[/tex]. Then any closed geodesic is unstable, that is, can
be shortened by a variation.
A compact, orientable, even-dimensional manifold with positive sectional curvatures is simply connected.
to remind you, sectional curvature is the Gaussian curvature determined by two tangent vectors X and Y | {"url":"http://www.physicsforums.com/showpost.php?p=98614&postcount=8","timestamp":"2014-04-18T15:52:27Z","content_type":null,"content_length":"7636","record_id":"<urn:uuid:ded04ccc-b4b3-44b8-8826-2d03bf4e8d8b>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00148-ip-10-147-4-33.ec2.internal.warc.gz"} |
Flux Integral
December 15th 2008, 04:43 PM #1
Dec 2008
Flux Integral
Let S be the part of the plane
I'm stuck
If i use the equation int int Fnds i get the int int of 11/(11)^(1/2)
if I use int int Fn |ruXrv|i get stuck at ruXrv because I don't know how to find my ru and rv.
I can't use the divergence therem because there are no values.
What method should I use? And how do I set up the bounds? Thank you so much!
how do you come up with that?
Sorry for my English is not very well.
The direction of the flux vector is A=(1,3,1),and the direction of the normal of the plane is B=(2,1,1).So the flux of the vector field across the surface need to calculate the flux vector
project on the normal of the plane.The angle x formed by two vector A and B is calculated by
so the angle x between the vector (1,3,1) and (2,1,1) is
so the length of the projection on the the plane's normal of the flux vector is
its direction as same as (2,1,1).
December 15th 2008, 05:43 PM #2
Dec 2008
December 15th 2008, 06:00 PM #3
Dec 2008
December 15th 2008, 07:48 PM #4
Dec 2008 | {"url":"http://mathhelpforum.com/calculus/65154-flux-integral.html","timestamp":"2014-04-18T04:28:24Z","content_type":null,"content_length":"35363","record_id":"<urn:uuid:8076bd18-ad79-4e1c-b542-f7c0c041e925>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00445-ip-10-147-4-33.ec2.internal.warc.gz"} |
Is a left invertible element of a group ring also right invertible?
up vote 14 down vote favorite
Given a group $G$ we may consider its group ring $\mathbb C[G]$ consisting of all finitely supported functions $f\colon G\to\mathbb C$ with pointwise addition and convolution. Take $f,g\in\mathbb C
[G]$ such that $f*g=1$. Does this imply that $g*f=1$?
If $G$ is abelian, its group ring is commutative, so the assertion holds. In the non-abelian case we have $f*g(x)=\sum_y f(xy^{-1})g(y)$, while $g*f(x)=\sum_y f(y^{-1}x)g(y)$, and this doesn't seem
very helpful.
If $G$ is finite, $\dim_{\mathbb C} \mathbb C[G]= |G|<\infty$, and we may consider a linear operator $T\colon \mathbb C[G]\to\mathbb C[G]$ defined by $T(h) = f*h$. It is obviously surjective, and
hence also injective. Now, the assertion follows from $T(g*f)=f=T(1)$.
What about infinite non-abelian groups? Is a general proof or a counterexample known?
ra.rings-and-algebras gr.group-theory
add comment
1 Answer
active oldest votes
A ring is called Dedekind-finite if that property holds. Semisimple rings are Dedekind finite, so this covers $\mathbb CG$ for a finite group $G$; this is easy to do by hand. It is a
up vote 20 theorem of Kaplansky that this also holds $KG$ for arbitrary groups $G$ and arbitrary fields $K$ of characteristic zero. See [Kaplansky, Irving. Fields and rings. The University of
down vote Chicago Press, Chicago, Ill.-London 1969 ix+198 pp. MR0269449] It is open, I think, for general fields.
"...if $R$ is an Artinian ring and $G$ is a sofic group, then the group ring $R[G]$ is stably finite..." -MR2362939 (2009a:16046) Ceccherini-Silberstein, Tullio(I-SAN-EN); Coornaert,
1 Michel(F-STRAS-I) Linear cellular automata over modules of finite length and stable finiteness of group rings. J. Algebra 317 (2007), no. 2, 743--758. See also MR2335561
(2008j:16077) Nasrutdinov, M. F.(RS-KAZA) Stable finiteness of group rings. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 2006, , no. 11, 29--32; translation in Russian Math. (Iz. VUZ)
50 (2006), no. 11, 27--30 (2007) – Jonas Meyer Mar 17 '10 at 18:42
It is worth noting that (AFAIK) there is no group which is known to be non-sofic. Also, in more or less the same breath, Kaplansky observes that the group von Neumann algebra of a
1 discrete group, like any finite von Neumann algebra, is Dedekind finite (also sometimes called directly finite). A slightly lower tech proof can be extracted from M. S. Montgomery,
Left and right inverses in group algebras, Bull. Amer. Math. Soc. 75 (1969) 539--540. (Might also shamelessly plug arxiv.org/abs/1003.1650 while I'm typing...) – Yemon Choi Mar 17
'10 at 20:16
One should add that the result for sofic groups was first shown by Elek and Szabo in "Sofic groups and direct finiteness", Journal of Algebra Vol 280, Issue 2 – Łukasz Grabowski Jun
15 '11 at 11:50
add comment
Not the answer you're looking for? Browse other questions tagged ra.rings-and-algebras gr.group-theory or ask your own question. | {"url":"http://mathoverflow.net/questions/18508/is-a-left-invertible-element-of-a-group-ring-also-right-invertible?sort=newest","timestamp":"2014-04-21T10:11:55Z","content_type":null,"content_length":"55315","record_id":"<urn:uuid:6bc37181-0357-45fb-8a45-ed850a44149e>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00265-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mirror of Flop?
up vote 5 down vote favorite
If two Calabi-Yau 3-folds are bi-rational to each other via a Flop , then what is the relation between their mirrors ?
mirror-symmetry complex-geometry ag.algebraic-geometry
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2 Answers
active oldest votes
I assume the question regards the coherent sheaves on these two CY's. These CY's should be regarded as the "same" complex manifold with two different choices of complexified symplectic
forms ("Kahler form," in physics terminology).
The mirrors are a "single" symplectic manifold with two different complex structures on it. There is a curve of complex structures relating the two.
up vote 6 down
vote accepted That's about it. The tricky part is to "parallel transport" the category of coherent sheaves along this curve, using a "flat family of categories" defined by stability conditions.
Doing so should provide a preferred isomorphism of the categories. Examples have been studied, but general statements (like the ones I have glibly been making) are not proven.
Exactly, but let me narrow my question more: Those two complexified kahler forms are connected via a path and somewhere in the middle of the path the contraction mentioned by "VA"
above happens which is a wall-crossing between Kahler cones of two Calabi_Yau's . Is there a similar wall-crossing for the curve connecting two complex structures of mirror? If yes
then what is the nature of that? – Mohammad F. Tehrani Sep 1 '10 at 17:40
No. The singularity is not a wall. It is complex codimension 1, real codimension 2. Around the singularity, you have a loop. The "monodromy" around this loop produces an
1 autoequivalence of the derived category. These autoequivalences -- originally conjectured by Kontsevich -- have been studied in many cases, first rigorously by Seidel-Thomas. – Eric
Zaslow Sep 1 '10 at 18:24
add comment
Small contractions are mirrors to degenerations, so: degenerate, then deform out.
up vote 2 down
I did not ask for the mirror of degeneration. Two CY which are related by Flop will contract to same singular CY but how do you deform the mirror of this singular CY in two ways! –
Mohammad F. Tehrani Sep 1 '10 at 14:54
A flop is just small contraction + the opposite of small contraction, right? – VA. Sep 1 '10 at 15:09
add comment
Not the answer you're looking for? Browse other questions tagged mirror-symmetry complex-geometry ag.algebraic-geometry or ask your own question. | {"url":"http://mathoverflow.net/questions/37384/mirror-of-flop/37403","timestamp":"2014-04-24T12:38:28Z","content_type":null,"content_length":"57189","record_id":"<urn:uuid:cd2d36e4-0c31-4a05-941b-5a1e21dd0635>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00254-ip-10-147-4-33.ec2.internal.warc.gz"} |
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contains a wide variety of illustrative nonlinear examples solved with Mathematica. There are also 130 annotated Mathematica files that may be used to solve and explore the text's four hundred
Part I: Theory: Introduction | Nonlinear Systems, Part I | Nonlinear Systems, Part II | Topological Analysis | Analytic Methods | The Numerical Approach | Limit Cycles | Forced Oscillators |
Nonlinear Maps | Nonlinear PDE Phenomena | Numerical Simulation | Inverse Scattering Method
Part II: Experimental Activities: Introduction to Nonlinear Experiments | Magnetic Force | Magnetic Tower | Spin Toy Pendulum | Driven Eardrum | Nonlinear Damping | Anharmonic Potential | Iron Core
Inductor | Nonlinear LRC Circuit | Tunnel Diode Negative Resistance Curve | Tunnel Diode Self-Excited Oscillator | Forced Duffing Equation | Focal Point Instability | Compound Pendulum | Damped
Simple Pendulum | Stable Limit Cycle | Van der Pol Limit Cycle | Relaxation Oscillations: Neon Bulb | Relaxation Oscillations: Drinking Bird | Relaxation Oscillations: Tunnel Diode | Hard Spring |
Nonlinear Resonance Curve: Mechanical | Nonlinear Resonance Curve: Electrical | Nonlinear Resonance Curve: Magnetic | Subharmonic Response: Period Doubling | Diode: Period Doubling | Five-Well
Magnetic Potential | Power Spectrum | Entrainment and Quasiperiodicity | Quasiperiodicity | Chua's Butterfly | Route to Chaos | Driven Spin Toy | Mapping
This book is available in the Wolfram Research bookstore.
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Triangular load on a two-point support beam
September 27th 2012, 05:57 AM
Triangular load on a two-point support beam
Attachment 24943
Ok this is going to be a little hard to explain since my "math english" isn't that good.
We know that the amount the two point support beam bends under the net force P of the triangular load is
Attachment 249440<=x<=L
where e and i are positive constants associated to the support beam's material and cross-section. In what point the bending on the support beam is largest when L=3 meters?
I hope you understand it :D. Thanks.
September 27th 2012, 06:42 AM
Re: Triangular load on a two-point support beam
You explained it pretty well. I'm not sure where you're having trouble, so I'll just give a few hints/suggestions:
1) I like to de-clutter such problems. I'd set u = x/L and k = PL^3/(180ei), then work the problem for f(u) = k(3u^5 - 10 u^3 + 7u).
Since 0<=x<=L, have 0<=u<=1. (In the end, of course, will have to convert back to x and use L = 3 m.)
2) Sometimes a 4th degree equation is a quadratic equation in a square power. i.e. z^4 - 6z^2 - 10 = 0 is solveable
It's (z^2)^2 - 6(z^2) - 10 = 0, so if w = z^2, it's w^2 - 6w - 10 = 0, so w = [-(-6) +- sqrt(36-(-40)]/2 = 3+- sqrt(19).
Then z = +- sqrt[ 3+ sqrt(19)] and z = +- sqrt[sqrt(19)-3] i
Can you say where you're having difficulty with this problem?
September 27th 2012, 06:45 AM
Re: Triangular load on a two-point support beam
You will have to clarify something: what do you mean by "where bending is greatest?" If you mean where the deflection of the beam is greatest that would be where the slope of the beam deflection
= 0, so take the derivative of y with respect to position x, set it to 0, and solve for x. But if you mean where the curvature of the beam is greatest that would be where the 2nd derivative of y
is at a max (which also happens to be where the moment in the beam is greatest, and where the shear is 0).
I assume this problem comes from a course you are taking in beam theory - a mechanical or civil engineering class, right? You may find it good practice work to develop the beam equation from
fundamentals. To do this you would need to;
1. Find the reaction forces for the two suports, F1 and F2
2. Develop the equation for shear V(x) = integral of p(x), using the boundary condition that V(0) = F1 and V(L) = F2. To find where bending curvature is greatest, solve for v(x)=0.
3. Integrate V(x) to get the equation for the moment in the beam, M(x), using the boundary condition for simple supports that M(0)=M(L)=0.
4. Integrate the M(x) equation and divide by EI to get the equation for the slope of the deflection. We don't have any convenient boundary conditions to apply for this step, so don't forget to
include the constant of integration.
5. Integrate the slope equation to get the deflection equation y(x), and using boundary conditions y(0)= y(L)= 0 you can solve for the unknown constant from step 4.
Like I said, it's good practice!
September 27th 2012, 08:01 AM
Re: Triangular load on a two-point support beam
From Handbook American Inst of Steel Construction for load increasing uniformerly to one end
R1 low end of load = W/3 W in kips
R2 high end of load = 2W/3
Max moment @ X from R1 = l (span in inches )/rad3 = 2Wl/9rad3
September 27th 2012, 12:21 PM
Re: Triangular load on a two-point support beam
Thanks I got it sorted out with all of your tips :)
@ebaines I meant the deflection :) I thought I'd have to do something like that but wasn't sure how to approach it so thanks for clarifying :) | {"url":"http://mathhelpforum.com/differential-geometry/204177-triangular-load-two-point-support-beam-print.html","timestamp":"2014-04-20T19:02:32Z","content_type":null,"content_length":"8051","record_id":"<urn:uuid:4c6a89ea-38ee-494e-be0a-2a429953ceb9>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00187-ip-10-147-4-33.ec2.internal.warc.gz"} |
Portability GHC only
Maintainer Simon Meier <iridcode@gmail.com>
Safe Haskell None
This module implements all rules that do not result in case distinctions and equation solving. Some additional cases may although result from splitting over multiple AC-unifiers. Note that a few of
these rules are implemented directly in the methods for inserting constraints to the constraint system. These methods are provided by Theory.Constraint.Solver.Reduction. | {"url":"http://hackage.haskell.org/package/tamarin-prover-theory-0.8.5.0/docs/Theory-Constraint-Solver-Simplify.html","timestamp":"2014-04-18T16:05:04Z","content_type":null,"content_length":"2962","record_id":"<urn:uuid:5e71bc70-6079-4584-a449-95df0326bd8d>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00607-ip-10-147-4-33.ec2.internal.warc.gz"} |
Longest common sequence
Join Date
Feb 2011
Rep Power
I have an assignment in my data structures class to find the longest common sequence of letters of 2 strings. It does not need to be done programatically, but rather with a table. For example
with the 2 strings 'MZJAWXU' and XMJYAUZ here is the table. I am not following the logic. Any help appreciated. I looked at the wikipedia page and its still not clicking.
| 0 1 2 3 4 5 6 7
| M Z J A W X U
0 | 0 0 0 0 0 0 0 0
1 X | 0 0 0 0 0 0 1 1
2 M | 0 1 1 1 1 1 1 1
3 J | 0 1 1 2 2 2 2 2
4 Y | 0 1 1 2 2 2 2 2
5 A | 0 1 1 2 3 3 3 3
6 U | 0 1 1 2 3 3 3 4
7 Z | 0 1 2 2 3 3 3 4
Join Date
Jan 2011
Richmond, Virginia
Blog Entries
Rep Power
Would you mind posting the assignment details verbatim?
Join Date
Feb 2011
Rep Power
Find longest common sequence of letters from the 2 strings:
X = "skullandbones" Y = "lullabybabies"
Use the matrix as done in class (referring to a matrix similar to the one found here: Longest common subsequence problem - Wikipedia, the free encyclopedia)
The 'Worked Example' section on that page shows the way to do it, I just dont follow it.
i Think i get it...
difference between subsequence and substring,
the longest common substring would be 'ulla',
but the longest common subsequence would be 'ullabes'
the matrix is just a technique to find the longest subsequence progressively
and each step in building the matrix is just a break-down of comparisons of subsequences
1. If we have a sequence called S which contained AGCA, all the possible subsequences would be:
S1 = A
S2 = AG
S3 = AGC
S4 = AGCA
2. When finding the Longest Common Subsequence (LCS) of Two sequences,
to simplify our calculations, we first check the END of the sequence and remove any common last elements.
The example given is two sequences: BANANA and ATANA.
Why bother going through the whole sequence when you can see they both end with 'ANA'. The idea is to save your calculation and confusions, so remove ANA from both sequences and you can find
the LCS of the shorter sequences (which would be BAN and AT).
the LCS of BAN and AT is just 'A', so append the last elements you removed and you get the final LCS of A+ANA = 'AANA'.
3. When you get two sequences that do not end in the same symbol,
then you'd never need the last element from one of them.
ATK and BTG don't have a common last element, so the LCS could never include those last elements (K or G). In that case, you're trying to find the LCS of two sequences: ATK and BT - or - the
LCS of AT and BTG
if ATK is X and BTG is Y, then you're looking for LCS(Xn,Ym-1) or LCS(Xn-1,Ym) in this case.
4. Make the Matrix Table (one sequence as row header, the other sequence as column header), and for each co-ordinate make comparisons between the two sequences.
5. The two sequences you were given both end in 'ES', so you can remove those from your table, and append 'ES' to your LCS later
Last edited by ozzyman; 04-26-2011 at 05:46 PM.
Here's the table as we start off
The first comparison is S and L. Not the same, so in this case the longer sequence is taken,
but they're both empty, so 0 is given instead, with arrows pointing to both.
The same goes for the next two comparisons; L against K and L against U, they're not the same and neither direction contains a longer subsequence since they are empty
See picture
Finally we have a few matches, L and L. In the first case, since there is no longer subsequence, we have to use a diagonal arrow pointing to an empty cell. The resulting subsequence is (0L) =
In the second case, the sequence to the left is longer (1 symbol) than the sequence to the top (empty), so we use an arrow pointing to the longer sequence (left).
See 1st pic.
Now we have a longest common subsequence of LL so far.
Continue this work for the rest of the row and notice that the remaining cells always point Left because the sequence there is longer (LL) against (0).
See 2nd pic.
By the end of the 1st row, we have the longest subsequence of (LL)+(ES) = (LLES)
Last edited by ozzyman; 04-26-2011 at 06:24 PM.
To be honest I found the table very confusing.
Maybe someone can explain to me why it would be needed, or is it to explain the process? It all works fine with recursion:
Java Code:
package lcs;
public class LCS_maker{
private String DNA1 = "";
private String DNA2 = "";
public LCS_maker( String dna1, String dna2 ){
DNA1 = dna1;
DNA2 = dna2;
public static void main( String[] args ){
new LCS_maker( "skullandbones", "lullabybabies" );
// new LCS_maker( "AGCAT", "GAC" );
private void printLCS(){
int len1 = DNA1.length();
int len2 = DNA2.length();
String lcs = LCS( len1, len2 );
System.out.println( "LCS_maker: \"" + lcs + "\"" );
//Don't use Strings, to save space
private String LCS( int len1, int len2 ){
char last1, last2;
String lcs1, lcs2;
int lenLcs1, lenLcs2;
if( len1 <= 0 || len2 <= 0 )return "";
last1 = DNA1.charAt( len1-1 );
last2 = DNA2.charAt( len2-1 );
if ( last1 == last2 ){
return LCS( len1-1, len2-1 ) + last1;
lcs1 = LCS( len1, len2-1 );
lcs2 = LCS( len1-1, len2 );
lenLcs1 = lcs1.length();
lenLcs2 = lcs2.length();
return lenLcs1 > lenLcs2 ? lcs1 : lcs2;//Could be throwing some solutions away here
(By the way: thanks for the link. I first did DNA-alignment by an other recursive algorithm, repeatedly chopping it in half, searching and extending.
It worked fine, but this is much more elegant!)
Last edited by Jodokus; 04-29-2011 at 02:02 AM. Reason: spelling | {"url":"http://www.java-forums.org/new-java/43067-longest-common-sequence.html","timestamp":"2014-04-20T05:27:03Z","content_type":null,"content_length":"98410","record_id":"<urn:uuid:d043a6be-49d6-4904-9306-5bee6350c37a>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00336-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Department, Princeton University
High Energy Theory Seminar - Bertrand Duplantier, Institute for Theoretical Physics, Saclay - Schramm-Loewner Evolution and Liouville Quantum Gravity
Liouville quantum gravity in two dimensions is described by the ``random Riemannian manifold'' obtained by changing the Lebesgue measure in the plane by a random conformal factor, the exponential of
the Gaussian free field. This ``random surface" is believed to be the continuum scaling limit of certain discretized random surfaces that can be studied with combinatorics and random matrix theory.
When boundary arcs of a Liouville quantum gravity random surface are conformally welded to each other (in a boundary quantum-length-preserving way) the resulting interface is a random curve described
by the Schramm-Loewner evolution (SLE). This allows to develop a theory of quantum fractal measures, consistent with the Knizhnik-Polyakov-Zamolochikov (KPZ) relation, and to analyze their evolution
under conformal welding maps related to SLE. As an application, one can construct quantum length and boundary intersection measures on the SLE curve itself.
(Joint work with Scott Sheffield, MIT- Math)
Location: PCTS Seminar Room
Date/Time: 02/27/13 at 3:30 pm - 02/27/13 at 4:30 pm
Category: High Energy Theory Seminar
Department: Physics | {"url":"http://www.princeton.edu/physics/events_archive/viewevent.xml?id=579","timestamp":"2014-04-18T09:26:25Z","content_type":null,"content_length":"10750","record_id":"<urn:uuid:f55fe591-bd55-472b-ac45-48ee9f765f19>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00108-ip-10-147-4-33.ec2.internal.warc.gz"} |
Body of Knowledge - Six Sigma Green Belt
The last administration of the current SSGB Body of Knowledge will be December 5, 2014. The first administration of the new SSGB Body of Knowledge will be December 6, 2014. Compare the new and old
SSGB Body of Knowledge.
Body of Knowledge
Included in this body of knowledge are explanations (subtext) and cognitive levels for each topic or subtopic in the test. These details will be used by the Examination Development Committee as
guidelines for writing test questions and are designed to help candidates prepare for the exam by identifying specific content within each topic that can be tested. Except where specified, the
subtext is not intended to limit the subject or be all-inclusive of what might be covered in an exam but is intended to clarify how topics are related to the role of the Certified Six Sigma Green
Belt. The descriptor in parentheses at the end of each subtext entry refers to the highest cognitive level at which the topic will be tested. A complete description of cognitive levels is provided at
the end of this document.
I. Overview: Six Sigma and the Organization (15 Questions)
A. Six sigma and organizational goals
1. Value of six sigma
Recognize why organizations use six sigma, how they apply its philosophy and goals, and the origins of six sigma (Juran, Deming, Shewhart, etc.). Describe how process inputs, outputs, and
feedback impact the larger organization. (Understand)
2. Organizational drivers and metrics
Recognize key drivers for business (profit, market share, customer satisfaction, efficiency, product differentiation) and how key metrics and scorecards are developed and impact the
entire organization. (Understand)
3. Organizational goals and six sigma projects
Describe the project selection process including knowing when to use six sigma improvement methodology (DMAIC) as opposed to other problem-solving tools, and confirm that the project
supports and is linked to organizational goals. (Understand)
B. Lean principles in the organization
1. Lean concepts and tools
Define and describe concepts such as value chain, flow, pull, perfection, etc., and tools commonly used to eliminate waste, including kaizen, 5S, error-proofing, value-stream mapping,
etc. (Understand)
2. Value-added and non-value-added activities
Identify waste in terms of excess inventory, space, test inspection, rework, transportation, storage, etc., and reduce cycle time to improve throughput. (Understand)
3. Theory of constraints
Describe the theory of constraints. (Understand)
C. Design for Six Sigma (DFSS) in the organization
1. Quality function deployment (QFD)
Describe how QFD fits into the overall DFSS process. (Understand) (Note: the application of QFD is covered in II.A.6.)
2. Design and process failure mode and effects analysis (DFMEA & PFMEA)
Define and distinguish between design FMEA (DFMEA) and process (PFMEA) and interpret associated data. (Analyze) (Note: the application of FMEA is covered in II.D.2.)
3. Road maps for DFSS
Describe and distinguish between DMADV (define, measure, analyze, design, verify) and IDOV (identify, design, optimize, verify), identify how they relate to DMAIC and how they help close
the loop on improving the end product/process during the design (DFSS) phase. (Understand)
II. Six Sigma – Define (25 Questions)
A. Process Management for Projects
1. Process elements
Define and describe process components and boundaries. Recognize how processes cross various functional areas and the challenges that result for process improvement efforts. (Analyze)
2. Owners and stakeholders
Identify process owners, internal and external customers, and other stakeholders in a project. (Apply)
3. Identify customers
Identify and classify internal and external customers as applicable to a particular project, and show how projects impact customers. (Apply)
4. Collect customer data
Use various methods to collect customer feedback (e.g., surveys, focus groups, interviews, observation) and identify the key elements that make these tools effective. Review survey
questions to eliminate bias, vagueness, etc. (Apply)
5. Analyze customer data
Use graphical, statistical, and qualitative tools to analyze customer feedback. (Analyze)
6. Translate customer requirements
Assist in translating customer feedback into project goals and objectives, including critical to quality (CTQ) attributes and requirements statements. Use voice of the customer analysis
tools such as quality function deployment (QFD) to translate customer requirements into performance measures. (Apply)
B. Project management basics
1. Project charter and problem statement
Define and describe elements of a project charter and develop a problem statement, including baseline and improvement goals. (Apply)
2. Project scope
Assist with the development of project definition/scope using Pareto charts, process maps, etc. (Apply)
3. Project metrics
Assist with the development of primary and consequential metrics (e.g., quality, cycle time, cost) and establish key project metrics that relate to the voice of the customer. (Apply)
4. Project planning tools
Use project tools such as Gantt charts, critical path method (CPM), and program evaluation and review technique (PERT) charts, etc. (Apply)
5. Project documentation
Provide input and select the proper vehicle for presenting project documentation (e.g., spreadsheet output, storyboards, etc.) at phase reviews, management reviews and other
presentations. (Apply)
6. Project risk analysis
Describe the purpose and benefit of project risk analysis, including resources, financials, impact on customers and other stakeholders, etc. (Understand)
7. Project closure
Describe the objectives achieved and apply the lessons learned to identify additional opportunities. (Apply)
C. Management and planning tools
Define, select, and use 1) affinity diagrams, 2) interrelationship digraphs, 3) tree diagrams, 4) prioritization matrices, 5) matrix diagrams, 6) process decision program (PDPC) charts, and
7) activity network diagrams. (Apply)
D. Business results for projects
1. Process performance
Calculate process performance metrics such as defects per unit (DPU), rolled throughput yield (RTY), cost of poor quality (COPQ), defects per million opportunities (DPMO) sigma levels and
process capability indices. Track process performance measures to drive project decisions. (Analyze)
2. Failure mode and effects analysis (FMEA)
Define and describe failure mode and effects analysis (FMEA). Describe the purpose and use of scale criteria and calculate the risk priority number (RPN). (Analyze)
E. Team dynamics and performance
1. Team stages and dynamics
Define and describe the stages of team evolution, including forming, storming, norming, performing, adjourning, and recognition. Identify and help resolve negative dynamics such as
overbearing, dominant, or reluctant participants, the unquestioned acceptance of opinions as facts, groupthink, feuding, floundering, the rush to accomplishment, attribution, discounts,
plops, digressions, tangents, etc. (Understand)
2. Six sigma and other team roles and responsibilities
Describe and define the roles and responsibilities of participants on six sigma and other teams, including black belt, master black belt, green belt, champion, executive, coach,
facilitator, team member, sponsor, process owner, etc. (Apply)
3. Team tools
Define and apply team tools such as brainstorming, nominal group technique, multi-voting, etc. (Apply)
4. Communication
Use effective and appropriate communication techniques for different situations to overcome barriers to project success. (Apply)
III. Six Sigma – Measure (30 Questions)
A. Process analysis and documentation
1. Process modeling
Develop and review process maps, written procedures, work instructions, flowcharts, etc. (Analyze)
2. Process inputs and outputs
Identify process input variables and process output variables (SIPOC), and document their relationships through cause and effect diagrams, relational matrices, etc. (Analyze)
B. Probability and statistics
1. Drawing valid statistical conclusions
Distinguish between enumerative (descriptive) and analytical (inferential) studies, and distinguish between a population parameter and a sample statistic. (Apply)
2. Central limit theorem and sampling distribution of the mean
Define the central limit theorem and describe its significance in the application of inferential statistics for confidence intervals, control charts, etc. (Apply)
3. Basic probability concepts
Describe and apply concepts such as independence, mutually exclusive, multiplication rules, etc. (Apply)
C. Collecting and summarizing data
1. Types of data and measurement scales
Identify and classify continuous (variables) and discrete (attributes) data. Describe and define nominal, ordinal, interval, and ratio measurement scales. (Analyze)
2. Data collection methods
Define and apply methods for collecting data such as check sheets, coded data, etc. (Apply)
3. Techniques for assuring data accuracy and integrity
Define and apply techniques such as random sampling, stratified sampling, sample homogeneity, etc. (Apply)
4. Descriptive statistics
Define, compute, and interpret measures of dispersion and central tendency, and construct and interpret frequency distributions and cumulative frequency distributions. (Analyze)
5. Graphical methods
Depict relationships by constructing, applying and interpreting diagrams and charts such as stem-and-leaf plots, box-and-whisker plots, run charts, scatter diagrams, Pareto charts, etc.
Depict distributions by constructing, applying and interpreting diagrams such as histograms, normal probability plots, etc. (Create)
D. Probability distributions
Describe and interpret normal, binomial, and Poisson, chi square, Student’s t, and F distributions. (Apply)
E. Measurement system analysis
Calculate, analyze, and interpret measurement system capability using repeatability and reproducibility (GR&R), measurement correlation, bias, linearity, percent agreement, and precision/
tolerance (P/T). (Evaluate)
F. Process capability and performance
1. Process capability studies
Identify, describe, and apply the elements of designing and conducting process capability studies, including identifying characteristics, identifying specifications and tolerances,
developing sampling plans, and verifying stability and normality. (Evaluate)
2. Process performance vs. specification
Distinguish between natural process limits and specification limits, and calculate process performance metrics such as percent defective. (Evaluate)
3. Process capability indices
Define, select, and calculate C[p] and C[pk], and assess process capability. (Evaluate)
4. Process performance indices
Define, select, and calculate P[p], P[pk], C[pm], and assess process performance. (Evaluate)
5. Short-term vs. long-term capability
Describe the assumptions and conventions that are appropriate when only short-term data are collected and when only attributes data are available. Describe the changes in relationships
that occur when long-term data are used, and interpret the relationship between long- and short-term capability as it relates to a 1.5 sigma shift. (Evaluate)
6. Process capability for attributes data
Compute the sigma level for a process and describe its relationship to P[pk]. (Apply)
IV. Six Sigma – Analyze (15 Questions)
A. Exploratory data analysis
1. Multi-vari studies
Create and interpret multi-vari studies to interpret the difference between positional, cyclical, and temporal variation; apply sampling plans to investigate the largest sources of
variation. (Create)
2. Simple linear correlation and regression
Interpret the correlation coefficient and determine its statistical significance (p-value); recognize the difference between correlation and causation. Interpret the linear regression
equation and determine its statistical significance (p-value). Use regression models for estimation and prediction. (Evaluate)
B. Hypothesis testing
1. Basics
Define and distinguish between statistical and practical significance and apply tests for significance level, power, type I and type II errors. Determine appropriate sample size for
various test. (Apply).
2. Tests for means, variances, and proportions
Define, compare, and contrast statistical and practical significance. (Apply)
3. Paired-comparison tests
Define and describe paired-comparison parametric hypothesis tests. (Understand)
4. Single-factor analysis of variance (ANOVA)
Define terms related to one-way ANOVAs and interpret their results and data plots. (Apply)
5. Chi square
Define and interpret chi square and use it to determine statistical significance. (Analyze)
V. Six Sigma – Improve & Control (15 Questions)
A. Design of experiments (DOE)
1. Basic terms
Define and describe basic DOE terms such as independent and dependent variables, factors and levels, response, treatment, error, repetition, and replication. (Understand)
2. Main effects
Interpret main effects and interaction plots. (Apply)
B. Statistical process control (SPC)
1. Objectives and benefits
Describe the objectives and benefits of SPC, including controlling process performance, identifying special and common causes, etc. (Analyze)
2. Rational subgrouping
Define and describe how rational subgrouping is used. (Understand)
3. Selection and application of control charts
Identify, select, construct, and apply the following types of control charts:
4. Analysis of control charts
Interpret control charts and distinguish between common and special causes using rules for determining statistical control. (Analyze)
C. Implement and validate solutions
Use various improvement methods such as brainstorming, main effects analysis, multi-vari studies, FMEA, measurement system capability re-analysis, and post-improvement capability analysis to
identify, implement, and validate solutions through F-test, t-test, etc . (Create)
D. Control plan
Assist in developing a control plan to document and hold the gains, and assist in implementing controls and monitoring systems. (Apply)
Six Levels of Cognition based on Bloom’s Taxonomy (Revised)
In addition to content specifics, the subtext detail also indicates the intended complexity level of the test questions for that topic. These levels are based on the Revised “Levels of Cognition”
(from Bloom’s Taxonomy, 2001) and are presented below in rank order, from least complex to most complex.
Be able to remember or recognize terminology, definitions, facts, ideas, materials, patterns, sequences, methodologies, principles, etc. (Also commonly referred to as recognition, recall, or rote
Be able to read and understand descriptions, communications, reports, tables, diagrams, directions, regulations, etc.
Be able to apply ideas, procedures, methods, formulas, principles, theories, etc., in job-related situations.
Be able to break down information into its constituent parts and recognize the parts’ relationship to one another and how they are organized; identify sublevel factors or salient data from a complex
Be able to make judgments regarding the value of proposed ideas, solutions, methodologies, etc., by using appropriate criteria or standards to estimate accuracy, effectiveness, economic benefits,
Be able to put parts or elements together in such a way as to show a pattern or structure not clearly there before; able to identify which data or information from a complex set is appropriate to
examine further or from which supported conclusions can be drawn. | {"url":"http://asq.org/cert/control/six-sigma-green-belt/bok","timestamp":"2014-04-19T17:26:04Z","content_type":null,"content_length":"37964","record_id":"<urn:uuid:ac10d5d8-eee7-4e58-a87d-139b075d0486>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00577-ip-10-147-4-33.ec2.internal.warc.gz"} |
Harwood Heights Prealgebra Tutor
Find a Harwood Heights Prealgebra Tutor
...A little bit about me: -Bachelor of Arts, Human Development from Prescott College, High Honors, 4.0 GPA -Associate in Arts, Associate in General Studies, College of DuPage, High Honors, 4.0 GPA
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Dark Calculus
In calculus, we are taught that radians are the best unit of measurement for angles. But technically, it is more accurate to say that radians are a best unit of measurement for angles. There is one
other unit which is exactly as good: The Dark Radian.
Whereas a radian is defined so that one revolution equals 2pi radians, a Dark Radian is defined so that one revolution equals −2pi radians. Thus, 1 Dark Radian is −1 radians, and 1 radian is −1 Dark
Dark Trigonometry
We define the dark trigonometric functions as follows, where x is a real number:
• Darksine. We define Darksin(x) = sin(−x), where sin is the usual sine function. Thus, Darksin(x) is the “opposite/hypotenuse” corresponding to an angle of x Dark Radians in a right triangle.
□ Equivalently, Darksin(x) is “opposite/hypotenuse” corresponding to an angle of x radians in a wrong triangle. A wrong triangle is, of course, a triangle one of whose angles is pi/2 Dark
• Darkosine. We define Darkos(x) = cos(−x), similar to the above.
• Darktangent. We define Darktan(x) = tan(−x), similar to the above.
The dark trigonometric functions differentiate as follows: (is differentiate an ergative verb?)
• (Darksin(x))’ = −Darkcos(x).
• (Darkcos(x))’ = Darksin(x).
• (Darktan(x))’ = −1/Darkcos²(x) (or −Darksec²(x) in some dialects).
The Dark trig functions have inverse functions, when suitably restricted. These we call Darkarcsine, Darkarccosine, and Darkarctangent. Their derivatives are left as an exercise to the reader.
Dark Exponentials
It is well-known that there is one base of exponents preferred above all others. I am speaking, of course, of the Dark Exponential Base, æ, also known as “Dark e”. It has a numerical value of
approximately 0.36787. It is related to its holier sister, e, by the formula æ=1/e.
The reason æ is preferred is because it is the unique number such that (æ^x)’ = −æ^x. In other words, it is the base of an exponential function which is its own negative derivative. Thus, the unique
(up to linear combinations) solutions to the differential equation y””=y are y=Darksin(x), y=Darkos(x), y=æ^x, and y=æ^−x, explaining why these four Dark Functions play such a key role in the deep
laws and corruptions of our Dark universe.
The Darksponential can be used to define the Dark-Hyperbolic Trigonometric functions:
• Hyperbolic Darksine: Darksinh(x) = ½(æ^x−æ^−x)
• Hyperbolic Darkosine: Darkosh(x) = ½(æ^x+æ^−x)
• Hyperbolic Darktangent: Darktanh(x) = Darksinh(x)/Darkosh(x)
Darkomplex Analysis
We can extend the Darxponential Function to have domain the set of complex numbers by means of the defining equation
• æ^x+yi = æ^x(Darkos(y)+iDarksin(y)).
Thus æ^yi is the point on the Darkomplex unit circle with an angle of y Dark Radians from the positive x-axis.
Plugging in x=0, y=pi yields the infamous Dark-Euler’s Formula: | {"url":"http://www.xamuel.com/dark-calculus/","timestamp":"2014-04-21T02:02:31Z","content_type":null,"content_length":"23875","record_id":"<urn:uuid:de42627f-8684-4f5a-bfea-7f408e78daf5>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00130-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Geometrical meaning of √2
February 15th 2011, 04:20 PM #1
Apr 2008
Geometrical meaning of √2
See the drawing at the attached file, the sides of the triangle are 1,1,√2
Ιwant a geometric relation for √2, i.e. something like that:
and thus
But a/1=b/ais probably wrong. Can you proove it if it’s correct?
Any other similar relation I could get?
A right triangle is one in which ( leg1)^2 + (leg 2)^2 = (hypothenuse)^2
1^2 + 1^2 = rad2^2
hypothenuse =rad2
Any equal legged right rt triangle has a hypothenuse= to leg* rad2
If x is the length of the leg of an isosceles right triangle, then (x)(√2) is the length of its hypotenuse ... we can prove it using pythagorean theroem .
geometric meaning of rad2
Hello Thodorisk,
I read your largest triangle as a right triangle with sides 1 and 1 and hyoothenuse =rad 2
1^2 + 1^2 =rad2^2 proving that it is a right triangle If the two legs of a right triangle are equal then the hypothenuse = leg times rad2 Call a leg a then the hypot is atimes rad2
No rad, sin, cos etc please.
a is not √2-1.
It's not clear exactly what you are trying to do. I can see that you have a right triangle with legs of length 1 and so hypotenuse of length $\sqrt{2}$. Then, I think, you have marked of a length
of 1 on the hypotenuse and drawn a line from the right angle to that point. But how is the second line, separting the segments marked "a" and "b" determined. Is it parallel to the first line from
the right angle?
That huge drawing shows what's parallel and what's not. As I am not hoping to get an answer but only warnings from the moderators, it's a waste of time for me too. Bann me and I am over with your
February 16th 2011, 06:13 AM #2
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February 17th 2011, 03:14 PM #3
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February 19th 2011, 07:14 AM #7
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February 21st 2011, 05:48 PM #10
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Compilers - Am I writing one?
Michael Torrie torriem at gmail.com
Mon Jun 30 09:58:57 MDT 2008
Hans Fugal wrote:
> In which way? They take ascii art as input? I had no trouble with the
> syntax diagrams in 431, but I found them hard to work with. I couldn't
> send one by email easily. I had to have a PDF viewer and scan through
> dozens of pages of diagrams (we know how much fun that can be) or lug
> the large book (which I printed out at Kinkos) around. I decided to
> write my own parser generator, and so I had to enter all those diagrams
> into BNF format (which seemed like a reasonable machine-readable
> choice), and while tedious I don't remember it being difficult. In fact,
> I seem to remember very quickly finding a few patterns, and converting
> to BNF was primarily a rote exercise.
Good point. I presume from a casual googling that ANTLR must have some
notation that allows one to easily define a syntax diagram using some
notation. I would guess (and that's all it is) that the syntax would be
similar to how PyParsing defines a grammar. Easy ways to say "one or
more" or "zero or more," for example. Given that format it is just as
clear and easy to run through a parser generator as BNF.
I am curious about your easy rote exercise to create BNF from a syntax
diagram. How do you do it? How do you properly convert an iterative
diagram into the recursive productions that BNF requires? It's not
always correct to convert an iterative production into right-recursive
format, yet left-recursive is problematic. How do you deal with this?
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the Exclusive OR operator
Author the Exclusive OR operator
Ranch Hand
Joined: Jun 28, I have a very basic question:
2003 Exclusive or means that if the two operand bits are different the result is 1; otherwise the result is 0
Posts: 547
I read in a book that 2^5 evaluates to 7. How is that possible ?
I like...
SCJP 1.4, SCWCD 1.4, SCBCD 1.3, SCBCD 5
Visit my blog
Ranch Hand
Joined: Dec 14,
2004 So, the result of an XOR would be
Posts: 464
ph34r my 133t j4v4 h4><0r1ng sk177z
Ranch Hand
Joined: Jun 28,
2003 I still dont get it: shouldn't the result be 1 or 0 ?
Posts: 547 Why are we adding the 2 operands here ?
[ September 22, 2005: Message edited by: Max longbeach ]
I like...
Ranch Hand
The operands 2 and 5 are not being added, although addition in this case produces the same result, which may be confusing you. The operands are the individual bits that represent
Joined: Jul 10, each number in binary notation, not decimal notation.
Posts: 214 Here's another example 2^3=1.
I like... The ^ XOR operator works at the bit level, so convert the decimal numbers to binary:
2 = 0010
3 = 0011
^ = 0001
SCJP, SCWCD
Ranch Hand
You are thinking of a single-bit, exclusive-or operation:
Joined: Sep 16,
2005 0 EXOR 0 is 0
Posts: 1780 0 EXOR 1 is 1
1 EXOR 0 is 1
1 EXOR 1 is 0
But the Java ^ operation on integral types is a bitwise operation.
The operation defined above is applied to all the bits of the ^ operands,
in parallel, as the previous poster demonstrated.
Also note than ^ is defined for booleans, and does want you wish: the
result is true iff the operand have different boolean values:
false ^ false is false
false ^ true is true
true ^ false is true
true ^ true is false
But then again, you get the same result with boolean's == operation.
There is no emoticon for what I am feeling!
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The Mathematics of Dominoes
Let's define a [n-n] domino set to be all possible dominoes between [0-0] and [n-n]. The traditional Western domino sets are the [6-6] or double six set, the [9-9] or double nine set, and the [12-12]
or double twelve set.
While is clear that the double six set is derived from dice, I have no idea where the larger sets started. The Eskimos and Inuit Indians of Canada have animal bone domino sets with 61 to 148 pieces
in a set used for gambling games. I do not know if these are of native origin or if they might have seen Western dominoes and attempted to copy them.
Size of a Set
The number of tiles in a set of [n-n] dominoes is given by the formula ((n^2 + 3n + 2)/ 2). For example the number tiles in a [18-18] set is (18*18 + 3*18 + 2)/2 = 190.
Type of Set Number of Tiles Comment
[0-0] 1
[1-1] 3
[2-2] 6
[3-3] 10
[4-4] 15
[5-5] 21
[6-6] 28 <= most common set
[7-7] 36
[8-8] 45
[9-9] 55 <= commercially available set
[10-10] 66
[11-11] 78
[12-12] 91 <= commercially available set
[13-13] 105
[14-14] 120
[15-15] 136 <= commercially available set
[16-16] 153
[17-17] 171
[18-18] 190 <= commercially available set
Important Properties of a Set
One of the most important properties of a [n-n] set of dominoes is that any number k (where 0 <= k <= n) will appear (n+2) times on the tiles. The reason that this is important to a player is that
when you want to know if you can block another player, you can quickly count the occurrences of a number in your hand and on the table. If the count is equal to (n+2), you know that no other player
can match that number. Look for a double, usually played as a spinner in most games, and you will have found four of the eight occurrences immediately.
The Single Train Problem
Given a set of [n-n] dominoes, is it possible to arrange all of the tiles into a single train? A train is a line of tiles each of whose ends match the end of the tiles to their immediate left and
right, with the exception of the two end tiles which match on one end only, of course.
Can you arrange a set into a single circular train? A circular train is ring of tiles laid end to end where both ends of each tile matches its left and right hand neighbors. Obviously, if a circular
train exists, it can be broken apart at any point to give a single train.
By simple trial and error, the answer for the zero set is "yes" for a single train because it is a trivial train itself. The answer is "no" for a circular train because the one member of the set
cannot bend around to touch itself.
The [1-1] set is made up of the tiles [0-0], [0-1] and [1-1] which is a train when played in that order, but it is not a circular train. As the value of (n) increases, answering this question by
trial and error is going to get to be much harder.
Instead, let's use a different approach; graph theory. While I do not have time to go into the mathematics of graph theory in detail, I hope I can cover enough of it to explain this solution.
A graph is a mathematical modeling structure made up of nodes (dots) and edges (lines). The number of edges coming in and out of node is the degree of the node.
Let's make the nodes represent the numbers 0 to (n) and edges between them be the tiles in a [n-n] domino set. A double is shown as an arc that starts and finishes on the same node. These graphs will
have the same number of edges as there are dominoes in the set they model. For example, here is the graph for a [2-2] domino set:
Now, we can transform the problem of finding a train into the problem of finding a path thru the graph of the domino set that touches all the edges. The circular train problem becomes that of finding
a path thru the graph of the domino set which includes all the edges and returns to the node from which it started. Fortunately, this is a graph theory problem that has already been solved by
Leonhard Euler hundreds of years ago and it is known as the "Bridges of Königsberg problem". For such a circular train to exist, either
1. Each node has an even degree, or
2. There must two and only two nodes with an odd edge.
Therefore, the [3-3] set cannot be put into a single train because its nodes have degree 5. The Double six set can be put into a train because its nodes have degree 8. Notice that the degree of the
nodes of a [n-n] domino set graph will always be (n+2), so it is easy to answer the train and circular train questions. If (n) is even, then a circular train exists for that domino set. If (n) is
odd, then there are (n+1)/2 trains, each ending in a digit between zero and (n).
A common question is how many different trains can build with a standard double six set. Without doing the math, the answer is 7,959,229,931,520 if you count reversals or half that amount if you do
Clark's Law
This is a rather interesting result that says that in a blocked game of single spinner dominoes, the sum of the four arms of the tableau must always total to an even number.
The first corollary of Clark's Law is that the sum of the four hands in a blocked game is always an even number. This is because the double six set has 168 pips in it, which is an even number and an
even number minus an even number is an even number.
The second corollary of Clark's Law is that the sum of the hands of the two partnership in a blocked game must both be even or both be odd.
Clark's law derives from the facts that doubles are always even, so the spinner will have four identical halves against it. Then in each arm, the ends of the tiles must be paired, so they are even
until you get to the end of an arm. If the arm ends in a double, then the exposed tile is even.
A block can be made with two arms, three arms or four arms on the tableau. This reduces further to four arms with the same number showing, which gives us an even count, or with fours with two dead
numbers. The situation with three arms in a blocked game requires that they all end with the same suit as the spinner.
Dominoes and Checkerboards
The "mutilated checkerboard" is a classic demonstration of a method of setting up sets in a certain class of problems. The puzzle is to ask if you can cover a checkerboard whose opposite corners have
been cut off with domino tiles which exactly cover two cells of the checkerboard.
By inspection, there are 64 - 2 = 62 cells on the mutilated checkerboard, so we will need 31 tiles, assuming a solution exists. Now look at a how a domino can sit on the board -- it either faces
North-South or East-West. But no matter how it is oriented, a tile always covers one black and one white cell. The mutilated checkerboard has 32 of black cells and 30 whites cells. Therefore, you can
never cover the board.
You can find other proofs in the literature. Stan Wagon published "Fourteen Proofs of a Result about tiling a Rectangle", American Mathematical Monthly, 14, 94-1987, pp. 601-617.
One would expect that if the removed squares were of different colors both the answer and solution would be quite different. Indeed, cover the whole checkerboard with domino tiles. Cut two squares
from under one of the pieces. The remaining portion of the board will still be covered with the remaining domino tiles. Thus, at least, sometimes the remaining board is "coverable." As a warm-up
problem you may consider this one:
Problem 1
Cover two arbitrary squares of a checkerboard with a domino tile. Is it always possible to cover the remaining portion of the board with more tiles without disturbing the original piece?
Now, you may want to go further and relax one of the conditions in Problem 1: what if the squares are not adjacent? We already have a solution if they have the same color. There remains then just one
Problem 2
Two arbitrary squares of different colors have been removed from a checkerboard. Is it possible to cover the remaining portion of the board with domino so that each domino tile covers exactly two
Yes, it's always possible. To see this, draw two forks as shown in red on the diagram.
This splits the checkerboard into a chain of alternating squares. It takes only a little experimentation to convince oneself that the chain can be traversed by starting at any square and covering two
squares at a time with a tile. This observation actually solves the problem.
What would be the next question to ask related to covering of a checkerboard with domino? My son David came up with the following question:
Problem 3
Assume at every step we remove a pair of squares of different colors. What is the maximum number of pairs that may be removed such that it's always possible to cover the remaining portion of the
board with domino tiles?
An Aside
It's a very legitimate question to ask because obviously, at one extreme, when we remove all but two nonadjacent squares, the problem of covering whatever remains will be unsolvable for at least some
configurations. On the other hand, as we just saw, removing only two squares we have a solvable problem. Then there must be a borderline number somewhere in between.
At first the problem may seem to be difficult to solve. But you have to start somewhere. As is often the case, experimenting with a few particular cases helps gain insight into a more general
problem. So let's start with the next simplest case of four squares (2 white and 2 black) being removed. The forks that worked so well for the previous case become useless. Indeed, following the
chain we can remove two consecutive white and two consecutive black squares. This would not yet prove that thus obtained configuration is unsolvable; but doubts that the problem may not have an easy
solution may start creeping into your mind. At this point, it may make sense to think of the possibility of a negative result. Can we think of a counterexample? In other words, perhaps it's possible
to leave "noncoverable" board by just removing two pairs of squares.
What would make the board noncoverable? Recollecting our original problem, if it were possible to isolate a region of the checkerboard that contained an odd number of squares, we would be able to
claim that the solution to Problem 3 is 1, just one pair.
It's possible to remove two pairs of squares as to leave noncoverable checkerboard. Indeed, remove two white squares adjacent to a black corner. This creates a region consisting of a single (corner)
square that has no adjacent white squares.
Problem 4
Assume we are allowed to remove 2 white and 2 black squares so that the board does not yet fall apart, i.e., does not split into two or more separate pieces. Is it always possible to cover whatever
remains of the board with domino?
Trying to answer Problem 4 I ran into one noncoverable configuration with 3 pairs of squares removed. The diagram depicts the lower right corner.
The grey outined squares are cut off (3 black squares - the three white squares are assumed to have been removed elsewhere.) The yellow rectangle represents a domino tile in the only position where
it can cover the corner square. This blocks the single white square marked with a cross to its left. Thus the configuration is indeed noncoverable. We arrive at the conclusion that a single argument
is going to solve both Problem 3 (with the extra requirement that the removed squares do not disconnect the board) and Problem 4.
Problem 5
This is to check your understanding of the solution to Problem 2.
A rectangular 2nx2m board has been covered with domino tiles. Prove that there exists another cover such that no domino tile belongs to both covers.
• P.J.Davis and R.Hersh, The Mathematical Experience, Houghton Mifflin Company, Boston, 1981
• R.Honsberger, Mathematical Gems II, MAA, New Math Library, 1976
• M.Kac and S.M.Ulam, Mathematics and Logic, Dover Publications, NY, 1968. | {"url":"http://www.pagat.com/tile/wdom/math.html","timestamp":"2014-04-21T00:31:14Z","content_type":null,"content_length":"19051","record_id":"<urn:uuid:c25ac883-6f59-4bc1-bce0-5b748b906a85>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00491-ip-10-147-4-33.ec2.internal.warc.gz"} |
vectors - find equidistant point
Using vectors: Find the point on the y axis that is equidistant from the points (2,-1,1) and (0,1,3)
Hi The point $P$ we're looking for is on the $y$ axis hence $P(0,\,y,\,0)$ The distance between $P(0,\,y,\,0)$ and $A(2,\,-1,\,1)$ is $\|\vec{AP}\|=\sqrt{(2-0)^2+(-1-y)^2+(1-0)^2}=\sqrt{5+(1+y)^2}$
The distance between $P(0,\,y,\,0)$ and $B(0,\,1,\,3)$ is $\|\vec{BP}\|=\sqrt{(0-0)^2+(1-y)^2+(3-0)^2}=\sqrt{9+(1-y)^2}$ We want these two distances to be equal thus $\|\vec{AP}\|=\|\vec{BP}\| \
Leftrightarrow \sqrt{9+(1-y)^2}=\sqrt{5+(1+y)^2}$ Squaring both sides gives $9+(1-y)^2=5+(1+y)^2$ which you can solve for $y$. Does it help ? | {"url":"http://mathhelpforum.com/calculus/38947-vectors-find-equidistant-point.html","timestamp":"2014-04-19T00:28:51Z","content_type":null,"content_length":"34390","record_id":"<urn:uuid:a5f9852f-86dd-42f2-ab56-51b38278405d>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00006-ip-10-147-4-33.ec2.internal.warc.gz"} |
Related Rates Problems
November 9th 2009, 08:32 PM
Related Rates Problems
Two cars start moving from the same point. One travels north at 60 mph and the other travels west at 25 mph. At what rate is the distance between the cars increasing two hours later?
What I've Done:
So I drew a right triangle like this:
I know I want to find $\frac{dx}{dt}$ at t = 2, and I know I have $\frac{dN}{dt}$ which is 60mph and $\frac{dW}{dt}$ which is 25mph.
I wrote the equation $N^2 + W^2 = x^2$. Is this correct? Am I doing this right?
Do I just differentiate this equation with respect to x, and then plug in the values for $\frac{dN}{dt}$ and $\frac{dW}{dt}$ ?
I really have no idea what I'm doing. I think I need t in my equation somewhere...
November 9th 2009, 08:54 PM
Two cars start moving from the same point. One travels north at 60 mph and the other travels west at 25 mph. At what rate is the distance between the cars increasing two hours later?
What I've Done:
So I drew a right triangle like this:
I know I want to find $\frac{dx}{dt}$ at t = 2, and I know I have $\frac{dN}{dt}$ which is 60mph and $\frac{dW}{dt}$ which is 25mph.
I wrote the equation $N^2 + W^2 = x^2$. Is this correct? Am I doing this right?
Do I just differentiate this equation with respect to x, and then plug in the values for $\frac{dN}{dt}$ and $\frac{dW}{dt}$ ?
I really have no idea what I'm doing. I think I need t in my equation somewhere...
Differentiate you relationship with respect to t, and then sub in the proper values.
November 9th 2009, 08:57 PM
So far so good.
Differentiate (implicitly) to find an expression for dx/dt.
Then plug in dN/dt=60, dW/dt=25, N=120 (which is the value of N when t=2), W=50 and x=sqrt(120^2+50^2).
+ not - under the square root, VonNemo19.
November 9th 2009, 09:16 PM
So I differentiated implicitly and got $\frac{dx}{dt} = (\frac{1}{2})*( \frac{(2N \frac{dN}{dt} + 2W \frac{dW}{dt})}{\sqrt{N^2 + W^2)}})$
Is this right?
I ended up getting the answer to be 65mph. Is that also right?
November 9th 2009, 09:20 PM
One travels north at 60mph, so after 2 hours N=120. You don't need a "formula" to work that out. Same idea for W.
Yes your derivative is correct. Now sub in the bits and pieces you know.
November 9th 2009, 10:16 PM
Did you get it? | {"url":"http://mathhelpforum.com/calculus/113600-related-rates-problems-print.html","timestamp":"2014-04-17T10:52:55Z","content_type":null,"content_length":"10923","record_id":"<urn:uuid:1ca01b6d-22b2-4345-ade9-4e7cb1dfe695>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00103-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Find the degrees.
February 16th 2010, 04:55 PM #1
Sep 2008
Find the degrees.
Hey all,
Got a question here that states:
ABCD is a quadrilateral inscribed in a circle; AB and CD are each equal to the radius. AC and BD meet in E. Find the number of degrees in angle AEB.
I think I have the picture correct, as shown below:
Any help would be greatly appreciated. Thanks guys.
There is not enough information to solve this problem. Any other rectangle so drawn, even a square, would have and angle indicated as it is here. If it was a square, that might be sufficient
additional information and the angle at the center would be 90. Otherwise it can't be found form this diagram.
If AB=CD=radius
then AB=AE=BE
then triangle ABE is equil triangle!
So, angle AEB is (you should know, right?)
(Or, am I reading the question incorrectly?)
This isn't a perfect drawing but I would take the question as this. I'm just not sure where to go from there.
February 16th 2010, 05:54 PM #2
Feb 2010
February 16th 2010, 10:34 PM #3
Feb 2010
February 17th 2010, 03:18 AM #4
Feb 2010
February 17th 2010, 05:41 AM #5
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July 21 - 25, 2008
Algebraic and Analytic Geometry Summary
Monday - Friday, July 21 - 25, 2008
In this final week of the PCMI summer session, we attended Professor Garrity's course for the first half of the week, where we worked to generalize our results on the topology of points on cubic
curves to curves of higher degree. We went through an intuitive argument for a formula relating the genus of these spaces to the degree of the curve, and discussed how to define a genus for cases
that involve curves over algebraically complete fields other than the complex numbers. Professor Garrity concluded the course with some statements, intuition, definitions and lemmas on the way to
proving the Riemann-Roch Theorem, which relates function theory (specifically, the dimensions of vector spaces defined by specific sets of poles and zeros, with multiplicity) to topology
(specifically, the genus of the topological spaces defined by polynomial curves). This is a big, non-intuitive result, and though we got that, the pace of the course made it challenging to appreciate
the import of the theorem. Still, we've come a long way since the beginning of the course.
Most of us did not attend Thursday's lecture, and none of us attended Friday's lecture, because we were working hard to bring together our work on our projects. Our nomographs (function diagram)
group explored the linear and quadratic cases more carefully, and used these examples to help motivate the real projective line and illuminate its structure, as well as making some interesting
connections between Mobius transformations and conics. They also created some lesson materials and a Sketchpad tool to help look at both the linear and quadratic cases. Our projective geometry group
created a structure for a sequence of lessons that examined the invariants and changes to shapes when projected, including some opportunities to use either manipulatives (especially the Línárt
sphere) or technology (especially Cabri 3D) to make these investigations more rich, leading up to some amazing animations from the Dimensions video series. Finally, we assembled some ideas from the
lectures and summaries of our projects into a coherent presentation for our fellow SSTP participants.
Overall, we enjoyed attending the lectures, working through the course material, and working together to devise worthwhile experiences for secondary students. A+++ would do again!
Back to Journal Index
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IAS/Park City Mathematics Institute is an outreach program of the School of Mathematics
at the Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540
Send questions or comments to: Suzanne Alejandre and Jim King
With program support provided by Math for America
This material is based upon work supported by the National Science Foundation under Grant No. 0314808.
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This Article
Bibliographic References
Add to:
Combinatorial Properties of Two-Level Hypernet Networks
November 1999 (vol. 10 no. 11)
pp. 1192-1199
ASCII Text x
Hui-Ling Huang, Gen-Huey Chen, "Combinatorial Properties of Two-Level Hypernet Networks," IEEE Transactions on Parallel and Distributed Systems, vol. 10, no. 11, pp. 1192-1199, November, 1999.
BibTex x
@article{ 10.1109/71.809576,
author = {Hui-Ling Huang and Gen-Huey Chen},
title = {Combinatorial Properties of Two-Level Hypernet Networks},
journal ={IEEE Transactions on Parallel and Distributed Systems},
volume = {10},
number = {11},
issn = {1045-9219},
year = {1999},
pages = {1192-1199},
doi = {http://doi.ieeecomputersociety.org/10.1109/71.809576},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - JOUR
JO - IEEE Transactions on Parallel and Distributed Systems
TI - Combinatorial Properties of Two-Level Hypernet Networks
IS - 11
SN - 1045-9219
EPD - 1192-1199
A1 - Hui-Ling Huang,
A1 - Gen-Huey Chen,
PY - 1999
KW - Best container
KW - container
KW - fault diameter
KW - graph-theoretic interconnection network
KW - hypernet network
KW - wide diameter.
VL - 10
JA - IEEE Transactions on Parallel and Distributed Systems
ER -
Abstract—The purpose of this paper is to investigate combinatorial properties of the hypernet network. The hypernet network owns two structural advantages: expansibility and equal degree. Besides, it
was shown efficient in both communication and computation. Since the number of nodes contained in the hypernet network increases very rapidly with expansion level, we emphasize the hypernet network
of two levels (denoted by HN$(d, 2)$) with a practical view. Recently, combinatorial properties such as container (i.e., node-disjoint paths), wide diameter, and fault diameter have received much
attention due to their increasing importance and applications in networks. In this paper, the following results are obtained for HN$(d, 2)$: 1) best containers with width $d-1$, 2) containers with
(maximum) width $d$, 3) the ($d-1$)-wide diameter, 4) the $d$-wide diameter, 5) the $(d - 2)$-fault diameter, and 6) the $(d-1)$-fault diameter. More specifically, between every two nodes of HN$(d,
2)$, $d$ (or $d-1$) packets can be transmitted simultaneously with at most $D + 2$ (or $D + 1$) parallel steps, where $D=2d+1$ is the diameter of HN$(d, 2)$. Besides, the diameter of HN$(d, 2)$ will
increase by at most two (or one), if there are at most $d-1$ (or $d-2$) node faults. Our results reveal that HN$(d, 2)$ is not only efficient in parallel transmission, but robust in fault tolerance.
[1] S.B. Akers, D. Harel, and B. Krishnamurthy, “The Star Graph: An Attractive Alternative to then-Cube,” Proc. Int'l Conf. Parallel Processing, pp. 393-400, 1987.
[2] F. Buckley and F. Harary, Distance in Graphs. Addison-Wesley, 1990.
[3] G.H. Chen, S.C. Hwang, and D.R. Duh, “A General Broadcasting Scheme for Recursive Networks with Complete Connection,” Proc. Int'l Conf. Parallel and Distributed Systems, pp. 148-255, 1998.
[4] K. Day and A. Tripathi, “Characterization of Node Disjoint Paths in Arrangement Graphs,” Technical Report TR 91-43, Computer Science Dept., Univ. of Minnesota, Minneapolis, 1991.
[5] M. Dietzfelbinger, S. Madhavapeddy, and I.H. Sudborough, "Three disjoint path paradigms in star networks," Proc. Third IEEE Symp. Parallel and Distributed Processing, pp. 400-406, 1991.
[6] D.R. Duh and G.H. Chen, “Topological Properties of WK-Recursive Networks,” J. Parallel and Distributed Computing, vol. 23, pp. 468-474, 1994.
[7] D.R. Duh, G.H. Chen, and J.F. Fang, “Algorithms and Properties of a New Two-Level Network with Folded Hypercubes as Basic Modules,” IEEE Trans. Parallel and Distributed Systems, vol. 6, no. 7,
pp. 714-723, July 1995.
[8] A. El-Amawy and S. Latifi, "Properties and Performance of Folded Hypercubes," IEEE Trans. Parallel and Distributed Systems, vol. 2, no. 1, pp. 31-42, 1991.
[9] D.F. Hsu, “On Container Width and Length in Graphs, Groups, and Networks,” IEICE Trans. Fundamental of Electronics, Comm., and Computer Sciences, vol. A, no. 4, pp. 668-680, 1994.
[10] http://www.csie.ntu.edu.tw/~ghchenhypernet.html .
[11] H.L. Huang and G.H. Chen, “Combinatorial Properties of Two-Level Hypernet Networks,” Technical Report NTUCSIE 96-03, Dept. of Computer Science and Information Eng., Nat'l Taiwan Univ., pp. 41,
[12] H.L. Huang and G.H. Chen, “Topological Properties and Algorithms for Two-Level Hypernet Networks,” Networks, vol. 31, no. 2, pp. 105-118, 1998.
[13] K. Hwang and J. Ghosh, "Hypernet: A Communication Efficient Architecture for Constructing Massively Parallel Computers," IEEE Trans. Computers, pp. 1,450-1,466, 1987.
[14] M.S. Krishnamoorty and B. Krishnamoorty, "Fault diameter of interconnection networks," Computers and Mathematics with Applications, vol. 13, no. 5-6, pp. 577-582, 1987.
[15] S. Latifi, “Combinatorial Analysis of the Fault Diameter of the$n$-Cube,” IEEE Trans. Computers, vol. 42, no.1, pp. 27-33, 1993.
[16] S. Latifi, “On the Fault-Diameter of the Star Graph,” Information Processing Letters, vol. 46, pp. 143-150, June 1993.
[17] F.J. Meyer and D.K. Pradhan, "Flip-trees: Fault tolerant graphs with wide containers," IEEE Trans. Computers, vol. 37, no. 4, pp. 472-478, Apr. 1988.
[18] E.T. Ordman, “Fault-Tolerant Networks and Graph Connectivity,” J. Combinatorial Math. and Combinatorial Computing, vol. 1, pp. 191-205, 1987.
[19] F.P. Preparata and J. Vuillemin, “The Cube-Connected Cycles: A Versatile Network for Parallel Computation,” Comm ACM, vol. 24, no. 5, pp. 300-309, 1981.
[20] M.O. Rabin, Efficient Dispersal of Information for Security, Load Balancing and Fault Tolerance J. ACM, vol. 36, no. 2, pp. 335-348, 1989.
[21] Y. Rouskov and P.K. Srimani, “Fault Diameter of Star Graphs,” Information Processing Letters, vol. 48, pp. 243-251, 1993.
[22] Y. Saad and M. Schultz, "Topological Properties of Hypercubes," IEEE Trans. Computers, vol. 37, no. 7, pp. 867-872, July 1988.
[23] M.R. Samatham and D.K. Pradhan, "The de Bruijn Multiprocessor Network: A Versatile Parallel Processing and Sorting Network for VLSI," IEEE Trans. Computers, vol. 38, no. 4, pp. 567-581, Apr.
[24] H.S. Stone, “Parallel Processing with the Perfect Shuffle,” IEEE Trans. Computers, vol. 20, no. 2, pp. 153-161, Feb. 1971.
[25] G.D. Vecchia and C. Sanges, “A Recursively Scalable Network VLSI Implementation,” Future Generation Computer Systems, vol. 4, no. 3, pp. 235-243, 1988.
Index Terms:
Best container, container, fault diameter, graph-theoretic interconnection network, hypernet network, wide diameter.
Hui-Ling Huang, Gen-Huey Chen, "Combinatorial Properties of Two-Level Hypernet Networks," IEEE Transactions on Parallel and Distributed Systems, vol. 10, no. 11, pp. 1192-1199, Nov. 1999, doi:10.1109
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Dynamics in S^1: rotation number
April 6th 2011, 03:35 PM #1
Apr 2011
Dynamics in S^1: rotation number
Suppose f,g in Hom_+(S^1),i.e, orientation-preserving homeomorphisms of the circle (deg(f) = deg(g) = 1), such that f o g = g o f
Show that rho(f o g) = rho(f) + rho(g) (mod 1), where rho is the rotation number.
This is where I got so far:
Let F and G be liftings to R of f and g, respectively.
Then F o G and G o F are both liftings of f o g = g o f.
So there is an integer k such that F o G = G o F + k
Then resolving the recurrence a_{n+1} = a_n + n,
one can check that (F o G)^n = F^n o G^n -(n(n-1)/2).k (I)
Then ((F o G)^n(x)-x)/n = (F^n(G^n(x)) - G^n(x))/n + (G^n(x)-x)/n - ((n-1)/2)k
The problem here is that when taking limit in n->inf, the last term of the right side of the equation goes to -inf, which doesnt make sense, since Poincare says that the limit in n of all the
other terms in that equation are finite
I probably messed up in calculating (I), but I believe that(n(n-1)/2) is the number of permutations required for getting from F(G(F(G(...(F(G(F(G(x)))))...)))) to F(F(F(F(...(G(G(G(G
((x))))))...)))) and in each one of those, we permute G o F by F o G, thus a -k appearing in each step.
I guess if I find liftings which conmmute, then the problem is solved. But I dont see why that should happen
Well, thanks in advance
Using commutativity and the fact that the rotation number is independent of $x$, we get:
$<br /> \[<br /> \begin{array}{ccl}<br /> \rho (f\circ g)&=&\displaystyle{\lim_{n\rightarrow \infty} \frac{(F\circ G)^n(x)-x}{n}}\\\\<br /> &=&\displaystyle{\lim_{n\rightarrow \infty} \frac{F^n(G
^n(x))-x}{n}}\\\\<br /> &=&\displaystyle{lim_{n\rightarrow \infty} \frac{F^n(y)-y}{n}}-\displaystyle{\lim_{n\rightarrow \infty}\frac{G^{-n}(y)-y}{n}}\\\\<br /> &=&\rho(f)-\rho(g^{-1})\> \text
{mod} \>1\\\\<br /> &=&\rho(f)+\rho(g)\> \text{mod} \>1<br /> \end{array}<br /> \]<br />$
April 12th 2011, 06:19 PM #2
Senior Member
May 2010
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scalar diffusions are reversible
up vote 2 down vote favorite
It is well known that under mild assumptions a scalar diffusion $dX_t = a(X_t) dt + \sigma(X_t) dW_t$ with invariant probability distribution $\pi$ is reversible. This is indeed not true for
multidimensional diffusions. The usual proofs consists in writing down generators, speed functions etc...
I am trying to intuitively understand this result, and the only (not very satisfying) argument that I have found is the following. It is straightforward to check that any Markov chain on $\mathbb{Z}$
that has an invariant probability $\pi$ and that can only make jumps of size $+1$ or $-1$ is reversible: notice for example that $$F(k) = \pi(k)p(k,k+1)-\pi(k+1)p(k+1,k)$$ is independent of $k$ and
is thus equal to $0$. If $a(\cdot)$ and $\sigma(\cdot)$ are regular enough, a diffusion can be seen as a limit of such Markov chains on $\epsilon \mathbb{Z}$ so that this makes the result plausible.
question: what are arguments/proofs/examples that could shed light on why a one dimensional ergodic diffusion is automatically reversible.
pr.probability markov-chains stochastic-processes
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1 Answer
active oldest votes
The identity you are referring to can be interpreted as saying that if you run your Markov chain on $\mathbb Z$ from the initial distribution $\pi$, then the probability of crossing the edge
$[k,k+1]$ in the positive direction is the same as the probability of crossing it in the negative direction (as otherwise the measure $\pi$ won't be stationary).
The same idea works in the general case as well. For any nice test function $f$ $$ \lim \frac1t \int [f(\xi_t) - f(\xi_0)] d{\mathbf P}(\xi) = \int Df(x) d\pi(x) , $$ where ${\mathbf P}$ is
up vote the measure in the space of sample paths $\xi$ of the diffusion process with the initial stationary distribution $\pi$, and $D$ is the diffusion generator. The left-hand side of the above
2 down equation is 0 by stationarity of the measure $\pi$, so that $\int Df(x)d\pi(x)=0$. In the same way it is true for the reverse generator $D^{\ast}$ as well. Now, the operators $D$ and $D^{\
vote ast}$ may differ by their first-order parts only (denote these vector fields by $v$ and $v^{\ast}$, respectively). Taking a step-like function $f$ whose derivative is non-zero only on a small
set where $v\neq v^*$ (it is at this place that dim $=1$ is used) then gives a contradiction.
thanks: this is exactly the kind of simple approaches that I am looking for. – Alekk Nov 8 '10 at 17:51
add comment
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What is the distance between Moy and belfast international airport ltd. in miles?
You asked:
What is the distance between Moy and belfast international airport ltd. in miles?
Assuming you meant
• Moy, the place in Northern Ireland
Did you mean?
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Wolfram Demonstrations Project
Collinear Classical Helium Atom
Collinear helium is a hypothetical system with two electrons arranged along a line on opposite sides of a nucleus of charge +2, with infinite nuclear mass. The electron positions then imply a
problem with two degrees of freedom. The plots show the dynamics projected onto the plane of the position of the two electrons . For a number of initial conditions, trajectories escape to infinity
after several reflections.
The Hamiltonian is
where the are the generalized momenta, and replacing
the Hamiltonian takes the form
and the equations of motion are
where prime denotes derivative with respect to time.
[1] P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner, G. Vattay, N. Whelan, and A. Wirzba, "Helium Atom," in
Chaos: Classical and Quantum
, Copenhagen: Niels Bohr Institute, 2012.
[2] K. Richter, G. Tanner, and D. Wintgen, "Classical Mechanics of Two-Electron Atoms,"
Physical Review A
(6), 1993 pp. 4182–4196. doi: | {"url":"http://demonstrations.wolfram.com/CollinearClassicalHeliumAtom/","timestamp":"2014-04-21T04:33:37Z","content_type":null,"content_length":"45493","record_id":"<urn:uuid:43584827-193d-45fd-a58a-7ab5481ead24>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00219-ip-10-147-4-33.ec2.internal.warc.gz"} |
determining the equation of tangent line given ex^2x at x=1
October 31st 2012, 10:14 AM #1
May 2012
determining the equation of tangent line given ex^2x at x=1
the function is $f(x)=xe^2x$
find equation of tangent line at x=1
this is what I did
$f'(x)=x*(e^2x)*2 + e^2x$
so the equation of tangent line is
is this correct?
Re: determining the equation of tangent line given ex^2x at x=1
First, a LaTeX tip: if your exponent contains more than 1 character, enclose it within braces, e.g., e^{2x}
You have correctly found $f'(1)$, now for the tangent line, you want to use the point-slope formula:
Re: determining the equation of tangent line given ex^2x at x=1
October 31st 2012, 11:41 AM #2
November 1st 2012, 06:26 AM #3
May 2012 | {"url":"http://mathhelpforum.com/calculus/206488-determining-equation-tangent-line-given-ex-2x-x-1-a.html","timestamp":"2014-04-19T11:09:56Z","content_type":null,"content_length":"37136","record_id":"<urn:uuid:23326869-7bae-42a9-bb01-6b89d372d994>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00532-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Round off the answer correctly. 8.7 g + 15.43 g + 19 g = 43.13 g a 43.130 g b 43.1 g c 43 g d None of the above.
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Oh, I got that one wrong then
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same reason from the last question i answered for you i think let me check my notes
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Yep...the addend with the fewest decimal places wins.
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ok yeah
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thats a good way to put it lol
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Thank you!!!
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yw :) For reference (from wiki): "For multiplication and division, the result should have as many significant figures as the *measured* number with the smallest number of significant figures. For
addition and subtraction, the result should have as many decimal places as the *measured* number with the smallest number of decimal places."
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This Article
Bibliographic References
Add to:
Proof of the Correctness of EMYCIN Sequential Propagation Under Conditional Independence Assumptions
March/April 1999 (vol. 11 no. 2)
pp. 355-359
ASCII Text x
Xudong Luo, Chengqi Zhang, "Proof of the Correctness of EMYCIN Sequential Propagation Under Conditional Independence Assumptions," IEEE Transactions on Knowledge and Data Engineering, vol. 11, no.
2, pp. 355-359, March/April, 1999.
BibTex x
@article{ 10.1109/69.761668,
author = {Xudong Luo and Chengqi Zhang},
title = {Proof of the Correctness of EMYCIN Sequential Propagation Under Conditional Independence Assumptions},
journal ={IEEE Transactions on Knowledge and Data Engineering},
volume = {11},
number = {2},
issn = {1041-4347},
year = {1999},
pages = {355-359},
doi = {http://doi.ieeecomputersociety.org/10.1109/69.761668},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - JOUR
JO - IEEE Transactions on Knowledge and Data Engineering
TI - Proof of the Correctness of EMYCIN Sequential Propagation Under Conditional Independence Assumptions
IS - 2
SN - 1041-4347
EPD - 355-359
A1 - Xudong Luo,
A1 - Chengqi Zhang,
PY - 1999
KW - Uncertainty
KW - expert system
KW - probability theory
KW - certainty factor
KW - sequential propagation.
VL - 11
JA - IEEE Transactions on Knowledge and Data Engineering
ER -
Abstract—In this paper, we prove that under the assumption of conditional independence, the EMYCIN formula for sequential propagation can be derived strictly from the definition of the certainty
factor according to probability theory. We already have known that Adams [1] and Schocken [15] have proved that the EMYCIN formula for parallel propagation is partially consistent with probability
theory. Our result supplements their contribution and, together with theirs, explains why the EMYCIN model has been working reasonably well.
[1] J.B. Adams, "Probabilistic Reasoning and Certainty Factor," Rule-Based Expert Systems, B.G. Buchanan and E.H. Shortliffe, eds., pp. 263-271, Addison-Wesley, 1984.
[2] D.I. Blockley, B.W. Pilsworth, and J.F. Baldwin, "Measures of Uncertainty," Civil Eng. Systems, vol. 1, pp. 3-9, 1988.
[3] R. Buxton, "Modeling Uncertainty in Expert Systems," Int'l J. Man-Machine Studies, vol. 31, no. 4, pp. 415-476, 1989.
[4] G.F. Cooper,"The computational complexity of probabilistic inference using Bayesian belief networks," Artificial Intelligence, vol. 42, pp. 393-405, 1990.
[5] A.Y. Darwiche, "Objection-Based Causal Networks," Proc. Eighth Conf. Uncertainty in Artificial Intelligence, D. Dubois, et al., eds., pp. 67-73, Morgan Kaufmann, 1992.
[6] D. Dubois and H. Prade, "Handling Uncertainty in Expert Systems: Pitfalls, Difficulties, Remedies," The Reliability of Expert Systems, E. Hollnagel, ed., pp. 64-118, Ellis Horwood Ltd., 1989.
[7] R.O. Duda, P.E. Hart, and N.J. Nillson, "Subjective Bayesian Methods for Rule-Based Inference Systems," Proc. AFIPS Conf., vol. 45, pp. 1,075-1,082, AFIPS Press, 1976.
[8] O. Heckerman, "Probabilistic Interpretations of MYCIN's Certainty Factors," Uncertainty in Artificial Intelligence, L.N. Kanal and J.F. Lemmer, eds., pp. 167-196,North-Holland, 1986.
[9] J. Ihara, "Extension of Conditional Probability and Measures of Belief and Disbelief in a Hypothesis Based on Uncertain Evidence," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 9,
no. 41, pp. 561-568, 1987.
[10] X. Luo, "A Study of Probability-Based Uncertain Reasoning Model in Rule-Based Expert Systems: Prospector-Type Scheme," Automated Reasoning, Z. Shi, ed., pp. 123-135,North-Holland, 1992.
[11] X. Luo and C. Zhang, "A Solution to the Problem of Prior Probabilities in the PROSPECTOR Uncertain Reasoning Model," Proc. Joint Pacific Asian Conf. Expert Systems/Singapore Int'l Conf.
Intelligent Systems, D. Patterson, C. Leedham, K. Warendorf, and T.A. Hwee, eds., pp. 306-313, 1997.
[12] X. Luo and C. Zhang, "ILSNCC: A PROSPECTOR-Like Uncertain Reasoning Model," Australia J. Intelligent Information Processing Systems, vol. 4, no. 1, pp. 1-11, 1997.
[13] W. Van Melle, “A Domain-Independent System that Aids in Constructing Knowledge-Based Consultation Programs,” PhD dissertation, Report No. STAN-CS-80-820, Computer Science Dept., Stanford Univ.,
[14] J. Pearl, Probabilistic Reasoning in Intelligent Systems. San Mateo, Calif.: Morgan Kaufman, 1988.
[15] S. Schocken, "On the Rational Scope of Probabilistic Rule-Based Inference Systems," Uncertainty in Artificial Intelligence, J.F. Lemmer and L.N. Kanal, eds., vol. 2, pp. 175-189, Elsevier,
North-Holland, 1988.
[16] E.H. Shortliffe and B.G. Buchanan, "A Model of Inexact Reasoning in Medicine," Math. Bioscience, vol. 23, pp. 351-379, 1975.
[17] C. Zhang and X. Luo, "Isomorphic Transformation of Uncertainties of Propositions Among the EMYCIN and PROSPECTOR Uncertain Models," Proc. Second Int'l Conf. Multi-Agent Systems, M. Tokoro, ed.,
p. 465, AAAI Press, 1996.
Index Terms:
Uncertainty, expert system, probability theory, certainty factor, sequential propagation.
Xudong Luo, Chengqi Zhang, "Proof of the Correctness of EMYCIN Sequential Propagation Under Conditional Independence Assumptions," IEEE Transactions on Knowledge and Data Engineering, vol. 11, no. 2,
pp. 355-359, March-April 1999, doi:10.1109/69.761668
Usage of this product signifies your acceptance of the
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(And why machine learning experts should care)
HLearn library for haskell that I’ve been working on for the past few months. The idea of the library is to show that abstract algebra—specifically monoids, groups, and homomorphisms—are useful not
just in esoteric functional programming, but also in real world machine learning problems. In particular, by framing a learning algorithm according to these algebraic properties, we get three things
for free: (1) an online version of the algorithm; (2) a parallel version of the algorithm; and (3) a procedure for cross-validation that runs asymptotically faster than the standard version.
We’ll start with the example of a Gaussian distribution. Gaussians are ubiquitous in learning algorithms because they accurately describe most data. But more importantly, they are easy to work with.
They are fully determined by their mean and variance, and these parameters are easy to calculate.
In this post we’ll start with examples of why the monoid and group properties of Gaussians are useful in practice, then we’ll look at the math underlying these examples, and finally we’ll see that
this technique is extremely fast in practice and results in near perfect parallelization. | {"url":"http://izbicki.me/blog/2012/11","timestamp":"2014-04-21T12:08:54Z","content_type":null,"content_length":"18815","record_id":"<urn:uuid:263ad335-fbca-496c-9d8b-bf92393aafdb>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00164-ip-10-147-4-33.ec2.internal.warc.gz"} |
Gradient and Maximum Increase of a Function
Date: 07/19/2005 at 04:58:42
From: Jill
Subject: Linear Algebra and Calculus
I am having difficulty explaining why the gradient points in the
direction of the maximum increase of a function. Several resources
make the statement, but no one explains it. It is probably something
simple I am overlooking. Can the dot product be used to justify it?
Date: 07/19/2005 at 05:54:56
From: Doctor Jerry
Subject: Re: Linear Algebra and Calculus
Hello Jill,
One explanantion uses the idea of "directional derivative," which
usually precedes the definition of the gradient.
If f is a function of two variables, if a = <a1,a2> is a point
(specified, for convenience, as a position vector) in the domain, and
if u = <u1,u2> is a unit vector (thought of as being based at a), then
the directional derivative of f at a and in the direction u is the
limit of the ratio
[ f(a+h*u) - f(a) ] / h
as h->0. If this limit exists it is often denoted as D_u f(a), where
"_" means subscript. This is the rate of change of f in the direction
u. It is easy to see that if u = <1,0> and <0,1>, the directional
derivatives are the partials f_x(a) and f_y(a) of f at a,
If f is differentiable at a, one can show that
D_u f(a) = <f_x(a), f_y(a)> dot u. (dot = dot product of vectors)
Of course, <f_x(a), f_y(a)> is the gradient of f at a. Recalling that
the dot product of two vectors is the product of their lengths and the
cosine of the angle between them, it follows that the maximum
directional derivative happens when u is the direction of the
gradient, that is, the cosine of the angle between them is 0.
Please write back if my comments are not clear.
- Doctor Jerry, The Math Forum
Date: 07/19/2005 at 11:48:09
From: Doctor George
Subject: Re: Linear Algebra and Calculus
Hi Jill,
Doctor Jerry gave a very nice answer to your question. Here is
another approach to it.
Using Doctor Jerry's notation, if we examine the Taylor expansion of
the function, the linear term is
h <f_x(a), f_y(a)> dot u
For sufficiently small values of "h" the linear term will dominate the
expansion, so the maximum increase in a small neighborhood about "a"
will occur when the linear term is maximized. By Doctor Jerry's
reasoning, this happens when "u" is the direction of the gradient.
Just to broaden the picture a bit, in optimization theory the goal is
often the minimization of a function. The direction opposite the
gradient is called the direction of "steepest descent."
- Doctor George, The Math Forum
Date: 07/19/2005 at 13:43:29
From: Jill
Subject: Thank you (Linear Algebra and Calculus)
Thanks a bunch! I appreciate you both taking the time to help! | {"url":"http://mathforum.org/library/drmath/view/68326.html","timestamp":"2014-04-19T00:42:50Z","content_type":null,"content_length":"8012","record_id":"<urn:uuid:7ad0f4c7-e2f4-42aa-82a5-ed37de23222f>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00066-ip-10-147-4-33.ec2.internal.warc.gz"} |
Numerical Simulation of Water-Based Alumina Nanofluid in Subchannel Geometry
Science and Technology of Nuclear Installations
Volume 2012 (2012), Article ID 928406, 12 pages
Research Article
Numerical Simulation of Water-Based Alumina Nanofluid in Subchannel Geometry
^1School of Mechanical Engineering, Shiraz University, Shiraz 71348-51154, Iran
^2Department of Nuclear Engineering, Seoul National University, Seoul 151-744, Republic of Korea
^3PHILOSOPHIA Inc., 1 Gwanak Road, Gwanak-gu, Seoul 151-744, Republic of Korea
Received 10 July 2012; Revised 7 September 2012; Accepted 11 September 2012
Academic Editor: Iztok Tiselj
Copyright © 2012 Mohammad Nazififard et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Turbulent forced convection flow of Al[2]O[3]/water nanofluid in a single-bare subchannel of a typical pressurized water reactor is numerically analyzed. The single-phase model is adopted to simulate
the nanofluid convection of 1% and 4% by volume concentration. The renormalization group k-ε model is used to simulate turbulence in ANSYS FLUENT 12.1. Results show that the heat transfer increases
with nanoparticle volume concentrations in the subchannel geometry. The highest heat transfer rates are detected, for each concentration, corresponding to the highest Reynolds number Re. The maximum
heat transfer enhancement at the center of a subchannel formed by heated rods is ~15% for the particle volume concentration of 4% corresponding to Re = 80,000. The friction factor shows a reasonable
agreement with the classical correlation used for such normal fluid as the Blasius formula. The result reveals that the Al[2]O[3]/water pressure drop along the subchannel increases by about 14% and
98% for volume concentrations of 1% and 4%, respectively, given Re compared to the base fluid. Coupled thermohydrodynamic and neutronic investigations are further needed to streamline the
nanoparticles and to optimize their concentration.
1. Introduction
The heat transfer coefficients in forced convection are governed by thermal conductivity of the fluid as well as by factors representing turbulence and the operating condition. These fluids,
including oil, water, and ethylene glycol mixture, are rather poor heat transfer media. Their thermal conductivity plays an important role in the heat transfer between the working fluid and the
heated surface.
An innovative way to improve the thermal conductivity of a fluid is to suspend nano-sized particles on the order of 1~100nm with high thermal conductivity in the base fluid with low thermal
Generally, the thermal conductivity of the particles, metallic or nonmetallic, is typically an order of magnitude higher than that of the base fluids even at low concentrations resulting in
significant increases in heat transfer (Table 1). Nanofluids thus lend themselves to potential candidates for next generation heat transfer media.
Literature exists [1–16] on the single-phase nanofluids forced convection flow in such various geometries as flat plates, circular tubes, noncircular channels, annuli, and cross flow over circular
tubes. The previous results have underscored the enhancement of heat transfer due to the presence of nanoparticles in the fluids. In particular, the heat transfer coefficient increases appear to go
beyond the mere thermal conductivity effect and cannot be predicted by such traditional pure fluid correlations as Dittus-Boelter’s [4–9]. Pak and Cho [1] experimentally investigated the convective
heat transfer behavior of the γ-alumina (Al[2]O[3]) and titanium dioxide (TiO[2]) water-based nanofluids heated in a circular tube with constant heat flux. The Reynolds number Re and Prandtl number
Pr varied in the ranges 10^4-10^5 and 6.5–12.3, respectively. They observed that the Nusselt number Nu of the dispersed fluids for fully developed turbulent flow increased with increasing volume
concentration as well as Re [1]. They showed that the Darcy friction factors for dilute dispersion fluids used in their study coincided well with the values predicted from Kays’ correlation for
turbulent flow of a single-phase fluid. Due to the increase in the viscosity of dispersed fluids, there is an additional pumping penalty of approximately 30% at a volume concentration of 3% [1].
Experimental investigation of fluid flow in fuel bundle geometry is generally costly, time consuming, and technically complicated, however. In contrast, computational fluid dynamics (CFD) is a
nonexpensive method which has seen dramatic growth over the last decade on numerical simulation of nanofluid turbulent convection [5–14, 17–19].
Bianco et al. [5–8] in a series of numerical study analyzed the turbulent forced convection flow of Al[2]O[3]/water nanofluid in a circular tube subjected to a constant and uniform heat flux at the
wall. They showed numerically that the convective heat transfer coefficient for a nanofluid is generally greater than that of the base liquid and that the heat transfer enhancement increases with the
particle volume concentration and Re.
Bianco et al. [6] employed single-phase and mixture models with constant thermophysical properties in order to simulate the Al[2]O[3]/water nanofluid. Their results from the single-phase and mixture
models showed that the wall and bulk temperatures were quite similar for %, while there was a deviation for higher concentrations. They stated that the accuracy of the model could be improved by
using a better description of nanofluid thermophysical properties. Bianco et al. [8] also numerically investigated the heat transfer enhancement and entropy generation minimization (EGM) of
nanofluids turbulent convection flow in square section tube subjected to constant and uniform wall heat flux. Their EGM analysis showed that, at low Re, the entropy generation on account of the
irreversibility of heat transfer dominates, whereas with increasing Re and particles’ concentration, the entropy generation due to friction losses becomes more important. They showed an optimal value
of Re decreases as particles’ concentration increases [8].
Maïga et al. [9] studied numerically the hydrodynamic and thermal behavior of turbulent flow in a tube using the Al[2]O[3]/water nanofluid under the constant heat flux boundary condition. In their
study a new correlation is proposed to calculate the fully developed heat transfer coefficient for the nanofluid considered.
Rostamani et al. [10] simulated the turbulent flow of copper oxide (CuO), alumina (Al[2]O[3]), and oxide titanium (TiO[2]) nanofluids with different volume concentrations of nanoparticles flowing
through a two-dimensional duct under constant heat flux condition. They emphasized that both Nu and the heat transfer coefficient of the nanofluid are strongly dependent on nanoparticles and increase
with the volume concentration of nanoparticles. Their results showed that the CuO/water and TiO[2]/water nanofluids have the highest and lowest shear stress values, respectively, due to the higher
viscosity of copper oxide (CuO) in comparison to other nanofluids.
Behzadmehr et al. [11] numerically studied turbulent forced convection heat transfer in a circular tube with a nanofluid consisting of water and 1vol. % of Cu nano particle. Applying the two-phase
mixture model they showed that adding 1% nanoparticles increases Nu by more than 15% while it does not substantially affect the skin friction. They stated that the accuracy of the mixture model could
be improved by using suitable effective physical properties for nanofluid instead of volume weighted average of particle and fluid properties, though.
Corcione et al. [12] theoretically analyzed the heat transfer of nanoparticle suspensions in turbulent pipe flow. They assumed nanofluids behave more like single-phase fluids than like conventional
solid-liquid mixtures. They showed the existence of an optimal particle loading for either maximum heat transfer at constant driving power or minimum cost of operation at constant heat transfer rate.
Rahimi-Esbo et al. [13] numerically studied the turbulent forced convection jet flow of nanofluid in a converging duct. They showed that by increasing the volume fraction from 0% to 5%, the average
Nu on the down wall is enhanced by more than 8%.
Ghaffari et al. [14] numerically investigated turbulent mixed convection heat transfer of the Al[2]O[3]/water nanofluid throughout a horizontal curved tube using a two-phase mixture model. They
showed that at the low Grashof number Gr, the turbulent intensity augments with the nanoparticle concentration. While at the higher Gr, where the effect of buoyancy-induced secondary flow becomes
important, using higher nanoparticle concentration decreases the flow turbulent intensity across the vertical plan. They showed that the nanoparticle volume fraction enhances Nu. However, its effect
on the skin friction coefficient is not significant. Massoudi and Phuoc [15] discussed briefly the constitutive modeling of the stress tensor for nanofluids.
Mansour et al. [16] investigated the effect due to the uncertainty in the values of the physical properties of the Al[2]O[3]/water nanofluid on their thermohydrodynamic performance for both laminar
and turbulent fully developed forced convections in a tube with a uniform heat flux. They analyzed two types of problems: replacement of a simple fluid by a nanofluid in a given installation and
design of an elementary heat transfer installation. They illustrated that the required pumping power for a fixed heat transfer rate and required tube length for fixed mass flow rate and bulk
temperature change sizably with the thermophysical properties of the nanofluid.
Palm et al. [20] numerically investigated heat transfer enhancement capabilities of coolants with suspended metallic nanoparticles inside typical radial flow cooling systems. Their results clearly
indicate that considerable heat transfer benefits are possible with the use of these fluid/solid particle mixtures.
Eastman et al. [21] reported on interesting properties of nanofluids in their review. They mentioned that although the potential for the use of nanofluids in a wide variety of applications is
promising, a key stumbling block seriously hindering the development of the field is that a detailed atomic-level understanding of the mechanisms responsible for the observed property changes remains
There has recently been an increasing interest in practical application of nanofluid in nuclear reactor technology [17, 18, 22–25]. Considering cooling application of nanofluid it seems that the
nanofluids can potentially be applied to power plant cooling and safety systems owing to their enhanced properties such as thermal conductivity for instance. The importance of heat generation and
heat transfer processes in nuclear reactors is probably best emphasized by the fact that the rate of heat release and consequently power generation in a given reactor core is limited by thermal
rather than by nuclear considerations. There is no limit to the neutron flux level attainable in a reactor core, but the heat generated must be removed well enough.
Kim et al. [22] have worked on application of nanofluid to nuclear reactor. They evaluated the feasibility of nanofluids in nuclear applications by checking on the performance of any water-cooled
nuclear system that is heat removal limited. The nanofluids may as well be used in the emergency cooling system, where they could cool down the overheated surfaces more quickly leading to improved
power plant safety. Buongiorno and Truong [23] have shown that that circulation of water-based nanofluid in the primary cooling loop of light water reactor (LWR) will improve the heat removal from
the core. However, using nanofluids as working fluids has a number of limitations because any change in the reactor core materials affects the criticality and hence the effective neutron
multiplication factor. Previous studies of the application of nanofluids to LWR predicted that among nanofluids at low volume concentrations, both the alumina and zirconia nanoparticles are basically
transparent to neutrons. They can be used in LWRs since their contribution to coolant activation is minimal [23–25].
Buongiorno et al. [24] assessed the feasibility of using nanofluids to enhance the in-vessel retention capability in light water reactors. They assessed the benefits of critical heat flux enhancement
by nanofluids for decay heat power removal. Their analysis indicates that the use of nanofluid can increase the in-vessel retention capability of nuclear reactors by as much as 40% [24].
Hadad et al. [25] have recently reported that Al[2]O[3] shows the lowest rate of multiplication factor dropoff in comparison to aluminum, copper oxide, copper, and zirconia nanoparticles with
different concentrations in nanofluids. As compared with other nanomaterials such as carbon nanotubes, Al[2]O[3] nanoparticle is cheaper and has been used extensively in cooling application of
nanofluids. Recently, Nazififard et al. [17] numerically evaluated characteristics of Al[2]O[3]/water nanofluids for prospective application to water-cooled research reactors. Their results showed
that the heat transfer is enhanced by about 4% using nanofluid of 1% by volume in comparison to the base fluid and that the pressure drop of nanofluids is only about 3% higher than that of the base
fluid [17]. They also showed that injection of nanofluid into the core could help circumvent a runaway nuclear reaction even under normal operation and increase the safety margins. Nazififard et al.
[18] numerically studied the water-based nanofluid coolant for a typical small modular reactor as well.
The scope of the present paper is to make a further contribution to nanofluids turbulent convection in subchannel geometry by numerically investigating developing turbulent forced convection flow of
Al[2]O[3]/water nanofluid in a subchannel.
2. CFD Methodology
2.1. Mathematical Modeling
The fluid flow in the water cooled reactor is parallel to the fuel rod bundle, and the unit channel is called subchannel. A single subchannel is modeled using the flow symmetry which has been modeled
extensively before [18, 19, 27–29]. Steady-state Reynolds-averaged Navier-Stokes, mass, energy, and turbulence equations were discretized and solved using FLUENT 12.1. The continuity, momentum, and
energy equations are as follows [29]:
The renormalization group (RNG) model is used to simulate turbulence in ANSYS FLUENT [28] proposed by Yakhot et al. [30] to renormalize the Navier-Stokes equations and to account for the effect of
smaller scales of motion. The RNG model is an alternative to the standard model. It generally offers little improvement over the standard model, though. Standard wall treatment was adopted near the
wall for the momentum and energy equations [28].
2.2. Computational Domain and Meshes
The problem under consideration consists of steady, forced turbulent convection flows and heat transfer of a nanofluid flowing inside a subchannel of a typical pressurized water reactor (PWR). The
computational model geometry and the grid were generated using GAMBIT, the preprocessing module for the FLUENT software [28]. The computational domain boundaries and meshes are shown in Figure 1. The
CFD domain consists of a subchannel with length of 605mm. The fuel rod diameter is 9.5mm, and the pitch-to-diameter ratio P/D is 1.32. The hydraulic diameter of the subchannel is 11.8mm. The
channel has appropriate length in order to obtain fully developed velocity and temperature profiles at the outlet. The condition of the constant wall heat flux is considered in this study.
2.3. Mesh Generation Sensitivity and Turbulence Model
The accuracy of finite volume method is directly related to the quality of the discretization used. In this study, structured hexahedral meshes are used which are known to provide higher accuracy and
reduce the CFD computational effort (Figure 1) [27]. A comprehensive mesh sensitivity study was done to check on the influence of the mesh resolution on the results and to minimize numerical
influences introduced by the size of meshes and their distributions. For mesh sensitivity analysis, four meshes of differing size were used ranging from 0.32 to 0.65 million for the single subchannel
(Table 2). Note from the cross-sectional view of meshes in Figure 1 that mesh refinement was improved in each mesh refinement process. The mesh refinement ratio (MRR) is defined as the ratio between
consecutive meshes of mesh refinement. Table 2 shows that, for the considered subchannel, the mesh M4 appears to be satisfactory to ensure the accuracy of numerical results as well as their
independency with respect to the number of nodes used.
2.4. Boundary Conditions
The fluid enters with uniform temperature of = 293K and velocity profiles at the subchannel inlet. Different inlet uniform velocities are applied which are listed in Table 3. In order to validate
the CFD model Re and thermal boundary condition were chosen to match Re of the available correlations [1, 9]. At the outlet of the computational model a relative average pressure equaling zero was
defined. The surfaces of the walls were assumed hydraulically smooth. A constant heat flux KW/m^2 is specified for the wall (rod surface). A similar approach with Bianco et al. [5–7] has been chosen
to calculate the Re () based on thermophysical properties of Al[2]O[3]/water nanofluid at different volume fractions of 1% and 4% at .
2.5. Numerical Method
The modeled cases were solved using ANSYS FLUENT software version 12.1 [28]. A segregated, implicit solver option was used to solve the governing equations. The first order upwind discrimination
scheme was employed for the terms in energy, momentum, and turbulence parameters. A standard pressure interpolation scheme and SIMPLE pressure velocity coupling were implemented. A residual
root-mean-square (RMS) target value of 10^−5(10^−8for energy equation) was defined for the CFD simulations.
The simulations are performed on the desktop computer with Intel Core I5 2.4GHz with 4GB installed RAM. The typical CPU time for each modeling was about 25,200 s.
2.6. Thermophysical Properties of the Nanofluids
The determination of nanofluid thermophysical properties is an increasingly important area in nanofluid cooling applications. A considerable amount of literature has been accumulated on the basic
nanofluid thermophysical properties over the past few years but at present there is no agreement within the nanofluid community about description of thermophysical properties [15, 21, 26]. The
available experimental data are rather controversial, and there is no systematic study on thermophysical properties of nanofluids.
The single-phase approach is chosen to calculate the thermophysical properties of nanofluids as it is widely used in the literature [5–9, 17, 18]. In this model the homogenous mixture is assumed
prior to solving the governing equations of continuity, momentum, and energy for the single phase fluid flow that the presence of nanoparticles is realized by modifying physical properties of the
mixture fluid. It is assumed that there is no velocity difference between fluids and the particles, and the fluids and the particles are in thermal equilibrium [5–8]. This assumption implies that all
the convective heat transfer correlations available in the literature for single-phase flows can be extended to nanoparticle suspensions, providing that the thermophysical properties appearing in
them are the nanofluid effective properties calculated at the reference temperature [11, 12].
Note that most nanofluids used in practical applications usually comprise the oxide particles finer than 40nm [9]. In the current CFD study the considered nanofluid is a mixture of water and
particles of Al[2]O[3] 38nm in mean particle diameter. Equations (2)–(5) were used to compute the thermophysical properties of the alumina nanofluids for the CFD simulation [5–9, 17, 18]. The
thermophysical properties of the Al[2]O[3] nanoparticle and water-based nanofluid are presented at temperature 293K in Tables 4 and 5, respectively,
In the absence of experimental data, classical formula for the two-phase mixture is used to calculate the nanofluid density which is a constant value independent of temperature [5–9, 17, 18].
Similarly, a couple of expressions are proposed for determining the nanofluids specific heat [1, 5–9, 17, 18]. Equation (3) is theoretically more consistent since the specific heat is a mass-specific
quantity whose effect depends on the density of the components of a mixture [4–6].
The nanofluid viscosity is an important parameter for practical applications since it directly affects the pressure drop in forced convection. Equation (4) is purely experimental and turns out to be
more apt than the classical models, such as Einstein or Brinkman, which drastically underestimate the nanofluid viscosity [5–8]. Equation (5) is based on a classical model, nonetheless yields good
estimation of the thermal conductivity in the present case [5–9].
3. Results
Results are reported in terms of the average Nu, convective heat transfer coefficient, and pressure drop as a function of Re ranging from 20 × 10^3 to 80 × 10^3, and particle volume concentrations of
0%, 1%, and 4%. Results were obtained by the single-phase approach with the constant heat flux KW/m^2 on the wall. In all cases the particles size was considered equal to 38nm. Results were
validated by comparing the obtained ANSYS FLUENT results against the Dittus-Boelter correlation [31] for the case of pure water For the Al[2]O[3]/water nanofluid the validation has been performed by
the available correlations considering a fully developed flow in terms of the average Nu and friction coefficients for smooth pipe. Nu is compared with correlations suggested by Pak and Cho [1] and
Maïga et al. [9] both of which have been widely used in the literature: Note that the above correlations calculate the heat transfer coefficient in a tube using the Al[2]O[3]/water nanofluid under
the constant heat flux boundary condition.
3.1. Velocity and Turbulence
Figure 2 shows the development of the velocity magnitude along the subchannel centerline for %. The results suggest the existence of a fully developed region for and , whereas the developing length
for . It is clear that the fully developed values of the nondimensional centerline velocity ) decrease as Re increases because the corresponding velocity profiles become more uniform as Re increases.
It is noted that, downstream of the channel inlet, the boundary layer growth pushes the fluid towards the centerline region, causing an increase of the centerline velocity in accordance with [1, 11].
This may attribute to due to the fact that the corresponding velocity profiles become more uniform as Re increases.
Figure 3 shows the profiles of velocity magnitude along the central line of channel for Re = 80 × 10^3 and φ = 0%, 1% and 4%, respectively. The numerical results show that, if the current assumptions
are used to model the fluid properties, the presence of the nanoparticles does not affect the velocity profile.
The contour of velocity (m/s) and turbulent kinetic energy (m^2/s^2) at the outlet for pure water at Re = 20 × 10^3 at outlet are depicted in Figure 4.
The stability of the suspensions in nanofluids is the single most critical issue for enhancing heat transfer. Nanoparticles generally tend to settle down and deposit on the wall. The nanoparticle
clogging is another problem which can change the thermo and hydrodynamic characteristics of the coolant.
An important flow parameter affecting the homogeneity and stability of nanofluid in forced turbulent flow is the level of flow turbulence intensity. The turbulence intensity,TI, often referred to as
turbulence level is defined as the ratio of the root-mean-square of the velocity fluctuations, , to the mean flow velocity U [28].
Figure 5 displays the turbulence intensity at for nanofluid % at the pressure outlet. The turbulent intensities are high at the walls and levels down away from the walls. Observe that there are
regions of relatively high axial turbulence intensity on the rods close to rod gap. These saddle points are located on both sides at about 45° from the gap between the rods. Behzadmehr et al. [11]
showed that the particles can absorb the velocity fluctuation energy and reduce the turbulent kinetic energy.
3.2. Temperature Distribution
Contours of temperature at the outlet for base fluid and nanofluids are shown in Figure 6. Observe that the inclusion of nanoparticles has a beneficial effect on the wall and bulk temperatures of the
nanofluid compared to base fluid.
Figure 7 illustrates the coolant temperature distribution along the centerline of the subchannel for different volume concentration of nanofluids. Note that there is a steady increase in the coolant
temperature distribution along the channel for all the cases. As the coolant moves along the channel, it absorbs heat. Note that the bulk temperature for % and% is lower than the case of base fluid.
The FLUENT results point to the temperature decrease due to the presence of the particles considering the constant heat flux at the wall (rod surfaces) for all cases.
3.3. Heat Transfer
Heat transfer calculations were made for different volume concentration of nanofluid by applying a constant temperature to the inlet of the channel and a constant uniform heat flux to rod walls along
the subchannel. In this study the local and average heat transfer coefficients are defined as follows: The average heat transfer coefficient for all the concentrations and considered Re is reported
in Figure 8. Notice the useful contribution to the heat transfer provided by the inclusion of nanoparticles in comparison to the case with just the base fluid. Also note that heat transfer increases
with the particles volume concentration and Re. The highest heat transfer rates are identified, for each concentration, at the highest Re. The observed increase in heat transfer coefficient could be
attributed to improved thermophysical properties of the mixture with respect to the base fluid. Thus, a nanofluid with higher thermal conductivity increases the heat transfer along the channel.
Moreover, the term increases; therefore, more energy is required to increase the bulk temperature with respect to the case of the base water [5–9, 17, 18]. However, the difference between the wall
and bulk temperatures decreases with respect to the case of the base fluid provoking the increase of the heat transfer coefficient. Previous studies [1–9] have yet shown that the increase in heat
transfer is generally higher than that of the thermophysical properties. These explanations are not convincing enough to explain the increase in the heat transfer due to a nanofluid.
A Nu study was carried out by taking averages over the wall surface in the region. The results are given in Table 6. The average Nu is defined as follows: Figure 9 shows the average Nu for all the
concentrations and Re compared against the experimental and numerical correlations available in the literature [1, 9]. The ANSYS FLUENT results are compared against the experimental correlations
proposed by Pak and Cho [1] and the numerical correlation proposed by Maïga et al. [9].
Bianco et al. [5, 6] reported that the Maïga et al. [9] correlation overestimates the values provided by Pak and Cho [1] by about 20% which is clearly shown in Figure 10. However, this overestimation
can be considered acceptable, as reported also by Buongiorno [4]. The Pak and Cho [1] correlation concurs with the FLUENT result at , while at higher Re = 80 × 10^3 the Maïga et al. [9] correlation
agrees with the FLUENT result.
3.4. Pressure Drop
The pressure drop of the coolant in the heated channel is one of the central parameters determining the efficiency of nanofluids application. The pressure drop and coolant pumping power are closely
related. The pressure distribution in the subchannel for different volume concentrations of nanofluid at different Re is shown in Figure 11. It is clear that there is a linear decrease in the
pressure drop along the channel for all the cases. Figure 11 clearly shows that the pressure drop of nanofluids increases with the increasing volume concentrations. This result may be explained by
the fact that the density and viscosity are the main thermophysical parameters which could influence the coolant pressure drop and coolants.
Several studies [1–9] have revealed that that the pressure drop of the nanofluids fairly matches with the values predicted by the conventional correlations of base fluid for both laminar and
turbulent flows. Hence, the conventional fraction factor correlation can be used for the pressure drop prediction. The experimental study by Xuan and Li [3] implied that the friction factor
correlation for the single-phase flow (base fluid) can be extended to the dilute nanofluids. Figure 12 shows the pressure drop along the channel compared with pressure drop estimated by Blasius
It assumed that the wall of the channel is smooth and the friction factors may as well be determined for turbulent flow as in a smooth pipe pursuant to the Moody diagram or alternative correlation.
For 2300 < Re < 100 × 10^3, the single-phase flow turbulent friction factor for a smooth tube may be estimated by the Blasius formula as follows:
The pressure loss due to the flow friction may be calculated as follows:
Figure 13 shows the performances of the Blasius correlations to predict the pressure drop of nanofluid. The Blasius correlation underestimates the result, but the differences are less than 3% at Re <
40 × 10^3. The maximum difference is found for Re = 80 × 10^3, which is about 5%.
For easy understanding of the pressure drop of the nanofluid, the differences in pressure drop along the subchannel for nanofluid are compared with base fluid which is defined as follows: The result
reveals that the /water pressure drop increases by about 14% and 98% for φ = 1% and φ = 4% of, respectively, given Re.
4. Conclusions
Numerical simulation has been presented on heat transfer characteristics and pressure drop of Al[2]O[3]/water nanofluid in subchannel geometry under steady state turbulent flow. The homogenous fluid
assumptions with modified thermophysical properties were taken into account in order to simulate the Al[2]O[3]/water nanofluid. The CFD predictions were compared against the available experimental
data and literature correlations. The following conclusions can be drawn from the present study.(i) CFD predictions were shown to reproduce the enhancement in heat transfer, with respect to the base
fluid, known to characterize nanofluids. Convective heat transfer and friction pressure drop were correctly predicted to increase with the Al[2]O[3] nanoparticle concentration.(ii) The Pak and Cho [1
] correlation concurs with the FLUENT result at Re = 20 × 10^3, while the Maïga et al. [9] correlation agrees with the FLUENT result at Re = 80 × 10^3.(iii) The Blasius correlation underestimates the
FLUENT result, but the differences are less than 3% at Re < 40 × 10^3. The maximum difference is found for Re = 80 × 10^3, which is about 5%.(iv) The result reveals that the pressure drop of
nanofluid along subchannel increases by about 14% and 98% for φ= 1% and φ= 4% of, respectively, given Recompared to the base fluid.A complete understanding of heat transfer performance of the
nanofluids is prerequisite to their practical application to a commercial nuclear reactor. It is recommended that further research be undertaken to analyze nanofluids for apt nanoparticle and its
optimum concentration in the base fluid. Coupled thermohydrodynamic and neutronic investigations are further needed to streamline the nanoparticles and to optimize their concentration.
: Specific heat, J/kg K
: Diameter, m
: Equivalent hydraulic diameter, m
E: Empirical constant in turbulence model (9.793)
: Friction factor
g: Gravity acceleration, m/
: Heat transfer coefficient, W/K
: Thermal conductivity, W/m K
: Nusselt number
: Mean pressure, Pa
: Prandtl number
Heat flux, W/
: Reynolds number
: Temperature, K
: Turbulence intensity
: Velocity vector, m/s
: Root-mean-square of the turbulent velocity fluctuations
: Axial bulk mean velocity
: Coordinates
: Mean velocity (Re averaged).
Greek Symbols
: Dynamic viscosity, Pa·s
: Density, kg/
: Nanoparticle volume fraction.
: Average
: Base fluid
: Bulk
: Fluid
: Inlet section
: Nanofluids
: Outlet section
: Particle
: Wall
0: Initial value.
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31. F. W. Dittus and L. M. K. Boelter, “Heat transfer in automobile radiators of the tubular type,” University of California Publications of Engineering, vol. 2, pp. 443–461, 1930. | {"url":"http://www.hindawi.com/journals/stni/2012/928406/","timestamp":"2014-04-20T01:57:00Z","content_type":null,"content_length":"193563","record_id":"<urn:uuid:573031e1-5de1-4b8d-8257-c2a7c766162d>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00148-ip-10-147-4-33.ec2.internal.warc.gz"} |
Waiting line model
1. 131572
As it is, models have been developed to assist managers understand and make better decisions concerning the operation of waiting lines. In management, a waiting line is a queue, and the body of
knowledge dealing with waiting lines is known as queueing theory. As I understand it, queueing theory has become more sophisticated with applications in a wide variety of waiting line situations. I
believe waiting line models consist of mathematical formulas and relationships that can be used to determine the operating characteristics (performance measures) for a waiting line.
Operating characteristics of interest include the following:
The probability that no units are in the system
The average number of units in the waiting line
The average number of units in the system (the number of units in the waiting line plus the number of units being served)
The average time a unit spends in the waiting line
The average time a unit spends in the system (the waiting time plus the service time)
The probability that an arriving unit has to wait for service
Managers with such information are better able to make decisions that balance desirable service levels against the cost of providing the service.
The situation...
The New York City franchise of U.S. Citibank operates approximately 250 banking centers. Each center provides one or more automatic teller machines (ATMs) capable of performing a variety of banking
transactions. At each center, a waiting line is formed by randomly arriving customers who seek service at one of the ATMs.
In order to make decisions on the number of ATMs to have at selected banking center locations, management needs information about potential waiting times and general customer service. Waiting line
operating characteristics such as average number of customers in the waiting line, average time a customer spends waiting, and the probability that an arriving customer has to wait would help
management determine the number of ATMs to recommend at each banking center.
For instance, one busy Midtown Manhattan center had a peak arrival rate of 172 customers per hour. A multiple-channel waiting line model with six ATMs showed that 88% of the customers would have to
wait, with an average wait time between 6 and 7 minutes. This level of service was considered unacceptable. How did this information help management to know that they should change their ATM system?
Is it a good model to use to provide guidelines for such a decision? What other information would you suggest that they monitor and why?
This is a problem regarding a waiting line model, which looks at people waiting for an ATM. | {"url":"https://brainmass.com/math/probability/131572","timestamp":"2014-04-17T16:20:08Z","content_type":null,"content_length":"27755","record_id":"<urn:uuid:0610dfe4-f5fe-4271-8616-2132a0cda62d>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00424-ip-10-147-4-33.ec2.internal.warc.gz"} |
Multi-dimensional Orthogonal Graph Drawing with Small Boxes(Extended Abstract)
Wood, David R. (1999) Multi-dimensional Orthogonal Graph Drawing with Small Boxes(Extended Abstract). In: Graph Drawing 7th International Symposium, GD’99, September 15-19, 1999, Štirín Castle, Czech
Republic , pp. 311-322 (Official URL: http://dx.doi.org/10.1007/3-540-46648-7_32).
Full text not available from this repository.
In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the D-dimensional (D \ge 2) orthogonal grid. In this model no two vertices lie in the same grid
hyperplane. We show that for D \ge 3, given an arbitrary layout and initial edge routing a crossing-free orthogonal drawing can be determined. We distinguish two types of algorithms. Our layout-based
algorithm, given an arbitrary fixed layout, determines a degree-restricted orthogonal drawing with each vertex having aspect ratio two. Using a balanced layout this algorithm establishes improved
bounds on the size of vertices for 2-D and 3-D drawings. Our routing-based algorithm produces 2-degree-restricted 3-D orthogonal drawings. One advantage of our approach in 3-D is that edges are
typically routed on each face of a vertex; hence the produced drawings are more truly three-dimensional than those produced by some existing algorithms.
Repository Staff Only: item control page | {"url":"http://gdea.informatik.uni-koeln.de/359/","timestamp":"2014-04-17T12:55:20Z","content_type":null,"content_length":"37091","record_id":"<urn:uuid:2aa817ce-7cba-4159-b401-0ff5d9f0b40a>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00082-ip-10-147-4-33.ec2.internal.warc.gz"} |
The World of Absolute Inequalities
Some GMAT problems present inequalities involving absolute value. Dealing with these questions is a multi-step process:
First, isolate the absolute value so that it stands alone on one side of the inequality.
Next, differentiate between two different cases: the number case, and the variable case.
1) The number case
In the number case, the other side is a number, i.e., the inequality has the form:
|something| < Number or |something| > Number
For example:
If |x-6|< 2, what is the range of values for x?
Solve absolute values of the number case by considering two possible scenarios:
First scenario – copy the inequality without the absolute value brackets and solve:
Second scenario – remove the absolute value brackets. Put a negative sign around the other side of the inequality, AND flip the sign:
2) The variable case
If the inequality has an absolute value on one side, and variables on the other side, you cannot use the two-scenario approach. We call this the variable case, and it requires finding out what the
inequality really means, rather than simply solving for x. Investigate the variable case by plugging in simple numbers for the variable(s) until you find a pattern.
In addition, remember that anything greater than an absolute value must be positive, regardless of the content of the absolute value. Many GMAT questions of the variable case depend on recognizing
this simple concept for easy solution.
For example:
What is the range of x for which |x+1| < x?
This inequality does not fit the |something | < Number form, so the two scenario approach is out. Instead, investigate the inequality – what can you learn? Which numbers can you plug in?
For x to be greater than an absolute value, x must be positive. Therefore, plug in simple positive numbers to find out when the inequality is true.
If x cannot be 1.
If x cannot be 2 either.
Can you find a pattern? In other words, do you think that the inequality would be true for
If x is positive, then x+1| = x+1. But then the inequality cannot be true because it reads x that fits in the inequality.
Solve inequalities with an absolute value on one side and a number on the other by considering two scenarios:
1. First scenario – copy the inequality without the absolute value brackets and solve.
2. Second scenario – remove the absolute value brackets. Put a negative sign around the other side of the inequality, AND flip the sign. Solve.
When solving inequalities with an absolute value on one side and a variable on the other, the two scenario approach does not help. Instead:
1. Investigate the variable case by plugging in simple numbers for the variable(s) until you find a pattern.
2. Remember anything greater than an absolute value must be positive, regardless of the content of the absolute value. Many GMAT questions of the variable case depend on recognizing this simple
concept for easy solution.
Now, try solving the following data sufficiency question:
Is m>n?
(1) |m+n| < |m| + |n|
(2) |m| > |n| + 1
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked;
D. EACH statement ALONE is sufficient to answer the question asked;
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
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7 comments
B isn;t right. M=10 & N=5, (m>n) M=-10 & N=-5, (m<n) both of these would satisfy statement B.
I think the answer is E.
1. m and n are of different signs,but we cannot determine whether m>n
2. From this, m and n could be of same sign or different sign, but cannot determine whether m>n
m = -5, n = +3
m = +5, n = -3
m = +5, n = +3
Combining 1 and 2, we only know that m and n are of different signs and absolute value of m is 1 more than absolute value of n, but still cannot determine whether m>n.
I close my case here
Hi Nitin
1. This inequality will be true only if m and n have different signs. You can take any value for m and n with different sign and plugging in the values.
When m is -ve, then m n
Hence we cannot determine whether m > n
2. In this case, absolute value of m is one more than absolute value of n..
Lets pick few
a. m = -5, n = +3
|m| = |-5| = 5
|n| + 1 = |3| + 1 = 4
5 > 4
m 4
m > n
c. m = +5, n = +3
|m| = |5| = 5
|n| + 1 = |3| + 1 = 4
5 > 4
m > n
From Above three cases we can see that (2) is insufficient to prove m > n
Combining (1) and (2): m and n are of different signs and absolute value of m and 1 more than absolute value of n.
We see that (a) and (b) in the analysis of 2 satisfy both cases, but still we cannot determine whether m > n.
Please let me know whether this makes sense.
Hi Nitin,
Seems copy and paste did not work properly in my previous reply. I am reposting my reply.
1. This inequality will be true only if m and n have different signs. You can take any value for m and n with different sign and plugging in the values.
When m is -ve, then m n
Hence we cannot determine whether m > n
2. In this case, absolute value of m is one more than absolute value of n..
Lets pick few
a. m = -5, n = +3
|m| = |-5| = 5
|n| + 1 = |3| + 1 = 4
5 > 4
m 4
m > n
c. m = +5, n = +3
m| = |5| = 5
|n| + 1 = |3| + 1 = 4
5 > 4
m > n
From Above three cases we can see that (2) is insufficient to prove m > n
Combining (1) and (2): m and n are of different signs and absolute value of m and 1 more than absolute value of n.
We see that (a) and (b) in the analysis of 2 satisfy both cases, but still we cannot determine whether m > n.
Hi Nitin,
Seems copy and paste did not work properly in my last two reples. I am reposting my reply, and hope this time it comes out ok. Sorry about this.
1. This inequality will be true only if m and n have different signs. You can take any value for m and n with different sign and plugging in the values.
When m is -ve, then m n
Hence we cannot determine whether m > n
2. In this case, absolute value of m is one more than absolute value of n..
Lets pick few numbers for m and n
a. m = -5, n = +3
|m| = |-5| = 5
|n| + 1 = |3| + 1 = 4
5 > 4
m 4
m > n
c. m = +5, n = +3
m| = |5| = 5
|n| + 1 = |3| + 1 = 4
5 > 4
m > n
From Above three cases we can see that (2) is insufficient to prove m > n
Combining (1) and (2): m and n are of different signs and absolute value of m is 1 more than absolute value of n.
We see that (a) and (b) in the analysis of 2 satisfy both cases, but still we cannot determine whether m > n.
Hope this makes sense.
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Pendulum Dangling from a Spring
This Demonstration models the dynamics of a pendulum suspended on a spring. The spring-pendulum system has three degrees of freedom: the length of the spring ; the angle between the spring's axis
and the vertical ; and the angle between the pendulum's rod and the vertical . Follow the trace of the pendulum's bob and/or pivot as you change the parameters of the system.
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Let ^ be a nonempty indexing set, let A* = {A subscript alpha such that alpha is an..
November 13th 2011, 08:06 PM #1
Junior Member
Nov 2011
Let ^ be a nonempty indexing set, let A* = {A subscript alpha such that alpha is an..
Let ^ be a nonempty indexing set, let A*={A (subscript alpha) such that alpha is an element of ^} be an indexing family of sets, and let B be a set. Use the results of the theorems for indeing
sets and indexing families of sets to prove :
B- ( Intersection of (alpha in the ^ <- below intersect symbol; A (subscript alpha) <- next to intersection symbol) = Union of (B-A (subscript alpha) <- next to Union symbol w/ (alpha in the ^
below intersect symbol)
How would I go about doing this proof, it is extra points on my quiz and I cannot figure it out, please help!
Re: Let ^ be a nonempty indexing set, let A* = {A subscript alpha such that alpha is
Let ^ be a nonempty indexing set, let A*={A (subscript alpha) such that alpha is an element of ^} be an indexing family of sets, and let B be a set. Use the results of the theorems for indeing
sets and indexing families of sets to prove :
B- ( Intersection of (alpha in the ^ <- below intersect symbol; A (subscript alpha) <- next to intersection symbol) = Union of (B-A (subscript alpha) <- next to Union symbol w/ (alpha in the ^
below intersect symbol)
How would I go about doing this proof, it is extra points on my quiz and I cannot figure it out, please help!
Why don't you post in LaTeX? I cannot read that.
In order to help you we need to see your work..
Post some of your own thoughts on this question.
Tell us where your are having difficulties.
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Thus far, scientific evidence is pointing to an actual Infinity existing
reply to post by dominicus
Originally posted by dominicus
Based on recent studies of the cosmic microwave afterglow of the Big Bang, with which our known universe began 13.7 billion years ago, many cosmologists now believe that this observable universe
is just a tiny, if relentlessly expanding, patch of space-time embedded in a greater universal fabric that is, in a profound sense, infinite. It may be an infinitely large monoverse, or it may be
an infinite bubble bath of infinitely budding and inflating multiverses, but infinite it is, and the implications of that infinity are appropriately huge.
This is awesome, because it just boggles the mind as far as trying to contemplate the very size or way in which this Infinity may exist as part of our fabric of reality.
Relativity and inflation theory, said Dr. Aguirre, “allow us to conceptualize things that would have seemed impossible before.” Time can be twisted, he said, “so from one point of view the
universe is a finite thing that is growing into something infinite if you wait forever, but from another point of view it’s always infinite.”
Great article. I've always had a hunch that existence/reality may be Infinite after I was introduced to Infinite Math via Georg Cantor (same guy in my signature) as part of set theory.
Cantor found that the decimal numbers between 0 and 1, or say 1 and 2, is a bigger unlistable infinite amount of values than counting from 1 to forever. So if we apply that to reality, say the
space between 2 people (let's say 10 feet) can also be infinitely sliced up and you end up in quantum states, possible non-locality, and so on.
Anyway, great article!!!!
Nice find…
Just started reading some of Buddha’s teachings, so can’t wait to find out what a Buddha field lol is… never even knew that multiverses were covered in Buddhism…learn something knew everyday... nice
to see it mentioned in the article.
There are 2 excellent BBC Documentaries, I can highly recommend. One is entitled “How Small is the Universe?”, which describes the science behind the LHC, and the search for some of the smallest
particles in the known universe. The other documentary is called, “How Big is the Universe?”.
In the later documentary, scientists located the light from 3 surpernovas, thousands of light years apart from each other, forming a triangle. Then using adavanced mathematics and equations, they
concluded that the universe is flat.
Currently the Infinte Flat Model of the Universe is the most popular model, and fits best with the observational data.
Incidentally many of the Gnostic texts and teachings talk about this “Totality”, which is “without beginning” and “without end”, sometimes also referred to as the un-knowable one, and the
incomprehensible one etc.
From the “Gospel of truth”
As for the incomprehensible, inconceivable one, the Father, the perfect one, the one who made the totality, within him is the totality, and of him the totality has need.
- JC | {"url":"http://www.abovetopsecret.com/forum/thread914788/pg2","timestamp":"2014-04-21T02:01:48Z","content_type":null,"content_length":"30215","record_id":"<urn:uuid:01c3ef94-034e-48f1-9ef6-38e9ba515617>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00227-ip-10-147-4-33.ec2.internal.warc.gz"} |
Atiyah-MacDonald, exercise 7.19 - "decomposition using irreducible ideals"
up vote 7 down vote favorite
An ideal $\mathfrak{a}$ is called irreducible if $\mathfrak{a} = \mathfrak{b} \cap \mathfrak{c}$ implies $\mathfrak{a} = \mathfrak{b}$ or $\mathfrak{a} = \mathfrak{c}$. Atiyah-MacDonald Lemma 7.11
says that in a Noetherian ring, every ideal is a finite intersection of irreducible ideals. Exercise 7.19 is about the uniqueness of such a decomposition.
7.19. Let $\mathfrak{a}$ be an ideal in a noetherian ring. Let $$\mathfrak{a} = \cap_{i=1}^r \mathfrak{b}_i = \cap_{j=1}^s \mathfrak{c}_j$$ be two minimal decompositions of $\mathfrak{a}$ as an
intersection of irreducible ideals. [I assume minimal means that none of the ideals can be omitted from the intersection.] Prove that $r = s$ and that (possibly after reindexing) $\sqrt{\mathfrak{b}
_i} = \sqrt{\mathfrak{c}_i}$ for all $i$.
Comments: It's true that every irreducible ideal in a Noetherian ring is primary (Lemma 7.12), but I don't think our result follows from the analogous statement about primary decomposition. For
example, here is Example 8.6 from Hassett's $\textit{Introduction to Algebraic Geometry}$.
8.6 Consider $I = (x^2, xy, y^2) \subset k[x,y]$. We have $$I = (y, x^2) \cap (y^2, x) = (y+x, x^2) \cap (x, (y+x)^2),$$ and all these ideals (other than $I$) are irreducible.
If my interpretation of "minimal" is correct, then this is a minimal decomposition using irreducible ideals, but it is not a minimal primary decomposition, because the radicals are not distinct: they
all equal $(x,y)$.
There is a hint in the textbook: Show that for each $i = 1, \ldots, r$, there exists $j$ such that $$\mathfrak{a} = \mathfrak{b}_1 \cap \cdots \cap \mathfrak{b}_{i-1} \cap \mathfrak{c}_j \cap \
mathfrak{b}_{i+1} \cap \cdots \cap \mathfrak{b}_r.$$ I was not able to prove the hint.
I promise this exercise is not from my homework.
Update. There doesn't seem to be much interest in my exercise. I've looked at various solution sets on the internet, and I believe they all make the mistake of assuming that a minimal irreducible
decomposition is a minimal primary decomposition. Does anyone know of a reference which discusses irreducible ideals? Some google searches have produced Hassett's book that I mention above and not
much else.
2 The radicals of the ideals participating in a decomposition need not be different. – Mariano Suárez-Alvarez♦ Jan 19 '10 at 17:34
The intersection of P-primary ideals is P-primary, so instead of using a bunch of ideals with the same radical, you can just use the intersection. This would contradict minimality. – Joel Dodge
Jan 20 '10 at 2:22
Thanks for the comments. My question is not explicitly about primary decomposition, it's about decomposition using irreducible ideals. I found some answers on the internet which said 7.19 was true
because the analogous statement was true for primary decomposition. I think 8.6 shows this argument is incorrect. It shows you may need to include multiple ideals with the same radical, which as
Joel says, you do not need to do in a minimal primary decomposition. – CJD Jan 20 '10 at 11:10
add comment
3 Answers
active oldest votes
Here is a solution to the hint:
First of all, note that since all the ideals in question contain $\mathfrak a$, we may replace $A$ by $A/\mathfrak a$, and so assume that $\mathfrak a = 0$; this simplifies the notation
Next, the condition that $\mathfrak b_1 \cap \cdots \cap \mathfrak b_r = 0$ is equivalent to the requirement that the natural map $A \to A/\mathfrak b_1 \times \cdot \times A/\mathfrak
b_r$ (the product ot the natural quotient maps) is injective, while the condition that $\mathfrak b_i$ is irreducible is equivalent to the statement that if $I$ and $J$ are non-zero
ideals in $A/\mathfrak b_i$, then $I \cap J \neq 0$ also.
Now suppose given our two irreducible decompositions of $0$. Choose $i$ as in the hint, and set $I_j := \mathfrak b_1 \cap \cdots \cap \mathfrak b_{i-1} \cap \mathfrak c_j\cap \mathfrak
b_{i+1} \cap \cdots \cap \mathfrak b_r,$ for each $j =1,\ldots,s$.
up vote 13 Then $I_1\cap \cdots \cap I_s = 0$ (since it is contained in the intersection of the $\mathfrak c_j$, which already vanishes).
down vote
accepted Now we recall that $A$ embeds into the product of the $A/\mathfrak b_{i'}$. Note that $I_j$ is contained in $\mathfrak b_{i'}$ for $i'\neq i$. Thus, if we let $J_j$ denote the image of
$I_j$ in $A/\mathfrak b_i$, then we see that the image of $I_j$ under the embedding $A \hookrightarrow A/\mathfrak b_1\times\cdots\times A/\mathfrak b_i \times \cdots \times A/\mathfrak
b_r$ is equal to $0 \times \cdots \times J_j \times\cdots \times 0$. Thus the intersection of the images of the $I_j$, which is the image of the intersection of the $I_j$ (since we
looking at images under an embedding) is equal to $0\times \cdots \times (\bigcap J_j) \times \cdots \times 0.$ Thus, since the intersection of the $I_j$ is equal to $0$, we see that $\
bigcap J_j = 0.$ But $\mathfrak b_i$ is irreducible, and so one of the $J_j = 0$. Equivalently, the corresponding $I_j = 0.$
This proves the hint.
(I think the exercise should be a fairly easy deduction from the hint. The statement that $r = s$ at least follows directly, using the hint together with minimality of the two
Excellent! Best 100 reputation points I've ever spent. If you don't mind, I'm going to add something below which I think finishes the proof. If you see an easier way to finish the
proof, I'd be happy to hear it! Lastly, I'm going to wait to accept your answer at least a day to see if anyone has any objection. (Accepted bounty answers last forever, I believe I
read in the FAQ.) – CJD Feb 15 '10 at 17:44
Dear CJD, Take your time. I remember seeing this question the first time around and not seeing how to solve it straight away, and so was happy to be reminded of it this second time
around. – Emerton Feb 15 '10 at 20:07
Today I tried to solve this problem. Now I've read this discussion, but I still don't understand, why example 8.6 does not contradict the statement of Atiyah MacDonald's 7.19?
Furthermore, how do you construct I_{j}, what is the choice of c_j corresponding to b_i? – user38485 Aug 11 '13 at 14:05
@Efim: Dear Efim, I don't have a copy of A&M, so I can' answer your question about the relationship b/w 8.6 and 7.19. As for the $I_j$, I don't really understand your question. The
ideal $I_j$ is defined as in my answer, i.e. we fix an $i$, and then for each $j$ the ideal $I_j$ is defined by a certain formula in terms of the corresponding ideal $\mathfrak c_j$. So
there is no particular "choice" of $\mathfrak c_j$; for each $\mathfrak c_j$ there is a corresponding $I_j$ (while the index $i$ is fixed throughout the discussion). Regards, – Emerton
Aug 12 '13 at 11:54
add comment
This is meant to finish the proof left by Professor Emerton. Thanks to him for explaining the part I was stuck on. If anyone sees an easier way to finish the proof, I'd be happy to hear it!
As stated, the claim r = s is clear. So it remains to show that $\sqrt{\mathfrak{b}_i} = \sqrt{\mathfrak{c}_i}$, after possibly reordering. First, note that we can get from a minimal
irreducible decomposition to a minimal primary decomposition by intersecting all the ideals with the same radical. So we know at least that the set of radicals which occur is uniquely
determined, though not necessarily how many times they occur.
Our proof will be by induction on the maximal length of a chain of prime ideals in $\text{Ass}(\mathfrak{a})$. Consider first the base case, in which every prime ideal in the set is minimal.
up vote Consider any $\mathfrak{p}$ in this set. The primary ideal to which it corresponds is uniquely determined by Theorem 4.10 in Atiyah-MacDonald: the crucial thing is that this is a minimal
2 down ideal in the set $\text{Ass}(\mathfrak{a})$. Thus, replacing $\mathfrak{a}$ by this primary ideal, which we have written as an intersection of irreducible ideals $\mathfrak{b}_i$ and
vote irreducible ideals $\mathfrak{c}_j$, our comment that $r=s$ above shows that the number of irreducible ideals $\mathfrak{b}_i$ with radical $\mathfrak{p}$ equals the number of irreducible
ideals $\mathfrak{c}_j$ with radical $\mathfrak{p}$.
Now assume the result for chains of maximal length $r-1$. Consider a chain of maximal length $r$. By Theorem 4.10 again, we know that the intersection of the primary ideals which correspond
to it is uniquely determined. Ditto for the intersection of the primary ideals corresponding to the first $r-1$ primes in the chain. Using our $r=s$ claim in both cases and subtracting, we
finish the proof.
add comment
See also Bourbaki, Algebra, Chapter II, exercise 17 to §2, for an alternative proof.
up vote 1 down vote
add comment
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Moment generating function for geometric
August 14th 2011, 04:29 PM #1
Aug 2011
Moment generating function for geometric
geometric distribution given by
$Pr(X=x) = (1/a)(1-(1/a)^{x-1})$ where a > 1.
Am I right in saying the distribution above is the same as
$Pr(X=x) = p(1-p)^{x-1}$ which is just the normal geometric pdf or does the change from p to 1/a change something else
Re: Moment generating function for geometric
It doesn't change anything, but I just don't understand where the MGF is in there ?
Re: Moment generating function for geometric
What is raised to the power (x-1) in each of these cases? (and is it a typo?)
August 15th 2011, 12:17 AM #2
August 15th 2011, 10:20 PM #3
Grand Panjandrum
Nov 2005 | {"url":"http://mathhelpforum.com/advanced-statistics/186154-moment-generating-function-geometric.html","timestamp":"2014-04-18T05:07:22Z","content_type":null,"content_length":"37797","record_id":"<urn:uuid:39c1478d-76b5-41de-98fa-9ca7c1facf0e>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00345-ip-10-147-4-33.ec2.internal.warc.gz"} |
Factor. A) x^2-4x+3 B)x^2+5x-24 - Homework Help - eNotes.com
Factor. A) x^2-4x+3 B)x^2+5x-24
We have given
A. `x^2-4x+3`
factorise 3 as 3.1
`` `=x(x-3)-1(x-3)`
`=(x-3)(x-1)` ( brackets are equal so factor out )
`B. x^2+5x-24`
`we write 24=8xx3`
`x^2+(8-3)x-24` ,
in bracket we take minus because last term has minus sign.
`=(x+8)(x-3)` (brackets are same so factor out)
A. (x-3)(x-1) b. (x+8)(x-3)
`A)` `x^2-4x+3` find soluton of `x^2-4x+3=0`
`Delta= 16-4(3)=4>0` has two solution
`x=(4+-sqrt(4))/2=(4+-2)/2` `x_1=3` `x_2= 1`
The poliniom is decomponed in:
`(x-x_1)(x-x_2)= (x-1)(x-3)`
`B)` `x^2+5x-24=0`
`Delta= 25-4(-24)=121>0` Two real different solutions.
`x_1=3` `x_2=-8`
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the first resource for mathematics
High-dimensional graphs and variable selection with the Lasso.
(English) Zbl 1113.62082
Summary: The pattern of zero entries in the inverse covariance matrix of a multivariate normal distribution corresponds to conditional independence restrictions between variables. Covariance
selection aims at estimating those structural zeros from data. We show that neighborhood selection with the Lasso is a computationally attractive alternative to standard covariance selection for
sparse high-dimensional graphs. Neighborhood selection estimates the conditional independence restrictions separately for each node in the graph and is hence equivalent to variable selection for
Gaussian linear models.
We show that the proposed neighborhood selection scheme is consistent for sparse high-dimensional graphs. Consistency hinges on the choice of the penalty parameter. The oracle value for optimal
prediction does not lead to a consistent neighborhood estimate. Controlling instead the probability of falsely joining some distinct connectivity components of the graph, consistent estimation for
sparse graphs is achieved (with exponential rates), even when the number of variables grows as the number of observations raised to an arbitrary power.
62H99 Multivariate analysis
62J07 Ridge regression; shrinkage estimators
05C90 Applications of graph theory
62F12 Asymptotic properties of parametric estimators
62H12 Multivariate estimation
65C60 Computational problems in statistics
62J05 Linear regression | {"url":"http://zbmath.org/?q=an:1113.62082&format=complete","timestamp":"2014-04-18T00:19:31Z","content_type":null,"content_length":"22292","record_id":"<urn:uuid:9d8b7eba-98ed-47d2-af75-b26fea335312>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00045-ip-10-147-4-33.ec2.internal.warc.gz"} |
West Coast Stat Views (on Observational Epidemiology and more)
(Some final thoughts on statistical significance)
The real problem with p-values isn't just that people want it to do something that it can't do; they want it to do something that no single number can ever do, fully describe the quality and
reliability of an experiment or study. This simply isn't one of those mathematical beasts that can be reduced to a scalar. If you try then sooner or later you will inevitably run into a situation
where you get the same metric for two tests of widely different quality.
Which leads me to the curse of large numbers. Those you who are familiar with statistics (i.e. pretty much everybody who reads this blog) might want to skip the next paragraph because this goes all
the way back to stat 101.
Let's take simplest case we can. You want to show that the mean of some group is positive so you take a random sample and calculate the probability of getting the results you saw or something more
extreme (the probability of getting exactly results you saw is pretty much zero) working under the assumption that the mean of the group was actually zero. This works because the bigger the samples
you take the more the means of those samples will tend to follow a nice smooth bell curve and the closer those means will tend to group around the mean of the group you're sampling from.
(For any teachers out there, a good way of introducing the central limit theorem is to have students simulate coin flips with Excel then make histograms based on various sample sizes.)
You might think of sampling error as the average difference between the mean of the group you're interested in and the mean of the samples you take from it (that's not exactly what it means but it's
close) . The bigger the sample the smaller you expect that error to be which makes sense. If you picked three people at random, you might get three tall people or three millionaires, but if you pick
twenty people at random, the chances of getting twenty tall people or twenty millionaires is virtually are next to nothing.
The trouble is that sampling error is only one of the things a statistician has to worry about. The sampled population might not reflect the population you want to draw inferences about. Your sample
might not be random. Data may not be accurately entered. There may be problems with aliasing and confounding. Independence assumptions may be violated. With respect to sample size, the biases
associated with these problems are all fixed quantities. A big sample does absolutely nothing to address them.
There's an old joke about a statistician who wakes up to find his room on fire, says to himself "I need more observations" and goes back to sleep. We do spend a lot of our time pushing for more data
(and, some would say, whining about not having enough), but we do that not because small sample sizes are the root of all of our problems but because they are the easiest problem to fix.
Of course "fix" as used here is an asymptotic concept and the asymptote is not zero. Even an infinite sample wouldn't result in a perfect study; you would still be left with all of the flaws and
biases that are an inevitable part of all research no matter how well thought out and executed it may be.
This is a particular concern for the corporate statistician who often encounters the combination of large samples and low quality data. It's not unusual to see analyses done on tens or even hundreds
of thousands of sales or customer records and more often than not, when the results are presented someone will point to the nano-scale p-value as an indication of the quality and reliability of the
As far as I know, no one reviewing for a serious journal would think that p<0.001 means that we're 99.9% sure that a conclusion is true, but that's what almost everyone without an analytic background
And that is a problem.
1 comment:
1. I'm pretty sure the administrative database crowd could compete with the corporate statistician for the combination of low data quality and big numbers. It really changes your intuition about
statistical inference to work in these settings.
Nice way of putting it, though! | {"url":"http://observationalepidemiology.blogspot.com/2010/03/curse-of-large-numbers-and-real-problem.html","timestamp":"2014-04-18T00:13:31Z","content_type":null,"content_length":"101818","record_id":"<urn:uuid:22f6d825-5f23-4e0f-a699-1315d4eb0ea1>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00030-ip-10-147-4-33.ec2.internal.warc.gz"} |
Reference Request: Steinberg's 1975 paper "On a paper of Pittie"(retrieved)
up vote 4 down vote favorite
I am currently work on a senior project trying to prove for semisimple Lie groups, $R(T)$ is a free module over $R(G)$ by computing an explicit basis for all the A,B,C,D cases. The canoical reference
is a paper by Pittie( H.V. Pittie: Homogeneous vector bundles on homogeneous spaces, Topology II (1972) 199-203), but I could not find it online or in any books available in the library. Steinberg
generalized Pittie's statement in his paper (Robert Steinberg, On a theorem by Pittie, Topology Vol. 14. pp. 173-177. Pergamon Press, 1975, Printed in Great Britain. Received 1 October 1974).
Since they already proved this in the past, I would like to see their papers before I finish my project, even if at some monetary cost. But I could not access either of them. Not knowing their work
would not hinder my research, for I work in a much more elementary level than they did, but I think their work might be related to my eventual results and I should acknowledge in case they proved
some formula I proved again on my own. So I want to ask where I can find them in paper or electronically. I can read parts of Steinberg's paper via google books, but I would like a pdf file or
something (so I may check).
With advisor's help and the links provided by all the people below, I retrieved the two papers.
Received Steinberg's replying email. He notes "A correction should be made on p.175, line 6 ( which starts with "Consider now ") by putting the exponent "n sub a" on the item over which the product
is being taken.The paper by Pittie appears in Topology, vol. 11, 1972, pp. 199-203, and, if I remember correctly, does not contain an explicit basis for the quotient. " This is important so I put in
reference-request rt.representation-theory lie-groups
1 Changwei, both are available online. Pittie: sciencedirect.com/science/article/pii/0040938372900079 Steinberg: sciencedirect.com/science/article/pii/0040938375900257 Hopefully, your institution
has a membership which will allow you to get them without paying full price... – B R Feb 1 '12 at 23:21
@BR: I accessed both websites via the link you provided, but when I press the button "view the full text", there is no output. Thanks for your help though. – Kerry Feb 1 '12 at 23:41
Changwei, I think you might have to either purchase the article or log in through an institutional account. I was able to download both through my institution (technically, I went through
MathSciNet). – B R Feb 2 '12 at 3:25
@BR: This is disappointing, but at least I can try something now. Thank you. – Kerry Feb 2 '12 at 3:54
3 Not to be too public about it, but I'm sure that if all else fails in retrieving these papers, various people on MO would be glad to download and email to you. – Steve D Feb 3 '12 at 20:46
add comment
2 Answers
active oldest votes
To supplement Barry's citations, I'd point out that the journal Topology was at that time managed by a company which eventually gave up on it after editors resigned partly in protest
against the high prices charged. While the online rights now belong to the ScienceDirect conglomerate, it's expensive to access. This can be frustrating because each paper discussed here
is only 4+ pages long.
On the other hand, Steinberg's paper is reprinted in the moderately priced one volume Collected Papers (AMS 1997). Though Steinberg is long retired from UCLA, he maintains an email link
there, and might be able to supply a reprint of his article. Pittie is an Indian mathematician who has taught at one of the colleges of City University of New York but has not published
for many years; his entry in the combined membership list CML (www.ams.org) does give a current mailing address in New York City.
Some users of MO including myself do have access to both papers and might be able to answer precisely stated questions about them.
up vote 5 ADDED: I hadn't heard previously about the recent death of Harsh Pittie. I was somewhat acquainted with him when we were both at NYU-Courant decades ago and recall hearing some of his
down vote lectures on topology of Lie groups. His paper from that period was grounded in topology and K-theory, but Steinberg's follow-up (in his typical concise style) rounded out the discussion
accepted of representation rings in a more algebraic framework. Moreover, Steinberg exhibits an explicit basis for $R(T)$ as a free $R(G)$-module in the crucial case where $G$ is a semisimple
simply connected compact Lie group and $T$ any maximal torus. In particular, the rank here is the order of the Weyl group $W$. (He also observes that the same ideas work for algebraic
groups over any algebraically closed field.)
Though I've never worked through the details of Steinberg's paper carefully, the underlying idea can be observed (in an oversimplified way) in the rank 1 case. Denoting the weight lattice
(character group of $T$ in additive notation) by $X$, the respective representation rings look like $\mathbb{Z}[X]$ and $\mathbb{Z}[X]^W$. Then Steinberg's basis elements, one for each
element $w \in W$, are defined by applying $w^{-1}$ to a product of symbols (in my notation $e^\lambda$) with $\lambda$ running over suitable fundamental weights. In rank 1, the basis
just consists of $e^0, e^{-\rho}$.
@Jim Humphreys: Thanks for the comment. I think I should contact them directly. My main question is whether their means to prove $R(T)$ is a free module over $R(G)$ related to the
action of the Weyl group on fundamental weights(thus the weight lattice), for that is the approach I am going to take. I also wish to express the gratitude as I learnt a lot by reading
your small book. – Kerry Feb 2 '12 at 1:50
@Jim Humphreys: Unfortunately I found Pittie has already passed away (see jxxcarlson.wordpress.com/2012/01/25/harsh-pittie-in-memoriam). But thank you for the information. – Kerry Feb 2
'12 at 2:04
add comment
If you're willing to pay, you can go to the Topology website and track the articles down. Here's a link that'll take you straight to the issue with the Pittie piece: http://
www.sciencedirect.com/science/journal/00409383/11/2 -- you can find a link to the Steinberg issue there too. (Caveat: I don't know for a fact the articles are actually available; it's
up vote 2 possible the site will say the order can't be filled. I didn't want to plunk down the coin to find out.)
down vote
@Barry Cipra: Hi, I will try to access the Topology website after dinner. For the link you provided (as well the links provided from above comments), I just do not know how to get the
full article as there is no output by pressing the button. – Kerry Feb 1 '12 at 23:45
Unfortunately, the button you have to press is the one that says "Purchase." Hopefully someone can arrange to get you reprints. – Barry Cipra Feb 2 '12 at 2:44
@Barry Cipra: I see, so essentially I need to log in to access both articles. Let me contact my school (Bard)'s librarian to see if there is something they can do in this situation. –
Kerry Feb 2 '12 at 11:11
add comment
Not the answer you're looking for? Browse other questions tagged reference-request rt.representation-theory lie-groups or ask your own question. | {"url":"http://mathoverflow.net/questions/87285/reference-request-steinbergs-1975-paper-on-a-paper-of-pittieretrieved?answertab=oldest","timestamp":"2014-04-19T17:43:11Z","content_type":null,"content_length":"74079","record_id":"<urn:uuid:164d5561-12fd-4cd2-bcd6-a597a34d29c0>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00351-ip-10-147-4-33.ec2.internal.warc.gz"} |
Brownian Motion
March 17th 2010, 11:00 AM
Brownian Motion
I don't understand this one small point about Brownian motion...
Here's the question, my answer and the part I always miss out...
Let $\{W_t\}_{t \geq 0}$ be a standard Brownian motion. Show that.
(a) $\{X_t\}_{t \geq0}$ is a standard Brownian motion, where $X_t = -W_t$
Question is easy enough. Can show that $X_0 = 0$ almost surely and get that for $0 \leq s \leq t$, $X_t - X_s = -(W_t - W_s)$ is distributed with mean $0$ and variance $t-s$ and is independent of
$F_s = \sigma (W_u: 0 \leq u \leq s)$. (Not ENTIRELY sure why this is though)
Then comes this small statement that I don't know how to show (and hence don't know if it's true).
$X_t$ has continuous paths.
How do I show that? It's only ever stated in answers so I don't need some in depth proof, just a simple, it has continuous paths because...
March 17th 2010, 11:13 AM
March 17th 2010, 02:11 PM
The reason for the law is because the normal distribution is symmetric. The independence follows rather easily if you use the weak definition (see below), that is for any continuous bounded
function f and $A \in \mathcal{F}_s$ you have that (by independent increments of BM)
Now if you consider any f that is bounded and continuous then $g(x)=f(-x)$ is bounded and continuous hence
A rather long and boring way of proving what your intuition tells you. | {"url":"http://mathhelpforum.com/advanced-statistics/134294-brownian-motion-print.html","timestamp":"2014-04-17T06:01:20Z","content_type":null,"content_length":"10943","record_id":"<urn:uuid:4a50c955-b8cb-42e9-933a-fed4354ec90b>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00434-ip-10-147-4-33.ec2.internal.warc.gz"} |
CGTalk - Finding Rotation Angle
11-28-2010, 01:25 AM
I am trying to aggregate a number of equilateral triangles recursively. I start with one triangle (triangle1) , duplicate it, place the pivot of the new triangle (triangle2) on vertex 0, run through
a number of conditional statements and then move triangle2 (with the pivot on vertex 0) to one of the 3 vertices of triangle1. This is not a 2-dimensional operation, therefore the rotational
operation cannot be the same variable all the time. So, if the new triangle2.vtx0 is placed on triangle1.vtx1, I need to rotate triangle2 (in 3-d) so that it's edge 1 meets triangle1.edge0.
Basically, I need to orient triangles so they meet on an edge to edge connection.
Any help would be greatly appreciated. | {"url":"http://forums.cgsociety.org/archive/index.php/t-938453.html","timestamp":"2014-04-20T06:02:40Z","content_type":null,"content_length":"9043","record_id":"<urn:uuid:8111a41c-39cf-47d5-b731-09ae9b8fe938>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00188-ip-10-147-4-33.ec2.internal.warc.gz"} |
he Quant
Einstein moved to the USA after the rise of nazism. He continued to be dissatisfied with the formulation of quantum mechanics and tried to find ways to challenge the theories. Seemingly unable to use
the uncertainty principle for this purpose, he developed the EPR paradox with two younger colleagues at Princeton, Boris Podolsky and Nathan Rosen. Their argument centered around a quantum system
consisting of a pair of particles (A and B) which are created with opposite spin, i.e.: the overall system has no net spin.
The particles move apart at the speed of light and when they are widely separated, the spin of one of the particles, say A, is measured and found to point 'up'. The classical interpretation is
simple: particle 'B' was always in a spin 'down' state. The quantum mechanical interpretation is more tricky as the Copenhagen Interpretation says that the spin of particle 'A' has no definite value
until it is measured. Or, to quote Bohr: '… the state of an atomic system before a measurement is not defined but only has the possibility of certain values with associated probabilities'. At this
point, it must produce an instantaneous effect on particle 'B' to give the opposite spin. This requires 'action at a distance' or faster than light communication! Faster than light action was
completely disallowed by Einstein's theory of special relativity. Einstein and his colleagues said that there must be hidden information which is not contained in the wave function of the system.
Einstein debated this at length with Bohr until his death in 1955. | {"url":"http://www.nobelprize.org/educational/physics/quantised_world/interpretation-3.html","timestamp":"2014-04-16T07:17:40Z","content_type":null,"content_length":"8158","record_id":"<urn:uuid:75da160b-3bae-4c91-afa0-a444a9e14085>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00144-ip-10-147-4-33.ec2.internal.warc.gz"} |
Scenario Analysis Provides Glimpse Of Portfolio Potential
Scenario analysis evaluates the expected value of a proposed investment or business activity. The statistical mean is the highest probability event expected in a certain situation. By creating
various scenarios that may occur and combining them with the probability that they will occur, an analyst can better determine the value of an investment or business venture and the probability that
the expected value calculated will actually occur.
Determining the probability distribution of an investment is equal to determining the risk inherent in that investment. By comparing the expected return to the expected risk and overlaying that with
an investor's risk tolerance, you may be able to make better decisions about whether to invest in a prospective business venture. This article will present some simple examples of various ways to
conduct scenario analysis and provide rationale for their use. (To learn more about probability distributions, read Find The Right Fit With Probability Distributions.)
Historical performance data is required to provide some insight into the variability of an investment's performance and to help investors understand the risk that has been borne by shareholders in
the past. By examining periodic return data, an investor can gain insight into an investment's past risk. For example, because variability equates to risk, an investment that provided the same return
every year is deemed to be less risky than an investment that provided annual returns that fluctuated between negative and positive. Although both investments may provide the same overall return for
a given investment horizon, the periodic returns demonstrate the risk differentials in these investments. (For more insight, read Measure Your Portfolio's Performance.)
Strict regulations over the calculation and presentation of past returns ensure the comparability of return information across securities, investment managers and funds. However, past performance
does not provide any guarantee about an investment's future risk or return. Scenario analysis attempts to understand a venture's potential risk/return profile. By performing an analysis of multiple
pro-forma estimates for a given venture and denoting a probability for each scenario, one begins to create a probability distribution (risk profile) for that particular business enterprise.
Scenario analysis can be applied in many ways. The most typical method is to perform multi-factor analysis (models containing multiple variables) in the following ways:
• Creating a Fixed Number of Scenarios
□ Determining the High - Low Spread
□ Creating Intermediate Scenarios
• Random Factor Analysis
□ Numerous to Infinite Number of Scenarios
Many analysts will create a multivariate model (a model with multiple variables), plug in their best guess for the value of each variable and come up with one forecasted value. The mean of any
probability distribution is the one that has the highest probability of occurrence. By using a value for each variable that is expected to be the most probable, the analyst is in fact calculating the
mean value of the potential distribution of potential values. Although the mean has informational value, as previously stated, it does not show any potential variation in the outcomes.
Risk analysis is concerned with trying to determine the probability that a future outcome will be something other than the mean value. One way to show variation is to calculate an estimate of the
extreme and the least probable outcomes on the positive and negative side of the mean. The simplest method to forecast potential outcomes of an investment or venture is to produce an upside and
downside case for each outcome and then to speculate the probability that it will occur. Figure 1 uses a three scenario method evaluating a base case (B) (mean value), upside case (U) and a downside
case (D).
For example a simple two factor analysis:
Value V= Variable A + Variable B, where each variable value is not constrained.
By assigning two extreme upside and downside values for A and B, we would then get our three scenario values. By assigning the probability of occurrence, let us assume:
50% for Value (B) = 200
25% for Value (U) = 300
25% for Value (D) =1 00
When assigning probabilities the sum of the probabilities assigned must equal 100%. By graphing these values and their probabilities we can infer a rather crude probability distribution (the
distribution of all calculated values and the probability of those values occurring). By forming the upside and downside cases we begin to get an understanding of other possible return outcomes, but
there are many other potential outcomes within the set bounded by the extreme upside and downside already estimated.
Figure 2 presents one method for determining the fixed number of outcomes between the two extremes. Assuming that each variable acts independently, that is, its value is not dependent on the value of
any other variable, we can conduct an upside, base and downside case for each variable. In the simplistic two factor model, this type of analysis would result in a total of nine outcomes. A
three-factor model using three potential outcomes for each variable would end up with 27 outcomes, and so forth. The equation for determining the total number of outcomes using this method is equal
to (Y^X), where Y= the number of possible scenarios for each factor and X= the number of factors in the model. (For more, see Modern Portfolio Theory Stats Primer.)
In Figure 2, there are nine outcomes but not nine separate values. For example, the outcome for BB could be equal to the outcome DU or UD. To finalize this study, the analyst would assign the
probabilities for each outcome and then add those probabilities for any like values. We would expect that the value corresponding to the mean, in this case being BB would appear the most times since
the mean is the value with the highest probability of occurring. The frequency of like values occurring increases the probability of occurrence. The more times values do not repeat, especially the
mean value, the higher the probability that future returns will be something other than the mean. The more factors one has in a model and the more factor scenarios one includes, the more potential
scenario values are calculated resulting in a robust analysis and insight into the risk of a potential investment.
Drawbacks of Scenario Analysis
The major drawback for these types of fixed outcome analyses are the probabilities estimated and the outcome sets bounded by the values for the extreme positive and negative events. Although they may
be low probability events, most investments, or portfolios of investments, have the potential for very high positive and negative returns. Investors must remember that although they don't happen
often,these low probability events do happen and it is risk analysis that helps determine whether these potential events are within an investor's risk tolerance. (For related reading, see
Personalizing Risk Tolerance and Risk Tolerance Only Tells Half The Story.)
A method to circumvent the problems inherent in the previous examples is to run an extreme number of trials of a multivariate model. Random factor analysis is completed by running thousands and even
hundreds of thousands of independent trials with a computer to assign values to the factors in a random fashion. The most common type of random factor analysis is called "Monte Carlo" analysis, where
factor values are not estimated but are chosen randomly from a set bounded by the variables own probability distribution. (To learn more about this analysis, read Introduction To Monte Carlo
Standards set for reporting investment performance ensure that investors are provided with the risk profile (variability of performance) for past performance of investments. Because past performance
does not have any bearing on future risk or return, it is up to the investor or business owners to determine the future risk of their investments by creating pro-forma models. The output of any
forecast will only produce the expected or mean value of that initiative; the outcome that the analyst believes has the highest probability of occurrence. By conducting scenario analysis an investor
can produce a risk profile for a forecasted investment and create a basis for comparing prospective investments.
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Not Even Wrong
This Week’s Hype
This week’s hype comes from an unusual source, John Baez and his ex-student John Huerta, who have a new article in Scientific American entitled The Strangest Numbers in String Theory.
The expository article about octonions by John (Baez) that appeared in the AMS Bulletin (copy here, a web-site here) is one of the best pieces of mathematical exposition that I have ever seen. The
octonions can be thought of as a system of numbers generalizing the quaternions. As with the quaternions, multiplication does not commute, and things are even worse, it’s not associative either. So,
probably best not to try and think of these as “numbers”, but they do give a very remarkable exotic algebraic structure, one that explains all sorts of other exotic structures occurring in different
areas of mathematics. The article beautifully explains a lot of the intricate story of how octonions connect up surprising phenomena in algebra, geometry, group theory and topology.
If you’re a mathematical physics mystic like myself, you’re susceptible to the belief that anything this mathematically deep, showing up in seemingly unrelated places, must somehow have something to
do with physics. The story of octonions is closely related to the story of Clifford algebras, which are definitely a crucial part of physics, but it seems to me we’re still a long ways from truly
understanding the role in physics of Clifford algebras, much less the more esoteric octonions. One thing that is fairly well understood is that the sequence of division algebras explains some of the
structure of low-dimensional spin groups in Minkowski signature, through the isomorphisms:
The octonion story is supposed to be the next in line, involving Spin(9,1), but made much trickier by the fact that SL(2,O) doesn’t really exist, since the octonions are non-associative.
Back in 1982, a very nice paper by Kugo and Townsend, Supersymmetry and the Division Algebras, explained some of this, ending up with some comments on the relation of octonions to d=10 super
Yang-Mills and d=11 super-gravity. Baez and Huerta in 2009 wrote the very clear Division Algebras and Supersymmetry I, which explains how the existence of supersymmetry relies on algebraic identities
that follow from the existence of the division algebras. Kugo-Townsend don’t mention string theory at all, and Baez-Huerta refers to superstrings just in passing, only really discussing
supersymmetric QFT. There’s also Division Algebras and Supersymmetry II by Baez and Huerta from last year, with intriguing speculation about Lie n-algebras and what these might have to do with
relations between octonions and 10 and 11 dimensional supergravity. For a nice expository paper about this stuff, see their An Invitation to Higher Gauge Theory.
In contrast to the tenuous or highly-speculative connections to string theory that appear in these sources, the Scientific American article engages in the all-too-familiar hype pattern. The headline
argument is that octonions are important and interesting because they’re “The Strangest Numbers in String Theory”, even though they play only a minor role in the subject. It wouldn’t surprise me at
all if octonions someday do end up playing an important role in a unified theory, but the rather obscure connection to the calculation of the critical dimension of the superstring that seems to be
the main point of the Scientific American article isn’t a very convincing argument for such a role.
Somehow I suspect that those string theorists who were upset by Scientific American’s decision to publish speculation by Garrett Lisi about E8 and wrote in to complain, won’t be similarly upset to
find this highly speculative material about the octonions appearing in the magazine.
27 Responses to This Week’s Hype
1. Hi Peter,
I don’t think it is sporting to say that octonions have played only a minor role in string theory. Any area where maximal supersymmetry, large exceptional Lie-Groups (F_4 and E_6-8), triality,
G_2, Exceptional Jordan algebra or a host of other constructs have cropped up is, in essence, an octonionic branch of mathematical or physical theory. Even Bott periodicity uses the octonions (in
disguise) to cycle through the other three division algebras.
While hardly a supporter of the string triumphalist movement, I think that string theorists have been at their best in calling the attention of the mathematics and traditional physics communities
to the possibility of integrating octonionic mathematics with core geometric and physical theory.
Octonionic mathematics used to be on the ‘down low’ but has now made the leap out of the closet and into mainstream research.
So: Two cheers for string theory!
Join me, won’t you?
2. Hi Eric,
It’s kind of like with E8: I’ve got mixed feelings about all these exotic structures, feeling I can’t be sure what’s a beautiful, fundamental structure, and what’s a complex piece of mathematical
junk that managed to just barely hold on to some interesting structure for kind of random reasons. The string theorists also did a great publicity job for E8, although they seem to have lost
interest in it.
At least with Garrett’s SciAm article on E8, you didn’t have to put up with a sales job implying string theory was the reason to take an interest in the subject.
I really did love John’s expository article on Octonions. It lays out beautifully the attractive part of the mathematics and how it hangs together. If that could be connected to physics, I’d be
interested, but the only connection advertised in the SciAm piece is the rather obscure one about getting the supersymmetry algebra to work out a certain way in a certain dimension. It’s not even
clear to me that this is what explains why 10 is the critical dimension, which is the claim repeatedly hammered home in the article.
3. Corinne Manogue, Tevian Dray, and others wrote some very illuminating papers where they spell out the properties of octonions and postulate about their applications to physics:
The one below by Manogue and Schray was especially enlightening. They even define structure matrices for the octonions in Appendix A:
4. I never understood the big deal with octonions. Sure, it is the last division algebra, but if you relax your axioms a little the Cayley-Dickson construction gives an infinite tower of
increasingly uninteresting algebras:
n=1: Reals.
n=2: Complex numbers.
n=4: Quaternions, not commutative.
n=8: Octonions, not associative.
n=16: Sedenions, not alternative but power associative.
n=32: 32-ions?
I can see why you need to give up commutativity – things can be done in different order – but why is the division algebra property important? There might be a reason that octonions have not made
it into physics in 150 years.
5. Peter wrote:
The octonion story is supposed to be the next in line, involving Spin(9,1), but made much trickier by the fact that SL(2,O) doesn’t really exist, since the octonions are non-associative.
PSL(2,O) does exist: you just have to be careful about it. The octonions aren’t associative, but the subalgebra generated by any two octonions is. So, you just need to avoid recklessly
multiplying lots of numbers when you don’t really need to.
You can define the octonionic projective line and the group PSL(2,O) acting on it. You can even go ahead and define the octonionic projective plane and PSL(3,O), which turns out to be a certain
real form of E6. But then the show stops: to define PSL(4,O) we’d really need the associative law. It’s fun to see how one “hits the wall” here.
This is explained in my octonions paper, though I could make it all much clearer now.
but the only connection advertised in the SciAm piece is the rather obscure one about getting the supersymmetry algebra to work out a certain way in a certain dimension.
It’s not at all “obscure” — within the narrow confines of string theory at least — that classical superstring Lagrangians rely for their supersymmetry on a magical identity that holds only in
spacetimes of dimensions 3, 4, 6 and 10. It’s explained in Green, Schwarz and Witten’s textbook, for example.
And it’s been known for quite some time that these special dimensions are 2 more than the dimensions of the reals, complexes, quaternions and octonions. And in fact this is no coincidence: the
existence of these number systems is the simplest explanation of what’s going on here.
So, don’t try to make it sound like an obscure yawn-inducing technicality about “some supersymmetry algebra working out a certain way in a certain dimension”. It’s a shocking and bizarre fact,
which hits you in the face as soon as you start trying to learn about superstrings. It’s a fact that I’d been curious about for years. So when John Huerta finally made it really clear, it seemed
worth explaining — in detail in some math papers, and in a popularized way in Scientific American.
But of course, none of this has anything to do with whether superstring theory is right as a theory of physics. The article says quite clearly that superstring theory makes no testable
At this point we should emphasize that string theory and M-theory have as of yet made no experimentally testable predictions. They are beautiful dreams — but so far only dreams. The universe
we live in does not look 10- or 11-dimensional, and we have not seen any symmetry between matter and force particles. David Gross, one of the world’s leading experts on string theory,
currently puts the odds of seeing some evidence for supersymmetry at CERN’s Large Hadron Collider at 50 percent. Skeptics say they are much less. Only time will tell.
I thought you’d quote that part. Oh well.
By way, I just won a case of scotch from Dave Ring: I’d bet him that the LHC wouldn’t discover “strong evidence for supersymmetry” in its first year of operation.
It’s not even clear to me that this is what explains why 10 is the critical dimension, which is the claim repeatedly hammered home in the article.
I don’t think the octonions explain why 10 dimensions is the critical dimension – at least, not now. It would be cool if they did. But so far, all I know is that classical superstrings favor
dimensions 3, 4, 6 and 10, thanks to the four normed division algebras. Other considerations pick out the 10-dimensional case, which happens to be the octonionic one.
It’s a bit odd to say we “repeatedly hammer home” something about the critical dimension – that was certainly not our intention, and we never even use the phrase “critical dimension”. But maybe I
can guess how you’d get that impression. It’s probably worth comparing what John Huerta and I originally wrote, to what came out in the final version.
We originally wrote:
For strings, when the number of extra directions is 1, 2, 4, or 8, we get supersymmetry. Why? Because then its vibrations can be described using numbers in a division algebra. But the total
number of dimensions of space and time is 2 more than the number of extra dimensions. So, we get supersymmetry when the total number of dimensions is 3, 4, 6, or 10. One of these dimensions
is time; the rest are space.
Curiously, when we fully take quantum mechanics into account, it appears that only the 10-dimensional theory is consistent. This is the theory that uses octonions. So, if string theory is
right, the octonions are not a useless curiosity: on the contrary, they play a fundamental role in understanding spacetime, matter, and the forces of nature!
Here it’s pretty clear, I hope, that quantum considerations pick out the 10-dimensional theory—not some special fact about octonions. The “curiously” makes it clear that we don’t understand the
connection to the octonions.
The final version says:
At any moment in time a string is a one-dimensional thing, like a curve or line. But this string traces out a two-dimensional surface as time passes. This evolution changes the dimensions in
which supersymmetry naturally arises, by adding two — one for the string and one for time. Instead of supersymmetry in dimension one, two, four or eight, we get supersymmetry in dimension
three, four, six or 10.
Coincidentally string theorists have for years been saying that only 10-dimensional versions of the theory are self-consistent. The rest suffer from glitches called anomalies, where computing
the same thing in two different ways gives different answers. In anything other than 10 dimensions, string theory breaks down. But 10-dimensional string theory is, as we have just seen, the
version of the theory that uses octonions. So if string theory is right, the octonions are not a useless curiosity: on the contrary, they provide the deep reason why the universe must have 10
dimensions: in 10 dimensions, matter and force particles are embodied in the same type of numbers—the octonions.
This “provide the deep reason why the universe must have 10 dimensions” makes it sound as if some special fact about octonions explains why the universe has 10 dimensions. So yeah, that’s bad.
But if you read the beginning of the paragraph you’ll see that’s not what’s going on: it’s quantum considerations that pick out the number 10. And the “coincidentally” makes it clear that we
don’t understand the connection.
If you saw how much editing and counter-editing were involved in, perhaps you’ll forgive us for letting that phrase slip past. I also don’t like the title, but I couldn’t think of a better one. I
had to admit that our original proposed title, “The Octonions”, would to most readers be about as appealing as “The Metaphraxis” or “The Sexadent”.
Of course, it would be cool if we could find a link between the calculation that singles out the number 10 and the fact that the normed division algebras give an identity that lets you write down
supersymmetric string theory Lagrangians in dimensions 3, 4, 6 and 10. It’s hard to find real “coincidence” of this magnitude in math. So, I suspect it’s just a matter of time before someone
finds a deeper link. I’ve made some nice progress on this but not enough to talk about.
By the way, I have permission to put the Sci Am article on my website about a month after it comes out, and I’ll also put up the various drafts, just for the amusement of people who wonder how
this sort of editing process works.
6. Thomas wrote:
… but why is the division algebra property important?
The division algebra property is not important. What’s important are two things.
First, you get a normed division algebra in dimension n if and only if you can find an n-dimensional spinor representation of Spin(n). This starts the interplay between spinors and vectors, which
provides the special features of Lorentzian geometry in dimension n+2: namely, dimensions 3, 4, 6 and 10.
Second, a normed division algebra is alternative. The alternative law gives the spinor identity that makes supersymmetry work for super-Yang-Mills theory and classical superstrings in dimensions
3, 4, 6 and 10. It also gives the Jacobi identity for the exceptional Lie algebras F4, E6, E7 and E8 — which contain the Lorentz Lie algebras for dimensions 3, 4, 6 and 10. It also gives a
3-cocycle on the superPoincare groups in these special dimensions, which let John Huerta build ‘categorified’ Lie supergroups relevant to describing superstring theory as a higher gauge theory.
So, a lot of math interacts in a marvelous way when you have a normed division algebra.
None of this stuff works anymore when we move further up the Cayley-Dickson tower. I bet something interesting does work, and I’d love to know what it is, but I don’t know and nobody seems to be
working on it.
Needless to say, I’m not talking about whether any of this stuff is useful for physics. I’m talking about math.
7. By the way, John Huerta is not my “ex-student”. His thesis defense is Wednesday May 18th and he should be working on his thesis right now — not wasting his time reading blogs.
8. @baez
you wrote “The universe we live in does not look 10- or 11-dimensional”
so what do octonions tell us about 11 and the M?
9. The split-octonions have already been used by Ferrara and Duff et al. in the description of black hole and string charge vectors in toroidally compactified M-theory (N=8 supergravity). See
arXiv:1002.4223 [hep-th]
10. John,
Thanks a lot for the detailed explanations. I should have made clear that this is a quite different case than the usual editions of “This Week’s Hype”, which just about always involve a claim
that “we’ve found a way to test string theory”. Thankfully, there’s nothing at all like that here, and the material in the article about the relationship of string theory/supersymmetry to the
real world was accurate.
11. I read the article and enjoyed it. From the SA headlines, it sounded hypey, but the article itself was not. I am guessing Baez and Huerta don’t control how SA chooses to headline the story, which
of course will be done in a way that catches the most attention.
12. I have done some reading of octonions, and I read John’s paper on it. It is very good, but technical to me. From what I understand about octonions is that they are anti-associative for three
oconions that are different (excluding a real scaling factor). Since, the ORDER of the octonions determines the value, has anyone tried to make a consistent theory of spacetime using octonions?
The order would determine different possible outcomes, but nature would only choose one. Also since the order does determine output, this gives a before and after quality if we build a length of
time by successive octonion multiplication.
13. also anon wrote:
so what do octonions tell us about 11 and the M?
Maybe you can try my blog article here… it’s a bit technical, but a lot less technical than our actual paper.
But here’s something really quick: the real numbers, complex numbers, quaternions and octonions don’t just let you write down the (classical) theories of supersymmetric strings in 3, 4, 6, and 10
– they also let you write down (classical) theories of supersymmetric 2-branes in dimensions 4, 6, 7 and 11! And the 11-dimensional octonionic case is believed to be relevant to “M-theory”,
whatever that is.
Once you hear this, you should wonder about 3-branes in dimensions 5, 7, 8 and 12. But if you look at the old “brane scan” picture on that blog entry I’m pointing you to, you’ll see it doesn’t
quite work as smoothly as that. In particular, John Huerta has done a bunch of calculations showing that the trick relating octonions to strings in 10 dimensions and 2-branes in 11 dimensions
does not continue to work one dimension higher. Apparently the nonassociativity messes things up!
(It’s long been known by string theorists that something “ends” at dimension 11. However, we are focusing on a limited portion of the math, not the whole story physicists are interested in. So,
the calculations John Huerta did may be new, or at least a little different than the usual story.)
By the way, John Huerta has gotten a postdoc position with Peter Bouwknegt at Australian National University in Canberra. That’s ‘close’ to where I’m working here in Singapore – meaning, only
about a 12-hour flight. (In fact Australia isn’t close to anything.) So, I hope to continue doing a bit of work with him on octonionic puzzles. There are not many people with a good intuition for
the octonions, and he’s one.
14. Pingback: Octonions in String Theory « viXra log
15. Okay, I know this is trite (and liable to get deleted) but I haven’t found the answer after a bit of survey. So how is “octonions” pronounced? Is it “Oc-’tone-ions” or is it “‘oct-’ung-ions”? (As
in the In-and-Out: do you want a _lot_ of onions with that?)
16. Everyone I know says “Oc-’tone-ions”. I’m assuming that little accent thing means the second syllable is accented – that’s what everyone does.
This is like “quaternions”, which also has the second syllable accented.
But the really evil people, who need to be brutally mocked, are the ones who write “octonian”, often in the same sentence as “quaternion”. I have no idea why.
17. Peter wrote:
“The string theorists also did a great publicity job for E8, although they seem to have lost interest in it.”
This has nothing to do with the E8 per se as a mathematical structure. They were forced to focus on Heterotic E8 since at that time it was the only road connecting String theory to phenomenology.
With the advent of D-branes and flux compactifications though they were able to construct realistic string vacua in IIB/F-theory with all the moduli stabilized and even to connect these models to
cosmology i.e. dS vacua (via KKLT), or warped D-Brane inflation (via KKLMMT) etc. On the other hand Heterotic has some know problems with respect to moduli stabilization and wasn’t able to
compete with IIB/F-theory on all these fronts of phenomenology.
Of course this doesn’t really matter since as is well know all these theories are connected via dualities. There is only one String theory.
18. Pingback: Baez & Huerta in Scientific American: “The Strangest Numbers in String Theory” « Francis' world inside out
19. Pingback: Los octoniones y el secreto de la teoría de supercuerdas « Francis (th)E mule Science's News
20. just heard from colleagues that Quillen is no more.
21. (Posted here because answering is disabled in: http://www.math.columbia.edu/~woit/wordpress/?p=942 )
H. Hironaka speaks about news on resolution of singularities in positive characteristics http://www.mpim-bonn.mpg.de/node/3357 in ca. two weeks.
22. “If you’re a mathematical physics mystic like myself, you’re susceptible to the belief that anything this mathematically deep, showing up in seemingly unrelated places, must somehow have
something to do with physics.”
This point of view is absolutely pervasive with particle/GUT theorists right now. There is an absolute disdain for non-rigorous mathematics in theory. The problem is that non-rigorous mathematics
has often been the tool to advance theory in the first instance. Newton’s theories depended on a very non-rigorous theory of calculus. Advancements in quantum mechanics were made for decades
before renormalization could be rigorously handled. Feynman’s initial methods for QED were an absolute mess of ad hoc rules and tricks. Only later were the rules formalized.
To make better theory, we don’t need better mathematics. We need better Physics. Sort out the math later.
23. Zathras,
This has nothing to do with rigorous vs. non-rigorous. The theories being studied by particle/GUT theorists are not rigorously well-defined, and there’s nothing wrong with that.
Many physicists believe that whatever mathematicians are working on, it’s irrelevant to them: they just need the right “physical idea”, with the only math needed maybe something like PDEs. Newton
was not so foolish. He was at the cutting edge of developing both new mathematics and new physics, well aware that the kind of new ideas about mathematics being developed by Leibniz and others
were going to be required to express the new ideas about physics that he was developing. If he had taken the attitude that “all these new-fangled ideas about math can’t be needed for physics”,
surely the dynamics of particles can be expressed using basic algebra and the theory of functions (e.g. today’s high school pre-calculus math), he would not have gotten far.
The problem with what has come out of string theory is not that it is based on too much abstract mathematics, but that it is based on a physical idea that turned out to be wrong.
24. “The problem with what has come out of string theory is not that it is based on too much abstract mathematics, but that it is based on a physical idea that turned out to be wrong.”
No, that was the problem with string theory. The problem now is that the people doing string theory are so raptured with the beauty of the mathematics in string theory, that they think the theory
must be correct. Which brings us back to your quote. People have stopped looking for other physical analogies because they think their rigorous math has to be correct.
25. Zathras,
String theory is not based on rigorous mathematics. No one has a consistent definition of the theory, even at a non-rigorous level. What people won’t give up on is not a piece of rigorous
mathematics, but a physical idea: space-time has 10/11 dimensions and strings/M-theory branes unify particle physics and gravity.
Some interesting mathematics has come out of string theory (e.g. mirror symmetry), but this happened not by physicists coming up with rigorous mathematics, but by physicists coming up with
non-rigorous, conjectural ideas about mathematics, which mathematicians then used as inspiration to come up with rigorous versions (of small parts of the original ideas).
26. Peter
Very good! I agree to your thought. I think you are right.
As you said, many physicists and mathematicians won’t give up string theory right now because that gives the inspiration to them, especially mathematicians.
I think you too don’t give up the string theory as mathematician
because that you and your people in mathematics can obtain the conjectural ideas from string theory.
Even if string theory is not good physics, it is good for mathematicians to long for getting conjectural ideas.
27. @Peter: To me it seems increasingly likely that string theory is the physics equivalent of what in markets we call a “speculative episode”. I think theoretical physics has suffered from badly
exagerated expected returns on investment of time in string theory. And as usual those who are stuck in the investment till their necks will not be the first ones to blow the whistle that a
market correction is coming. But I think for the rest of us … we might start to think about how we’re going to bail our colleagues out …
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Ben S.
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Go Figure!
One of the comments on the posting
Faux Diamonds
was quite intriguing. One teacher wrote,
"Would you believe on the NY state 4th grade math test this year, they would not accept "diamond" as an acceptable answer for a rhombus, but they did accept "kite"!!!!! Can you believe this? Since
when is kite a shape name? Crazy."
Well, believe it or not, a kite is a geometric shape! The figure on the right is a kite. In fact, si
nce it has four sides, it is classified as a quadrilateral. It has two pairs of adjacent sides that are congruent (the same length). The dashes on the sides of the diagram show which side is equal to
which side. The one dashed sides are equal to each other, and the two dashed sides are equal to each other.
A kite has just one pair of equal angles. These congruent angles are a light orange on the illustration at the left. A kite also has one line of symmetry which is represented by the dotted line. (A
line of symmetry is an imaginary line that divides a shape in half so that both sides are exactly the same. In other words, when you fold it in half, the sides match. It is like a reflection of
yourself in a mirror.)
The diagonals of the kite are perpendicular because they meet and form four right angles. In
other words, one of the diagonals bisects or cuts the other diagonal exactly in half. This is shown on the diagram on the right. The diagonals are green, and one of the right angles is represented by
the small square where the diagonals intersect.
There you have it! Don't you think a geometric kite is very similar to the kites we use to fly as a kids?
Well, maybe you didn't, but I do remember observing Ben Franklin flying one!
Anyway, as usual, the wind is blowing strong here in Kansas, so I think I will go fly that kite!
Have you ever wondered why a negative number times a negative number equals a positive number? As my mathphobic daughter would say, "No, Mom. Math is something I never think about!" Well, for all of
us who tend to be left brained people, the question can be answered by using a pattern. Have you noticed a reoccurring theme in my articles? All Math is Based on Patterns!
Let's examine 4 x -2 which means four sets of -2. Using the number line above, start at zero and move left by twos - four times. Voila! The answer is -8. Locate -8 on the number line above. Now try
3 x -2. Again, begin at zero on the number line, but this time move left by twos - three times. Ta-dah! We arrive at -6. Therefore, 3 x -2 = -6.
Here is what the mathematical sequence looks like. Moving down the sequence, observe that the farthest left hand column decreases by one each time, while the -2 remains constant. Simultaneously, the
right hand answer column increases by 2 each time. Therefore, based on this mathematical pattern, we can conclude that a negative times a negative equals a positive!!!!
I wish to thank my long time friend, Barbara, who teaches mathematics at a University in North Carolina, for sharing this pattern with me.
Sometimes my college students like to ask me what seems to be a difficult question. (In reality, they want to play Stump the Teacher.) I decided to find out what sort of answers other mathematicians
give; so, I went to the Internet and typed in the infamous question, "Why is any number to the zero power one?" It was no surprise to find numerous mathematically correct answers, most written in
what I call "Mathteese" - the language of intelligent, often gifted math people, who have no idea how to explain their thinking to others. I thought, "Wow! Why is math always presented in such
complicated ways?" I don't have a response to that, but I do know how I introduce this topic to my students.
Since all math, and I mean all math, is based on patterns and not opinions or random findings, let's start with the pattern you see on the right. Notice in this sequence, the base number is always
3. The exponent is the small number to the right and written above the base number, and it shows how many times the base number, in this case 3, is to be multiplied by itself. (Side note: Sometimes I
refer to the exponent as the one giving the marching orders similar to a military commander. It tells the base number how many times it must multiply itself by itself. For those students who still
seem to be in a math fog and are in danger of making the grave error of multiplying the base number by the exponent, have them write down the base number as many times as the exponent says, and
insert the multiplication sign (×) between the numbers. Since this is pretty straight forward, it usually works!) Notice our sequence starts with 3^1 which means 3 used one time; so, this equals
three; 3^2 means 3 × 3 = 9, 3^3 = 3 × 3 × 3 = 27, and so forth. As we move down the column, notice the base number of 3 remains constant, but the exponent increases by one. Therefore, we are
multiplying the base number of three by three one additional time.
Now let's reverse this pattern and move up the column. How do we get from 243 to 81? That's right! We divide by three because division is the inverse operation (the opposite) of multiplication.
Notice as we divide each time, the exponent by the base number of 3 is reduced by one. Let's continue to divide by three as we move up the column. 27÷ 3 = 9; 9 ÷ 3 = 3. Now we are at 3^1 = 3 which
means we must divide 3 by 3 which gives us the quotient of 1. Notice, to fit our pattern, the 3 in the left hand column would have to be 3^0^ ; so, 3^0 must equal one! This works for any number you
wish to put in the left hand column. Try substituting the base number of three with two. Work your way up the sequence dividing by two each time. You will discover that two divided by two equals one
(2^0 = 1). Therefore, we can conclude that any natural number with an exponent of 0 is equal to one. (Zero is not included; it's another mystery to solve.)
What happens if we continue to divide up the column past 3^0^ ? (Refer back to the sequence on the left hand side.) Based on the pattern, the exponent of zero will be one less than 0 which gives us
the base number of 3 with a negative exponent of one or 3^-1 .^ ^ To maintain the pattern on the right hand side, we must divide 1 by 3 which looks like what you see on the left. Continuing up the
column and keeping with our pattern, 3 must now have a negative exponent of 2 or 3^-2 ^ and we must divide 1/3 by 3 which looks like what is written on the right.
As a result, the next two numbers in our pattern are...........................
Isn't it amazing how a pattern not only answers the question: "Why is any number to the zero power one?" But it also demonstrates why a negative exponent gives you a fraction as the answer. (By the
way math detectives, do you see a pattern with the denominators?)
Mystery Solved! Case Closed!
In some preschool and kindergarten classes across the country, the geometric shape formerly known as a diamond is now being called a rhombus. Why? Does it matter?
To be honest, a diamond is not technically a mathematical shape whereas a rhombus is. When someone says the word rhombus, you know they are referring to a quadrilateral that has all four sides the
same length; the opposite sides are parallel, and the opposite angles are equal. (Mathematical Warning: A rhombus is not thinner than a diamond. AND the plural form, rhombi, is not a dance performed
on the program Dancing With the Stars.)
But what comes to mind when you hear the word diamond? If you are a woman, you might envision a large sparkling gem setting on the ring finger of your left hand. If you are a guy, you might think of
a baseball infield. (The distance between each base is the same, making the shape a diamond.) If you play cards, the word might bring to mind a suit of playing cards, OR you might recall a line in
the song, Twinkle, Twinkle, Little Star. Calling a rhombus a diamond is similar to calling a child a "kid" (could be a baby goat), or a home your "pad" (might be a notebook). The first is an accurate
term, the second one is not.
So how does this affect you as a teacher? It doesn't, unless rhombus is on a local benchmark or state test. But if you are an elementary grade teacher, use the correct mathematical language because a
middle school math teacher will thank you; a high school geometry teacher will sing your praises, (see song below) and a college math teacher, like me, will absolutely love you for it!
Rhombus, Rhombus, Rhombus
(sung to the "Conga" tune)
(The song where everyone is in a line with their hands on each other's shoulders)
Rhombus, rhombus, rhombus;
Rhombus, rhombus, rhombus
Once it was diamond;
Now it's called a rhombus. | {"url":"http://gofigurewithscipi.blogspot.com/2011_06_01_archive.html","timestamp":"2014-04-21T15:48:36Z","content_type":null,"content_length":"204711","record_id":"<urn:uuid:81a1bed1-9205-4cf0-9100-26714b777cae>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00284-ip-10-147-4-33.ec2.internal.warc.gz"} |
= Preview Document = Member Document = Pin to Pinterest
six pages with answer sheet Common Core: Geometry 6.G.A1, 5.G.B3, 4.MD.3
Describes (with pictures) congruent figures, and then provides several worksheets that can be used as quizzes or review pages.
Describes (with pictures) two dimensional figures, from line segments to angles to polygons, and then provides worksheets that can be used as quizzes or review pages.
• Describes (with pictures) two dimensional figures, and then provides worksheets that can be used as quizzes or review pages.
• Graphic chart to help students study volumes and areas of geometric shapes. Common Core: Geometry 6.G.A1, 5.G.B3, 4.MD.3
Eight colorful math posters that help teach the concepts of area, perimeter and dimensional figures. Common Core: Geometry 6.G.A1, 5.G.B3, 4.MD.3
This mini-unit is a complete and clear introduction to the concepts of perimeter and area, with worksheets to practice. Includes teaching suggestions.
Common Core: Geometry 6.G.A1, 5.G.B3, 4.MD.3
• Help students review geometry terms with this fun bingo game. Includes game boards and calling cards.
Use geometrical shapes and vocabulary to design a map.
• [member-created with abctools] From "angle" to "vertex". These vocabulary building word strips are great for word walls.
• [member-created with abctools] From "acute angle" to "vertical angles". These vocabulary building word strips are great for word walls.
• [member-created using abctools] From "circle" to "rectangle". These vocabulary building word strips are great for word walls.
Describes line relationships (line segments, and rays; intersecting, parallel, perpendicular) and then provides worksheets that can be used as quizzes or review pages. Common Core: Geometry:
4.G.A.1, 5.GA.1
Describes lines (line segments, rays, points, and more) and then provides worksheets that can be used as quizzes or review pages.
Describes rays and angles, and then provides worksheets that can be used as quizzes or review pages.
• Includes circle, oval, square, rectangle, triangle, star and diamond Common Core: Geometry K.3.1, 1.G.1 2.G.1, 3.G.1
• Includes pentagon, hexagon, octagon, trapezoid cone,and right triangle. Common Core: Geometry K.3.1, 1.G.1 2.G.1, 3.G.1
• Includes circle, oval, square, rectangle, triangle, star, and diamond. Common Core: Geometry K.3.1, 1.G.1 2.G.1, 3.G.1
• Includes pentagon, hexagon, octagon, trapezoid, cone, and right triangle. Common Core: Geometry K.3.1, 1.G.1 2.G.1, 3.G.1
Includes line, line segment, points, end points, ray, intersecting, parallel and perpendicular line posters. Common Core: Geometry: 4.G.A.1, 5.GA.1
Brief introductions to basic shapes: quadrilateral shapes, triangles, and curved shapes; one set per page, each with a picture and a definition.
All 20 of our shape posters in one easy download: quadrilaterals (square, rectangle, rhombus, parallelogram), triangles (equilateral, isoceles, scalene, right, obtuse, acute), curved shapes
(circle, oval, crescent), other polygons (pentagon, hexagon, octagon); one per page, each with a picture and a definition. Common Core: Geometry K.3.1, 1.G.1 2.G.1, 3.G.1
This math mini office contains information on determining perimeter and area in metric and standard measurement for a variety of geometric shapes.
Common Core: Geometry 6.G.A1, 5.G.B3, 4.MD.3
This math mini office contains information on determining perimeter and area in metric and standard measurement for a variety of geometric shapes.Common Core: Geometry 6.G.A1, 5.G.B3, 4.MD.3
Type your own text on each face of the cube, then glue the edges together for a cube with a personal message.
Explanation of different triangles according to their angles, followed by two practice page: one labeling triangles, and one creating triangles to match their descriptions.
Poster showing four different triangles according to their angles, with two activities to test concept.Common Core: Geometry K.3.1, 1.G.1 2.G.1, 3.G.1
Explains solid figures, and then reviews the concept with four worksheets. | {"url":"http://www.abcteach.com/directory/middle-school-junior-high-math-geometry-3312-5-1","timestamp":"2014-04-18T21:18:39Z","content_type":null,"content_length":"151485","record_id":"<urn:uuid:4886a83e-0416-4159-9651-a918bde7a744>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00118-ip-10-147-4-33.ec2.internal.warc.gz"} |
Subclasses for SPMD
The package Thyra includes subclasses for common serial and SPMD vector spaces, vectors, and multi-vectors which are described here. More...
class Thyra::ConstDetachedSpmdVectorView< Scalar >
Create an explicit detached non-mutable (const) view of all of the local elements on this process of an VectorBase object. More...
class Thyra::DetachedSpmdVectorView< Scalar >
Create an explicit detached mutable (non-const) view of all of the local elements on this process of an VectorBase object. More...
Thyra Operator/Vector Base Support Subclasses for SPMD Thyra Implementations
The package Thyra contains base subclasses that support a common type of SPMD implementation of vectors and multi-vectors that are described here. The base subclasses described here provide
support for a simple, yet general, category of Serial and SPMD vectors, multi-vectors and vector spaces.
Efficient Generic SPMD Concrete Thyra Operator/Vector Subclass Implementations
The package Thyra contains highly efficient concrete implementations of SPMD vector space, vector and multi-vector subclasses.
Official utilities for accessing local data in SPMD vectors and multi-vectors.
These non-member helper functions provide the standard way by which clients can get access to local data in VectorBase and MultiVector base objects for the standard SPMD objects. They are the
primary interoperability mechanism for clients to get at local SPMD VectorBase and MultiVectorBase data.
Detailed Description
The package Thyra includes subclasses for common serial and SPMD vector spaces, vectors, and multi-vectors which are described here.
When interfacing to most other well designed packages, there should be no need to create new concrete serial or SPMD vector space, vector or multi-vector subclasses given the subclasses described
Subclasses for SPMD Thyra Implementations
1. Thyra Operator/Vector Base Support Subclasses for SPMD Thyra Implementations
Click here if you want to know about the basic support base subclasses for a common type of SPMD vector and multi-vector implementation that all of the concrete implementations described below
depend on.
2. Concrete SPMD subclasses
Below are some concrete implementations of Thyra classes that are derived from the basic SPMD-support base subclasses mentioned above.
1. Efficient Generic SPMD Concrete Thyra Operator/Vector Subclass Implementations
Click here if you want to know about some general, yet very efficient, concrete implementations of SPMD vector spaces, vectors and multi-vectors.
2. Epetra to Thyra Operator/Vector Adapters (separate doxygen collection)
Click here if you want to know about some general code that takes Epetra objects and creates Thyra wrappers for them. | {"url":"http://trilinos.sandia.gov/packages/docs/dev/packages/thyra/doc/html/group__Thyra__Op__Vec__spmd__adapters__grp.html","timestamp":"2014-04-17T06:50:59Z","content_type":null,"content_length":"12780","record_id":"<urn:uuid:9c36bf90-0d4e-4b1c-bf33-96e660ce4d9e>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00126-ip-10-147-4-33.ec2.internal.warc.gz"} |
Indices problem
Can someone look at this and tell me where ive slipped up? Thanks R
Unless some of your given values have been read wrong, I can see nothing wrong with the calculation. By the way, a helpful comment. Those X's you have in there to represent multiplication are very
distracting. (Remember we typically use x as a variable.) I'd recommend writing somthing like "25 X 5.9" as (25)(5.9) or even 25*5.9. -Dan | {"url":"http://mathhelpforum.com/algebra/24756-indices-problem.html","timestamp":"2014-04-17T10:13:26Z","content_type":null,"content_length":"32942","record_id":"<urn:uuid:649a36c7-fa38-4768-af9a-4f367cc45822>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00451-ip-10-147-4-33.ec2.internal.warc.gz"} |
Exercises 8
Define the function f[n](x) = nx^n(1 - x).
The graphs of f[n] for n = 2, 4, 6 are shown on the right.
Prove that the sequence (f[n]) converges to the 0-function in the metric d[1] on the space C[0, 1] of continuous functions on [0, 1].
Use the usual method for finding the turning point of a differentiable function to find the maximum value of f[n] on the interval [0, 1].
What is the limit of this maximum value as n
Deduce that (f[n]) does not converge to the 0-function in the norm d[].
Which of the following real-valued functions on the open interval (0, 1) are continuous?
a) Define a function f on a real-number x by taking the decimal expansion of x (terminating in infinitely many 0's rather than infinitely many 9's if it is an exact decimal) and discarding the first,
third, fifth and so on, decimal places.
So, for example, f( 0.1234) = 0.24, f(0.1415926536...) = 0.45256... .
b) Define a function g on a real-number x by taking the decimal expansion of x and replacing 0's by 1's, replacing 1's by 2's and so on except that 9's are replaced by 0's.
So, for example, g(0.1298) = 0.23091111... (since the infinitely many 0's at the end all get replaced), g(0.1415926536...) = 0.1526037647... .
c) Define a function h on a real-number x by taking the decimal expansion of x and replacing 0's by 9's, replacing 1's by 8's , 2's by 7's, 3's by 6's, 4's by 5's and vice-versa.
So, for example, h(0.1298) = 0.9701999999... = 0.9702 (since the infinitely many 0's at the end all get replaced), h(0.1415926536...) = 0.9594073463... .
[Hint: Observe that the sequence (0.49, 0.499, 0.4999, ...) converges to 0.5 and use the sequential definition of convergence.] | {"url":"http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/Tutorials/T8.html","timestamp":"2014-04-17T15:47:15Z","content_type":null,"content_length":"6244","record_id":"<urn:uuid:4c2723a0-4062-4842-9f85-db6777a88351>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00363-ip-10-147-4-33.ec2.internal.warc.gz"} |
Integer dynamics hitting infinitely many primes
up vote 5 down vote favorite
I am wondering if there are any rigorous results telling that some dynamical system hits infinitely many primes (except for the case when orbits are just arithmetic progressions). To make it
specific, is there a polynomial $f(x)$ such that its iterations $f^n(x)$ are prime for infinitely many $n$ given that the integer $x$ is fixed. A simple observation here is that if $f(x) = 2x + 1$
then asking if iterations of $f(1)$ contain infinitely many primes is equivalent to the Mersenne primes conjecture.
P. S. This question seems to be related to the Bunyakovsky conjecture, so maybe somebody knows about any partial results in this direction?
nt.number-theory ds.dynamical-systems
1 Pedantic comment alert: Presumably this is not exactly what you meant: $f(x)=2x-7$ satisfies $f^n(7)$ is prime for all $n$. – Anthony Quas Apr 26 '13 at 1:18
3 @Anthony - Actually, in arithmetic dynamics your sequence 7,7,7,... is considered an arithmetic progression, with initial term 7 and common difference 0. This is a convenient convention, for
example, for stating the dynamical Mordell-Lang conjecture, which says that $\{n : f^n(x)\in Y\}$ is a finite union of arithmetic progressions. Here $f:X\to X$ is a morphism of a (smooth)
projective variety, and $Y\subset X$ is a (smooth) subvariety. – Joe Silverman Apr 26 '13 at 2:40
(I) $f(x):=(x−1)\cdot x+1$, for $f(1):=3$, gives what can be called Euclid numbers $f(1)\ f(2)\ \ldots$. (One can throw in $f(0):=2$, so that $f(n)=f(0)\cdot \ldots\cdot f(n−1)+1$ for every $n=1 2
\ldots$). -o-#-o- (II) $f(x):=(x−2)\cdot x+2$ gives Fermat numbers $f(1)\ f(2)\ \ldots$ for $f(1):=3$ (so that $f(n)=f(1)\cdot \ldots\cdot f(n−1)+2$ for every $n=2 3 \ldots$). -o-#-o- (III) Etc –
Wlodzimierz Holsztynski May 2 '13 at 18:15
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3 Answers
active oldest votes
I'm fairly certain that nothing along these lines is known.
Until the late 1990's there were no known `elementary' polynomial sparse sequences that contained infinitely many primes(*). In 1998 Friedlander and Iwaniec proved that the sequence of
integers of the form $x^2 +y^4$ contains infinity many primes. The number of elements of this sequence less than $N$ is $N^{3/4}$. In 2001, Heath-Brown proved that the sequence of
integers of the form $x^3+2y^3$ contain infinitely many primes. The number of elements of this sequence less than $N$ is $N^{2/3}$. This seems the best result in this direction to date.
Nothing seems to be known (other than upper bounds from sieve theory) about polynomials in a single variable (other than linear equations / arithmetic progressions, of course).
If there was a polynomial whose iterates contained infinity many primes this would give an exponentially sparse sequence with infinitely many primes. This seems far from anything anyone
up vote 10 can prove at present. In particular, there appears to be no polynomial time (in the bit length) constructable sequence of integers which is proven to be prime infinitely often and whose
down vote intersection with the first $N$ integers is less than $N^{1/2}$. Indeed such a result would yield new results on the problem of deterministically constructing large primes (see the
accepted Polymath 4 project).
On the other hand, there is some remarkable work in which nontrivial estimates have been obtained for the number of prime divisors of sequences obtained by iterates of various quadratic
polynomials. See: R. Jones, The density of prime divisors in the arithmetic dynamics of quadratic polynomials. J. Lond. Math. Soc. (2) 78 (2008), no. 2, 523–544
Update: (*) Indeed, I had forgotten about the Piatetski-Shapiro prime number theorem when I wrote this. However, the best exponent on this result is gives a set of density around $N^
{.861}$ which is inferior to Heath-Brown's result for the purposes discussed here.
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In 1947, American William . H. Mills proved that there is a real number $A$, greater than 1 but not an integer, such that integer part of
$$ A^{3^n} $$
is prime for all $n =1, 2, 3, \ldots$. The numnber $A$ is known as the Mill's constant. Its value is unknown, but if the Riemann hypothesis is true then,
up vote 8 down vote $$ A \approx 1.3063778838630806904686144926... $$
[1] Mills, W. H. (1947) A prime representing function. Bull. Amer. Math. Soc., 53: 604: MR 8, 567.
[2] [Mill's constant]1
1 thanks, Nilothal, but it looks like it difficult to prove anything useful about this number, so it hardly helps for integer dynamics – DmitryZ Apr 28 '13 at 9:38
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As Mark says, nothing is known about infinitely many primes in dynamical sequences, other than ones that contain an arithmetic progression. An easier, but still useful, question, is that of
primitive prime divisors. A prime $p$ is a primitive prime divisor of $f^n(x)$ if $p\mid f^n(x)$ and $p\nmid f^m(x)$ for all $m\lt n$. There's a vast literature on primitive prime divisors in
various sorts of sequences. In general, if $\mathcal{A}=(a_n)$ is a sequence of integers, or more generally, rational numbers, the Zsigmondy set of $\mathcal{A}$ is $$ \mathcal Z(\mathcal A)
= \{ n : \hbox{the numerator of $a_n$ has no primitive prime divisors} \}. $$ Patrick Ingram and I [1] showed that under suitable hypotheses on $f\in\mathbb{Q}(T)$ and $x,y\in\mathbb{Q}$, the
Zsigmondy set $\mathcal{Z}(f^n(x)-y)$ is finite. In addition to some obvious conditions needed to avoid trivial counterexamples, we needed to assume that $y$ is preperiodic for $f$. Recently,
up vote Gratton, Nguyen, and Tucker [2] removed this restriction on $y$, conditional on the $abc$ conjecture.
7 down
vote [1] Ingram, Patrick; Silverman, Joseph H.; Primitive divisors in arithmetic dynamics. Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 289–302 (MR2475968)
[2] Chad Gratton, Khoa Nguyen, Thomas J. Tucker, ABC implies primitive prime divisors in arithmetic dynamic, preprint, http://arxiv.org/abs/1208.2989
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Not the answer you're looking for? Browse other questions tagged nt.number-theory ds.dynamical-systems or ask your own question. | {"url":"http://mathoverflow.net/questions/128780/integer-dynamics-hitting-infinitely-many-primes/128800","timestamp":"2014-04-18T08:05:04Z","content_type":null,"content_length":"64534","record_id":"<urn:uuid:9bab016a-f939-4c0c-affc-c352fb51764c>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00484-ip-10-147-4-33.ec2.internal.warc.gz"} |
Progress of Monte Carlo methods in nuclear physics using EFT-based NN interaction and in hypernuclear systems.
Armani, Paolo (2011) Progress of Monte Carlo methods in nuclear physics using EFT-based NN interaction and in hypernuclear systems. PhD thesis, University of Trento.
PDF (Progress of Monte Carlo methods in nuclear physics using EFT–based NN interaction and in hypernuclear systems) - Doctoral Thesis
Available under License Creative Commons Attribution.
Introduction In this thesis I report the work of my PhD; it treated two different topics, both related by a third one, that is the computational method that I use to solve them. I worked on
EFT-theories for nuclear systems and on Hypernuclei. I tried to compute the ground state properties of both systems using Monte Carlo methods. In the first part of my thesis I briefly describe the
Monte Carlo methods that I used: VMC (Variational Monte Carlo), DMC (Diffusion Monte Carlo), AFDMC (Auxiliary Field Diffusion Monte Carlo) and AFQMC (Auxiliary Field Quantum Monte Carlo) algorithms.
I also report some new improvements relative to these methods that I tried or suggested: I remember the fixed hypernode extension (§ 2.6.2) for the DMC algorithm, the inclusion of the L2 term (§
3.10) and of the exchange term (§ 3.11) into the AFDMC propagator. These last two are based on the same idea used by K. Schmidt to include the spin-orbit term in the AFDMC propagator (§ 3.9). We
mainly use the AFDMC algorithm but at the end of the first part I describe also the AFQMC method. This is quite similar in principle to AFDMC, but it was newer used for nuclear systems. Moreover,
there are some details that let us hope to be able to overcome with AFQMC some limitations that we find in AFDMC algorithm. However we do not report any result relative to AFQMC algorithm, because we
start to implement it in the last months and our code still requires many tests and debug. In the second part I report our attempt of describing the nucleon-nucleon interaction using EFT-theory
within AFDMC method. I explain all our tests to solve the ground state of a nucleus within this method; hence I show also the problems that we found and the attempts that we tried to overcome them
before to leave this project. In the third part I report our work about Hypernuclei; we tried to fit part of the Î N interaction and to compute the Hypernuclei Î -hyperon separation energy.
Nevertheless we found some good and encouraging results, we noticed that the fixed-phase approximation used in AFDMC algorithm was not so small like assumed. Because of that, in order to obtain
interesting results, we need to improve this approximations or to use a better method; hence we looked at AFQMC algorithm aiming to quickly reach good results.
Repository Staff Only: item control page | {"url":"http://eprints-phd.biblio.unitn.it/479/","timestamp":"2014-04-19T05:08:29Z","content_type":null,"content_length":"20480","record_id":"<urn:uuid:617a6d65-0e18-4a4c-ac27-818b0dbc9c2e>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00263-ip-10-147-4-33.ec2.internal.warc.gz"} |
Ice Cream Extravaganza Giveaway!
UPDATE: The winner of the Ice Cream Extravaganza Giveaway is:
#5,323 – Kristin C.: “follow you on pinterest”
Congratulations Kristin! Be sure to reply to the email you’ve been sent to confirm your mailing address, and your goodies will be shipped off to you!
A couple of weeks ago I remembered that July is National Ice Cream Month and set about having a fabulous time churning up tons of fun frozen treats. You might remember my ice cream roundup of
favorite flavors, a whole week dedicated to homemade ice cream toppings, and a crazy awesome banana split ice cream cake. While I enjoy ice cream all year round and don’t need much encouragement to
bust out the ice cream bowl and try a new flavor, summer is so full of bright and fresh ingredients that it’s the perfect time to experiment. Since getting my ice cream maker attachment nearly 4
years ago, I can count on one hand how many times I’ve purchased store-bought ice cream. Homemade is infinitely better; the difference is astonishing. Plus, you can play around with different flavors
and experiment ’til your heart’s content! Everyone should know the joy of churning ice cream at home, so in an attempt to further that goal, I’m giving away a fun package of ice cream goodies to get
your homemade ice cream-making off the ground! Read below for the details!
Giveaway Details
One (1) winner will receive the following:
1. Your choice of either:
The KitchenAid Ice Cream Maker Attachment (compatible with all KitchenAid stand mixers)
Cuisinart Pure Indulgence 2-Quart Automatic Frozen Yogurt, Sorbet & Ice Cream Maker
2. One (1) copy of each of the following books:
Jeni’s Splendid Ice Creams at Home
3. One (1) Rösle Ice Cream Scoop
How to Enter
To enter this giveaway, just answer the following question in the comments section of this post:
“What’s the last ice cream flavor you ate?”
Additional (Optional) Entries
To up your chances of winning, you can receive up to FIVE additional entries to win by doing the following (these are optional, not required):
1. Subscribe to Brown Eyed Baker by either RSS or email. Come back and let me know you’ve subscribed in an additional comment.
2. Become a fan of Brown Eyed Baker on Facebook. Come back and let me know you became a fan in an additional comment.
3. Follow @browneyedbaker on Twitter. Come back and let me know you’ve followed in an additional comment.
4. Tweet the following about the giveaway: “Enter to win an ice cream extravaganza package from @browneyedbaker! http://wp.me/p1rsii-4SO”. Come back and let me know you’ve Tweeted in an additional
5. Follow Brown Eyed Baker on Pinterest. Come back and let me know you’ve followed in an additional comment.
The Fine Print
Deadline: Thursday, July 26, 2012 at 11:59pm EST.
Winner: The winner will be chosen at random using Random.org and announced at the top of this post. The winner will also be notified via email; if the winner does not respond within 48 hours, another
winner will be selected.
Disclaimer: This giveaway is provided by Brown Eyed Baker.
GOOD LUCK!!
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100. Ben and Jerry’s Chocolate Fudge Brownie and Mint Chocolate Chip, it was delicious!! | {"url":"http://www.browneyedbaker.com/2012/07/23/ice-cream-extravaganza-giveaway/comment-page-56/","timestamp":"2014-04-18T03:35:34Z","content_type":null,"content_length":"124156","record_id":"<urn:uuid:bc8db60f-2367-4951-8e39-6f551ccdefdb>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00104-ip-10-147-4-33.ec2.internal.warc.gz"} |
system of equations
July 13th 2008, 10:28 AM #1
Jul 2008
system of equations
First off my last question was answered in such an awesome way! I am so thankful for this site- You see I'm in the hospital and am trying to complete my math homework for my class via laptop, I
only have minimal calculator functions on this thing and no book to reference the how to's- so I can understand it. The hw is do monday regardless of my condition as says my teacher so this site
is a heaven sent. Anyway need help with this one:
y= x^3-x
y= e^x
solve the system of equations using any method
i just want to make sure the exponents are correct here.
the first equation is:
$y = x^{3} - x$
yes the first has an exponent of 3 and the second is e to the x power
First off my last question was answered in such an awesome way! I am so thankful for this site- You see I'm in the hospital and am trying to complete my math homework for my class via laptop, I
only have minimal calculator functions on this thing and no book to reference the how to's- so I can understand it. The hw is do monday regardless of my condition as says my teacher so this site
is a heaven sent. Anyway need help with this one:
y= x^3-x
y= e^x
solve the system of equations using any method
First seeing that there are two solutions to this equation, we first note that
if $f(x)=x^3-x-e^x$
$f(2)<0$ and $f(2.5)>0$
Thus since this function is continuous we can see by the Intermediate Value Theorem that
Using the Intermediate Value Theorem again but this time for the second zero we see that
So now using $2.25$ and $4.25$ as intital guesses in Newton's method we see that
$\text{As }n\to\infty~~f_n(x_n)\to2.3217$
$f_{n_1}(x_{n_1})\to{4.3717}$ respectively
Therefore $f(4.3717)\approx{f(2.3217)}\approx{0}$
July 13th 2008, 10:44 AM #2
July 13th 2008, 10:50 AM #3
Jul 2008
July 13th 2008, 11:01 AM #4 | {"url":"http://mathhelpforum.com/algebra/43590-system-equations.html","timestamp":"2014-04-17T03:52:41Z","content_type":null,"content_length":"40233","record_id":"<urn:uuid:dbbe8788-a1cb-454b-803c-a09d0fe41fdb>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00220-ip-10-147-4-33.ec2.internal.warc.gz"} |
24bit unsigned integer
There may be more to this question than this, I don't think the full details of the original challenge have been stated clearly.
A bit of googling has turned up this, which I think might be the actual question:
Let us take the following three decimal numbers:
199, 77, 202
Convert each one into a byte. (Even though 77 does not require all 8 bits to express itself, when dealing with a group of data, we usually keep it in a consistent form.) Now, take those three bytes
and combine them to form a 24-bit unsigned integer. The 199 byte is the high byte (most significant) and so forth. Please enter that 24-bit integer in decimal form, and that is your answer. (Hint:
your answer will not be '19977202'!)
If that is the actual question, then the solutions I gave so far have been wrong, for two reasons. Firstly, it used integer rather than byte values, and secondly the byte order in the result was
For a byte value one might use an
unsigned char
type. Similarly the result might be an
unsigned int
Include header
for the latter types, which are guaranteed to occupy the specified number of bits.
Topic archived. No new replies allowed. | {"url":"http://www.cplusplus.com/forum/beginner/110857/","timestamp":"2014-04-19T22:13:02Z","content_type":null,"content_length":"13072","record_id":"<urn:uuid:e59421e2-a25f-4bd3-8649-f1d9f1799134>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00579-ip-10-147-4-33.ec2.internal.warc.gz"} |
PGS Behind The Scenes: Executive Editor Vickie Kearn and the Science of Loving Math
You and math – one of the greatest love/hate relationships of all time. What is it about the subject that excites us yet sends a chilling tingle down our spine at the same time? How can it be so
precise, yet so fickle? We may never know the answers to these questions, but we do know that math is ubiquitous, though some of us may try to hide from it.
We also know that there are those who thrive off the subject, who can’t get enough of it. PUP Executive Editor Vickie Kearn is one of those people. After all, since 2001, her job here at the Press
has been acquiring the best titles in mathematics – and even before she came to PUP, she spent her whole life surrounded by numbers and equations. While math may sometimes cause us to cry tears of
despair, it has caused Vickie to cry tears of joy. Her love of math started as a natural childhood talent, became a pleasantly surprising college benefit, and eventually grew into a career – one
that she has dedicated to making the pursuit of mathematical knowledge easier and more enjoyable.
As part of our continuing series of Q&As with our Science and Reference Editors for Princeton Global Science, we found out more about Vickie’s publishing and math background. Read her truly
illuminating answers below:
What is your background in the sciences and particularly mathematics?
For as long as I can remember, I have loved math, whether it was counting things, looking at patterns, or solving logic problems. I grew up in Venezuela and the American school was very small. There
were only three of us in my grade so we got a lot of attention and our teacher really loved math. Because the school only went through the 9th grade, I went to boarding school in the states for the
rest of high school. Salem Academy only had 100 girls at the time I went. I had no idea that girls weren’t supposed to like math. Elsie Nunn taught the upper level math classes so I had her for three
years. She was amazing. She did not have any fancy equipment but she taught a ton of math. We had math club every day after school and she always came up with something amazing. She knew all about
the lives of the mathematicians so I had the benefit of knowing who the people were behind the math. I went to the University of Richmond where I studied math. At the time I went there, the campus
was split and men and women were on separate sides of a lake. Since math was taught on the men’s side, I got to take all of my classes with them. Men and women were only allowed to talk on Wednesday
afternoon and Saturday and Sunday. But since I was a math major and there was only one other woman math major, I was allowed to talk to the men all the time. Who would have thought math would have
such great benefits?
When I graduated, I taught school for 8 years. I initially taught elementary school and then moved to the junior high to teach math. I taught in a rural open space school (no internal walls) near
Richmond and later in an inner city school in Norfolk, VA. My kids taught me that not everyone learns in the same way and that if math is difficult for some students, you just need to find a
different way to teach it.
What got you interested in publishing, and when did you become a science editor?
During the time I taught, I served on a lot of textbook adoption committees and found that the textbooks got worse and worse. I had the opportunity to move to New York with a friend and decided it
was time to leave teaching and try to improve math teaching and learning in a bigger way. I thought I would try publishing. My first job was at Academic Press. I was a developmental editor. For three
years I edited all the undergraduate textbooks. I made sure all the problems could be worked and wrote the solution’s manuals. After three years and about six different calculus textbooks, I decided
that acquiring books would give me more of an opportunity to have an impact on the content so I moved to Marcel Dekker. I initially was the math/statistics editor but as editors came and went, I also
did engineering and food science. Although I really liked acquiring, I was required to sign 60 books a year. I no longer had the time to do any development work so after 8 years I went to the Society
for Industrial and Applied Mathematics. There I helped to establish the book program and also got to work with journals, membership, and marketing. The staff was a small hard working group of people
dedicated to supporting the members of the society. After 13 years, (and a four hour roundtrip commute) I moved to PUP.
How long have you been in publishing, and when did you come to PUP?
I have been in publishing for 33 years. I came to PUP in March 2001.
Tell us about the books and authors that you’ve worked with. Are there any authors or titles that stand out from the rest? Why?
I have been lucky enough to meet many of the greatest mathematicians. There are so many that stand out but I will pick one from each place I have worked. The first book I edited at Academic Press was
Gil Strang’s Introduction to Linear Algebra (now in its 4th edition). I thought it was the best book I had ever read. That was in 1977. My son is a student at Virginia Tech and I gave him that book
when he took linear algebra his freshman year. I still see Gil at math meetings. I had the amazing opportunity to work with David and Gregory Chudnovsky at Marcel Dekker on A. D. Sakharov: Collected
Scientific Works (1982). During the time we were working on the book, Sakharov was kept tight under Soviet police surveillance and all correspondence had to be smuggled in and out. All of his notes
were handwritten. This was probably the most exciting of my projects. One more that I have to mention from this era is The Shape of Space by Jeffrey Weeks. I signed this book when he was a PhD
student at Princeton University. This in itself was pretty unusual but this was also the first popular math title that I signed. I recently met up with Jeff again when he was attending a seminar at
Princeton. One project that really stands out from SIAM is Matrix Analysis and Applied Linear Algebra by Carl Meyer (2000). This was the first undergraduate textbook that SIAM published and we added
all the whistles and bells we could think of. We also included a really neat CD with all kinds of fun math facts and history about the folks mentioned in the book. I have since published another book
with Carl and we are working on a third. The book at PUP that is extra special for me is Steve Strogatz’s The Calculus of Friendship: What a Teacher and a Student Learned about Life while
Corresponding about Math. This is a wonderful heartwarming story (which also teaches a lot of math) and Steve’s teacher reminds me so much of my high school teacher, Elsie Nunn. This is the only math
book that has ever made me cry. Although I have mentioned a few books that stand out I must say that each time I get a message that one of my books has come in, I race to my mailbox to see it. Taking
an idea to a pile of papers and then a bound book is an amazing process. It takes the cooperation of so many people. Being an editor is a bit like a bartender or therapist. You need to be a good
listener at times, a cheerleader at others, and always a compassionate friend. So many things can happen over the course of writing and publishing a book. I look back over the past three decades and
am so happy to say that I have hundreds of friends–most of them mathematicians.
What makes mathematics publishing different from other subject areas?
In short–equations. When determining how big a book will be, compositors want to know how many words there are. I still don’t know how to count equations as words. We now work with final book pages
which is much easier to calculate. In many disciplines, academics need to publish books to get tenure. In math (and many of the other sciences) academics need to publish papers; books do not add to
their vita. My colleagues in the humanities and social sciences get numerous manuscripts submitted to them each year. I learn of new book projects through networking with mathematicians in academia
and industry.
A surprising fact is that some of PUP’s best-selling books are math books. What is your take on popular math? Is this a trend that will continue to grow?
I believe that the attitude toward math is changing. Many people still “hate” it or find it “really hard.” I agree that much of the math in the books we publish at PUP is way beyond my understanding
but everyone can learn math and appreciate it at some level. That is what our popular math books are all about. They are about finding out really neat facts and how things work. We want people to see
the connections to other areas such as biology and economics. We want them to understand the history of math and how it was developed. And, we want them to understand how mathematicians work and what
they do when they are not doing math. I have known mathematicians who are wonderful painters and singers. I know one who used to drop the starting flag at the Indianapolis 500. Another I know is an
accomplished break dancer. One used to play for the Boston Red Sox until he was injured. Math was his backup plan. We get terrific reviews of the popular math books. Two quotes that stand out for me
are the following. One reviewer stated that if he had to be stranded on a deserted island, he would want to have a book by Paul Nahin with him. One reader of Fearless Symmetry by Avner Ash and Robert
Gross said it was like climbing a mountain. You might not get all the way to the top but the view was just as good. We get lots of fan mail about the popular math books, from kids as young as middle
school. My hope is that everyone who is a math hater will pick up a popular math book and give it a try.
The Calculus Lifesaver by Adrian Banner is a monumental mathematics course book. What is the story behind it?
Adrian Banner developed a study course for non-math majors who were struggling with calculus. At first, only a few students showed up but it wasn’t long before the lecture hall was packed, especially
right before tests. He developed notes as he taught the course. After I badgered him for a few years, he polished them and even did all the typesetting. The book is written for 18-year-old students
and includes examples that are fun for them to work through. Adrian wrote it so that the students would understand the process and not learn just how to get the right answer. The book has its own
MySpace and Facebook page. Adrian gets a lot of fan mail from students who profess that his book “Saved my life!”. We taped all of the lectures which are available for free. They have had more than
60 million downloads. The interesting thing is that although this book was written just as a study guide, and not a textbook, schools are starting to adopt it because students can really learn from
it. It also costs only $25.00.
What are some of the most outstanding PUP mathematics series, and why? What plans do you have for these series?
The cornerstone of our mathematics program is the Annals of Mathematics Studies book series. It was started in 1940 and includes books by John Tukey, Hermann Weyl, Paul Halmos, Alonzo Church, John
von Neumann, John Milnor and many other outstanding mathematicians. I sometimes look at my bookshelf and am amazed at all of those great books in one place. We publish about 4 books each year in this
series. The books are rigorously refereed even though the authors are at the top of their careers. We plan to continue this series. In the past 10 years we have added an applied math monographs
series and are also publishing undergraduate and graduate level textbooks.
Are there any authors or academics who haven’t worked with PUP before that you’re dying to work with?
This is an interesting question. Of course I would love to work with anyone who has won a Fields Medal or the Abel Prize. These are people who have amazing mathematical minds and who will take
mathematics to the next highest level. Then there are those dedicated teachers who are able to bring out the best in their students, who may one day be Fields Medalists themselves. Most often it is
these teachers who write our popular math books and our textbooks which are so important to exciting young people. I would love to work with more women authors. When I started as a math editor in
1977 I never had to describe myself before a meeting because there were so few women. That has really changed but we still need more women mathematicians and authors.
1. Henry Ricardo says:
Keep up the great work, Vickie. I am responsible for one of the reviews you mention: “One reviewer stated that if he had to be stranded on a deserted island, he would want to have a book by Paul
Nahin with him.”
I stand by that review and look forward to seeing other great books from PUP in the future. | {"url":"http://blog.press.princeton.edu/2010/08/31/executive-editor-vickie-kearn-and-the-science-of-loving-math/","timestamp":"2014-04-20T06:02:56Z","content_type":null,"content_length":"93408","record_id":"<urn:uuid:57c9f203-5dfd-41cc-a322-f73c8c93071b>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00062-ip-10-147-4-33.ec2.internal.warc.gz"} |
Prealgebra for Two-Year Colleges/Workbook AIE/Properties of multiplication
From Wikibooks, open books for an open world
Materials Needed: None
Learning Objectives[edit]
1. Use the distributive property to estimate products, and then to correct your estimate.
2. Find and use patterns in the multiplication table.
You have $80 in your wallet. You want to buy 3 DVD's for $19.95 each. (Fortunately, you are in Oregon where there is no sales tax.)
1. Approximate the price a DVD to the nearest whole dollar.
2. Estimate the cost of the three DVD's.
3. Estimate how much money you will have in your wallet after the purchase.
4. What do you need to add or subtract from your approximate price of a DVD to get the actual price?
5. What do you need to add or subtract from your estimate of the cost of the three DVD's to get the actual cost?
6. Exactly how much money you will have in your wallet after the purchase?
7. Explain how your estimation and calculation is related to the following application of the distributive property. Be specific about why you are adding or subtracting.
\begin{align} 80 - \underbrace{3(20 - 0.05)} & = 80 - \underbrace{3(20)} + \underbrace{3(0.05)} \\ & = \underbrace{80 - 60} + 0.15 \\ & = \underbrace{20 + 0.15} \\ & = 20.15 \\ \end{align} | {"url":"http://en.wikibooks.org/wiki/Prealgebra_for_Two-Year_Colleges/Workbook_AIE/Properties_of_multiplication","timestamp":"2014-04-18T00:30:35Z","content_type":null,"content_length":"28810","record_id":"<urn:uuid:2c009055-0399-4d00-9cf1-13a2f755c84f>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00451-ip-10-147-4-33.ec2.internal.warc.gz"} |
nag_elliptic_integral_complete_E (s21bjc)
NAG Library Function Document
nag_elliptic_integral_complete_E (s21bjc)
1 Purpose
nag_elliptic_integral_complete_E (s21bjc) returns a value of the classical (Legendre) form of the complete elliptic integral of the second kind.
2 Specification
#include <nag.h>
#include <nags.h>
double nag_elliptic_integral_complete_E (double dm, NagError *fail)
3 Description
nag_elliptic_integral_complete_E (s21bjc) calculates an approximation to the integral
$Em = ∫0 π2 1-m sin2θ 12 dθ ,$
$m\le 1$
The integral is computed using the symmetrised elliptic integrals of Carlson (
Carlson (1979)
Carlson (1988)
). The relevant identity is
$Em = RF 0,1-m,1 - 13 mRD 0,1-m,1 ,$
is the Carlson symmetrised incomplete elliptic integral of the first kind (see
nag_elliptic_integral_rf (s21bbc)
) and
is the Carlson symmetrised incomplete elliptic integral of the second kind (see
nag_elliptic_integral_rd (s21bcc)
4 References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Carlson B C (1979) Computing elliptic integrals by duplication Numerische Mathematik 33 1–16
Carlson B C (1988) A table of elliptic integrals of the third kind Math. Comput. 51 267–280
5 Arguments
1: dm – doubleInput
On entry: the argument $m$ of the function.
Constraint: ${\mathbf{dm}}\le 1.0$.
2: fail – NagError *Input/Output
The NAG error argument (see
Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
for assistance.
On entry, ${\mathbf{dm}}=〈\mathit{\text{value}}〉$; the integral is undefined.
Constraint: ${\mathbf{dm}}\le 1.0$.
7 Accuracy
In principle nag_elliptic_integral_complete_E (s21bjc) is capable of producing full machine precision. However round-off errors in internal arithmetic will result in slight loss of accuracy. This
loss should never be excessive as the algorithm does not involve any significant amplification of round-off error. It is reasonable to assume that the result is accurate to within a small multiple of
the machine precision.
You should consult the
s Chapter Introduction
, which shows the relationship between this function and the Carlson definitions of the elliptic integrals. In particular, the relationship between the argument-constraints for both forms becomes
For more information on the algorithms used to compute
, see the function documents for
nag_elliptic_integral_rf (s21bbc)
nag_elliptic_integral_rd (s21bcc)
, respectively.
9 Example
This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.
9.1 Program Text
9.2 Program Data
9.3 Program Results | {"url":"http://www.nag.com/numeric/CL/nagdoc_cl23/html/S/s21bjc.html","timestamp":"2014-04-19T07:08:08Z","content_type":null,"content_length":"12133","record_id":"<urn:uuid:c5825854-6b05-4686-a759-62924674d5ea>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00558-ip-10-147-4-33.ec2.internal.warc.gz"} |
FOM: f.o.m. and the working mathematician; Quinean holism
Stephen G Simpson simpson at math.psu.edu
Tue Oct 21 17:57:02 EDT 1997
Neil Tennant writes:
> Is the phrase 'foundations of' different from the prefix 'meta'?
> Is foundations of physics different from metaphysics? My guess is
> not--or at least, that many a metaphysician would not accept the
> identity claim in question.
It's not clear to me that the prefix `meta' has any specific meaning.
For instance, I don't think metamathematics is to mathematics as
metaphysics is to physics. I would however insist that the term
`foundations of' has a specific meaning. Namely, `foundations of X'
means the systematic study of the most basic concepts of subject X,
the logical structure of subject X, etc., with an eye to the unity of
human knowledge. See my short essay at
> Question: on either proposal, how is that extra part of foundations of
> mathematics that cannot itself be mathematized to be made
> intellectually accessible, arresting and worthwhile to the working
> mathematician?
I don't know of any way to compel working mathematicians to be
interested in foundations of mathematics. In fact, many working
mathematicians of the late 20th century are notoriously narrow-minded
and take no interest in any mathematical or scientific topic outside
their own specialized branch of mathematics.
> Is it "merely" philosophy of mathematics?
Why "merely"? Philosophy of mathematics is an important subject.
Foundations of mathematics and philosophy of mathematics are very
closely related.
> Does the WM have any intellectual responsibility to consider it?
Yes, obviously, it's part of their responsibility as mathematicians.
But many of them routinely evade that responsibility. They can get
away with it, because nothing more is demanded of them.
> Would the ordinary work of the WM be enhanced by paying any
> attention to it?
In many cases, yes. For example, Atiyah's Bakerian lecture (in front
of a general scientific audience) might have gained a lot by taking
account of the foundational perspective.
> Could it guide and shape, or help set the agenda, of the community
> of WMs?
Yes, it could. For example, most number theorists are completely
uninterested in the foundational issues raised by Matiyasevic's
theorem. If they took an interest in those issues, number theory
might have a different shape.
> I note with interest that many of the mathematicians who have
> contributed to the list so far write as though the discussion of
> foundations of mathematics cannot proceed without a precise technical
> definition of its methods and concerns.
Well, I think "precise technical definition" is going a little too
far. But I would say this: If we are going to discuss f.o.m., then
obviously it's important to have some sort of working definition of
f.o.m. on the table, so that we know what we are talking about, to
avoid talking at cross purposes about everything under the sun. I
think my concept of f.o.m. given in
www.math.psu.edu/simpson/Hierarchy.html is sufficiently precise to
give us a basis for discussion.
> The very phrase 'foundations of...' carries with it the implication
> that the field in question can be built, or reconstructed, on some
> solid base on which the foundationalist can focus.
> The Quinean holist rejects such a picture for knowledge in general.
> According to the holist, all knowledge is interdependent, none of it
> privileged, every 'known' proposition enjoying that status only
> because of how it lends support to, and finds support from, other
> propositions in the system.
OK, this is interesting. Neil seems to be saying that there is an
alternative view of human knowledge, which he dubs Quinean holism,
according to which my concept of f.o.m. makes no sense, because there
is no hierarchy of mathematical concepts. In other words, according
to this holistic view, no mathematical concept is more basic than any
other mathematical concept. Is that right? Under this holistic view,
the concept "positive integer" is not more basic than the concept
"symplectic manifold". Is that right? I'm not trying to ridicule
Neil's brand of holism; I'm only trying to understand it and its
implications for f.o.m.
> Would one respectable 'foundational' enterprise be a definitive
> articulation of the hierarchy of *definitions* of concepts in all
> the main fields of mathematics?
In my view, that activity would be foundational (according to my
concept of foundational) only insofar as it would focus on the nature
and role of the most basic mathematical concepts. When you get into
higher level mathematical concepts such as symplectic manifolds, you
are moving away from f.o.m.
A modest proposal:
Let's make a tentative list of what might be considered the most basic
mathematical concepts. In this list I would tentatively include
concepts such as the following: number, shape, set, axiom, proof,
algorithm, .... I would not include specialized mathematical concepts
such as "symplectic manifold", which are to be defined and explained
in terms of more basic concepts. Can we FOM'ers agree on this much?
-- Steve Simpson
More information about the FOM mailing list | {"url":"http://www.cs.nyu.edu/pipermail/fom/1997-October/000049.html","timestamp":"2014-04-19T09:26:00Z","content_type":null,"content_length":"7733","record_id":"<urn:uuid:20ed6b77-99e6-4514-98e5-73f509bc2a10>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00152-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: names, trivial ops, variadic extensions & bitwise-=
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
Re: names, trivial ops, variadic extensions & bitwise-=
> From: shivers@xxxxxxxxxxxxx
> From: Aladdin Petrofsky <aladdin@xxxxxxxxxxxxx>
> The fact that (= x1 x2 x3 ...) means (and (= x1 x2) (= x2 x3 ...))
> creates a reasonable expectation that (bitwise-eqv x1 x2 x3) might
> mean (bitwise-and (bitwise-eqv x1 x2) (bitwise-eqv x2 x3 ...)). For
> this reason, perhaps xnor would be a better name than eqv.
> Yes, as a name, XNOR generalises well to the n-ary case: I'm
> confused completely independent of the number of arguments passed to
> the function.
Right. Clearly, NXOR would be the more logical name, since we're
NOTing the result of XOR, rather than excluding something from the
result of NOR. In the n-ary case, NXORC would more fully indicate
that it is the NOT of the XOR of the complements of all the arguments.
(Similarly, OR and AND could be named OR and NORC, or NANDC and AND.)
Perhaps the most symmetric names would be XOR and IAND: XOR is like OR
with the both-true case excluded, and IAND is like AND with the
both-false case included. I suggested the backwards XNOR only because
it seems that the hardware logic folks long ago settled on it. I
suppose this is because "not or" has a handy contraction that happens
to already be in the dictionary, but getting people to accept
"nexclusive or" would be difficult. I guess they weren't as shy about
coining the word NAND because at least "and" is a conjunction, like
"or". (There are precedents for adding an n to an adjective, but
that's a whole nother story.)
> I hope, in passing, that anyone who ever does anything with dyadic
> boolean functions in the future, uses these names.
If one of the things that person does with dyadic boolean functions is
to extend them to be variadic, he won't want to use ARG2 as the name
of the associative LAST function, so your hope will more likely be
realized if you choose the more generic name.
> This is getting way too complicated. I guess the issue is that,
> while there are only 16 dyadic boolean functions, there are just
> lots and lots and lots of variadic boolean functions, and once we
> start considering variadics, we go right off the side of a
> cliff. Consider, for example, if we went with BITWISE-FIRST and
> BITWISE-LAST. Then mightn't we also want BITWISE-SEVENTH, and maybe
> a few more? Same reasoning applies to n-ary variants of andc1,
> andc2, orc1 and orc2 -- maybe we need andc7 and a few more. Just say
> no.
For each of the 16 dyadic functions, there are multiple ways to extend
it to be variadic. For example, EQV can be extended to test if all
the arguments are the same; XOR can be extended to test if there is
exactly one true argument; AND can be extended to test if there is
more than one true argument; OR can be extended to test if there is no
more than one false argument. That doesn't mean we can't choose the
single most sensible variadic extension for each dyadic function. For
the eight associative functions, the best choice would seem to be the
result of folding the dyadic function across the arguments. For the
other eight (which include NOR and NAND), I think the best choice
would be the simplest combination of a single not gate with a single
variadic associative gate. Those rules lead to these specifications
for the four associative trivial operations and the eight
non-associative operations:
(define (bitwise-const0 . args) 0)
(define (bitwise-const1 . args) -1)
(define (bitwise-first n . rest) n)
(define (bitwise-last . args) (apply bitwise-first (reverse args)))
(define (bitwise-nor . args) (bitwise-not (apply bitwise-or args)))
(define (bitwise-nand . args) (bitwise-not (apply bitwise-and args)))
(define (bitwise-nfirst . args) (bitwise-not (apply bitwise-first args)))
(define (bitwise-nlast . args) (bitwise-not (apply bitwise-last args)))
(define (bitwise-or-cfirst n . rest) (apply bitwise-or (bitwise-not n) rest))
(define (bitwise-and-cfirst n . rest) (apply bitwise-and (bitwise-not n) rest))
(define (bitwise-or-clast . args) (apply bitwise-or-cfirst (reverse args)))
(define (bitwise-and-clast . args) (apply bitwise-and-cfirst (reverse args)))
The practice of inverting one argument of many has precedent in the
standard - and / functions. (Okay, those functions actually invert
*all but* one argument, which is a little different. If they had been
correctly designed to invert one argument of many then the desired
result for the one-argument case (inversion) would have naturally
resulted, rather than having to be a special case.)
> Where do these more exotic dyadic boolean functions get used? One
> example is the graphical bitblt operation, which is parameterised by
> a dyadic boolean that says how to combine the source bit with the
> destination bit.
In such systems, I thought the trivial operations (i.e. those that
ignore one or both arguments) were used at least as often as the
nontrivial asymmetric operations. | {"url":"http://srfi.schemers.org/srfi-33/mail-archive/msg00018.html","timestamp":"2014-04-21T12:09:38Z","content_type":null,"content_length":"8673","record_id":"<urn:uuid:08044811-a595-42f7-a8a2-decd4d134046>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00180-ip-10-147-4-33.ec2.internal.warc.gz"} |
Number of results: 216
Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully absorbed, the quantity of drug in the body decreases at a rate proportional to the amount left in the body, with
constant of proportionality . Suppose that the half-life of hydrocodone bitartrate ...
Sunday, March 6, 2011 at 12:43am by calculas
thank you so muchh :)
Tuesday, December 7, 2010 at 9:51pm by Anonymous
2 seconds
Saturday, September 10, 2011 at 3:59pm by =P
is it x^3 -(2x+5)/(x+4) ?
Saturday, March 16, 2013 at 8:35am by black_widow
I am still lost :(
Wednesday, February 15, 2012 at 3:33pm by Anonymous
pre -calculas
No answer.
Wednesday, April 25, 2012 at 2:59pm by Anonymous
pre -calculas
Wednesday, April 25, 2012 at 2:59pm by bobpursley
Is your question is f(x)= x^3-2x+(5/x)+4 ?
Saturday, March 16, 2013 at 8:35am by black_widow
Xcube -2x +5 divide by x+4
Saturday, March 16, 2013 at 8:35am by no
If f(t) = square root of 7/t^5. Find f'(x) and f'(2)
Thursday, October 21, 2010 at 3:56pm by sandy
see me after class!!
Sunday, December 4, 2011 at 11:07pm by schroeder
square root of 2x
Tuesday, July 24, 2012 at 9:47am by Drew
Can someone please help me?
Tuesday, September 18, 2012 at 10:29pm by ladybug
ap calculas
derivative of (-x^3+3)(3x+2)^3
Monday, March 25, 2013 at 12:18pm by Rebecca
is 4 square root of 12 right?
Tuesday, December 7, 2010 at 10:00pm by Anonymous
Yes i did but tht was wrong so i am confused now...
Tuesday, February 14, 2012 at 11:56pm by Anonymous
Never mind I got it thnks
Tuesday, February 14, 2012 at 11:56pm by Anonymous
Find the length of the curve y=(2x), from x=1 to x=4.
Tuesday, July 24, 2012 at 9:47am by Drew
msybe, you have to ask a question
Tuesday, September 18, 2012 at 10:29pm by bobpursley
f(x-h) is f(x) shifted right h units. So, you want y = (x+6)^2
Tuesday, September 18, 2012 at 10:50pm by Steve
ap calculas
derivative of (2x^3+5)cos2x^5
Monday, March 25, 2013 at 11:47am by Rebecca
Calculas HS
If the average (159/2) is greater than the speedlimit...
Thursday, November 6, 2008 at 8:05pm by bobpursley
Saturday, February 13, 2010 at 6:57am by YAMNA
Pre Calculas
1 + 1/(csc^2 - 1) 1 + 1/cot^2 1 + tan^2 sec^2
Wednesday, November 2, 2011 at 7:26pm by Steve
find the compute derivative of f(x)=sinh^2(5x)
Thursday, April 12, 2012 at 12:58pm by ASB
let p=(1,1,1), Q=(2,3 5), and R=(-1,3,1). find the area A of thr triangle with vertices P,Q, and R
Tuesday, December 11, 2012 at 10:18am by Anonymous
Solve following equations tan^(2 ) x-5tanx=-6
Tuesday, March 5, 2013 at 7:40pm by preet
if f(x) = a cos(kx) amplitude = a period = 2pi/k
Wednesday, April 3, 2013 at 10:21am by Steve
The function f(x)= (8x-7)e^2x has one critical number. Find it
Thursday, November 5, 2009 at 5:08pm by jenny
Expand the contents of the logarithm, then use the property that [d/du]ln(u) = u'/u.
Monday, March 29, 2010 at 5:28pm by Marth
int( absolute(x-1) + sqrt(9-x) ) Hint: you may interpret the integral as an area.
Friday, February 24, 2012 at 3:25pm by Calculas
I assume your temperatures are in Fahrenheit. What is k supposed to be? It should have dimensions
Monday, March 12, 2012 at 10:11pm by drwls
Write following equations in polar coordinate then graph them Y=4X
Tuesday, March 5, 2013 at 7:37pm by preet
Find the area of the region under the curve y = 12 e ^{3 x} between x = -2.6 to x =2.6
Saturday, March 8, 2014 at 5:43pm by ryan
Question # 01 Solve the inequality and sketch the solution on a coordinate line.
Tuesday, March 20, 2007 at 5:04pm by khan
Go with Damon ... Calculas
sorry, I should really read more carefully,
Tuesday, October 12, 2010 at 7:17pm by Reiny
F(X)=9 csc6(X) it is continous from -infity to infinity except what value?
Wednesday, October 24, 2012 at 4:58pm by Mohammad Al-Abdullah
I would graph it, frankly. I think it has only one real root, near x=-2
Monday, March 23, 2009 at 8:40pm by bobpursley
I stumble a question I cannot answer please help up. Find value of limit k for equation:
Sunday, May 9, 2010 at 5:25am by Mike
did you look at the answer by MATH - Reiny, Monday, February 13, 2012 at 6:07pm ?
Tuesday, February 14, 2012 at 11:56pm by Anonymous
Find equation of all lines having slop -4 that are tangent to the curve y=16/(x-2)
Thursday, October 24, 2013 at 3:57pm by Anood
correction of problem. Should be: -2x squared + 6x + 5 = 0 Sorry about that! Thanks for your help!
Sunday, February 16, 2014 at 10:34am by Marina
P(4) = 100(4) - 3(16)ln(4) = .. I assume you have a calculator to get appr 333.46
Friday, April 4, 2014 at 10:02pm by Reiny
11th grade-Calculas
wait to take the derivative, should i multiple out the stuff inside the ()?
Sunday, April 5, 2009 at 8:54pm by Anonymous
Pre-Calculas 11
There is no diagram given its just what the question says. There is no other info
Tuesday, October 1, 2013 at 12:42am by ravneet gill
Pre-Calculas 11
There is no diagram given its just what the question says. There is no other info
Tuesday, October 1, 2013 at 12:42am by ravneet gill
find the LCD and solve: x+3 12x + 4 1 - 2x ____ + _________ = _______ 3-x x squared - 9 x + 3
Sunday, February 16, 2014 at 9:50am by Marina
Use "completing the square" to find the zeros of the polynomial. =2x squared + 6x + 5 = 0
Sunday, February 16, 2014 at 10:34am by Marina
the marginal profit is dP/dx = 100 - 6x lnx - 3x so, now plug in x=4
Friday, April 4, 2014 at 10:02pm by Steve
This is simple substitution and arithmetic It is also not clear if you mean g(x) = (2/3)x or g(x) = 2/(3x) I will do a) using g(x) = (2/3)x f(-3) + g(6) = 3(-3) +1 + (2/3)(6) = -9+1+4 = 14 do b) the
same way.
Sunday, June 5, 2011 at 10:24pm by Reiny
take the derivative of cost with respect to x. SEt it equal to zero. Solve for x. I will be happy to critique your thinking.
Friday, December 16, 2011 at 3:56pm by bobpursley
do a long algebraic or a synthetic division to show that (x^2 - 2x + 15)/(x+7) = x-9 + 78/(x+7) the slant asymptote is y = x-9
Sunday, March 10, 2013 at 4:54pm by Reiny
if the box is all cardboard, and AB = 193 and h=2, then something doesn't add up, since the surface area is 4(A+B)+2AB
Monday, April 7, 2014 at 10:20pm by Steve
as usual, divide the fencing equally among lengths and widths. So, I'll go ahead and predict that the solution will be 7500 x 15000
Wednesday, February 19, 2014 at 4:47pm by Steve
Find the volume of the solid that is obtained when the region under the curve y=x+3 over the interval [5, 24] is revolved about x-axis.
Thursday, June 28, 2012 at 9:25am by raza
If cone not cylinder then V = (1/3) h pi r^2 V = 10 pi r^2 The rest is the same
Tuesday, October 12, 2010 at 7:17pm by Damon
Pre Calculas
verify the following 1+1/csc^2x-1=sec^2x
Wednesday, November 2, 2011 at 7:26pm by Raven
Evaluate lim x→0 10x/√(9+x)-√(9-x)
Saturday, March 16, 2013 at 8:39am by Radical Limits!
Let f(x) = 1+3sin2x be a function, determin the amplitude and the period of the function,
Wednesday, April 3, 2013 at 10:13am by mathews
Let f(x) = 1+3sin2x be a function, determin the amplitude and the period of the function,
Wednesday, April 3, 2013 at 10:21am by mathews
Let f(x) = 1+3sin2x be a function, determin the amplitude and the period of the function,
Wednesday, April 3, 2013 at 10:21am by mathews
how to find the x intercepts for f(x) = 3x^3+7x^2+3x+1 please show your steps
Monday, March 23, 2009 at 8:40pm by Sandra
how to find the x intercepts for f(x) = 3x^3+7x^2+3x+1 please show your steps
Monday, March 23, 2009 at 9:42pm by Sandra
Math ( calculas)
find the intervals on which the curve y=x/(x+1)^2 is concave upward or concave downward.
Sunday, March 27, 2011 at 11:08am by Tina
Alegbra 1
This is not a calculas question, this is an ALGEBRA 1 question it is for my review. That answer you gave me is not right.
Friday, December 30, 2011 at 1:39pm by Hello
what is the function whose graph is the graph of y=+4, but is reflected about the the x-axis. y=
Wednesday, September 19, 2012 at 12:16am by ladybug
multiply top and bottom by (x-5)(x+5) which is (x^2-25) x^2 - 9 x - 10 = (x-10)(x+1) by the way
Sunday, February 16, 2014 at 9:34am by Damon
Find the equations for the tangent line and normal line to the curve for f(x)=(x^4)+(2e^x) at the point (0,2)?
Tuesday, September 28, 2010 at 12:16am by john
AP Calculas
Does f(x)=(2x^3-2x)/x^3+x^2-4x-4 have any horizontal asymptotes? How can you tell?
Thursday, October 7, 2010 at 12:10pm by Yishan shan
Pre Calculas
how would you varify the following sinx+sinx cot^2x=cscx
Wednesday, November 2, 2011 at 7:25pm by Raven
write a function whose graph is the graph of y=x^2, but is shifted to the left 6 units.
Tuesday, September 18, 2012 at 10:50pm by ladybug
how to find the x intercepts for f(x) = 3x^3+7x^2+3x+1
Monday, March 23, 2009 at 8:40pm by Sandra
f(x)=ln[(1+e^2x)/(e^2x-4) find the derivative
Monday, March 29, 2010 at 5:28pm by Lori
amplitude = 3 period = 2π/2 = π or 180
Wednesday, April 3, 2013 at 10:13am by Reiny
Applied calculas
Let y=〖tan〗^(-1) x Rewrite it as an equation involving a trig function. Use implicit differentiation to determine an expression for y.
Monday, April 8, 2013 at 5:31pm by jack
high school calculas
Find the (a) the local extrema, (b) the intervals on which the function is increasing, (c) the interval on which the function is decreasing h(x)=2/x
Thursday, November 6, 2008 at 8:27pm by pipi
The slope of the tangent line to the graph of a function, h(t) is 7/2t The function passes through the point (5,3) Find h(4)
Sunday, March 2, 2014 at 9:42pm by ryan
rate= 3F/hr= 3(5/9) C/hr
Thursday, April 29, 2010 at 11:02pm by bobpursley
The x intercepts are where y = f(x) = 0 I don't see any easy solutions here. Whatever roots (x-values) there are will be negative. I suggest a graphical or an iteration solution. One root is
approximately x = -1.90
Monday, March 23, 2009 at 9:42pm by drwls
Find any critical numbers of the function. (Enter your answers as a comma-separated list.) g(t) = t sqrt6 − t , t < 13/3 I don't even know how to start out!
Sunday, February 19, 2012 at 12:58pm by Mary
differential calculas
Find the instantaneous velocity and acceleration at the given time for the straight-line motion described by the equation: s= (t+2)^4- (t+2)^(-2) at t=1 Round to one decimal place.
Monday, November 26, 2012 at 8:22pm by Preet
Can someone walk me through the following questions: Partial Derivatives f(x,y) = x^7 -8y -3 find x and y f(x,y) = 7 / 5x + 4y, find x and y
Wednesday, June 11, 2008 at 12:26pm by Annette
simplify the following complex fraction: x ____ -2 x-5 _____________________________________ x squared - 9x - 10 ____________________ x squared - 25
Sunday, February 16, 2014 at 9:34am by Marina
Suppose that the box height is h = 2 in. and that it is constructed using 193 \text{in.}^2 of cardboard (i.e., AB = 193). Which values A and B maximize the volume?
Monday, April 7, 2014 at 10:20pm by MEMO
Can someone please walk me through the steps for the following Partial derivatives problems: f(x,y) = x^7 - 8y - 3, find x and y f(x,y) = 7 / 5x+4y, find x and y
Wednesday, June 11, 2008 at 12:29pm by Annette
A cylindrical can is made from tin.If it can be contain liquid inside it then what is the parameter of design if we are oblige use the minimum amount of tin.
Monday, December 2, 2013 at 12:25am by mash
If f(x)=3x-1 and g(x)=2/3x+2 find: a) f(-3)+g(6) b)f(3)-g(2)
Sunday, June 5, 2011 at 10:24pm by Genevieve
I stumble a question I cannot answer please help up. Find value of lower and upper limit of k for equation: (3x^3-5)/x^2 dx = 10 limit: -k , k
Sunday, May 9, 2010 at 5:29am by Mike
I a lighthouse is 75 feet tall and boat is out at sea 40 min away, how far is the boaat from the lighthouse. I'm stump on this problem..Help!
Thursday, September 1, 2011 at 11:08pm by Carlos
differential calculas
The angular displacement in radians is given by: θ=8√(t^2+2) Find the angular velocity and angular acceleration at t = 0.25 s
Monday, November 26, 2012 at 8:20pm by Preet
Why don't you read your textbook and figure it out? There are examples. but anyways: y=x^2 is a parabola, so after you draw a standard parabola, just count 6 units left for each points.
Tuesday, September 18, 2012 at 10:50pm by Luo
do the first derivative, set equal to zero. Then at each zero, test the second derivative. Post back if you are lost.
Thursday, November 5, 2009 at 6:11pm by bobpursley
i did however my answer is still wrong. can you show me step by step. this way i can see what i am doing wrong. thank you
Thursday, October 21, 2010 at 3:53pm by sandy
okay so after i find y and its 8 square root of 12 = 6(1/3)dy/dt how do you get rid of the square root?
Tuesday, December 7, 2010 at 10:00pm by Anonymous
Given f(x)=x^3−2x+5/x+4 and f′(3)=a/b, where a and b are coprime positive integers, what is the value of a+b? Details and assumptions f′(x) denotes the derivative of f(x).
Saturday, March 16, 2013 at 8:35am by Derivative of f(x)
Consider the function f(x) = 7 xe^{-7.5 x}, 0<_x<_2. This function has an absolute minimum value equal to: which is attained at x=
Wednesday, February 19, 2014 at 6:08pm by ryan
The profit obtained when x barbecues are sold is P(x) = 100 x − 3x^2 (lnx) dollars . Find the marginal profit when 4 are sold.
Friday, April 4, 2014 at 10:02pm by jake
f= (7t^-5)^1/2 f'= 1/2 ( )^-1/2 * (-5*7t^-6)
Thursday, October 21, 2010 at 3:56pm by bobpursley
if the length is decreasing at a rate of 2 in./min. while width is increasing at a rate of 2 in./min., what must be true about area of the rectangle
Tuesday, December 7, 2010 at 9:39pm by Anonymous
rationalize the denominator: 5 + the sq. rt. of 3 ________________________ 4 + the sq. rt. of 3
Sunday, February 16, 2014 at 12:40am by Marina
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Youngwhan's Simple Latex
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Discrete maths
August 8th 2006, 10:40 AM #1
Aug 2006
Discrete maths
suppose that f:N->N and g:N->N are defined by f(n)=nto the power of 3 and g(n)=n to the power of 2 for each natural number n.Show that gof not equal to fog.
suppose that f:N->N and g:N->N are defined by f(n)=nto the power of 3 and g(n)=n to the power of 2 for each natural number n.Show that gof not equal to fog.
$f=\{ (n,n^3)|n\in \mathbb{N} \}$
$g=\{ (n,n^2)|n \in \mathbb{N}\}$
$fg=f(n^2), n\in \mathbb{N}=n^6, n\in \mathbb{N}$
$gf=g(n^3), n\in \mathbb{N}=n^6, n\in \mathbb{N}$
Thus, saying that they are no equal is wrong.
Is that the actual answer and how it should be wrtten?
Is that the actual answer and how it should be wrtten?
It should be writen as PerfectHacker wrote it!
Suppose that f:N->N and g:N-> are defined by f(n)=n^3 and g(n)=n^2 for each natural number n.Show that gof is not equal fog????
Hello, Colette!
Is there a typo in the problem? . . . The statement is wrong.
Suppose that $f\!:\!N \to N$ and $g\!:\!N \to N$ are defined by
$f(n)=n^3$ and $g(n)=n^2$ for each natural number $n.$
Show that $g\circ f eq f\circ g$
We have: . $\begin{array}{ccc}g\circ f\:=\:g(f(n)) \:=\:g(n^3)\:=\<img src=$n^3)^2\:=\:n^6 \\ \\ f\circ g\:=\:f(g(n)) \:=\:f(n^2)\:=\
oops sorry i left out part of the q
Suppose that f:N->N and g:N-> are defined by f(n)=n^3 and g(n)=n^2-2 for each natural number n.Show that gof is not equal fog???? thats the correct version
Ahh there's a -2 that cropped up this time.
Let $f(n) = n^3,\,g(n)=n^2-2$
Well the domain for both is the natural numbers, but I noticed that the you left off the codomain for the function g. I don't know why, but I would venture to guess because its codomain has to
include -1, which isn't in the natural numbers.
$g(1) = 1^1-2 = -1$
So let $D$ be the codomain of g so that $g:N \to D$.
So in other words, the function g spits out something in $D$ but f can only handle things from $N$. So $f \circ g$ doesn't make sense because $D ot\subseteq N$.
(Note $g \circ f$ still can make sense, but it's certainly not possible that $f \circ g = g \circ f$)
Hello, Colette!
Ah, much better . . .
Suppose that $f:N\to N$ and $g:N \to N$ are defined by
$f(n)=n^3$ and $g(n)=n^2-2$ for all natural numbers $n$.
Show that $g\circ f eq f\circ g$
$g\circ f\;=\;g(f(n))\;=\;g(n^3)\;=\;(n^3)^2 - 2\;=\;n^6 - 2$
$f\circ g\;=\;f(g(n))\;=\;f(n^2-3) \;=\;(n^2-2)^3$$\;=\;n^6 - 6x^4 + 12n^2 - 8<br />$
Obviously, these two expression are equal for only specific values of $n$.
As it turns out, there are no real values of $n$ for which the two composites are equal.
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The notion of locale is a “point-less” version of that of topological space. A localic group is much like a topological group, but there are some differences.
A localic group is a group object in the category of locales.
Localic groups versus topological groups
Localic groups are similar to topological groups, and many examples can be considered as either one. For instance, the real numbers $\mathbb{R}$ under addition can be considered as either a
topological group or a localic group. Since the “space of points” functor $Loc \to Top$ is a right adjoint, it preserves limits and hence group objects, so every localic group has an underlying
topological group.
However, the “locale of opens” functor $Top\to Loc$ does not preserve products, so not every topological group is a localic group—even if its underlying topological space is sober (hence is the space
of points of some locale). In particular, the locale $\mathbb{Q}$ of rational numbers (with topology induced from that of $\mathbb{R}$) is not a localic group under addition, because the locale
product $\mathbb{Q}\times_l \mathbb{Q}$ is “bigger” than the topological-space product (and in particular is not spatial), and the addition map $\mathbb{Q}\times \mathbb{Q}\to \mathbb{Q}$ cannot be
extended to the locale product. However, if $G$ is a locally compact topological group (such as $\mathbb{R}$), then the space product $G\times G$ does agree with the locale product (using the
ultrafilter principle in the proof), and hence $G$ is also a localic group.
• Another important source of localic groups is from progroups: cofiltered limits of discrete groups.
Localic subgroups are closed
A remarkable fact about localic groups is the following (which also proves that $\mathbb{Q}$ cannot be a localic group):
Any subgroup of a localic group is closed.
Sketch of Proof
Details can be found in C5.3.1 of the Elephant, in the more general case of localic groupoids. The basic idea of the proof is to use the fact that the intersection of any two dense sublocales is
again dense (a fact which very much fails for topological spaces).
If $H\rightarrowtail G$ is a localic subgroup, we construct its closure $\bar{H}$, which is also a localic subgroup in which $H$ is dense. By pullback, it follows that $H\times \bar{H} \to \bar{H} \
times \bar{H}$ is fiberwise dense? over $\bar{H}$ via the second projection. Applying the automorphism $(g,h) \mapsto (g,g^{-1}h)$ of $G\times G$, we conclude that $H\times \bar{H} \to \bar{H} \times
\bar{H}$ is also fiberwise dense over $\bar{H}$ via the “composition” map. Dually, $\bar{H}\times H \to \bar{H} \times \bar{H}$ is also fiberwise dense over $\bar{H}$ via the “composition” map, and
thus (by the basic fact cited above), so is their intersection, which is $H\times H$. Since $\bar{H}\times \bar{H}\to \bar{H}$ is an epimorphism, so is $H\times H\to\bar{H}$. But this map factors
through $H\rightarrowtail \bar{H}$ (since $H$ is itself a subgroup of $G$), so that inclusion is also epic. But it is also a regular monomorphism, and hence an isomorphism; thus $H$ is closed.
Localic groupoids
An important generalization of localic groups is to localic groupoids , i.e. internal groupoids in the category of locales. Localic groupoids are important, among other reasons, because every
Grothendieck topos can be presented as the topos of equivariant sheaves on some localic groupoid. This fact is due to Joyal and Tierney. For more see classifying topos of a localic groupoid. | {"url":"http://www.ncatlab.org/nlab/show/localic+group","timestamp":"2014-04-18T11:27:05Z","content_type":null,"content_length":"25295","record_id":"<urn:uuid:2d2af7b8-bd04-417f-afa8-7eed042b9608>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00187-ip-10-147-4-33.ec2.internal.warc.gz"} |