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Modeling Procedures
Modifying Link-Level Emissions Modeling Procedures for Applications within the MOVES Framework
Appendix B
Additional Drive Cycle Development Documentation
Figure B-1a. Square of the Length of T-C as Micro-Trips Are Added: Restricted 0-20 MPH Bin
Figure B-1b. Square of the Length of T-C as Micro-Trips Are Added: Restricted 20-30 MPH Bin
Figure B-1c. Square of the Length of T-C as Micro-Trips Are Added: Restricted 30-40 MPH Bin
Figure B-1d. Square of the Length of T-C as Micro-Trips Are Added: Restricted 40-50 MPH Bin
Figure B-1e. Square of the Length of T-C as Micro-Trips Are Added: Restricted 50-60 MPH Bin
Figure B-1f. Square of the Length of T-C as Micro-Trips Are Added: Restricted 60+ MPH Bin
Figure B-1g. Square of the Length of T-C as Micro-Trips Are Added: Unrestricted 0-15 MPH Bin
Figure B-1h. Square of the Length of T-C as Micro-Trips Are Added: Unrestricted 15-20 MPH Bin
Figure B-1i. Square of the Length of T-C as Micro-Trips Are Added: Unrestricted 20-25 MPH Bin
Figure B-1j. Square of the Length of T-C as Micro-Trips Are Added: Unrestricted 25-28 MPH Bin
Figure B-1k. Square of the Length of T-C as Micro-Trips Are Added: Unrestricted 28-32 MPH Bin
Figure B-1l. Square of the Length of T-C as Micro-Trips Are Added: Unrestricted 32+ MPH Bin
Figure B-2a. Speed Versus Time: Restricted 0-20 MPH
Figure B-2b. Speed versus Time: Restricted 20-30 MPH
Figure B-2c. Speed versus Time: Restricted 30-40 MPH
Figure B-2d. Speed versus Time: Restricted 40-50 MPH
Figure B-2e. Speed Versus Time: Restricted 50-60 MPH
Figure B-2f. Speed versus Time: Restricted 60+ MPH
Figure B-2g. Speed Versus Time: Unrestricted 0-15 MPH
Figure B-2h. Speed versus Time: Unrestricted 15-20 MPH
Figure B-2i. Speed versus Time: Unrestricted 20-25 MPH
Figure B-2j. Speed versus Time: Unrestricted 25-28 MPH
Figure B-2k. Speed versus Time: Unrestricted 28-32 MPH
Figure B-2l. Speed versus Time: Unrestricted 32+ MPH
Figure B-3a. Acceleration versus Speed for Cycle: Restricted 50-60 MPH
Figure B-3b. Acceleration versus Speed for Target: Restricted 50-60 MPH
Figure B-3c. Acceleration versus Speed for Cycle: Unrestricted 25-28 MPH
Figure B-3d. Acceleration versus Speed for Target: Unrestricted 25-28 MPH
Figure B-4a. Frequency Distribution of Speeds in Cycle: Restricted 50-60 MPH
Figure B-4a Table
Figure B-4b. Frequency Distribution of Speeds in Target: Restricted 50-60 MPH
Figure B-4b Table
Figure B-4c. Frequency Distribution of Speeds in Cycle: Unrestricted 25-28 MPH
Figure B-4c Table
Figure B-4d. Frequency Distribution of Speeds in Target: Unrestricted 25-28 MPH
Figure B-4d Table
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Calc 3 question - change of variables and computing the jacobian?
November 25th 2012, 11:52 AM #1
Nov 2012
Calc 3 question - change of variables and computing the jacobian?
The problem asks to use a change of variables compute the double integral of y^4dA over R, where R is the region bounded by the hyperbolas xy=1 and xy=4 and the lines y/x=1 and y/x=3.
I was able to find that u=xy and v=y/x, and that u goes from 1 to 4 and v goes from 1 to 3. However, I can't compute the Jacobian determinant for the life of me! When I try to solve for x and y,
I get expressions that have both u and v and x and y; I can't seem to isolate x and y. Please help me! Thanks.
Re: Calc 3 question - change of variables and computing the jacobian?
Looks like all you need is x = x(u,v), y = y(u,v).
Well try multiplying your two expressions together. That is,
u * v = (xy) * (y/x) = y^2.
Similarly, dividing gives
u / v = (xy) / (y/x) = x^2.
Now you have,
x = (u/v)^1/2, y = (u*v)^1/2.
Hope that's enough to get you on the right track. Let me know otherwise and I'll try and clear it up.
November 27th 2012, 06:22 AM #2
Junior Member
Nov 2012
Jacksonville, FL
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Necessity vs. Sufficiency
I am still confused on what Kaplan means by this whole "necessity vs. sufficiency" idea behind flaw questions. Can someone explain this, using examples, in very simple terms?
Also, can someone give a run down of the wrong answer choices most frequently used by lsat?
Thanks, I love this FORUM! I finally have friends to "freak out"/obsess about the LSAT with...
P.S.S.- who's going to be spending their family Thanksgiving studying for the LSAT? Don't you just love it! not!
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Similarity of Triangles
October 1st 2009, 10:59 PM #1
Sep 2009
There are two problems based on similarity of triangles in RD Sharma for 10th
1. D is the mid point of side BC of a triangle ABC.AD is bisected at point E and BE produced cuts AC at X. Prove that BE:EX=3:1.
2.ABCD is parallelogram and APQ is a straight line meeting BC at P and DC produced at Q.Prove that the rectangle obtained by BP and DQ is equal to rectangle obtained by AB & BC.
NO FIGURES HAVE BEEN PROVIDED FOR BOTH.
1) rotate the triangle AEX around the point E by 180 degrees and you'll obtain triangle DEX', as on the picture. Now DX' and CX are parallel (because DX' and AX are) and D is the midpoint of BC,
so X' is the midpoint of BX (a basic result you should understand). Now since the length of XE is 1/2 of the length of XX' which is 1/2 of the lenght of XB, we have the length of XE is 1/4 of the
length of XB.
As for 2) it would be polite to define what "two rectangles are equal" mean. If they mean that the rectangles have equal area, then since triangles ABP and QCP are similar, we have
BP/AB = CP/CQ
CQ.BP = AB.CP
AB.BP + CQ.BP = AB.BP + AB.CP
(AB+CQ).BP = AB.(BP+CP)
DQ.BP = AB.BC
October 2nd 2009, 06:39 AM #2
Aug 2009
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Chemistry 231
Homework Set 1 (this includes the data set needed for processing)
Homework Set 2 (this includes the link to the Excel example)
Set 1 HPLC Instructions (note: the equation given for calculation of a 95% uncertainty in a linear least squares derived unknown concentration was incorrect in the version of this handout passed out.
The correct uncertainty should be +tSx not +tSx/n^0.5 (where Sx = the standard deviation in x). This is now corrected - see top of p. 5.
List of Books and Journals of Use to Class updated 4/17/13
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adhesive category
adhesive category
Category theory
Universal constructions
Adhesive categories
An adhesive category is a category in which pushouts of monomorphisms exist and “behave more or less as they do in the category of sets”, or equivalently in any topos.
The following conditions on a category $C$ are equivalent. When they are satisfied, we say that $C$ is adhesive.
1. $C$ has pullbacks and pushouts of monomorphisms, and pushout squares of monomorphisms are also pullback squares and are stable under pullback.
2. $C$ has pullbacks, and pushouts of monomorphisms, and the latter are also (bicategorical) pushouts in the bicategory of spans in $C$.
3. (If $C$ is small) $C$ has pullbacks and pushouts of monomorphisms, and admits a full embedding into a Grothendieck topos preserving pullbacks and pushouts of monomorphisms.
4. $C$ has pullbacks and pushouts of monomorphisms, and in any cubical diagram:
if $X\to Y$ is a monomorphism, the bottom square is a pushout, and the left and back faces are pullbacks, then the top face is a pushout if and only if the front and right face are pullbacks. In
other words, pushouts of monomorphisms are van Kampen colimits.
In an adhesive category, the pushout of a monomorphism is again a monomorphism.
E.g. (Lack, prop. 2.1) Notice that generally monomorphisms (as discussed there) are preserved by pullback.
• Any topos is adhesive (Lack-Sobocisnki). For Grothendieck toposes this is easy, because $Set$ is adhesive and adhesivity is a condition on colimits and finite limits, hence preserved by functor
categories and left-exact localizations. For elementary toposes it is a theorem of Lack and Sobocinski.
• The fact that monomorphisms are stable under pushouts in toposes plays a central role for Cisinski model structures such as notably the standard model structure on simplicial sets, where the
monomorphisms are cofibrations and as such required to be closed under pushout (in particular).
Adhesiveness is an exactness property, similar to being a regular category, an exact category, or an extensive category. In particular, it can be phrased in the language of “lex colimits”.
Revised on May 2, 2013 04:35:16 by
John Baez
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Posts about renewal process on Xi'an's Og
I was attending a lecture this morning at CREST by Patrice Bertail where he was using estimated renewal parameters on a Markov chain to build (asymptotically) convergent bootstrap procedures.
Estimating renewal parameters is obviously of interest in MCMC algorithms as they can be used to assess the convergence of the associated Markov chain: That is, if the estimation does not induce a
significant bias. Another question that came to me during the talk is that; since those convergence assessments techniques are formally holding for any small set, choosing the small set in order to
maximise the renewal rate also maximises the number of renewal events and hence the number of terms in the control sequence: Thus, the maximal renewal rate þ is definitely a quantity of interest:
Now, is this quantity þ an intrinsic parameter of the chain, i.e. a quantity that drives its mixing and/or converging behaviour(s)? For instance; an iid sequence has a renewal rate of 1; because the
whole set is a “small” set. Informally, the time between two consecutive renewal events is akin to the time between two simulations from the target and stationary distribution, according to the Kac’s
representation we used in our AAP paper with Jim Hobert. So it could be that þ is directly related with the effective sample size of the chain, hence the autocorrelation. (A quick web search did not
produce anything relevant:) Too bad this question did not pop up last week when I had the opportunity to discuss it with Sean Meyn in Gainesville!
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CSCI 700 - Algorithms I
This class will cover the fundamentals of the design and analysis of algorithms. A focus on implementation in out-of-class assignments will accompany the theoretical discussions of the material in
class. Solid grasp of the design and analysis of algorithms is foundational to being a successful computer programmer and computer scientist.
Class Policy
Come to Class. It will be difficult to do well in the class without regular attendance. There is no additional penalty for missing class.
Cell phones must be on silent, and are not to be checked or used during class - if you are expecting an urgent call, tell the instructor at the start of class.
Laptops are fine for taking notes. No internet, no chat, no games.
Cell phone and Laptop policy: One warning, after that 5 points off the next homework or exam for each issue. Same policy for the instructor. One warning, after that, everyone gets 5 points on the
next homework or exam.
Grading Policy
Assignments: 48% (12 x 4%)
Written Midterm: 15%
Coding Midterm: 12%
Final: 25%
The Final Letter Grade will be based on a scaled adjustment of the Final Numeric Grade. When the scale has been determined, the class will be informed either in class or over email, and it will be
posted to the course webpage (here).
Assignment Policy
Do not cheat. You may discuss assignments with your classmates, but write or program your assignment alone. Do not ask for or offer to share code, or written assignments. If you discuss an assignment
with a classmate, or on an online forum, include the name of the classmate or URL of the forum on your assignment or in the documentation of your code. The first instance of cheating results in an
automatic zero for the assignment (or midterm or final). A second instance of cheating results in a zero (F) for the course. The Computer Science Department will be notified in writing of all
instances of cheating. On a second instance a report will be submitted to the Office of Academic Integrity.
Assignments will be posted to the website (here) after class on Tuesdays.
All assignments will be scored out of 100 points.
There are 12 assignments. 7 are written assignments, 5 are coding assigments.
Assignments will be due just after the start of class, 6:35pm, on the following Monday. Written assignments should be emailed or hard-copies should be delivered in class.
Deliver assignments at the start of class or email with a timestamp before 6:35pm to avoid a late penalty. If an extension is needed let me know as soon as possible. I will do my best to be
reasonable to you and fair to the rest of class. Delivering an assignment while being more than 5 minutes late for class will be make the assignment considered Late. There is a 5 point Late Penalty
for each 12 hours late the assignment is delivered. Due Date (DD) 5:00pm - DD+1 5:00am -5 points. DD+1 5:00am - DD+1 5:00pm -10 pts. DD+1 5:00pm - DD+2 5:00am: -15pts. DD+2 5:00am - DD+2 5:00pm
Grades will be posted at 5:00pm on Wednesdays, just before class. After 5:00, 2 days after an assignment was due, no assignments will be accepted. Assignments that were delivered on time will be
returned during Thursday classes.
After each assignment and the midterm is graded, anonymous mean, median, maximum and minimum scores will be distributed.
Coding Assignments
Assignments will be written in C++ or java, and must compile using g++ or javacon venus.cs.qc.cuny.edu with no external libraries.
In general, grading will be 15% Compilation, 15% Execution, 35% Correctness (15% passing tests, 20% implementational details), 35% Documentation and Style. This may be adjusted for some assignments.
Always read the assignment for the grading breakdown.
Testing will be performed automatically. Sample tests will be delivered with each assignment. If code does not operate using the published and distributed testing format, the assignment will be
considered Incorrect and zero "Correctness" points will be awarded.
Detailed requirements will accompany each assignment. The instructions and requirements on a particular assignment always take precedence over the general guidelines on the course website.
Submission of coding assignments should be performed over email. Don't forget to attach your files. Submitting multiple times is fine. The latest assignment submitted on time will be graded. If you
submit an assignment late, after submitting an assignment on time, you must let me know, via email, that you would like the late submission graded for the assignment.
Written Assignments
Written Assignments can be delivered electronically by email, or hard copies can be delivered by hand either in class, or dropped at my office NSB A330.
Electronic copies must be in one of the following formats: .pdf, Microsoft Word .doc, Google Docs.
Hand written copies are acceptable, but be very careful that the work is clear. If I can't read that an answer is correct, it is wrong.
Points for each question will be described in each assignment.
Exam Policy
The Midterm will be held during class on October 21nd. If you will not be able to make this date, let me know as early as possible, and I will do my best to schedule another time for you to take the
During both the Midterm and Final exam, you will be allowed to bring a single (8.5in x 11in) sheet of paper with notes. No other material will be allowed during the exams.
The Final Exam will be cumulative, with a focus on material covered in the second half of the semester.
When the Final Exam is scheduled, time and location information will be posted here.
Text Book
Introduction to Algorithms 3rd Edition by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein. MIT Press. 2001. ISBN: 978-0262033848
This should be available through the bookstore, but may be found through other outlets at a discount.
Date Material Assignments
August 29 No Class.
August 31 No Class.
September 5 No Class - Labor Day
Introduction. Class Overview. Review of Basic Data Structures and Math.
September 7 Sequential v. Binary Search HW 1 Assigned
Slides [pptx]
September 12 Asymptotic Notation. Recursion. Review of Inductive Proofs. HW 1 Due. HW 2 Assigned
Slides [pdf]
September 14 Runtime of Recursive Algorithms Part I
Slides [pdf]
Runtime of Recursive Algorithms Part II HW 2 Due. HW 3 Assigned
September 19 Slides [pdf] Sample input files
Sample output files
September 21 Linear-time Sorting -- Counting Sort, Radix Sort, Bucket Sort
Slides [pdf]
September 26 Binary Search Trees HW 3 Due. HW 4 Assigned
Slides [pdf]
September 28 No Class - Rosh Hashannah
October 3 Balanced Binary Search Trees: AVL Trees HW 4 Due. HW 5 Assigned
Slides [pdf]
October 5 Balanced Binary Search Trees: Red Black Trees
Slides [pdf]
October 10 No Class - Columbus Day
October 12 Heaps HW 5 Due.
Slides [pdf]
October 17 Midterm Review
Slides [pdf]
October 19 Midterm Exam
October 24 Coding Midterm Exam
October 26 Operations on Streams of Data HW 6 Assigned
Slides [pdf]
October 31 Dynamic Programming
Slides [pdf]
November 2 Greedy Algorithms HW 6 Due. HW 7 Assigned
Slides [pdf]
November 7 Huffman Coding
Slides [pdf]
November 9 Introduction to Graphs HW 7 Due. HW 8 Assigned
Slides [pdf]
November 14 Graph Algorithms: Breadth-first-search. Depth-first-search. Dijkstra's Algorithm
Slides [pdf]
November 16 Strongly Connected Components. Minimum Spanning Trees. Prim's Algorithm HW 8 Due. HW 9 Assigned
Slides [pdf]
November 21 Kruskal's Algorithm. Bellman-Ford
Slides [pdf]
November 23 Hashing I HW 9 Due.
Slides [pdf]
November 28 Hashing II HW 10 Assigned
Slides [pdf]
November 30 Multi-core Algorithms
Slides [pdf]
December 5 Multi-core Algorithms ctd., NP-Completeness HW 10 Due. HW 11 Assigned
December 7 NP-Completeness
Slides [pdf]
December 12 Final Review HW 11 Due.
Slides [pdf]
December 21 @ 4:00-6:00 Final Exam. Note the different start time.
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Re: Introduction
Hi catty3979;
Welcome to the forum. I find that math makes me so aggravated I can not see straight.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
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Did you know that by default you see only ratings for your marketplace? I didn’t, and was very amazed to find out my apps have more reviews and ratings than what shows up in my phone marketplace or
Zune window.
Using App Ratings, you can now view the ratings by market on a single screen and access reviews in all markets. Search the marketplace for “App Ratings” or click here to get the app
See how fast I can type on that keyboard:
App Ratings uses Atom feed provided by Zune. For more info see the forum posts below:
The other app I’d like to mention is called “dev screen saver” and is made by a colleague and friend of mine. It can help prevent your phone from locking – great to use during long hour sessions of
debug and deploy in Visual Studio to avoid having to manually unlock your phone. It also moves an image on screen to prevent burnout/burn-in effects and increase screen life.
I hope these apps will be usable to some people.
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Internet Study Guide To Nonlocal Quantum Physics And Solution To Einstein Podolsky Rosen Paradox
The proof against local realism is humankind’s the most relevant finding in the realm of science and philosophy; there is no excuse to be ignorant about it while pretending to be interested and
knowledgeable about cool science and its implications for philosophy. It was a difficult topic once, but from now on, there is no excuse anymore: Any good high school student, certainly any
university undergraduate is able to follow the arguments as long as there is no religious/obsessive hurdle to overcome internally.
Here is a study guide to the disproof of 'local realism' and the solution of the Einstein Podolsky Rosen paradox. UPDATE: What is written in this post is now mostly much better accessed in this,
extremely pedagogically written article: http://arxiv.org/abs/1311.5419
The following posts help to understand the preprints:
1) Disproving Local Realism and Hidden Variable Madness. The experimental setup is introduced and the ground set: Quantum physics disproves local realistic models. It may be helpful to also read
about the Quantum Randi Challenge, which will make it yet clearer that no locally realistic model can be quantum mechanical. However, the Quantum Randi Challenge is a side issue here.
2) Spooky World or Crazy Mind: Nonlocality versus Antirealism is somewhat of an introduction that should help to understand that there are two sides to local realism, namely (Einstein) locality and
(direct/naive) realism. This post may help to understand the introduction and thus the general problematic, but it is not necessary to understand the main arguments.
3) A classical model is introduced in Many Worlds by Splitting a Wiener Sausage. That model is simplified further in Many World Sausage with Correct Quantum Factors. The latter article shows that the
model is able to come up with desired factors needed in a quantum model (UPDATE: Such models have now also been found).
4) Classical Parallel Worlds and Local Non-Locality yet again clarifies that although the correct factors may be found in the initial sausage model, the model is still classical.
5) Gambling in the Multiverse: Empirical Probability versus Classical Fair Meta-Randomness explains the difference between classical probability and quantum probability and why the probability that
we empirically see in experiments is the quantum probability. This post contains a tree that was too off-topic for the preprint article yet it should be very helpful to most readers!
6) Einstein Podolsky Rosen Paradox Resolved by Local Modal Realism finally explains how the sausage model is turned into a quantum model. It introduces the growing branches instead of cutting the
sausage and discusses the crucial step that turns the direct realism into a modal realism.
7) Why Einstein Could Not Solve EPR Though He Could Have explains that modal relativity is a necessary paradigm that was already indicated by special relativity.
[1] S. Vongehr: "Many Worlds Model resolving the Einstein Podolsky Rosen paradox via a Direct Realism to Modal Realism Transition that preserves Einstein Locality." arXiv:1108.1674v1 [quant-ph]
(2011). UPDATE: This reference is the first paper on the possibility of such models, but they have now actually been constructed and are much better explained in S. Vongehr: “Against Absolute
Actualization: Three "Non-Localities" and Failure of Model-External Randomness made easy with Many-Worlds Models including Stronger Bell-Violation and Correct QM Probability” http://arxiv.org/abs/
1311.5419 (2013)
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Prime numbers in asm
03-21-2011 #1
Registered User
Join Date
Mar 2011
Prime numbers in asm
a little background info. Im a complete and utter noob when it comes to programming. I am still in uni and through the semester ive learned enough to somewhat know what i am doing but am far from
great like you guys
yea, first post of many.
heres thee question:
Write a procedure named IsPrime that sets the Zero flag if the 32-bit integer passed in the EAX register is prime. Optimize the program’s loop to run as efficiently as possible. Write a test
program that prompts the user for an integer, calls IsPrime, and displays a message indicating whether or not the value is prime. Continue prompting the user for integers and calling IsPrime
until the user enters -1.
heres what i have so far:
include 'irvine 32'
org 100h ; set location counter to 100h
jmp CodeStart
max dw " "
space db " ", 0
mov bx, 1
call IsPrime
cmp dx, 0
; must be a prime
mov ax, bx
call print_num
; print a space
mov si, offset space
call print_string
add bx, 1
cmp bx, max
jle LoopStart
IsPrime PROC
; uses a loop to determine if number in bx is prime
; upon return if bx not prime dx will be 0, otherwise dx > 0
; we only have to test divisors from 2 to bx/2
; prepare to divide dx:ax / 2
mov ax, bx
mov dx, 0
mov cx, 2
div cx
; move result into si for loop
mov si, ax
; assume the value is prime
mov dx, 1
; start loop at 2
mov cx, 2
; compare loop count(in cx) and max loop value (in si)
cmp cx, si
; jump out of loop if count(cx) > si
ja StopLabel
; divide test value (in bx) by loop count (in cx)
mov ax, bx
mov dx, 0
div cx
; check remainder (in dx), if zero then we found a divisor
; and the number cannot be prime
cmp dx, 0
; if dx = 0 then we found a divisor and can stop looking
je StopLabel
; increment count
add cx, 1
jmp PrimeLoop
IsPrime ENDP
I am so lost so thanks for your patience
Based on the code, you don't look lost.
The code doesn't seem to match the person who posted the code. Red flag. Also, this should be in Tech.
if (a) do { f( b); } while(1);
else do { f(!b); } while(1);
Nice assembler code... why are you asking in a C forum?
Oh wait... it's scoop and poop, isn't it?
This is one of the older versions of that code. According to the OP there, it's DeVry's custom assembly. From other posts I've seen, it looks like the prof gives them this code, then they go
modify it.
@aestetics: Your best bet here is probably your course materials, notes, books, etc. I'm not familiar with that assembly language. But for starters, somewhere in the main loop you actually need
to call IsPrime. You also need to call whatever function you use to get input, and check if it's -1, and if so exit.
When I find myself coding assembly, I often write out what I want to do in pseudo code, then write some C code to do it, then translate to assembly. Since it's so easy to make mistakes in
assembly and so hard to debug them, work in very small chunks, and test often.
It's an impressive list of screw ups for a first post.
Crappy title
How To Ask Questions The Smart Way
No code tags, wrong forum, some others
Anyway, lazy mode, all I'm gonna do is fix the title and move to a better forum
If you dance barefoot on the broken glass of undefined behaviour, you've got to expect the occasional cut.
If at first you don't succeed, try writing your phone number on the exam paper.
I support http://www.ukip.org/ as the first necessary step to a free Europe.
03-21-2011 #2
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Aug 2010
Ontario Canada
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Nov 2010
Long Beach, CA
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focusing on how to give students the tools they need so that they can READ Latin with ease. I also filled in as a substitute teacher for a University of Michigan introductory Latin class.
20 Subjects: including ACT Math, English, reading, writing
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A systems biology approach to analyse leaf carbohydrate metabolism in Arabidopsis thaliana
Plant carbohydrate metabolism comprises numerous metabolite interconversions, some of which form cycles of metabolite degradation and re-synthesis and are thus referred to as futile cycles. In this
study, we present a systems biology approach to analyse any possible regulatory principle that operates such futile cycles based on experimental data for sucrose (Scr) cycling in photosynthetically
active leaves of the model plant Arabidopsis thaliana. Kinetic parameters of enzymatic steps in Scr cycling were identified by fitting model simulations to experimental data. A statistical analysis
of the kinetic parameters and calculated flux rates allowed for estimation of the variability and supported the predictability of the model. A principal component analysis of the parameter results
revealed the identifiability of the model parameters. We investigated the stability properties of Scr cycling and found that feedback inhibition of enzymes catalysing metabolite interconversions at
different steps of the cycle have differential influence on stability. Applying this observation to futile cycling of Scr in leaf cells points to the enzyme hexokinase as an important regulator,
while the step of Scr degradation by invertases appears subordinate.
Systems biology; carbohydrate metabolism; Arabidopsis thaliana; kinetic modelling; stability analysis; sucrose cycling
Plant metabolic pathways are highly complex, comprising various branch points and crosslinks, and thus kinetic modelling turns up as an adequate tool to investigate regulatory principles. Recently,
we presented a kinetic modelling approach to investigate core reactions of primary carbohydrate metabolism in photosynthetically active leaves of the model plant Arabidopsis thaliana [1] with an
emphasis on the physiological role of vacuolar invertase, an enzyme that is involved in degradation of sucrose (Scr). This model was developed in an iterative process of modelling and validation. A
final parameter set was identified allowing for simulation of the main carbohydrate fluxes and interpretation of the system behaviour over diurnal cycles. We found that Scr degradation by vacuolar
invertase and re-synthesis involving phosphorylation of hexoses (Hex) allows the cell to balance deflections of metabolic homeostasis during light-dark cycles.
In this study, we investigate the structural and stability properties of a model derived from the Scr cycling part of the metabolic pathway described in [1]. Based on the existing model structure,
model parameters were repeatedly adjusted in an automated process applying a parameter identification algorithm to match the measured and simulated data. A method for statistical evaluation of the
parameters and simulation results is introduced, which allows for the estimation of parameter variability. Statistical evaluation demonstrates that the same nominal concentration courses are
predicted for different identification runs, while small variability in fluxes and larger variability in parameters can be observed. Further, the parameter identification results were analysed
applying a principal component analysis (PCA). This leads to a more extensive investigation with respect to the extension and alignment of the parameter values in the parameter space. In addition,
this allows for conclusions concerning the identifiability of the parameters and the confirmation that the cost function is sensitive along parameter combinations. An investigation of structural
stability properties of Scr cycling showed feedback inhibition of Hex on invertase and sugar phosphates (SP) on hexokinase likely to be involved in stabilisation of the metabolic pathway under
consideration. Feedback inhibition of hexokinase was more efficient in stabilising Scr cycling than inhibition of invertase, indicating that, at this step of the cycle, a superior contribution to
stabilisation of homeostasis can be achieved.
The central carbohydrate metabolism in leaves of A. thaliana
Within a 24-h light/dark cycle, two principal modes of metabolism can be distinguished for plant leaves: photosynthesis (day), and respiration (night). During the day, carbon dioxide is taken up, and
storage compounds like starch (St) accumulate, while this stock is in part respired during the night. Under normal conditions, a certain proportion of carbon is fixed as new plant biomass. However,
typical source leaves as considered here are mature, and thus carbon use for growth can be neglected. Therefore, the carbon balance is completely determined by photosynthesis, respiration and carbon
allocation to associated pathways or heterotrophic tissues that are not able to assimilate carbon on their own. Based on this information and known biochemical reactions, a simplified model structure
for the interconversion of central metabolites was created (Figure 1).
Figure 1. Model structure of the central carbohydrate metabolism in leaves of A. thaliana. SP, sugar phosphates; St, starch; Scr, sucrose; Glc, glucose; Frc, fructose. v represent rates of metabolite
The compounds SP, St, Scr, glucose (Glc) and fructose (Frc) are derived from photosynthetic carbon fixation and linked by interconverting reactions. The flux v[St]. The reaction v[SP]→[Scr ]
represents a set of reactions leading to Scr synthesis. Among them, the reaction of Scr phosphate synthase is considered the rate-limiting step [2]. Scr can either be exported, for example, by a
transport to sinks v[SP]→[Sinks], or cleaved into Glc and Frc by invertases, v[Inv]. The free Hex can be phosphorylated by v[Glc]→[SP ]and v[Frc]→[SP], respectively. These reactions are catalysed by
the enzymes glucokinase and fructokinase.
Mathematical model structure
Time-dependent changes of metabolite concentrations during a diurnal cycle can be described by a system of ordinary differential equations (ODE). With c being the m-dimensional vector of metabolite
concentrations, N being the m × r stoichiometric matrix and v being the r-dimensional vector of fluxes, the biochemical reaction network can be described as follows:
with v(c,p) indicating that the fluxes are dependent on both, metabolite concentrations c and kinetic parameters p. Thus, based on the model structure (Figure 1) of our system, the concentration
changes of the five-state variables: SP, St, sucrose, Glc and Frc are defined as:
The stoichiometric coefficients account for the interconversions of species with a different number of carbon atoms. For example, the reaction ν[SP]→[Scr ]has a stoichiometric coefficient value of 1
in the SP state equation, while in the Scr state equation, this value is 0.5 because SP contains 6 carbon atoms and Scr contains 12 carbon atoms. The stoichiometric coefficients for the reaction
catalysed by invertase are 1 in all the respective state equations because this reaction represents the cleavage of the disaccharide Scr into two monosaccharides: Glc, and Frc. St content is
expressed in Glc units, i.e. a carbohydrate with six carbon atoms. The rates of the ODE system (Equation 2) are determined in three ways: by measurements (model inputs), carbon balancing and kinetic
rate laws.
Model input and carbon balancing
The rate of net photosynthesis [1]. Interpolated values of the measurements were applied to the SP state equation.
For modelling St synthesis and carbohydrate export, we used the following phenomenological approach. Although based on experimental data, the rate of net St synthesis was still subject to the
identification process. It was defined as
with v[St, min ]and v[St, max ]being derived from the measured concentration changes, i.e. the derivatives of the interpolated minimal and maximal concentrations. The parameter p[1 ]varied between 0
and 1 and was determined in the process of parameter identification.
The rate of carbohydrate export
was dynamically determined by balancing the external flux v[St ]and measured minimal and maximal total concentration changes of soluble carbohydrates v[C, min ]and v[C, max], respectively. v[C, min ]
and v[C, max ]were calculated as already described for v[St, min ]and v[St, max ]by interpolating and differentiating with respect to time. In this way, the mechanistically and quantitatively unknown
carbohydrate export can be calculated using measurement data of one flux v[St], v[C, min/max]). As with p[1], the parameter p[2 ]varied between 0 and 1 and was determined in the process of parameter
This balancing formed the boundary condition for the system in Equation 2 and the model described the distribution of overall carbon flux through the internal reactions. The experimental setup as
well as results of experimental data on carbohydrates and net photosynthesis are presented explicitly in [1].
Kinetic rate equations
The rate of Scr synthesis (v[SP]→[Scr]) was assumed to follow a Michaelis-Menten enzyme kinetic:
Rates of Scr cleavage (v[Inv]), Glc phosphorylation (v[Glc]→[SP]) and Frc phosphorylation (v[Frc]→[SP]) were defined by Michaelis-Menten kinetics including terms for product inhibition (Equations
6-8) as described in [3] and [4]:
where V[max](t) values represent time-variant maximal velocities of enzyme reactions, K[m ]are the Michaelis-Menten constants representing substrate affinity of the enzyme and K[i ]are the inhibitory
constants. Changes in maximal velocities of enzyme reactions were described over a whole diurnal cycle by a cubic spline interpolation for V[max](t). This course is defined by the sample t[k ]=
{3,7,11,15,19,23} h and values for V[max](t[k]), which are subject to parameter identification. This description reflects changes of enzyme activity, mainly resulting from changes in enzyme
concentration. Measurements of enzyme activities supported this assumption [1]. The kinetic rate law for the invertase reaction included a mixed inhibition by the products Glc and Frc, while hexose
phosphorylation (v[Glc]→[SP], v[Frc]→[SP]) was assumed to be inhibited non-competitively by SP. The model description, simulation and parameter identification was performed using the MATLAB
SBToolbox2 [5].
Parameter identification
Parameters were automatically adjusted applying a parameter-identification process representing the minimization of the sum of squared errors between measurement and simulation outputs by changing
the parameter values within their bounds. For an overview of the formulation of such problems, see, e.g. [6]. In this context, the outputs which correspond to the model states are the concentration
values of SP, St, Scr, Glc and Frc measured over a whole diurnal cycle at chosen time points. For a more detailed description of the quantification procedure and time points, refer to [1].
Measurements and simulations were carried out for A. thaliana wild type, accession Columbia (Col-0), and a knockout mutant inv4 defective in the dominating vacuolar invertase AtßFruct4 (At1G12240).
The final parameters have been identified using a particle swarm algorithm [7] that minimizes the sum of quadratic differences between measurement and simulation. This identification algorithm
contains a stochastic component that enables overriding of local minima. We used the algorithm provided by the MATLAB/SBToolbox2 with its default options. The possible parameter ranges were
constrained by different lower and upper bounds known from our own experiments (V[max]) and the literature (K[m], K[i]). The model and the complete set of parameters and the best-fit comparison plots
can be found in [1].
Statistical fit analysis
The model was intended not only to reproduce experimental data but also to allow predictions of variables and parameter values, for which no data were obtained. Therefore, the model was analysed for
the variability of parameters and fluxes, which both are used for predictions. In [1], we performed 75 parameter-identification runs for the wild-type and the mutant. Within the chosen numerical
accuracy, the algorithm converged to the same nominal cost function values in N[i,Col0 ]= 72 and N[i,inv4 ]= 71 cases, respectively. To give an impression of the fitting quality of the metabolite
concentrations, all the N[i ]simulation runs and measurements for both genotypes' Frc concentration are shown in Figure 2 exemplarily. The measurement error bars, i.e. the measurement standard
deviations, are calculated from N[r ]= 5 replications. The comparisons of measurements with simulations for the whole set of metabolites are shown in [1].
Figure 2. Comparison of measurements (error bars: standard deviations; N[r ]= 5 replicates) and simulations (lines; N[i,Col-0 ]= 72 and N[i,inv4 ]= 71 identification runs) of Frc concentrations in
leaf extracts. (a) Wild-type (black), (b) mutant (grey). Time 0 h = 06:00 a.m. daytime. Concentrations are given in μmol per gFW (leaf fresh weight).
We were able to identify significant differences in carbohydrate interconversion rates, which were not obvious and could not be determined by intuition [1]. For instance, one finding highlights the
robustness of the considered system in spite of a significant reduction of the dominating activity of invertase in inv4. During the whole diurnal cycle, the calculated flux rates for the invertase
reaction in wild-type and inv4 mutant differed considerably less than did the corresponding V[max ]values for invertase (Figure 3). This observation indicated a possible stabilizing contribution of
feedback mechanisms, for example, by product inhibition of invertase activity. In section "Stability properties of Scr cycling", this aspect is investigated further.
Figure 3. Diurnal dynamics of (a) measured maximal invertase activity and (b) simulated rates of Scr cleavage (v[Inv]) for wild-type (black lines) and mutant (grey lines). Values in (a) represent
means ± SD (N[r ]= 5 replicates), values in (b) represent means ± SD (identification runs: N[i,Col-0 ]= 72, N[i,inv4 ]= 71). Time 0 h = 06:00 a.m. daytime. Concentrations are given in μmol per gFW
(leaf fresh weight).
Further, for displaying the variability of parameters, we chose boxplots that are superior in displaying distributions for skewed data sets, see, e.g. [8]. To compare identification results for
different parameters, we scaled the identified values represented by their median and plotted distributions as box-and-whisker plots. The resulting graphs for all the parameters and flux values at
the time points defined by the time-variant V[max ]are shown in Figures 4 and 5. Outliers are displayed as dots. For a comparison of the parameter quality, values were sorted by their box width in
the ascending order.
Figure 4. Boxplots of identified kinetic parameters for wild-type (left side; N[i,Col-0 ]= 72) and mutant (right side; N[i,inv4 ]= 71). Numbers in brackets indicate time points (in hour) of
time-variant parameters. Black dots represent outliers. The parameter K[i, Frc, Inv ]of Col-0 has outliers at 21.7, 58.5 and 58.6. The upper quartile of the parameter K[i, Frc, Inv ]of inv4 is at
37.6. V[max,SP]→[Scr ](23) of inv4 has outliers at 10.0.
Figure 5. Boxplots of the simulated metabolite fluxes for wild-type (left side; N[i ]= 72) and mutant (right side; N[i ]= 71). Numbers in brackets indicate time points in h. Black dots represent
outliers. The flux v[SP]→[Scr](23) of inv4 has outliers at 10.3 and 10.5.
The parameter with the largest variability is the inhibition coefficient of fructokinase in both, the wild type and the mutant. Still, complete omission of inhibition structures leads to inferior
simulation results (data not shown). Apart from the variability within the parameters, it can be observed that fluxes, such as v[Inv], have smaller boxes than some of the associated kinetic
parameters (here: K[i,Frc,Inv]), and that the wild type is less variable than the mutant (Figures 4 and 5). Further, the simulated concentrations show a relatively small variation (Figure 2). The
result may be influenced by the number of runs, the algorithm's internal parameters, the algorithm itself or by the estimation bounds and should not be taken as confidence intervals of the parameter
values. Therefore, the presented results only give an impression as to how the parameter variability is distributed for the chosen statistical setup.
Our observation that some parameter values have a much higher variability than the corresponding concentration and flux simulations is consistent with that of Gutenkunst et al. [9] in which many
systems' biological models show the so-called sloppy parameter spectrum. Gutenkunst et al. [9] analysed several models with a nominal parameter vector p^o leading to nominal concentration courses.
They studied the set of parameter values p, which lead to similar concentration courses as the nominal parameter values. For this purpose, they computed an ellipsoidal approximation of this set using
the Hessian matrix of the χ^2 function, which is a measure for the deviation of the concentration courses from the nominal concentration courses. They found that in all the studied models, the
lengths of the principal axes of this ellipsoid span several decades and are not aligned to the coordinate axes. Since parameters may vary along the long principal axes of the ellipsoid without
significantly affecting the concentration courses, this means that many parameter values cannot be determined reliably by fitting the model to experimental data. At the same time, the model
predictions may nevertheless be reliable.
We analysed whether an analogous property is found in our N[i ]parameter sets. For this purpose, we performed a PCA [10]. PCA identifies the principal axes of a set of vectors. We applied a PCA to
the set of vectors of logarithmic parameters that resulted from the convergent identification runs. For this purpose, we computed the covariance matrix C of the logarithmic parameter vectors such
that C[ij ]= cov(log(p[i]), log(p[j])) corresponding to the ith and jth parameters, p[i ]and p[j], respectively. The eigenvectors of this matrix give the directions of the principal axes of the set
of logarithmic parameter vectors. The eigenvalues correspond to the variances of the logarithmic parameters along the principal axes and present a measure for the lengths of the principal axes. An
ellipsoid with these properties is given by Δp^T·C^-1·Δp ≤ 1, where Δp = log(p)-log(p°) is the deviation of the logarithmic parameter vector from its nominal value.
The longest principal axis of the mutant is approximately four times longer than the longest axis of the wild-type. This observation reflects the comparatively large boxes of the mutant box plots.
For the mutant, the covariance matrix C is singular, with six eigenvalues being equal to zero within numerical tolerance. Two of those six eigenvalues correspond to the parameters describing the
maximal velocity of the invertase reactions at two different time points (V[max,Inv ]at t = 11 and 23 h) i.e. parameters directly connected to the mutation. These two parameters do not show a
variation but are always at their bounds, which are much lower than in the wild-type. The analysis of the other four eigenvectors with eigenvalue zero revealed linear combinations of 29 parameters
(all parameters except V[max,Inv](11), V[max,Inv](23) and V[max,SP]→[Scr](23)), and their intuitive interpretation is not obvious.
The above observations indicate that the parameter-identification problem for the mutant does not have a unique optimum, and the optima are on the border of the allowed area. For further analysis, we
only analyse the principal axis with a non-zero variance. We removed six parameters from the parameter vector and computed the non-singular matrix C for the remaining parameters. The spectrum of the
lengths of the principal axes is shown in Figure 6. The lengths were scaled such that the longest axis has a length of unity (10°). As expected for a sloppy system, the lengths of the principal axes
span several orders of magnitude.
Figure 6. Results of the principal component analysis. Spectra of the principal components' variances (= eigenvalues of the covariance matrix) for wild-type (a) and mutant (b). (Displayed values were
scaled by the maximal variance. Some values are outside the displayed range). Spectra of the intersection/projection ratio (I/P) for wild type (c) and mutant (d).
Next, we verified whether the principal axes are aligned with the coordinate axes. Gutenkunst et al. [9] suggest the use of the I[i]/P[i ]ratio to quantify the alignment of the principal axes with
the coordinate axes. Here, I[i ]is the intersection of the ellipsoid with the ith coordinate axis and P[i ]is the projection onto ith coordinate axis. A perfectly aligned principal axis has I[i]/P[i
]= 1, whereas a skewed axis will lead to a deviation of unity. Gutenkunst et al. [9] give an expression to compute the I[i]/P[i ]ratio on the basis of a quadratic form defining the ellipsoid. With
our symbols, this expression is
The I[i]/P[i ]ratios span several orders of magnitude (Figure 6). This means that most principal axes are not aligned with the coordinate axes, as expected for a sloppy system.
In conclusion, the statistical analysis of the parameter vectors revealed three important properties of the system:
1. Different parameter-identification runs for the mutant converge to different edges of the allowed area. This fact reveals a problem with the identifiability of the model parameters for the mutant
and explains the relatively large variation of the parameter values. In order to get a unique optimum, more experimental data of the previously unmeasured variables and a critical reassessment of the
lower and upper bounds are needed.
2. The N[i ]parameter sets show a sloppy parameter spectrum. This means that many parameter values cannot be reliably determined by parameter-identification algorithms that fit the model to
experimental data.
3. The box plots in Figures 4 and 5 suggest which parameters and fluxes are likely to be determined reliably and which are not.
Stability properties of Scr cycling
As mentioned above, the knockout mutation of the dominant vacuolar invertase AtßFRUCT4 showed a dramatic reduction of cellular invertase activity, whereas the corresponding flux v[Inv ]did not
decrease in a corresponding manner (Figure 3). This finding indicated that the behaviour of the metabolic cycle of Scr degradation and re-synthesis are strongly determined by strong regulatory
effects, as the product inhibition of invertase activity and of the synthesis of SP, as well as the activation of the synthesis of Scr by the Hex. Steady states in such strongly regulated systems are
prone to instability, leading to effects as bi-stability or oscillations. The model defined by Equations 2, 5, 6, 7 and 8 approaches a stable steady state for given values of the in- and out-going
reactions v[St ]and v[Scr]→[Sinks ]if the overall carbon balance is fulfilled, i.e. [11,12]. SKM is a specific application of generalized modelling [13] in which normalized parameters replace
conventional parameters such as V[max ]or K[m ]in the modelling of metabolic networks. SKM in conjunction with a statistical analysis of the parameter space was used to determine whether a given
steady state of a metabolite is always stable or whether it may be unstable for certain values of the normalized parameter [12]. We applied this methodology to our metabolic cycle of Scr degradation
and synthesis, i.e. the central part of the system in consideration. In order to simplify the analysis, we summarised Glc and Frc as Hex. With this simplification, we obtained the network shown in
Figure 7. Hex can activate v[2 ]as described in [14]. Hex can also act as feedback inhibitors on v[4], and v[5 ]can be inhibited by the reaction product SP (Figure 7).
Figure 7. Schematic representation of the metabolic cycle of Scr synthesis and degradation. Inhibitory instances are indicated by red lines; activation is indicated by green lines. SP, sugar
phosphates; Scr, sucrose; Hex, hexoses; F, reference flux; α, scaling parameter to describe fluxes as proportions of F.
SKM allows analysing models with respect to given steady-state concentrations c[0], and fluxes v[j](c[0]). In this study, these values are subject to diurnal changes. However, the relative changes in
concentration are small. Thus, we assumed steady-state concentrations of the metabolites, which we computed as the mean value of the concentrations over a whole day/night cycle. In steady state, flux
v[1 ]equals flux v[3 ]= 6v[Scr]→[Sinks]. We set v[1 ]= v[3 ]= αF, where F represents the invertase flux. The parameter α can take values between 0 and 1 and determines the degree of Scr cycling. For
α = 1, no cycling occurs. For α = 0, the cycling of carbon becomes maximal, and no carbon enters or leaves the cycle.
SKM defines normalised parameters with respect to the steady-state concentrations c[0 ]and fluxes v[j](c[0]):
with i = 1...m (number of metabolites) and j = 1...r (number of reactions). The vector x describes the metabolite concentrations normalised based on their steady-state concentrations, the matrix Λ is
the stoichiometric matrix normalised with respect to steady-state fluxes and steady-state metabolite concentrations, and μ represents the fluxes normalised related to steady-state flux values.
As described in [12], x[0 ]= 1 represents the steady state of the system and the corresponding Jacobian J can be written as
Each element of the matrix [j ]with respect to the normalised substrate concentration x[i]:
thus indicating the degree of change in a flux as a particular metabolite is increased [11]. For irreversible Michaelis-Menten kinetics, as used in our kinetic model, the values in θ can assume
values in the interval of [0,1]. In the case of allosteric inhibition by a product, as, for example, feedback inhibition of Hex on invertase enzymes, the corresponding element in θ assumes values
within the range [-1,0]. Further details on θ for Michaelis-Menten kinetics can be found in [11]. The power of this approach lies in the ability to analyse the stability of the system by sampling
combinations of the elements of θ which again represent combinations of the original kinetic parameters.
Considering the metabolic cycle shown in Figure 6 that contains three metabolites and five reactions, the following Λ (m × r) and θ (r × m) matrices can be developed:
The Jacobian matrix J[x ]was calculated according to Equation 12. The system is guaranteed to be locally asymptotically stable if all eigenvalues of J[x ]have negative real part. It is unstable, if
one or more eigenvalues have positive real parts. The stability of nonlinear systems where all eigenvalues have non-positive real parts, but one which has a real part of zero, cannot be analysed with
this approach. In the present setting, the latter case can be ignored since it occurs only for a lower dimensional subset of the parameter space. To explore stability properties of the considered Scr
cycle, we performed computational experiments, in which the parameters in θ and α were set randomly following a standard uniform distribution on the open interval [0-1]. We analysed different
modifications of the metabolic cycle by varying modes of activation and inhibition. Each particular metabolic cycle was simulated for 10^6 different sets of parameters, and resulting maximal real
parts of the eigenvalues were plotted in histograms (Figures 8 and 9).
Figure 8. Histograms of values of the maximal real part of eigenvalues for the metabolic system described in Figure 6. (a) Histogram of the system without instances of activation or feedback
inhibition; (b) histogram of the system with activation of v[2 ]by Hex without feedback inhibition; and (c) histogram of the system with activation of v[2 ]by HexHexHex and feedback inhibition of Hex
on v[4 ]and SP on v[5].
Figure 9. Histograms of values of the maximal real part of eigenvalues for the metabolic system described in Figure 6. (a) Histogram of the system with activation of v[2 ]by Hex, weak feedback
inhibition of SP on v[5 ]and strong feedback inhibition of Hex on v[4]; (b) histogram of the system with activation of v[2 ]by Hex, strong feedback inhibition of SP on v[5 ]and weak feedback
inhibition of Hex on v[4].
First, we analysed the stability properties of a system without instances of activation and inhibition (Figure 8a), i.e. by setting θ[2], θ[5 ]and θ[6 ]to zero. All real parts of eigenvalues were
negative, indicating stability for all the samples. Yet, if we considered v[2 ]to be activated by Hex (θ[2 ]> 0), positive real parts occurred, suggesting that the system may become unstable for
certain parameter sets (Figure 8b). When additional instances of strong feedback inhibition (θ[5 ]= θ[6 ]= -0.99), e.g. by Hex or SP [1] were included, no positive eigenvalues appeared any more, and
the system became stable again for all the tested parameter values (Figure 8c).
To determine whether feedback inhibition by Hex and SP contributed equally to stabilisation, we further analysed systems with (i) weak feedback inhibition of v[5 ]by SP (θ[6 ]= -0.01) and strong
inhibition of v[4 ]by Hex (θ[5 ]= -0.99), and (ii) strong feedback inhibition of v[5 ]by SP (θ[6 ]= -0.99) and weak inhibition of v[4 ]by Hex (θ[5 ]= -0.01). The histograms representing the
corresponding results showed that stability of the system for all the samples was only achieved when v[5 ]was assumed to be inhibited strongly by SP (Figure 9a,b). Applying this theoretical model to
a physiological context, reaction v[5 ]would be represented by hexose phosphorylation through hexokinase enzymes, which have been shown to play a central role in sugar signalling, hormone signalling
and plant development [15]. Our findings point to a strong influence of hexokinase on system stability and establishment of a metabolic homeostasis, supporting a crucial role in plant carbohydrate
metabolism. In addition, a prevailing role of hexokinase in regulating Scr cycling would explain why a strong reduction of invertase activity caused only minor changes in the magnitude of Scr cycling
in the inv4 mutant as already outlined in [1] (see Figure 3).
Recently, we presented a kinetic modelling approach to simulate and analyse diurnal dynamics of carbohydrate metabolism in A. thaliana. Based on simulated fluxes in leaf cells, we could assign
possible physiological functions of vacuolar invertase in carbohydrate metabolism. Here, we explicate this model in more detail and perform a statistical evaluation that proves reproducibility of the
prediction of cellular metabolite concentrations and fluxes. The PCA revealed that the identifiability of the mutant parameters could be improved by more measurements. In addition, it was shown that
this system's biology model exhibits the property of sloppiness [9], allowing for good predictions while some parameters show larger variability. The analysis of stability properties of Scr cycling
indicated an important role of feedback inhibition mechanisms in stabilisation of futile metabolic cycles, and application of this concept to plant carbohydrate metabolism supported a role for
hexokinase as a crucial regulator of Scr cycling.
Frc: fructose; Glc: glucose; Hex: hexoses; ODE: ordinary differential equations; PCA: principal component analysis; Scr: sucrose; SKM: structural kinetic modelling; SP: sugar phosphates; St: starch.
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: tridiagonal
F01LEF LU factorization of real tridiagonal matrix
F04BCF Computes the solution and error-bound to a real tridiagonal system of linear equations
F04BGF Computes the solution and error-bound to a real symmetric positive-definite tridiagonal system of linear equations
F04CCF Computes the solution and error-bound to a complex tridiagonal system of linear equations
F04CGF Computes the solution and error-bound to a complex Hermitian positive-definite tridiagonal system of linear equations
F04LEF Solution of real tridiagonal simultaneous linear equations (coefficient matrix already factorized by F01LEF)
F06RNF 1-norm, ∞-norm, Frobenius norm, largest absolute element, real tridiagonal matrix
F06RPF 1-norm, ∞-norm, Frobenius norm, largest absolute element, real symmetric tridiagonal matrix
F06UNF 1-norm, ∞-norm, Frobenius norm, largest absolute element, complex tridiagonal matrix
F06UPF 1-norm, ∞-norm, Frobenius norm, largest absolute element, complex Hermitian tridiagonal matrix
F07CAF Computes the solution to a real tridiagonal system of linear equations
F07CBF Uses the LU factorization to compute the solution, error-bound and condition estimate for a real tridiagonal system of linear equations
F07CDF LU factorization of real tridiagonal matrix
F07CEF Solves a real tridiagonal system of linear equations using the LU factorization computed by F07CDF
F07CGF Estimates the reciprocal of the condition number of a real tridiagonal matrix using the LU factorization computed by F07CDF
F07CHF Refined solution with error bounds of real tridiagonal system of linear equations, multiple right-hand sides
F07CNF Computes the solution to a complex tridiagonal system of linear equations
F07CPF Uses the LU factorization to compute the solution, error-bound and condition estimate for a complex tridiagonal system of linear equations
F07CRF LU factorization of complex tridiagonal matrix
F07CSF Solves a complex tridiagonal system of linear equations using the LU factorization computed by F07CDF
F07CUF Estimates the reciprocal of the condition number of a complex tridiagonal matrix using the LU factorization computed by F07CDF
F07CVF Refined solution with error bounds of complex tridiagonal system of linear equations, multiple right-hand sides
F07JAF Computes the solution to a real symmetric positive-definite tridiagonal system of linear equations
F07JBF Uses the modified Cholesky factorization to compute the solution, error-bound and condition estimate for a real symmetric positive-definite tridiagonal system of linear equations
F07JDF Computes the modified Cholesky factorization of a real symmetric positive-definite tridiagonal matrix
F07JEF Solves a real symmetric positive-definite tridiagonal system using the modified Cholesky factorization computed by F07JDF
F07JGF Computes the reciprocal of the condition number of a real symmetric positive-definite tridiagonal system using the modified Cholesky factorization computed by F07JDF
F07JHF Refined solution with error bounds of real symmetric positive-definite tridiagonal system of linear equations, multiple right-hand sides
F07JNF Computes the solution to a complex Hermitian positive-definite tridiagonal system of linear equations
F07JPF Uses the modified Cholesky factorization to compute the solution, error-bound and condition estimate for a complex Hermitian positive-definite tridiagonal system of linear equations
F07JRF Computes the modified Cholesky factorization of a complex Hermitian positive-definite tridiagonal matrix
F07JSF Solves a complex Hermitian positive-definite tridiagonal system using the modified Cholesky factorization computed by F07JRF
F07JUF Computes the reciprocal of the condition number of a complex Hermitian positive-definite tridiagonal system using the modified Cholesky factorization computed by F07JRF
F07JVF Refined solution with error bounds of complex Hermitian positive-definite tridiagonal system of linear equations, multiple right-hand sides
F08FEF Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form
F08FFF Generate orthogonal transformation matrix from reduction to tridiagonal form determined by F08FEF
F08FSF Unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form
F08FTF Generate unitary transformation matrix from reduction to tridiagonal form determined by F08FSF
F08GEF Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form, packed storage
F08GFF Generate orthogonal transformation matrix from reduction to tridiagonal form determined by F08GEF
F08GSF Unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form, packed storage
F08GTF Generate unitary transformation matrix from reduction to tridiagonal form determined by F08GSF
F08HEF Orthogonal reduction of real symmetric band matrix to symmetric tridiagonal form
F08HSF Unitary reduction of complex Hermitian band matrix to real symmetric tridiagonal form
F08JAF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix
F08JBF Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix
F08JCF All eigenvalues and optionally all eigenvectors of real symmetric tridiagonal matrix (divide-and-conquer)
F08JDF Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix (Relatively Robust Representations)
F08JEF All eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from real symmetric matrix using implicit QL or QR
F08JFF All eigenvalues of real symmetric tridiagonal matrix, root-free variant of QL or QR
F08JGF All eigenvalues and eigenvectors of real symmetric positive-definite tridiagonal matrix, reduced from real symmetric positive-definite matrix
F08JHF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a matrix reduced to this form (divide-and-conquer)
F08JJF Selected eigenvalues of real symmetric tridiagonal matrix by bisection
F08JKF Selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in real array
F08JLF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a symmetric matrix reduced to this form (Relatively Robust Representations)
F08JSF All eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from complex Hermitian matrix, using implicit QL or QR
F08JUF All eigenvalues and eigenvectors of real symmetric positive-definite tridiagonal matrix, reduced from complex Hermitian positive-definite matrix
F08JVF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (divide-and-conquer)
F08JXF Selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in complex array
F08JYF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (Relatively Robust Representations)
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Necessity vs. Sufficiency
I am still confused on what Kaplan means by this whole "necessity vs. sufficiency" idea behind flaw questions. Can someone explain this, using examples, in very simple terms?
Also, can someone give a run down of the wrong answer choices most frequently used by lsat?
Thanks, I love this FORUM! I finally have friends to "freak out"/obsess about the LSAT with...
P.S.S.- who's going to be spending their family Thanksgiving studying for the LSAT? Don't you just love it! not!
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Travel Diary of Makoto Sakurai
Gaitsgory's talk was entitles as "(Overview of the work of Beilinson and Drinfeld)". His talk was on the D-schemes, horizontal sections, critical levels, formal punctured disc, Hecke eigensheaves,
and Beilinson-Bernstein. I asked Gaitsgory about the equivalence of jet schemes and n-jets of motivic integration. Loeser was on cell decompositions theorem of Denef-Pas, Tarski' theorem, theorem of
Cluckers-Loeser, or such kind of model theoretical stuffs. Miles Reid talked during 12:00 to 12:20 about the history from 31 years ago in Arcadia (a name of village) near Vancouver. His story
included Deligne, Griffiths, Dixon and football on TV. And then Deligne-Mumford, computational algebra, MacPherson, moduli of curves, 3-folds of Mori, and Kawamata. Something like music school was
stated, and he said they had only tiny black boards and dispatched to buy a bigger black board.
I selected Mirkovic, Bloch, and Kaledin. Milkovic was on unramified global geometric conjecture (Beilinson and Drinfeld), loop Grassmannians (Drinfeld, Ginzburg, Lustzig, Vilonen), affine flag
vaeirty of G and dual{G} (Bezrukavnikov,Arkhipov,Ginzburg), application to Lie algebras for p>0, exotic coherent sheaves on the tangent bundles and flag variety, and Springer fibers (Bezrukavnikov,
Milkovic, and Rumynin). Bloch was on the motives for graphs, which was similar to the Feynmann diagrams of "Schwinger trick". The rests were on Kontsevich conjecture & motives associated ot graph
hypersurface, and disproof by Belkale-Brosaa of Kontsevich conjecture (Hopf algebras for renormalization and for mixed Tate motives). Kaledin was again on the deformation of structure sheaf for flag
varieties. During the afternoon session, I could talk to Mirkovic and he said he knew some of the physics works. I show the poster presentation and my drafts of research paper to Mirkovic and
Today was the end of my long trip to Canada (Toronto) and America (Seattle).
Gaitsgory was on the central extension of loop algebras, ind-scheme (D-modules for the affine Grassmann), criticality, and Beilinson-Bernstein for the flag variety. Loeser was on motivic Milnor
fiber, starting from the theorem of Denef-Loeser, monodromy conjecture, Steenbrink's mixed Hodge structure, and motivic zeta function (and a little on Voevodsky?). At the last of morning session, one
organizer talked about some history of the Summer Institute in Algebraic Geometry, which started at 1954. It was after the WW2 that Kunihiko Kodaira was invited to the IAS by Spencer (Kodaira-Spencer
theory and Spencer-Zariski) and he mantioned something like Seminar Bourbaki, Thom's paper, the lecture by Hironaka on resolution of singularity (just sketch of proofs?), and Grothendieck, something
about going to the beach.
The afternoon was Olsson, Vistoli, and Kaledin. Olsson was on the nonabelian p-adic Hodge theroy, which was similar to the recent work \pi_1 (\mathbb{P}^1 \ {0,1,\infty}, b) of Deligne-Terasoma from
e'tale, de Rham to crystalline. It also started from Hain (mixed Hodge structure) and relative Malcev completion. The organization of talk was 1. abelian p-adic HT (Theorem of Fortaine Messng,
Faltings, Tsuji, Niziol and definition of crystalline), 2. p-adic HT for \pi_1 (Tannaka duality), 3.Computing Lie algebras, 4.Idea of proofs (Toeu's theory: Higher Tannaka duality). Vistoli was on
tame Artin (with Abramovich) stacks and he said he thought nobody would come for the title of Artin stacks. It was a generalization of "tame"-ness to Artin stacks, which is defined as the order of
the action group to the Deligne-Mumford stack is prime to the char k, where k is an algebraically closed field. He then talked about the linearly reductive finite group schemes, relation of tameness
to properness (Theorem of Abramonich-Corti-Vistoli), and the final remark on a problem: "Is there a modular description of the closure?"Kaledin was on Fedosov quantization in X: smooth over k:
positive characteristic p, especially for p>2. The details were on the Poisson brackets (Poisson manifolds and Frobenius structure), central quantization, Frobenius-constant quotient, (restricted)
Poisson structure, canonical filgration, and extension of Poisson structure to Frobenius quantization, and the Azumaya algebras.
At the BBQ, I heard that there is an international conference: "Algebraic geometry and beyond" in Kyoto in the middle of December. Then I went to the drop-in center to ask Arinkin about the paper of
Drinfeld (2003).
Today I heard only the talk of Gaitsgory and skipped the rest (Loeser and Conrad). His talk was on 1. the classical local Langlands correspondence, 2. geometric Langlands, and 3. abelian categories
over stacks. First one was on the class field theory of Takagi and Weyl group (rather than Galois group). Second one was on the geometric representation of loop group (probably the affine D-module).
Third was something on the base change. I could asked to Gaitsgory about the Beilinson-Drinfeld and local / global geometric Langlands personally.
Griffiths' talk was based on the recent book of AMS 157 with Green. It has a background of Spencer Bloch on Chow groups, Duisequx series, Mumford (infinite dimensionality), and Bloch-Suslin. Conrad
wore a T-shirt with a brief proof of Fermat-Wiles-Taylor last theorem proved 10 years ago when I was a high school student. His talk was on an elementary explanation of modularity, p-adic Galois
representations, and Hecke rings.
Bondal, Hain, and Kaledin were what I chose today. Bondal was on the derived category of toric varieties with a background of homological mirror symmetry with superpotential for Landau-Ginzburg. I
asked him whether we need the DERIVED Fukaya category, but he said the Fukaya category is already a triangulated category. He seemed not to use symplectic sides, so that it was not a serious problem
anyway. He also mentioned that the SCFT before topological twists does not have a categorical formulation yet. The method was essentially exceptional collections for the toric actions. Hain was on
the elliptic cohomology theory: stable homotopy theory, topological modular forms, which was done by Landweber and Hopkins et.al. Kaledin was on the first lecture of his series of three talks. It is
essentially close d string noncommutative algebraic geometry. It started from 3-dimensional McKay correspondence by Miles Reid (1997), the conjecture solved by Bridgeland-King-Reid 1999 and its
generalization by Bezrukavnikiov to any dimension, any resolution, and symplective group G contained in Sp(V), and then Bridgeland-Van den Bergh. p-adic version was also mentioned with a question
from the audience on the recent work of Kontsevich on p-adic D-modules and Jacobian conjecture.
From 7:30, we had an extended talk by Miles Reid on K3s and Fano 3-folds, which was postponed last week. He reviewed the Mori category of Q-factorial terminal singularity, the Hilbert series
(Syzyny), the orbifold Riemann-Roch theorem, and the Gorenstein rings. Half of his talk was done by transparency (OHP) and available at his website, where we can try his computer programme of Fano
data base with Type I projection / unprojection and Tom / Jerry.
I heard the talks of Griffiths and Loeser for the morning session. Griffiths was on the Hodge cycles, generalized Hodge conjectures, Bloch-Beilinson conjecture, and , after assuming GHC and BB, he
thought of some problems arising from, for example, the codimension 2 case. He finished his talk 15 minutes before the schedule. Loeser was an introduction to the definition and history of motivic
integration from the birational viewpoint of Denef-Loeser (1987) and Batyrev (1995). I could talk to him a little after the presentation.
For the afternoon session, I chose Arinkin, Hoboush, and Nadler. Arinkin was on the quantum Liouville theory from the (polarized) deformation quantization and some problems of quantum Hitchin system
of affine curve (fibers not connected, not smooth, no Lagrangian sections). Hoboush was on representation theroy for flag varieties including the Kazhdan-Lusztig conjecture. Nadler was on the
perverse sheaves and introduction of the notion of tilting and some application to G=GL_2 and X=P^1 constructing the tiltings via Morse theory.
At night, I summarized the article of Drinfeld (2003) at the terrace to understand his notion of closed string heterotic CFT.
On Saturday, I took a strong nup until 4:00 pm to be refleshed. I went to the Ave to take two pieces of Pizzas with Coke and then to the Starbacks to write an e-mail. Sunday was also a good holiday
without going out and just reviewing the materials last week.
Morning session was performed by V.V.Shokurov and C.Voison. Sokorov's slides were too small to read so that I can not follow all. His talk consists of 1.Flips and flops, 2.Functional algebras,
3.Reductions (char=0) following the book of Ambro, Corti, Fujino, Mckernan and Takagi (Oxford). The first was on the existence and uniqueness of (log) flips and MMP for 4-folds. The second was on the
canonical embedding as well as mobile and characteristic system, and discrepancies / FGA conjectures. The third was on flliping algebras. Voison was on the examples of Kaehler manifolds which cannot
be polarized utilizing torus / Kummer surfaces. Her final remark was on Tsunoda-Campana's question.
Afternoon session, I went to hear the talks of Yum-Tong Siu, Lev Borisov, and Mikhail Kapranov. The talk of Siu was on the techniques towards the conjecture of finite generation of canonical rings;
from the viewpoint of complex analysis of several variables such as Skoda's estimate, irreducible Lelong sets, finiteness of Lelong numbers, Shokorov's theorem, Pemailly's observation, and Fujita's
conjecture. Borisov started from very elementary level of cones in connection with Hartshorne, Griffith-Harris, and string theory (Batyrev: normal curve). The original work was on non-normal toric
varieties and Eisenbud-Goto conjecture. Kapranov was on the definition of formal loop space in Zariski topology (locally compact ind-schemes) and its application to chiral de Rham complex
(Kapranov-Vasserot) and small quantum cohomolgoy (Arkhipov-Kapranov) with a few comments on Beilinson-Drinfeld chiral / factorization algebras and J-function of Iritani. I could talk to him
personally a little after his talk on possible application towards non-toric target space other than flag manifolds, which I cannot write here.
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still ont to ont problem
August 26th 2009, 09:21 PM #1
Aug 2009
(a). Define the function f:
by f(x,y) = (x+2y, 3x+4y). Is f one-to-one?
(Justify your answer)
can i answer this question like follow
?i know sounds stupid but this is all i can remember from the lecture.
anyone can give me some ideas? thanks^0^
one to one problem~~~... i mean
Have you studied matrices?
What you can do is express the simultaneous equations in matrix form. As it's a simple 2x2 it's then straightforward to determine whether the determinant is zero.
If it is not zero, then the matrix is invertible, and therefore the function is one-to-one and onto.
Any help?
Do you remember the definition of "one to one function"? A function is "one to one" if and only if f(a)= f(b) only when a= b. Since here, f is applied to two numbers, that is "f(a,b)= f(x,y) if
and only if (a,b)= (x,y)" which is the same as saying a= x and b= y.
From f(a,b)= (a+ 2b, 3a+ 4b)= (x+ 2y, 3x+ 4y)= f(x,y), which is the same as saying a+ 2b= x+ 2y and 3a+ 4b= 3x+ 4y, can you show that a= x and b= y?
Do you remember the definition of "one to one function"? A function is "one to one" if and only if f(a)= f(b) only when a= b. Since here, f is applied to two numbers, that is "f(a,b)= f(x,y) if
and only if (a,b)= (x,y)" which is the same as saying a= x and b= y.
From f(a,b)= (a+ 2b, 3a+ 4b)= (x+ 2y, 3x+ 4y)= f(x,y), which is the same as saying a+ 2b= x+ 2y and 3a+ 4b= 3x+ 4y, can you show that a= x and b= y?
thanks ..help me out^.^
August 26th 2009, 09:36 PM #2
August 27th 2009, 12:45 AM #3
MHF Contributor
Apr 2005
August 27th 2009, 03:40 AM #4
Aug 2009
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Payroll vs. wins -- still significant in 2006
Payroll vs. wins -- still significant in 2006
The “Wages of Wins” blog approvingly quotes a news story from the Southern Ledger arguing a weak relationship between MLB payroll and performance.
First, the news story. Its prime exhibit is a chart of “dollars spent per win,” which is exactly what it sounds like: payroll divided by wins. It shows the Marlins leading, at $220,000 per win (as of
this week), and the Yankees trailing, at $2.4 million per win. Ranked by dollars per win, the chart seems a nearly-random mix of good teams and poor teams, which writer Bill Freehling uses to imply
that salaries don’t matter all that much.
However, dollars per win is a poor measure of payroll efficiency, because the relationship between pay and performance isn't linear. A team could, in theory, have 25 players making the minimum
$327,000, which would make its payroll about $8.2 million. That team could probably still win a few games. Even if it only won 20 games out of 140, that would still be only $410,000 per win, which
would appear to lead the league.
For the Yankees to match that kind of efficiency with their $195 million payroll, they’d have to win 475 games out of 140. That would be difficult, even if Derek Jeter's defense improved
So, clearly, the relationship isn’t linear. Some have suggested “marginal dollars per marginal win” as an appropriate measure. Even that one isn’t perfect, because of diminishing returns on
increasingly high salaries. But at least it’s better. In any case, the chart isn’t very informative.
Freehling is on more solid ground when he points out that the teams in “the top-third [of payroll] spent about 117 percent more than the bottom-third but had won just 14 percent more games.”
But even that statistic doesn’t mean much out of context. Is 14% a lot? A little? What about 117%? What kind of relationship do these numbers imply, and do they really mean that salary doesn’t matter
14% more wins, in a 162-game season, is about 11 games in the standings. That’s a fair bit of difference between the high-spending teams and the low-spending teams. Not as much as one would think,
perhaps, but it’s still 1.5 standard deviations from equal – a low-payroll team has a long way to go to catch up to the richer clubs.
Second, the blog entry itself. David Berri uses the article to confirm “The Wages of Wins’” claim that payroll is not an economically significant factor in predicting wins, because the r-squared of
the relationship is only 17%. (As I argued here, the important number is the r, not the r-squared. An r-squared of 17% is an r of 41%, which implies that, in a certain sense, payroll actually
explains 41% of wins.)
Berri notes that so far in 2006, the r-squared is 24%, which is statistically significant but doesn't carry much "oomph."
But again, I argue that would make the r around 49%, which is pretty significant.
How signficant? Here’s another way of looking at it.
Start by observing that payroll doesn’t buy wins directly – it can only buy talent. Do a “what if.” Suppose that GMs could evaluate talent perfectly, and always spent exactly the correct amount for
the player’s ability. Suppose we also were able to find a function to translate increased payroll into increased ability, whatever that function might be.
In that case, the correlation between salary and wins would be exactly the same as the correlation between ability and wins. So what would the correlation be between ability and wins?
It wouldn't be 100%, because teams don’t always win exactly in accordance with their ability. Just as a fair coin might have more or less than 50% heads, just by luck, a .500 team might go 90-72 or
77-85 just by chance alone. So r would be less than 1.
How much less? Tangotiger tells us here (see comments) that the variance of team ability is .06^2. Based on that, I ran a simulation, and came up with a wins/talent correlation around .83.
(This number happens to be SD(ability)/SD(wins) … which might not be a coincidence. A result from Tangotiger tells us that var(ability)/var(wins) is the correlation between two independent sets of
wins with the same ability distribution … this might be a variation of that result.)
So even if payroll completely determined talent, the best we could get would be an r of 0.83.
Stated in point form:
--If payroll had no relationship to talent, we’d get r=0.
--If payroll had 100% relationship to talent, we’d get r=0.83.
--For 2006, we actually get r=0.49.
Or, put into baseball terms:
-- Suppose there were a perfect relationship between payroll and ability. You’d find that a one SD increase in payroll led to a .83 SD increase in wins.
--In real life, a one SD increase in payroll leads to a .49 SD increase in wins.
All things considered, the relationship between payroll and wins seems pretty oomphy to me.
3 Comments:
At Monday, September 11, 2006 9:19:00 PM, said...
Good job, as usual.
I found that there might be a close connection between wins and salaries. I made a comment at
Also, teams don't play identical schedules. The Red Sox and Yankees play each other alot. That might cut into their win total since they play each other more than they play teams in other
At Friday, January 19, 2007 10:14:00 AM, said...
Links to this post:
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MathGroup Archive: February 2008 [00480]
[Date Index] [Thread Index] [Author Index]
Re: Re: "Assuming"
• To: mathgroup at smc.vnet.net
• Subject: [mg85686] Re: [mg85649] Re: "Assuming"
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Tue, 19 Feb 2008 01:59:06 -0500 (EST)
• References: <fp0m11$5u7$1@smc.vnet.net> <fp192b$gju$1@smc.vnet.net> <fp2khg$pka$1@smc.vnet.net> <fp3ui0$9jt$1@smc.vnet.net> <200802171221.HAA01121@smc.vnet.net>
In fact there is a perfectly clear mathematical sense in which it is
true that (a^2 - 1)/(a - 1) is exactly equal to a-1. Consider the
field of fractions of the ring of polynomials in F[a], where F can be
the field of rationals or reals or complexes and a is what is
sometimes called "an indeterminate". In other words, its elements are
rational functions of the form p(a)/q(a). Then, in this field of
fractions (a^2 - 1)/(a - 1) is exactly equal to a+1. If you are
algebraically minded (like myself) than you may like to think of
Mathematica itself as a (admittedly rather unusual) algebraic
structure generated by all syntactically correct symbols and the n-ary
operations Times and Plus, and subject to certain relations. Note that
almost anything in Mathematica can be "added" to or multiplied by
almost anything else and we get things such as:
In[1]:= Simplify[("dog"^2 - 1)/("dog" + 1)]
Out[1]= "dog" - 1
Well, clearly "dog" being a string, can't be set equal to 1 or
anything else, it is simply a Mathematica expression (in this case a
string) but the rules for Plus, Times and Simplify apply all the same.
This is what I mean when I say that I like to think or Mathematica in
algebraic sense - to put it somewhat crudely all that Mathematica does
is to perform certain "dumb" algebraic transformations using certain
rules. It certainly does not "take limits" or do any other of the sort
of more sophisticated mathematical thinking that is required when
dealing with limits (you can't really talk of limits unless you have
some sort of topology and "dumb algebraic manipulation" is not taking
limits even if it looks like it.
So I think there is no need to consider the fact that a^2/a returns 1
as any kind of "misdemeanor", even the most minor one.
However, things get somewhat more subtle when assumptions are
involved. In the case,
In[2]:= Assuming[x == 1, Simplify[(x^2 - 1)/(x - 1)]]
Out[2]= 2
it would, ideally, be preferable to get some other answer, since now
one could argue that under the assumption that x is the real number 1,
we are no longer working in a field of rational functions but rather
in the field of real numbers, where the expressions is not defined
(or,arguably, could be defined as Indeterminate). Mathematica does not
do this because it is much more efficient to perform any algebraic
cancellations that can be performed before substituting values. So the
practical considerations win over strict mathematical correctness
(perhaps). But in any case, once this behaviour is understood as a
"principle" of "computer algebra", one can use it just as reliably as
any other formal algebraic rule.
Andrzej Kozlowski
On 17 Feb 2008, at 13:21, David W.Cantrell wrote:
> dh <dh at metrohm.ch> wrote:
>> Hi David,
>> we all learned not to divide by zero,
>> but things are a bit more subtle than this.
> Of course.
>> You claim that (a^2 - 1)/(a - 1) should simplify to If[a == 1,
>> Indeterminate, a + 1] and not to (a+1),
> I said that it should simplify to _something equivalent to_
> If[a == 1, Indeterminate, a + 1]. Furthermore, when I said "should", I
> should have made it clear that I was talking about "in an ideal
> sense",
> rather than practicality. Sorry that I failed to make that clear.
> I do not claim that Mathematica should be rewritten so that
> Simplify[(a^2 - 1)/(a - 1)] yields anything other than what it
> currently
> does. Perhaps you noticed what I said at the end of my response:
> "But such
> a result [like If[a == 1, Indeterminate, a + 1] here] is normally
> considered to be too cumbersome to be practical; the 'misdemeanor'
> simplification is considered preferable. All CASs known to me make
> such
> simplifications." Thus, if the developers consider
> If[a == 1, Indeterminate, a + 1] to be too cumbersome to be
> practical, I
> accept their judgement, despite the fact that (a + 1) is not
> equivalent to
> (a^2 - 1)/(a - 1).
> Also, note that "simplify to something equivalent to
> If[a == 1, Indeterminate, a + 1]" leaves open the possibility that
> Simplify[(a^2 - 1)/(a - 1)] would yield (a^2 - 1)/(a - 1) unchanged.
>> but you do not give a good reason for this.
> [P.S. Thanks to Mariano for having already addressed this point.]
> I didn't mention a reason; I thought it was obvious. Namely,
> (a^2 - 1)/(a - 1) and (a + 1) are not equivalent. In Mathematica, when
> a is 1, the former expression is Indeterminate while the latter is 2.
>> Obviously (a^2 - 1)/(a - 1) is not defined for a==1,
> Well, in Mathematica, it's not defined as anything I'd normally call a
> "number", but it is defined to be Indeterminate. [You may consider
> that
> comment to be mere pedantry. However, Indeterminate in Mathematica
> corresponds fairly well to NaN in standard floating-point
> arithmetic. IIRC,
> Kahan is quite adamant that NaN is not the same as "undefined".]
>> but note that the left and right limit exist and are identical.
> Sure, the singularity is removable.
>> Now what harm is done if
>> we replace the first function by a second one (a+1) that agrees with
>> the first one where it is defined, but that is also defined for a==1?
> A singularity has <poof!> vanished all of a sudden. Although that
> required
> us to invoke Simplify in this case, there are simpler cases when
> this can
> happen without even asking for any simplifcation, such as
> In[7]:= x/x
> Out[7]= 1
> Whatever potential harm this "misdemeanor" might do is outweighed,
> in the
> minds of most (perhaps, all) CAS developers, by the utility of
> having a
> simpler result.
> For all I know, some future version of Mathematica might return 0 and
> Sinc[x] for FullSimplify[0^Abs[z]] and FullSimplify[Sin[x]/x],
> resp.; in
> the current version, FullSimplify leaves those arguments unchanged.
>> This seems to me a better aproach than your proposal, especially if
>> we consider the implications on practical calculations.
> As I said above, I am not proposing that Mathematica should be
> rewritten so that Simplify[(a^2 - 1)/(a - 1)] yields anything other
> than (a + 1), despite the fact that (a^2 - 1)/(a - 1) should not, in
> an ideal sense, simplify to (a + 1).
> This response should also have addressed a point raised by David
> Bailey.
> David W. Cantrell
>> David W.Cantrell wrote:
> [snip]
>>> Here's a portion of a response of mine which appeared some years
>>> ago in this newsgroup. It's relevant to what Markus was asking
>>> about.
>>> -------------------------------------------------------
>>> In a sense, there is something wrong. These simplifications are
>>> what David Stoutemyer called "misdemeanors" in his article "Crimes
>>> and
>>> Misdemeanors in the Computer Algebra Trade", _Notices of the
>>> American
>>> Mathematical Society_ 38:7 (1991) 778-785.
>>> In case anyone doesn't see why simplifying, say, x^0 to 1 is a
>>> misdemeanor: In Mathematica, 0^0 is regarded as being Indeterminate.
>>> Thus, without knowing that x is nonzero, Mathematica should not
>>> bluntly
>>> simplify x^0 to 1. Rather, it would seem that, in Mathematica, x^0
>>> should simplify to, say, If[x == 0, Indeterminate, 1]. But such a
>>> result is normally considered to be too cumbersome to be
>>> practical; the
>>> "misdemeanor" simplification is considered preferable. All CASs
>>> known
>>> to me make such simplifications.
>>> -------------------------------------------------------
• References:
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Course Listings
MAT115 Precalculus (3)
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MAT121 Concepts of Math for Elementary Teachers I (3)
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solving and topics from theory of numbers.
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problem solving and topics from theory of numbers.
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Fundamental Group and Etale Cohomology
up vote 1 down vote favorite
I encountered the following statement without a reference many times. For a smooth variety $X$ over a perfect field $k$.
$Hom(H^1_{et}(X, \mathbb{Z}/n), \mathbb{Z}/n) \cong \pi^{ab}_1(X)/n$
Is there any reference for this? Why this is true?
fundamental-group etale-cohomology ag.algebraic-geometry
5 Dear Grilo, As for "why is this true", it is the analogue in the etale world of the analogous statement for topological spaces and usual cohomology, namely that $Hom(H^1(X,\mathbb Z/n),\mathbb Z/
n)=\pi_1(X)^{ab}/n$, which follows from Hurewicz's theorem relating $\pi_1$ and $H_1$. Regards, Matthew – Emerton Dec 20 '12 at 15:33
1 @Emerton: but saying that "it is the analogue" doesn't answer "why"! – John Pardon Dec 20 '12 at 18:50
@unknown For characteristic zero the result actually follows from Hurewicz (and the Lefschetz principle and comparison between étale and simplicial cohomology) – Felipe Voloch Dec 20 '12 at 21:28
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1 Answer
active oldest votes
See Milne's online course notes on Étale Cohomology, Example 11.3, or Lei Fu's Étale Cohomology Theory, Proposition 5.7.20. (By passing to the direct limit over all $n$, you can even
prove it for $\mathbf{Q}/\mathbf{Z}$.)
You only need $X$ to be connected Noetherian.
up vote 4
down vote I am interested in alternative proofs not using torsors and Cech cohomology.
BTW, for $X$ normal, it is also true for $\mathbf{Z}$- or $\mathbf{Q}$-coefficients---both sides are $0$ in this case.
It may be worth noting that the main content lies in the implicit use of Proposition 10.6 in those notes (valid for every abelian sheaf, such as constant coefficients in any abelian
group) and that this is all valid for any connected scheme, whereas the identification of ${\rm{H}}^1(X,A)$ with the group of continuous homomorphisms from $\pi_1(X)$ into $A$ fails if
we do not assume $A$ is finite (e.g., for the nodal cubic $X$ and $A = \mathbf{Z}$, there are nontrivial etale $\mathbf{Z}$-torsors over $X$), the "problem" being due to finiteness
aspects built into the definition of $\pi_1(X)$. – user29720 Dec 20 '12 at 15:49
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The General Theory of Relativity, Metric Theory of Relativity and Covariant Theory of Gravitation. Axiomatization and Critical Analysis
Authors: Sergey G. Fedosin
The axiomatization of general theory of relativity (GR) is done. The axioms of GR are compared with the axioms of the metric theory of relativity and the covariant theory of gravitation. The need to
use the covariant form of the total derivative with respect to the proper time of the invariant quantities, the 4-vectors and tensors is indicated. The definition of the 4-vector of force density in
Riemannian spacetime is deduced.
Comments: 14 Pages.
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Lateral Stability
Lateral Stability & Control
Lateral Static Stability
1. What is the effect of wing dihedral on the lateral stability of an airplane?
2. What is the effect of wing sweep on the lateral stability of an airplane?
3. Does it make a difference if the sweep is aft or forward?
4. How do we usually compensate for the sweep effect on high performance airplanes?
5. What is the effect of flaps (employed during takeoff and landing) on the lateral stability of an airplane?
6. What is the effect of the fuselage on the lateral stability of an airplane?
7. What is the effect of the vertical stabilizer on the lateral stability of an airplane?
8. What is the effect of a dorsal fin on the lateral stability of an airplane?
9. What is the effect of a ventral fin on the lateral stability of an airplane?
10. What is the effect of wing placement (high vs. low) on the lateral stability of an airplane?
Lateral Dynamic Stability
1. Describe damping in roll.
2. Describe autorotation.
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[FOM] inconsistency of P
Harvey Friedman friedman at math.ohio-state.edu
Mon Oct 3 01:47:00 EDT 2011
I am reasonably aware of Nelson on this, but my main point is that
mathematicians (including Nelson, by the way) seem to regularly use -
and teach - facts which logically imply the consistency of at least
significant fragments of PA, including PRA, or single quantifier PA.
At least, the former Nelson - and what about Nelson's current math
Even this claim of mine above requires serious justification, and
needs the new SRM = strict reverse mathematics, to fully justify.
On the other hand, it appears that a very large amount of mathematics
is "innocent" and does not imply the consistency of, say, PRA, or
single quantifier PA.
This leads to the program of coming up with very comprehensive
conservative extensions of, say, RCA0 (and weaker!) that have lots of
abstract objects - even uncountable objects - and support flexible
reasoning involving such.
Harvey Friedman
> On Oct 2, 2011, at 8:42 PM, Timothy Y. Chow wrote:
> Harvey Friedman wrote:
>> As I indicated before on the FOM, there is a proof that any given
>> finite fragment of PA is consistent, using "every infinite sequence
>> of
>> rationals in [0,1] has an infinite Cauchy 1/n subsequence".
> Perhaps you haven't read any of Nelson's philosophical writings. He
> doesn't believe in infinity, except "potential infinity," and regards
> even so-called "finitary" reasoning (let's say, PRA) as having hidden
> infinitary assumptions in it. Thus it is suspect.
> Here's another way to put it. You [Friedman] have suggested before
> that
> mathematics is "essentially" Pi^0_1. For example, if someone were to
> prove P != NP, then we'd immediately want a more quantitative
> version of
> it that gives us bounds, and we'd search for a Pi^0_1
> strengthening. For
> Nelson, math is essentially Pi^0_0. Nelson will accept statements
> of the
> form "T is a theorem of X" as immediately meaningful because they're
> Pi^0_0. (At least, he'll accept them if the proof has feasible
> length; my
> guess is that he'd do the old "wait 2^n when asked if 2^n exists"
> trick if
> you asked him about large finite numbers.) But any infinitary
> statement T
> is at best a convenient fiction for helping us find Pi^0_0
> statements, or
> is perhaps a shorthand for "T is provable in X."
> It's not clear to me whether there is anything that Nelson would
> accept
> as settling the consistency of first-order Peano arithmetic in the
> affirmative. So when he says that "the consistency of P remains an
> open
> problem" I think he just means that nobody has yet found an explicit
> proof
> of a contradiction in P.
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Math Age Word Problem
Date: 03/20/2001 at 20:45:11
From: Rosalynd Nicholl
Subject: Math Age Word Problem
My mom and I searched your archives and even looked at your many
examples. We are still stuck! Mom spent all night but had no success.
My fourth grade teacher gave us this problem.
Bob is as old as John will be when Bob is twice as old as John was
when Bob's age was half the sum of their present ages. John is as old
as Bob was when John was half the age he will be ten years from now.
How old are John and Bob?
My teacher gave us the answer, but she asked us to figure out how
they came up with it. I cannot break down the sentences.
Bob is 40
John is 30
How can this BE?
Date: 03/21/2001 at 17:01:21
From: Doctor Peterson
Subject: Re: Math Age Word Problem
Hi, Rosalynd.
This is a challenging problem even for adults, and I can't imagine
doing it without at least a basic knowledge of algebra. I'll interpret
it for you and show you the algebraic approach, and maybe I'll think
of a simpler way while I work.
We have two unknown ages, and also three unknown times ("when ...") in
addition to the present. I'll call their current ages B and J, and the
times +X, -Y, and -Z. Time +X, the first one mentioned, is in the
future, X years from now, when everyone is X years older than they are
now; times -Y and -Z are in the past, when everyone was Y or Z years
younger, respectively. You'll see where those fit in in a moment.
Now we can clarify the problem:
Bob is [now] as old as John will be [at time +X] when Bob is
twice as old as John was [at time -Y] when Bob's age was half
the sum of their present ages [now]. John is [now] as old as
Bob was [at time -Z] when John was half the age he will be ten
years from now [at time +10].
We can break it down into separate sentences to make it even clearer:
Bob is now as old as John will be at time +X.
At time +X, Bob will be twice as old as John was at time -Y.
At time -Y, Bob's age was half the sum of their present ages [now].
John is now as old as Bob was at time -Z.
At time -Z, John was half the age he will be at time +10.
Finally, we can translate each new sentence into an equation:
Bob is now as old as John will be at time +X.
B = J+X
At time +X, Bob will be twice as old as John was at time -Y.
B+X = 2(J-Y)
At time -Y, Bob's age was half the sum of their present ages [now].
B-Y = (B+J)/2
John is now as old as Bob was at time -Z.
J = B-Z
At time -Z, John was half the age he will be at time +10.
J-Z = (J+10)/2
Now here comes the hard work. Don't try to follow everything I say; I
promise to come back down to earth at the end, so skim until you see
something that says I'm back with you. Your mom may or may not want to
try to wade through it all; my main point is to show that it really is
far beyond you!
All we've done so far is to figure out what everything means, and put
it into algebraic terms. Now we have to do actual algebra, eliminating
the unknowns we don't care about, until we can find J and B. First
I'll simplify the equations:
B = J + X
B + X = 2J - 2Y
2(B - Y) = B + J --> B - 2Y = J
J = B - Z
2(J - Z) = J + 10 --> J - 2Z = 10
The last equation gives me Z, which I can put into the fourth
Z = J/2 - 5
J = B - (J/2 - 5)
3J/2 = B + 5
3J = 2B + 10
Now I can set that aside, and put Y from the third equation into the
Y = (B - J)/2
B + X = 2J - (B - J)
2B + X = 3J
I can put X from this into the first equation:
X = 3J - 2B
B = J + (3J - 2B)
3B = 4J
Finally, I can put J from this into the equation I set aside:
J = 3B/4
3*3B/4 = 2B + 10
9B/4 - 2B = 10
B/4 = 10
B = 40
J = 3*40/4 = 30
Whew! We're done, but you surely couldn't follow all that algebra. Is
there any solution you could follow? Not that I can see. But what you
can do - and perhaps what you were really expected to do - is just to
check that the answer is right, and see that it really makes sense.
Even that is hard if you haven't solved the problem so as to know the
values of my X, Y, and Z. I'll just start that process, by looking at
the (easier) second statement.
John is [now] as old as Bob was [at time -Z] when
John was half the age he will be ten years from now [at time +10].
How old will John be ten years from now? 30 + 10 = 40. What is half of
that? 20. When was John that age? 30 - 20 = 10 years ago. (That's my
Z.) How old was Bob then? 40 - 10 = 30. That's John's age now, so it
That's still not easy, but it's much more reasonable to expect you to
do this than to actually solve the problem.
- Doctor Peterson, The Math Forum
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The Substitution Method
Even relatively untrained algebra learners can find the answer to a simpler system of equations using the guess and check method. If time permits, graphing the lines is an alternate way to find the
It is important to familiarize yourself with the above two methods of solving systems before moving on to tackle more heavy duty methods. Almost every system of two equations can be solved using the
algebraic methods of substitution or elimination. This lesson demonstrates how to substitute to find the solutions. Obviously, this method is called the substitution method.
Substituting is a concept that has been presented many times since the beginning of algebra.
First, you learned how to substitute for a single variable:
Next, you learned how to substitute for several variables:
In this lesson, the first step to solving a system of equations is to substitute for an entire side of the equation.
When there are many ways to do a particular problem, it makes sense to choose a method that gives you the best chance to get the answer. If two methods both give you a “best” chance at the correct
answer, then choose the one that requires the least amount of work.
The difficulty level of a systems of equations may range from “easy” up to “very hard.” When doing an easy problem, guess and check is a good method to use. More difficult problems may be nearly
impossible to guess, though, so they are best done using another method. When one of the equations of a system has a variable that can be easily isolated, a good method to use is substitution.
Substituting in a system of equations simply means that you will replace a variable in one of the equations with whatever that variable equals. The substitution in example 1 is clearly marked so that
you can see how the substitution worked.
Example 1: Find the solution to the system
So basically the first equation was substituted into the second equation, resulting in an answer of y = 4. Since there are two variables, you must also find the value of x in your solution. Since y =
4, you can substitute this value into the equation x = 2y – 1. Once you have found the value of both variables, rewrite them in a single location… this is your solution.
When both equations already have a variable isolated, then the problem can be done in much the same way. This type of problem is confusing to many people learning how to solve systems because there
is a choice of how to substitute. You can actually do the problem by substituting either equation into the other one.
Example 2: Find the solution to the system
It would be fine to use the a = 400 – 2b and substitute the (400 – 2b) in for the value of a into the second equation. The answer works out to be the same this way. The only advantage that (b + 100)
has over (400 – 2b) is that it is a little simpler to work with.
There are times when neither equation has a term that is isolated. In this case, determine whether one of the equations can be manipulated to isolate one term. If so, then substitution may be the
best method for solving the system. If it is not easy to isolate a term, then substitution is not a good option and you should consider another method (like elimination.)
Example 3: Find the solution to the system
Since there are so many work steps in these equations, it may be hard to tell if you made a mistake. Luckily, it is very easy to check your work on these problems… just plug your solution into each
equation and see if it works. It is worth the time and effort needed to check your answer. Here is a check to the answer of example #3.
Another tip is to make sure you leave yourself enough room for all your work. Efficient students can do about 6 of these problems on one piece of paper. Avoid the temptation to squeeze in work or
leave out steps in order to save paper. After you have spent all that time doing the problem, be sure you got it right by checking to see if your answer is right. If not, try the problem again. Only
after you have tried a second and third time and still are coming up with a solution that doesn’t work should you consider getting a second opinion on how to do the problem. Remember that you must
think for yourself in order to improve in mathematics and feel really confident about your ability. Don’t rob yourself of that opportunity by being impatient as you complete your assignment.
Related Links:
Looking for a different lesson on systems of equations? Try the links below.
Related Lessons
Before attempting to learn systems of equations, one should be comfortable solving a variety of (single) equations.
Equation Lessons
Looking for something else? Explore our menu of general math or algebra lessons.
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the encyclopedic entry of coprime
, the
integers a
are said to be
relatively prime
if they have no common
other than 1 or, equivalently, if their
greatest common divisor
is 1. The notation
is sometimes used.
For example, 6 and 35 are coprime, but 6 and 27 are not coprime because they are both divisible by 3. The number 1 is coprime to every integer.
A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm.
Euler's totient function (or Euler's phi function) of a positive integer n is the number of integers between 1 and n which are coprime to n.
There are a number of conditions which are equivalent to a and b being coprime:
• There exist integers x and y such that ax + by = 1 (see Bézout's identity).
• The integer b has a multiplicative inverse modulo a: there exists an integer y such that by ≡ 1 (mod a). In other words, b is a unit in the ring Z/aZ of integers modulo a.
As a consequence, if a and b are coprime and br ≡ bs (mod a), then r ≡ s (mod a) (because we may "divide by b" when working modulo a). Furthermore, if a and b[1] are coprime, and a and b[2] are
coprime, then a and b[1]b[2] are also coprime (because the product of units is a unit).
If a and b are coprime and a divides the product bc, then a divides c. This can be viewed as a generalisation of Euclid's lemma, which states that if p is prime, and p divides a product bc, then
either p divides b or p divides c.
The two integers a and b are coprime if and only if the point with coordinates (a, b) in a Cartesian coordinate system is "visible" from the origin (0,0), in the sense that there is no point with
integer coordinates between the origin and (a, b). (See figure 1.)
The probability that two randomly chosen integers are coprime is 6/π^2 (see pi), which is about 60%. See below.
Two natural numbers a and b are coprime if and only if the numbers 2^a − 1 and 2^b − 1 are coprime.
Cross notation, group
≥1 is an
, the numbers coprime to
, taken
modulo n
, form a
with multiplication as operation; it is written as (
Two ideals A and B in the commutative ring R are called coprime if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals (a) and (b) in the ring of integers Z are
coprime if and only if a and b are coprime.
If the ideals A and B of R are coprime, then AB = A∩B; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about
coprime ideals.
The concept of being relatively prime can also be extended any finite set of integers S = {a[1], a[2], .... a[n]} to mean that the greatest common divisor of the elements of the set is 1. If every
pair of integers in the set is relatively prime, then the set is called pairwise relatively prime.
Every pairwise relatively prime set is relatively prime; however, the converse is not true: {6, 10, 15} is relatively prime, but not pairwise relative prime. (In fact, each pair of integers in the
set has a non-trivial common factor.)
Given two randomly chosen integers $A$ and $B$, it is reasonable to ask how likely it is that $A$ and $B$ are coprime. In this determination, it is convenient to use the characterization that $A$ and
$B$ are coprime if and only if no prime number divides both of them (see Fundamental theorem of arithmetic).
Intuitively, the probability that any number is divisible by a prime (or any integer), $p$ is $1/p$. Hence the probability that two numbers are both divisible by this prime is $1/p^2$, and the
probability that at least one of them is not is $1-1/p^2$. Thus the probability that two numbers are coprime is given by a product over all primes,
$prod_p^\left\{infty\right\} left\left(1-frac\left\{1\right\}\left\{p^2\right\}right\right) = left\left(prod_p^\left\{infty\right\} frac\left\{1\right\}\left\{1-p^\left\{-2\right\}\right\} right\
right)^\left\{-1\right\} = frac\left\{1\right\}\left\{zeta\left(2\right)\right\} = frac\left\{6\right\}\left\{pi^2\right\}$ ≈ 0.607927102 ≈ 61%.
Here ζ refers to the Riemann zeta function, the identity relating the product over primes to ζ(2) is an example of an Euler product, and the evaluation of ζ(2) as π^2/6 is the Basel problem, solved
by Leonhard Euler in 1735. In general, the probability of $k$ randomly chosen integers being coprime is $1/zeta\left(k\right)$.
There is often confusion about what a "randomly chosen integer" is. One way of understanding this is to assume that the integers are chosen randomly between 1 and an integer $N$. Then for each upper
bound $N$, there is a probability $P_N$ that two randomly chosen numbers are coprime. This will never be exactly $6/pi^2$, but in the limit as $N to infty$, $P_N to 6/pi^2$.
See also
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Calculating the new x position - Java-Gaming.org
Hi ive been looking on the internet for a post from 1999 about 3d graphics algorithms if anyone could link to that it would be great and would solve the entire issue.
here is the issue.
Say I have my camera and x1 y1 z1 and i have a vector (just a pixel) and x2 y2 z2. How would I figure out the position I draw the x and y if Im drawing it orthagonally?
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When is the product of two ideals equal to their intersection?
up vote 20 down vote favorite
Consider a ring $A$ and an affine scheme $X=SpecA$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds to the union
$V(I\cap J)=V(I)\cup V(J)$. But a product $I.J$ gives a new subscheme $V(I.J)$ which has same support as the union but can be bigger in an infinitesimal sense. For example if $I=J$ you get a scheme
$V(I^2)$ which is equal to "double" $V(I)$.
Vague Question : What is geometric interpretation of $V(I.J)$ in general?
Precise question : When is $I\cap J=I.J$? Everybody knows the case $I+J=A$ but this is absolutely not necessary. For example if $A$ is UFD and $f,g$ are relatively prime then $(f).(g)=(f)\cap(g) $
but in general $(f)+(g)\neq A$ (e.g. $f=X, g=Y \in k[X, Y]$)
Thank you very much.
ac.commutative-algebra ag.algebraic-geometry
add comment
3 Answers
active oldest votes
To add to David Speyer's answer, since this story continues with a rather interesting and illustrious history:
When $A$ is regular, the Tor functor satisfies the following property:
(1) $\text{Tor}_1^A(M,N) = 0$ implies $\text{Tor}_i^A(M,N) = 0$ for $i>0$ for any two finitely generated modules.
(this is a theorem by Auslander in the geometric and unramified case and Lichtenbaum in the ramified case. (1) is called the rigidity of Tor).
It turns out that when $A$ is regular and local (so one can talk about depth), (1) implies
(2) $\text{depth} (M) + \text{depth}(N) = \dim A + \text{depth} {M\otimes_AN}$
This stunning formula looks exactly the same as the property of "proper intersection" in intersection theory, except that one uses depth instead of dimension. Note that if $M=A/I, N
up vote 21 down =A/J$ then $M\otimes N = A/(I+J)$, which represents the intersection of $V(I)$ and $V(J)$, so this is very geometric.
vote accepted
(3) Talking about intersection theory, by Serre formula for intersection multiplicity, as all the Tors vanish, one can compute the intersection multiplicity of $V(I), V(J)$ by
counting the length at the minimal components (i.e. the naive way). So you will have a generalization of Bezout theorem.
Finally, if $V(I)$ and $V(J)$ only intersect at isolated closed points, (2) implies (1) locally on the support of the intersection, so
(4) If $V(I) \cap V(J)= \{m_1, \cdots, m_n \}$ then $I\cap J = IJ$ if and only if $A/I, A/J$ are locally Cohen-Macaulay at the points $m_i$s.
You can find the last statement in Serre's Local Algebra book, V.6, Theorem 4, p 110 of the English version.
PS: Also, David did not mention his own interesting contribution, here.
Nice exhaustive answer, so let me ask a stupid reference. I do not want to prove that if $f,g$ have no common factor in a UFD then $(f)\cap(g)=(f\cdot g)$ in a paper I am writing,
but I found no explicit reference: do you know any? Adapting Serre's criterion seems a bit overkilling, for a UFD...Thanks. – Filippo Alberto Edoardo Aug 27 '13 at 10:54
Filippo: Any thing in $(f)\cap (g)$ would be of the form $fx=gy$. By writing both sides a product of irreducibles one concludes that $g$ divides $x$... – Hailong Dao Aug 28 '13 at
...hmm, I guess I agree and that NO reference is by far the best option. I was probably a bit puzzled when I asked, sorry. – Filippo Alberto Edoardo Aug 28 '13 at 4:38
add comment
Answer to the precise question: When $\mathrm{Tor}^1(A/I, A/J)=0$.
Proof: We have the exact sequence $$0 \to I \to A \to A/I \to 0$$ Tensoring with $A/J$, we get $$0 \to \mathrm{Tor}^1(A/I, A/J) \to I/(I \cdot J) \to A/J \to A/(I+J) \to 0.$$ The left hand
up vote term is $0$ because $A$ is flat as an $A$-module.
31 down
vote Now, what is the kernel of $I \mapsto A/J$? Clearly, it is $I \cap J$. So the kernel of $I/(I \cdot J) \to A/J$ is $(I \cap J)/(I \cdot J)$. We see that $I \cap J = I \cdot J$ if and only
if $\mathrm{Tor}^1(A/I, A/J)=0$.
3 The condition with Tor is looking more complicated than the question. – Mark Sapir Dec 13 '10 at 14:22
1 But it is more "geometric" since only $V(I)$ and $V(J)$ are involved. – Martin Brandenburg Dec 13 '10 at 14:35
1 @Martin: So you think it is better than the question? If we have Groebner bases of $I$, and $J$, can we decide whether $IJ=I\cap J$? I think that can be an interesting question. In fact
I am not sure that David's answer gives any algorithm to decide $IJ=I\cap J$. It must be decidable, though. – Mark Sapir Dec 13 '10 at 14:47
4 I agree that the condition with Tor is more geometric --- it can be viewed as a kind of `purity' of intersection (for instance, two smooth subvarieties of a smooth variety have this
property if and only if their intersection has the expected dimension). Is there an accepted name for this condition? – t3suji Dec 13 '10 at 15:38
@t3suji. Let V and W be closed integral subschemes of a nonsingular quasi-projective irreducible variety. Then, for any irreducible component Z of VcapW, it holds that codim Z <= codim V
2 + codim W. (See Serre's Local Algebra.) We say that V and W intersect properly in Z if equality holds. A stronger condition is being in general position. If V and W are in general
position all the higher Tor's vanish. The cycle [VcapW] associated to VcapW is then equal to the product cycle [V][W]. As far as I know, this is standard language in intersection theory
for algebraic varieties. – Ari Dec 13 '10 at 17:49
show 3 more comments
A vague answer to the vague question:
When you want the union of $V(I)$ and $V(J)$ to behave well under deformations and to `count with multiplicity', then you may prefeer to use the ideal $IJ$ rather than $I\cap J$.
Let me give an example:
Take $V=V(x)$, $W=V(x-t)$ and $T=V(t)$ denote $V_0:=V\cap T=V(x,t)=W\cap T=:W_0$. If you use intersection of ideals for the union of varieties you will get:
up vote 0 down $(V\cup W)\cap T=V(x^2)$, and
$(V_0\cup W_0)\cap T=V(x)$.
While using product you will get:
$(V\cup W)\cap T=V(x^2)=(V_0\cup W_0)\cap T$.
add comment
Not the answer you're looking for? Browse other questions tagged ac.commutative-algebra ag.algebraic-geometry or ask your own question.
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logical reasoning for class 4
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Event Details
Nuclear Physics & RIKEN Theory Seminar
"On holographic thermalization and gravitational collapse of scalar fields"
Presented by Bin Wu, iPhT Saclay
Friday, April 19, 2013, 2 pm
Small Seminar Room, Bldg. 510
In this talk, I will discuss the thermalization of a strongly coupled system via AdS/CFT. Two non-equilibrium initial states are studied, which are created respectively by switching on a marginal or
relevant perturbation in the CFT vacuum. The thermalization on the CFT side corresponds to the black hole formation process from the gravitational collapse of a massless or tachyonic scalar field in
the Poincare patch of AdS_5. In all the cases our results show that the system thermalizes in a typical time 1/T with T being the thermal equilibrium temperature.
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How the inverted classroom saves students time
February 23, 2011, 5:19 pm
Our semester is into its third full week, and most of my time (as you know from checking my Twitter or Facebook feed) is being spent, it seems, on making screencasts for the MATLAB class. I feel like
I’ve learned a great deal from a year’s worth of reflection on the first run of the class last spring, and it’s showing in the materials I’m producing and the work the students are giving back.
The whole idea of the inverted classroom has gotten a lot of attention in between the current version of the course and the inaugural run — the time period I think of as the “MATLAB offseason” —
through my blogging, conference talks, and everyday conversations at my work. One of my associate deans, off of whom I’ve bounced a number of ideas about this course, related a conversation he
recently had with someone about what I’m doing.
Associate Dean: So, Talbert is using this thing called the inverted classroom.
Other person: What’s that all about?
AD: He puts the lectures all online, and instead of lecturing in class he has them do group assignments on various kinds of problems.
OP: Doesn’t that double the amount of time students have to spend on the class?
I’ve never encountered that exact reaction before, although I did mention once that the biggest negative comment from students last year in the MATLAB course was that it took too much time relative
to the credit load (1 credit). I liked how my associate dean put the answer:
AD: Well… think about it this way. You are still doing both lecture and “homework”. But which part of that are going to need the most amount of help on?
OP: OK, now I get it.
Exactly. Students are going to need a lot more guidance on the difficult task of assimilating information than they will need on the relatively easy — incredibly easy, in fact — task of receiving a
transmission of information. Both phases of the game need to take place in some form, but assimilation is harder, and the probability of sinking massive amounts of time into work that goes nowhere is
a lot higher, than in transmission.
I’ve seen some great examples of where the inverted classroom method has actually saved students possibly hours of fruitless labor in the last two weeks.
Today, for instance, we were doing a lab problem set on command line plotting. In one of the tasks, students are asked to produce a 1×2 subplot illustrating the behavior of a two-parameter family of
functions. One team was stuck because their M-file wouldn’t execute properly even though their code looked correct. The problem: They used a dash (-) in the title, which causes MATLAB to think that
the stuff preceding the dash is a variable name, which wasn’t in the workspace. It’s an innocent error but not one that students with just two weeks of MATLAB under their belts could easily debug
themselves. Had they run into this problem outside of class, who knows how much time would have been wasted getting nowhere? But inside class, it was solved in the amount of time it took for them to
raise their hands and for me to come over and look.
Another example from today: A team had entered this code:
[sourcecode language="matlab"]
x = linspace(0,10);
y = 100 – exp(-2*x);
axis([0 15 90 105])
They had entered the code without line 3 already but didn’t like the look of the plot, so they added the axis command to try and change the viewing window. But nothing changed. Why? To the trained
eye, it’s simple — you have to have something plotted first before you can change the axis. So just reverse lines 3 and 4. But to the untrained eye, again, who knows how much time would be lost in
trying to figure this out? Instead I was able to instruct them directly on this, at the conceptual level (How is MATLAB thinking its way through your code?) and they got it. (It wasn’t just me
telling them, “You need to switch lines 3 and 4.”)
So above and beyond being more instructionally effective, I’m realizing — and I hope students are too — that the inverted classroom makes student time a lot more efficient, and there’s a much higher
success-to-effort ratio than in the traditional mode of teaching.
This entry was posted in Education, Educational technology, Inverted classroom, MATLAB, Teaching, Technology and tagged Classroom, Inverted classroom, Linear algebra, matlab, Source code, student,
time management. Bookmark the permalink.
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Balancing Brooms: It’s Not About the Planets | Science Blogs | WIRED
• By Rhett Allain
• 03.07.12 |
• 8:30 am |
This in’t new, but it is popular. Balancing a broom on its brushes. Cool trick, but the big problem is what people say.
“Hey, today is special because the planets are aligned and you can balance a broom!”
Well, today may indeed be special (it could be your birthday or something) but the position of the planets really have no effect on stuff. One more note. I am almost certain that others have shown
very similar calculations to what I will show. However, I just can’t remember where. If I had to guess, I would say it was Ethan at Starts With a Bang. But all of this has happened before and all of
it will happen again.
Let me start with gravity. Not your dad’s “mass times g” gravity, no the REAL stuff. Newton’s gravity (unless your dad was Newton, then these two are the same thing). Gravity is an interaction
between objects with the property mass. It is not just an interaction between things and the Earth. That just happens to be the thing with the most obvious interaction. Suppose I have two objects,
mass 1 and mass 2 that are separated by a distance r(as measured from the centers of the objects).
The magnitude of the gravitational force between these two would be:
Where M[1] and m[2] are the masses of the two objects and G is the gravitational constant with a value of 6.67 x 10^-11 N*m^2/kg^2. Yes, both masses have the same forces on them because forces are an
interaction between two objects.
Some Sample Calculations
Let me look at the broom and estimate its mass at around 1 kg. What objects could be interacting with this broom? Well, obviously the Earth. The Earth has a mass of 5.97 x 10^24 kg and the broom is
6.38 x 10^6meters from the center (the radius of the Earth). Using these values, the gravitational force on the broom from the Earth is:
You know why that looks the same as your “mass times g” formula? Because it is. Where do you think g = 9.8 N/kg comes from? Now, how about a couple of planets? Right now, Venus is fairly bright in
the night sky. But how far away is it? This is a perfect job for WolframAlpha. It says the distance to Venus is 1.292 x 10^11 meters. Since Venus has a mass of 4.87 x 10^24, this means the magnitude
of the gravitational force on the broom will be 1.94 x 10^-8 Newtons. This force is tiny compared to the gravitational force from the Earth. Why? Because the mass of Venus is around the same mass of
the Earth, but its center is WAY farther away. Ok, how about a planet with a little more mass. What about Jupiter? It has a mass of 1.90 x 10^27 kg and is currently 8.29 x 10^11 meters away. This
will create a gravitational force of 1.8 x 10^-7 Newtons – still tiny. One more object. What is the gravitational force between YOU and the broom? Let’s say you have a mass of 65 kg with a distance
of maybe 0.3 meters between your centers. This would create a gravitational force of 4.8 x 10^-8Newtons. Yes, this is also tiny. But look, the gravitational force from you is greater than the
gravitational force from Venus. So, here is your answer. How could the alignment of the planets matter when there are people around the broom that could matter almost as much (or maybe more)?
Then How Do You Balance a Broom?
It isn’t difficult. Really, there are two important things. First, the center of mass of a broom is quite low. Much closer to the ground than many people would estimate. Since the “brushes” part is
at the bottom and bigger than the handle, the center of mass is low. Here is a picture of me with my hands at the center of mass for a broom.
As a quick note, finding the center of mass for objects like broom is quite fun and simple. Here is a demo of how you can do that. What does the center of mass have to do with balancing a broom?
Well, if the center of mass for the broom is not directly over some part of the support for the broom, it will fall over. In this case, the support area of the broom is covered by the brushes. There
is another thing that is probably important. The brushes bend and act like a springy-type restoring force. This means that you don’t EXACTLY have to get the thing balanced before you let go. You just
have to be close. Let describe a similar situation. Suppose you have a completely spherical bowl turned upside down. Try to balance a marble on the top of this inverted bowl and you will find it
quite difficult. I guess theoretically, it is possible – but it will be tough. Now imagine a marble on top of an inverted bowl that looks like this:
I know, not my best drawing. Sorry, I will try better in the future. But here you can see that there are several places you can put this marble such that it will stay near the top. Of course, you
can’t put it just anywhere. The broom is sort of like this. That is why it can stay up. I guess the next thing would be for me to plot the restoring force on the broom as a function of angle. Maybe
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Torque Problem, acceration of mass. Help!
… lets say the cylinder rotates that same amount, but this time C[x] also moves, then the equation would predict the same result, but obviously it's not the same because now the cylinder is closer to
the mass...
no, because the mass has
moved, in this case the same amount as the cylinder (if the rotation is the same)
Sorry, but what is my constraint equation?
- C
= k - rθ …
a constraint equation is a geometrical equation rather than a force equation … like a body being constrained to move along a particular surface, or in this case the free length of string being
constrained to be connected to the position and rotation of the bodies
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Higher Categories and Their Applications
Posted by John Baez
The Fields Institute workshop on n-categories was a lot of fun. If you couldn’t make it, you can still see what it was like. Here’s a webpage with abstracts, transparencies and photos of lots of the
Someday soon I want to describe some of these talks — but not tonight.
Here are a couple of the photos that Dan Christensen took at the workshop.
André Joyal:
Urs Schreiber, John Baez and Toby Bartels:
There are a lot more photos in disorganized form here. If you went there and took some pictures, please pass them on to me, or at least give me links!
Posted at January 23, 2007 5:23 AM UTC
Re: Higher Categories and Their Applications
So how many of our questions got answered?
Posted by: David Corfield on January 23, 2007 12:16 PM | Permalink | Reply to this
Re: Higher Categories and Their Applications
So how many of our questions got answered?
I had long and very interesting discussions with Nils Baas, that I am very grateful for. I tried to attract Nils Baas to our first $n$-Café Millenium Prize Question. But I am not sure if I did
Posted by: urs on January 23, 2007 12:31 PM | Permalink | Reply to this
Re: Higher Categories and Their Applications
So how many of our questions got answered?
As it goes with good conferences, there is always more to talk about than time permits.
And that even holds if one is crazy enough to visit Toronto and be content with College Street being the only part of the city one visits.
Among the things that I was looking forward to talk about was Toby’s reconstruction of 2-bundles from local data.
We did find time for that, if maybe not as much as the topic deserves.
Interestingly, it turns out that, as far as I could figure out, Toby’s reconstruction does produce a $G_2$-2-bundle that is trivial as a fiber bundle of categories $B \to X$, i.e. where the category
$B$ is of the form $B = X \times F$. All the nontriviality is pushed into the $G_2$-action $B \times G_2 \to B$.
In fact, something roughly similar happens for the reconstruction of global transport 2-functors that I am working on with Konrad Waldorf.
Posted by: urs on January 23, 2007 9:10 PM | Permalink | Reply to this
Answers without Questions
So how many of our questions got answered?
One nice thing was that I received a couple of answers to questions that I did not even know I should ask! :-)
One nice insight was this:
André Joyal, in his talk, mentioned various important concepts in category theory. One of the them was related to factorization systems.
In that business, one considers collections $E$ and $M$ of morphisms that (are supposed to) behave like epi- and monomorphisms.
In particular, one considers squares $\array{ A &\stackrel{\mathrm{epi}}{\to}& B \\ \downarrow && \downarrow \\ C &\stackrel{\mathrm{mono}}{\to}& D }$ where the top morphism is an epimorphism and the
bottom morphisms is a monomorphism.
Of special interest, then, is the situation where a square of the above kind admits a diagonal $\array{ A &\stackrel{e}{\to}& B \\ \downarrow &\swarrow& \downarrow \\ C &\stackrel{m}{\to}& D } \,,$
for every choice of vertical arrows, such that everything still commutes
If that is the case, one says that the epimorphism $e$ is orthogonal to the monomorphism $m$.
There are more or less obvious generalizations of this concept for the case where everything lives in $n$-categories.
(I am grateful to Igor Bakovic for helpful discussion of this and of the related literature. In as far as I have forgotten the details again, that’s completely my own fault.)
What made all this interesting for me was that I recognized my own pet structures in this general structure that people are considering:
In my talk in Toronto I described how we can put local structure on an $n$-functor $\array{ P_n(X) \\ \;\;\downarrow \mathrm{tra} \\ T }$ by putting it into a square $\array{ P_n(U) &\stackrel{p}{\
to}& P_n(X) \\ \mathrm{tra}_U \downarrow\;\; &\Downarrow \sim& \;\;\downarrow \mathrm{tra} \\ T' &\stackrel{i}{\to}& T } \,,$ where $p$ is epi and $i$ is mono.
If $p$ happens to be “orthogonal” to $i$ in the sense of factorization system theory, i.e. if we can fill this diagram $\array{ P_n(U) &\stackrel{p}{\to}& P_n(X) \\ \mathrm{tra}_U \downarrow\;\; &\
swarrow& \;\;\downarrow \mathrm{tra} \\ T' &\stackrel{i}{\to}& T } \,,$ then this means, in the context I consider, that $\mathrm{tra}$ has the given structure not just locally, but already globally.
By itself that’s not deep or anything. But I enjoyed seeing this contact to a larger context.
Posted by: urs on January 30, 2007 6:11 PM | Permalink | Reply to this
Re: Higher Categories and Their Applications
Thanks for the report.
Will try to do something similar for
the coherent ;-) one in Paris
but I don’t know if anyone took photos.
Thanks for all those - who was the photographer?
Posted by: jim stasheff on January 30, 2007 5:02 PM | Permalink | Reply to this
Re: Higher Categories and Their Applications
I took most of the photos; Dan Christensen took a few.
Posted by: John Baez on January 30, 2007 6:35 PM | Permalink | Reply to this
Re: Higher Categories and Their Applications
For the Higher Structures … IHP fest
talks, some of which are now available,
go to
and click on Speakers (lower right side of page)
Posted by: jim stasheff on February 2, 2007 2:45 PM | Permalink | Reply to this
Dorette Pronk’s talk
I was asked for notes taken in Dorette Pronk’s talk #.
Unfortunately, I don’t have any reasonable notes to offer. Partly because I was quickly lost in the technical details.
Does anyone have notes he or she would share (say, scan in)?
Posted by: urs on February 6, 2007 1:36 PM | Permalink | Reply to this
Re: Dorette Pronk’s talk
I don’t, alas. If someone does, they should give me a copy, so I can add them to my webpage for this workshop. Right now for her talk I only have a link to the most relevant available paper of hers,
Entendues and stacks as bicategories of fractions. This does not include her new work on tricategories of fractions.
Posted by: John Baez on February 7, 2007 9:35 PM | Permalink | Reply to this
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Three more polynomial questions
May 28th 2008, 03:56 PM #1
Three more polynomial questions
hello again
"Simplify each of the following fractions. If they are already simplified say so.
1) a^2-2ab+b^2/a-b[my answer was: already simplified]
2) a^2-2ab+b^2/a+b[my answer was: already simplified]
"Some students simplify 2x^2-32/x-4 as 2x-8.
Choose values of x to show that this is not true.Then simplify it correctly."
I answered: Any value of x proves them incorrect. my simplification was:2x+8
thanks for all your help!
Do you mean $\frac{a^2 - 2ab + b^2}{a\pm b}$? If so, please mind the standard order of operations and use parentheses where appropriate to make your notation clearer.
Number one can be simplified, but you will need to place a restriction on $a \text{ and } b$. Here's a hint: factor the numerator.
$\frac{2x^2-32}{x-4} = \frac{2(x + 4)(x - 4)}{x - 4} = 2x + 8,\;xeq4$
So, your answer is correct, provided that you make the restriction that $xeq4$.
May 28th 2008, 05:52 PM #2
May 28th 2008, 06:21 PM #3
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On colouring the nodes of a network
Results 1 - 10 of 139
- in Surveys in Combinatorics 1993, London Math. Soc. Lecture Notes Series 187 , 1993
"... The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of
attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, al ..."
Cited by 76 (15 self)
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The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention.
Here we describe the techniques applied in the study of this subject, which combine combinatorial, algebraic and probabilistic methods, and discuss several intriguing conjectures and open problems.
This is mainly a survey of recent and less recent results in the area, but it contains several new results as well.
- Studia Sci. Math. Hung , 1966
"... by ..."
, 1994
"... ... for short, is one of the ~implest, most efficient, and most thoroughly studied methods for finding independent sets in graphs. We show that it surprisingly achieves a performance ratio of
(A+ 2)/3 for approximating inde-pendent sets in graphs with degree bounded by A. The analysis directs us tow ..."
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... for short, is one of the ~implest, most efficient, and most thoroughly studied methods for finding independent sets in graphs. We show that it surprisingly achieves a performance ratio of (A+ 2)/
3 for approximating inde-pendent sets in graphs with degree bounded by A. The analysis directs us towards a simple parallel and dis-tributed algorithm with identical performance, which on
constant-degree graphs runs in O(log ” n) time us-ing linear number of processors. We also analyze the Greedy algorithm when run in combination with a frac-tional relaxation technique of Nemhauser
and Trotter, and obtain an improved (2Z + 3)/5 performance ratio on graphs with average degree ~. Finally, we introduce a generally applicable technique for improving the approximation ratios of
independent set algorithms, and illustrate it by improving the per-formance ratio of Greedy for large ∆.
- In Proc. of Workshop on Algorithms and Data Structures , 1995
"... . The main problem we consider in this paper is the Independent Set problem for bounded degree graphs. It is shown that the problem remains MAX SNP--complete when the maximum degree is bounded
by 3. Some related problems are also shown to be MAX SNP--complete at the lowest possible degree bounds. N ..."
Cited by 39 (0 self)
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. The main problem we consider in this paper is the Independent Set problem for bounded degree graphs. It is shown that the problem remains MAX SNP--complete when the maximum degree is bounded by 3.
Some related problems are also shown to be MAX SNP--complete at the lowest possible degree bounds. Next we study better poly--time approximation of the problem for degree 3 graphs, and improve the
previously best ratio, 5 4 , to arbitrarily close to 6 5 . This result also provides improved poly--time approximation ratios, B+3 5 + ffl, for odd degree B. 1 Introduction The area of efficient
approximation algorithms for NP--hard optimization problems has recently seen dramatic progress with a sequence of breakthrough achievements. Even when restricted only to the area of constant bound
approximation the following remarkable results have been obtained in the last few years. The subclass of NP optimization problems, called MAX SNP, consisting solely of constant ratio approximable
problems ...
, 2000
"... Two natural classes of counting problems that are interreducible under approximation-preserving reductions are: (i) those that admit a particular kind of ecient approximation algorithm known as
an \FPRAS," and (ii) those that are complete for #P with respect to approximationpreserving reducibili ..."
Cited by 36 (12 self)
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Two natural classes of counting problems that are interreducible under approximation-preserving reductions are: (i) those that admit a particular kind of ecient approximation algorithm known as an \
FPRAS," and (ii) those that are complete for #P with respect to approximationpreserving reducibility. We describe and investigate not only these two classes but also a third class, of intermediate
complexity, that is not known to be identical to (i) or (ii). The third class can be characterised as the hardest problems in a logically dened subclass of #P. Research Report 370, Department of
Computer Science, University of Warwick, Coventry CV4 7AL, UK. This work was supported in part by the EPSRC Research Grant \Sharper Analysis of Randomised Algorithms: a Computational Approach" and by
the ESPRIT Projects RAND-APX and ALCOM-FT. y dyer@scs.leeds.ac.uk, School of Computer Studies, University of Leeds, Leeds LS2 9JT, United Kingdom. z leslie@dcs.warwick.ac.uk, http://www.dcs.warw...
- Combinatorics, Probability and Computing , 1995
"... Let G be a graph with maximum degree ∆(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfies χ(G) ≤ (1 + o(1)) ∆(G) log ∆(G) where o
(1) goes to zero as ∆(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same ..."
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Let G be a graph with maximum degree ∆(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfies χ(G) ≤ (1 + o(1)) ∆(G) log ∆(G) where o(1)
goes to zero as ∆(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same bound for the list-chromatic (or choice) number: provided g(G)> 4. 1
- In Proceedings of the 28th Annual ACM Symposium on Theory of Computing , 1996
"... In a seminal paper Leighton, Maggs, and Rao consider the packet scheduling problem when a single packet has to traverse each path. They show that there exists a schedule where each packet
reaches its destination in O(C + D) steps, where C is the congestion and D is the dilation. The proof relies o ..."
Cited by 34 (2 self)
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In a seminal paper Leighton, Maggs, and Rao consider the packet scheduling problem when a single packet has to traverse each path. They show that there exists a schedule where each packet reaches its
destination in O(C + D) steps, where C is the congestion and D is the dilation. The proof relies on the Lov'asz Local Lemma, and hence is not algorithmic. In a followup paper Leighton and Maggs use
an algorithmic version of the Local Lemma due to Beck to give centralized algorithms for the problem. Leighton, Maggs, and Rao also give a distributed randomized algorithm where all packets reach
their destinations with high probability in O(C +D log n) steps. In this paper we develop techniques to guarantee the high probability of delivering packets without resorting to the Lov'asz Local
Lemma. We improve the distributed algorithm for problems with relatively high dilation to O(C) + (log n) O(log n) D + poly(log n). We extend the techniques to handle the case of infinite streams of
, 1998
"... It is shown that the chromatic number of any graph with maximum degree d in which the number of edges in the induced subgraph on the set of all neighbors of any vertex does not exceed d²/f is at
most O(d/log f). This is tight (up to a constant factor) for all admissible values of d and f. ..."
Cited by 32 (17 self)
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It is shown that the chromatic number of any graph with maximum degree d in which the number of edges in the induced subgraph on the set of all neighbors of any vertex does not exceed d²/f is at most
O(d/log f). This is tight (up to a constant factor) for all admissible values of d and f.
- in Probabilistic Methods for Algorithmic Discrete Mathematics , 1998
"... 7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this
chapter. 1. Introduction The classical Monte Carlo method is an approach to estimating quantities that a ..."
Cited by 30 (1 self)
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7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this
chapter. 1. Introduction The classical Monte Carlo method is an approach to estimating quantities that are hard to compute exactly. The quantity z of interest is expressed as the expectation z = ExpZ
of a random variable (r.v.) Z for which some efficient sampling procedure is available. By taking the mean of some sufficiently large set of independent samples of Z, one may obtain an approximation
to z. For example, suppose S = \Phi (x; y) 2 [0; 1] 2 : p i (x; y) 0; for all i \Psi<F12
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Number of views: 1796 Article added: 2 February 2011 Article last modified: 9 February 2011 © Copyright 2010-2014 Back to top
Welcome to the Thermopedia A-to-Z Index!
This Index is organized alphabetically by subject areas and key words. Click on an item to the left in order to see the information associated with that item.
Each item will contain one of the following:
• A complete article
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• If there is noarticle associated with an item, a link is provided for more information
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You can also search Thermopedia by entering a term in the Search box above.
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word problems
November 12th 2012, 01:05 PM #1
Nov 2012
word problems
1. A steel drum in the shape of a right circular cylinder isrequired to have a volume of 100 cubic feet.
a. Expressthe amount A of material required to make the drum as a function of the radiosr of the cylinder.
b. How muchmaterial is required if the drums radius is 3 feet?
c. How muchmaterial is required if the drums radius is 5 feet?
d. Graph andidentify the radius that would require the least amount of material.
2. Suppose the population of species of fish (in thousands)is modeled by F(x)=x+10/0.5x^2+1, where x≥0is in years.
a. findthe equation of the horizontal asympytote.
b. Determinethe initial population,
c. Whathappens to the population of the fish as the number of years increase?
d. Interpretthe horizontal asymptote in terms of the fish population.
3. The Doppler effect is the change in pitch (frequency) ofsound from a source (s) as heard by an observer (o) when one or both are inmotion. If we assume both the source and the observer are
moving in the samedirection, the relationship is f^(')=f[a]((v-v[o])/(v-v[s]))
Where: f^(‘)=perceivedpitch by the observer
F[a]=actualpitch of the source
V=speedof sound in air (assume 772.4 mph)
V[o]=speedof observer
V[s]=speedof source
Suppose that you are traveling down the road at 45mph andyou hear an ambulance (with siren) coming toward you from the rear. The actualpitch of the siren is 600 hertz (Hz).
A. Writea function f^(‘)(v[s]) that describes this scenario.
B. Iff^(‘)=620 Hz find the speed of the ambulance.
C. Graphthe function and determine using the trace function what is the pitch perceivedby the observer when the speed of the source is 80mph.
Re: word problems
so ... what have you done to solve any part of these three problems?
November 12th 2012, 02:57 PM #2
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Dan Meyer
#NCTMNOLA is almost on us and it remains to be seen how much recapping will happen. I’ve posted an excerpt of my own schedule on my personal blog and I intend to recap a lot of it. I have conflicts
with great sessions in every slot, though, so it’d be great to get some of you to volunteer to recap even one session for this site.
Leave your info in the volunteer page if you’d like to help out and I’ll send along some brief instructions. Thanks, everybody, and safe travels to NOLA.
[Avery Pickford] Proof Doesn’t Begin With Geometry
Avery wins the prize for Best Session Description by sneaking in the totally droll line, “All hail CCSSM MP3.”
He briefly dinged two-column proofs, the Disaster Island where we usually sequester conjecture and argument. I’m all for broading and deepening the definition of proof but I think Avery stretched it
too far to include skills like estimation and asking us to justify our answers to the Locker Problem. Is “justify your answer” any different than “prove your conjecture”?
Later, he got into territory I found challenging and really well-thought-out. He showed how simple puzzles like Shikaku could be used to introduce the difference between axioms and theorems. He
showed a Shikaku puzzle and its answer (below) and asked us, “What are the rules here?”
“The numbers define the area of a rectangle” and “the side lengths of those rectangles are integers” are axioms, without which the game wouldn’t make any sense. Theorems are the consequences of the
axioms, like “Prime-numbered areas result in long, skinny rectangles with side-length 1.”
He also used a variation on the old game Mastermind to create a class-wide context for conjecturing, contradicting, and proving. Great stuff. Wish you were here.
[Michael Serra] Polygon Potpourri
Five interesting investigations with polygons [pdf]. Michael spent ten minutes prefacing the set, then let us investigate them for twenty minutes, and then asked a volunteer to debrief each one at
the end.
If nothing else, it was a nice morning moment to talk about math with Internet friends. That was enough. But I’ve been struck also by how hard it is to make a given math concept more challenging for
students and more interesting at the same time. We use bigger numbers. We mix in fractions and decimals. We lengthen the problem set. We time it.
For instance, once students understand how to find the sum of the interior angles of a polygon, it’s like, what do you do to make this more challenging and more interesting?
Michael introduced donut polygons:
Finding the interior angle sum of a donut polygon makes the original task more challenging and more interesting at the same time. In particular, it has a great stinger at the end when you find out
whether or not a triangle inside of a pentagon has the same angle sum as a pentagon inside of a triangle.
Michael had two questions at the end that asked, basically, “Do your conclusions hold if there’s a dent in the polygon?” Then, “What about two dents in the polygon?” This messed me up a little bit,
because, no, it shouldn’t matter, but then why would Serra include the two questions? Basically, Serra had your correspondent feeling briefly but completely off balance.
[Allan Bellman] Manipulatives vs. Technology: Bring Your Bias and an Open Mind
Great premise for a session:
Pose two lesson objectives. For instance:
1. Students will be able to understand why the angles in a triangle always add to 180 degrees.
2. Students will be able to understand how to calculate the shortest distance from a point to a line and then back to another point.
Allan then brought any resource you’d want, from low-tech to high-tech, everything from tracing paper and scissors to a class set of TI-Nspire’s. We used what we wanted to explore those objectives
and then debated the merits of the analog and digital technologies.
The debate hit a high register pretty quickly. For the 180 degrees question, people tended to favor the diverse ways you could demonstrate it with paper. (Cutting, tracing, drawing, etc.) With the
NSpire, you had one. You downloaded an applet from Allan and moved vertices around, watching the angle sum stay constant.
For the shortest path problem, most people preferred the calculator because you were able to set up the constraints and then drag a point around to see where the distance bottomed out. The only
analog method we discussed was to draw the scenario and then use string to measure the different possibilities, slowly narrowing in on the answer.
For my part, I was bothered that we never discussed a) any other digital technology, aside from Texas Instruments calculators, or b) the cost of a class set of any kind of technology.
Granted, I probably make sport of Texas Instruments too much (and I’m hardly unbiased here) but truly I just find the user experience miserable. From the untouchable, low-DPI screens, to the time it
takes me to find the right button out of the millions, to the mindless, button-pushing worksheets teachers have to pass out just to make the devices comprehensible, to the lurching way the cursor
moves across the screen in Cabri, I find the whole experience pretty painful.
It would have been great to see the same problems approached with Geogebra or Desmos, for instance. Or even an Excel spreadsheet.
Then there’s the cost. All other things held equal, the high-tech solution will still cost thousands of dollars more per classroom. So we shouldn’t be talking about which solution comes out barely
ahead of the other. Technology should shoulder the greater burden of proof here.
[Breedeen Murray] Telling Stories, Teaching Math
Bree claimed that stories are a useful medium for student learning. She backed this up with lots of citations that I hope she’ll share somewhere. [Update: She has.] She posed ideas for filtering our
own lessons through the logic of stories – setting context, adding conflicts, etc. She closed by asking us try to create story-based lessons for exponentials and fraction division.
My thoughts went to St. Matthew Island, which I’ll link without elaboration.
[Robert Kaplinsky] Real-World Problem-Based Learning Using Perplexing Tasks
Robert showed us this image and asked us to figure out how much it cost.
I’ve seen his lesson plan before but it didn’t prepare me for how interesting the math became.
We used the In-N-Out prices for a hamburger, a cheeseburger (a hamburger + cheese), and a Double-Double (a cheeseburger twice over) to figure out the cost of the 100×100 burger.
Our calculation was exactly right but we arrived at it differently. I used a system of three linear equations. My seatmates used a bit more conceptual creativity and got the same answer with a lot
less computation. Robert highlighted all of these methods.
My takeaway: it’s really, really hard to describe in a text-based lesson plan all the awesome, heady moments it may provoke. I knew the lesson would be fun and productive. I didn’t know it would be
this fun and productive. How do we create lesson plans that convey all those highs, rather than just the nuts and bolts of their implementation?
[Uri Treisman] Keeping Our Eyes on the Prize
On April 19, 2013, the third day of NCTM’s annual meeting in Denver, Uri Treisman gave a forty-minute address on equity that Zal Usiskin, director of the University of Chicago’s School Mathematics
Project, called the greatest talk he’d ever heard at the conference in any year. Stanford math professor Keith Devlin would later call it our “I have a dream” speech. At least one participant left in
I’ve personally seen it three times. I got the video feed from NCTM and the slides from Treisman. I then spent some time stitching the two together, resulting in this video. His message is important
enough that I’d like to use whatever technical skills I have, whatever time I have, whatever soapbox I can stand on, to help spread it.
If you’re interested in equity, you should watch it.
If you’re interested in teacher evaluation, you should watch it.
If you’re interested in school reform, you should watch it.
If you’re interested in charter schools, you should watch it.
If you’re interested in understanding which student outcomes teachers can control and which they can’t, you should watch it.
If you’re interested in the trajectory of math education in the era of the Common Core State Standards, you should watch it.
If none of those conditions apply to you, well, I can’t imagine the series of misclicks that brought you to my blog. Watch it.
Here’s a fair enough summary from Treisman himself:
There are two factors that shape inequality in this country and educational achievement inequality. The big one is poverty. But a really big one is opportunity to learn. As citizens, we need to
work on poverty and income inequality or our democracy is threatened. As mathematics educators … we need to work on opportunity to learn. It cannot be that the accident of where a child lives or
the particulars of their birth determine their mathematics education.
That was his destination and the talk took only three stops along the way:
1. What did education reform groups like Achieve, the Gates Foundation, et al, recommend in their “Benchmarking for Success” document in 2008?
2. How does TIMSS and NAEP data contradict or clarify those recommendations?
3. What should we actually do about equity, as teachers and citizens, if those recommendations prove unfounded?
Highly Quotable
• [A]s math people we know that if we’re going to work on a problem, we have to formulate it clearly. And as math people are wont, we need to swaddle ourselves in the numbers and the data because
that’s what gives math people direction, strength, and courage.
• Let’s look at “Benchmarking for Success” and see its analysis of the problem. Then let’s look at the data and see how it actually lines up with what we know today. And then let’s see where we
need to go to really enact the vision of NCTM for equity.
• So the notion was: “Let’s focus on teachers as the central driver of reform and rethink how we evaluate teachers.” They had the view that teachers were the single most important in-school factor
in student achievement. And math people know that was just an artifact of the way they modeled the problem.
• I’m now going to show you two graphs that I don’t believe anyone in the math community has seen. It’s the PISA data disaggregated by child poverty rates.
• About one half of students who go from high school to college are referred to remediation and mostly developmental math. Fewer than a quarter of those students will ever get a credential. Those
students are more likely to end up with debt than a credential. [..] Those remedial programs are burial grounds for the aspirations of students. And it’s mostly math that’s the key trigger.
35,000 students in California two years ago repeated a developmental course for the fifth or greater number of times. So no one can say those students don’t have persistence.
• So states – where you go to school – are a profound influence on what you actually get to know.
• Low income student scores in Texas were the top in the country in 2011. It’s really good for Texas to be the top of the country. Because whenever Texas does something well, everyone else is
positive that they can do better. When Massachusetts is at the top, people go, “Ah, it’s just Massachusetts.”
• Again, two and a half years difference in opportunity depending on where you happen to go to school. This is something that, as a math teaching profession, we can influence. Poverty is something
we need to work on as citizens. Opportunity to learn is something we need to work on as math educators. That’s a core message for this talk.
• So you would think that charters would fix this. Almost all the charters in Texas produced 0% of students who are college-ready. There are a few of them – one KIPP, one YES Prep, one IDEA, one
Harmony – that are pretty good. Most of them are well below the public schools. So this theory of Achieve, NGA, CCSSO, Race to the Top, that charters were the answer? Not so clear when you
actually climb into the numbers. The reverse looks true.
• When you visit most math classrooms it’s like you’re in a Kafkaesque universe of these degraded social worlds where children are filling in bubbles rather than connecting the dots. It’s driven by
a compliance mentality on tests that are neither worthy of our children nor worthy of the discipline they purport to reflect. That is the reality. That’s something that we as math educators can
• What this shows is that the current theory about school improvement – that charters, Common Core, value-added measures of teaching are going to solve the problem – is profoundly wrong. That
doesn’t mean we can’t use the Common Core powerfully to reboot our systems but it’s not the solution to the basic problems of schooling.
• Guess what? Poverty really sucks. It’s incredibly hard. All the lifespan studies going back to the 1920s show that poverty and youth is a very hard force. We need to build fault-tolerant schools
and systems if we’re actually going to address equity.
• Just think about it. The great majority of our children finish our schools positive that there’s a whole list of things they’re not. They come out of schooling believing they’re not mathematical,
they’re not artistic, they’re not philosophical, they’re not athletic. And these self-imposed beliefs undermine your sense of personal freedom, the font from which all freedoms come.
• You have to remember that when the Common Core was created, they didn’t come to NCTM. They got David Coleman to write it and he brought his friend Jason Zimba to do the math. They did not come to
NCTM. It’s time for us now – the professional societies – to talk about what standards should be and how to reshape the Common Core so that it reflects our best practice knowledge of schooling.
Hard message, but a necessary message.
• What is the determinant of whether you have a high-skill job in the US? Overwhelmingly it’s mathematics. It’s the single biggest factor in upward social and economic mobility. It’s our beloved
subject. It would be wonderful if it were music instead of math. Think how great the country would be if everyone were striving to learn to play an instrument instead of factor quadratic
equations but the fact is it is our discipline that is the primary determinant.
Dan Meyer is the editor of MathRecap.com. He blogs at dy/dan and tweets @ddmeyer.
[Help Wanted] Recappers Needed For NCTM13
NCTM 2013 is on us in two weeks.
1. It’s a banner year for speakers. I’ll post a few recommendations shortly but you have all the usual institutions plus a few new upstarts from the blogosphere. I’m looking forward to it.
2. It’s a very expensive ticket. No two ways about that.
So if you’ll be attending NCTM, consider recapping a session or two here at MathRecap. A photo and a few paragraphs is all it takes to open the conference up to the 99% of math teachers worldwide who
can’t attend. Leave your details at the volunteer page if you’d like to help them out.
Recaps Around the Web
• Henri Piccioto recaps Avery Pickford’s session on student-posed problems at CMC-North 2012. (Part one & part two.)
• Tom Ward recaps the best of NCTM 2012 in Chicago, with posts on sessions by Sarah Greenwald, Adam Poetzel, Ron Lancaster, and Jen Szydlik.
[Michael Shaughnessy] Reasoning and Sense-Making: Keys to Student Engagement
Shaughnessy’s morning keynote covered Reasoning and Sensemaking (RSM). He spent the first ten minutes reviewing the different resources NCTM offers. Then he illustrated RSM with four tasks he had us
work on and debrief. You can find all the problems in his slidedeck but the fourth one interests me most.
Problem 4
A “data detective” exploration. What do you notice in these tables? What do you wonder about?
As we noticed sums between rows and columns and speculated what they could be describing, he progressively added column and row headings, which in turn sent our reasoning down new, different paths.
His point was that RSM can and does occur in “naked number” contexts.
• Shaughnessy’s slidedeck [pdf]
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[Numpy-discussion] Determine if numpy is installed from an extension
[Numpy-discussion] Determine if numpy is installed from an extension
David Cournapeau cournape@gmail....
Wed Feb 3 03:41:54 CST 2010
On Wed, Feb 3, 2010 at 5:38 PM, Peter Notebaert <peno@telenet.be> wrote:
> >From an extension? How to import numpy from there and then test if that
> succeeded and that without any annoying message if possible...
One obvious solution would be to simply call PyImport_Import, something like:
#include <Python.h>
PyMODINIT_FUNC initfoo(void)
PyObject *m, *mod;
m = Py_InitModule("foo", NULL);
if (m == NULL) {
mod = PyImport_ImportModule("numpy");
if (mod == NULL) {
But I am not sure whether it would cause some issues if you do this
and then import the numpy C API (which is mandatory before using any C
functions from numpy). I know the python import system has some dark
areas, I don't know if that's one of them or not.
More information about the NumPy-Discussion mailing list
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Geometry in Art & Architecture Unit 6
The PLATONIC SOLIDS
"Let no one destitute of geometry enter my doors."
Plato (c. 427 - 347 B.C.E.)
Slide 6-1: RAPHAEL: School of Athens
American Catalgo, p. 126, #21061. Fresco, Vatican, Stanza della Signurata, the Pope's Private Library
We now move forward in time about 150 years, still staying in Greece, from Pythagoras to Plato, himself a Pythagorean.
In our last unit we studied some polygons, and I said that one of these, the triangle, was thought by Plato to be the building block of the universe. He presented that idea and others about creation,
such as the universe being created to resemble a geometric progression, in one of his books, the Timaeus.
In the Timaeus, we'll see how Plato describes how triangles make up five solids, now called the Platonic Solids, and how these solids make up the four elements and heaven. We'll look at regular
polyhedra in general, and see why only five are possible.
Finally we'll see how the Platonic solids were used as art motifs even before Plato, how they were used later, and how the served to tie together three Renaissance mathematicians and artists, Piero
della Francesca, Luca Pacioli, and Leonardo da Vinci.
Outline: Plato
The Timaeus
Music of the Spheres
The Elements
The Platonic Solids
Polyhedra in Art & Architecture
Slide 6-2: RAPHAEL: School of Athens. Center section
Profile: Plato (c.427-347 B.C.E.) was born to an aristocratic family in Athens. As a young man Plato had political ambitions, but he became disillusioned by the political leadership in Athens. He
eventually became a disciple of Socrates, accepting his basic philosophy and dialectical style of debate, the pursuit of truth through questions, answers, and additional questions. Plato witnessed
the death of Socrates at the hands of the Athenian democracy in 399 BC. In Raphael's School of Athens we see Socrates prone, with cup nearby.
Plato's most prominent student was Aristotle, shown here with Plato in Raphael's School of Athens, Aristotle holiding his Ethics and Plato with his Timaeus.
Plato's Academy
In 387 BCE Plato founded an Academy in Athens, often described as the first university. It provided a comprehensive curriculum, including astronomy, biology, mathematics, political theory, and
Plato's final years were spent lecturing at his Academy and writing. He died at about the age of 80 in Athens in 348 or 347.
Over the doors to his academy were the words
meaning, "Let no one destitute of geometry enter my doors."
Plato on Art and Geometry
Although Plato loved geometry, he would not have been good at teaching a course in Art & Geometry because he had a low opinion of art. He taught that, since the world is a copy or image of the real,
then a work of art is a copy of a copy, at third remove from reality.
He writes in his Republic(p. 603),
"... painting [and] ... the whole art of imitation is busy about a work which is far removed from the truth; ... and is its mistress and friend for no wholesome or true purpose. ... [it] is the
worthless mistress of a worthless friend, and the parent of a worthless progeny."
But on geometry he wrote in his Republic (p. 527),
"[Geometry is] . . . persued for the sake of the knowledge of what eternally exists, and not of what comes for a moment into existence, and then perishes, ...
[it] must draw the soul towards truth and give the finishing touch to the philosophic spirit."
The Timaeus
Plato left lots of writings. We've mentioned his Republic in our unit on number symbolism in which he gave the four cardinal virtues, but his love of geometry is especially evident in the Timaeus.
Written towards the end of Plato's life, c. 355 BCE, the Timaeus describes a conversation between Socrates, Plato's teacher, Critias, Plato's great grandfather, Hermocrates, a Sicilian statesman and
soldier, and Timaeus, Pythagorean, philosopher, scientist, general, contemporary of Plato, and the inventor of the pulley. He was the first to distinguish between harmonic, arithmetic, and geometric
In this book, Timaeus does most the talking, with much homage to Pythagoras and echos of the harmony of the spheres, as he describes the geometric creation of the world.
Music of the Spheres
Plato, through Timaeus, says that the creator made the world soul out of various ingredients, and formed it into a long strip. The strip was then marked out into intervals.
First [the creator] took one portion from the whole (1 unit)
and next a portion double the first (2 unit)
a third portion half again as much as the second (3 unit)
the fourth portion double the second (4 unit)
the fifth three times the third (9 unit)
the sixth eight times the first (8 unit)
and the seventh 27 tmes the first (27 unit)
They give the seven integers; 1, 2, 3, 4, 8, 9, 27. These contain the monad, source of all numbers, the first even and first odd, and their squares and cubes.
Plato's Lambda
Slide 8-72 : Arithmetic Personified as a Woman
Lawlor, Robert. Sacred Geometry. NY: Thames & Hudson, 1982. p. 7.
These seven numbers can be arranged as two progressions
Monad 1 Point
First even and odd 2 3 Line
Squares 4 9 Plane
Cubes 8 27 Solid
This is called Plato's Lambda, because it is shaped like the Greek letter lambda.
Divisions of the World Soul as Musical Intervals
Relating this to music, if we start at low C and lay off these intervals, we get 4 octaves plus a sixth. It doesn't yet look like a musical scale. But Plato goes on to fill in each interval with an
arithmetic mean and a harmonic mean. Taking the first interval, from 1 to 2, for example,
Arithmetic mean = (1+2)/2 = 3/2
The Harmonic mean of two numbers is the reciprocal of the arithmetic mean of their reciprocals.
For 1 and 2, the reciprocals are 1 and 1/2, whose arithmetic mean is 1+ 1/2 ÷ 2 or 3/4. Thus,
Harmonic mean = 4/3
Thus we get the fourth or 4/3, and the fifth or 3/2, the same intervals found pleasing by the Pythagoreans. Further, they are made up of the first four numbers 1, 2, 3, 4 of the tetractys.
Filling in the Gaps
He took the interval between the fourth and the fifth as a full tone. It is
3/2 ÷ 4/3 = 3/2 x 3/4 = 9/8
Plato then has his creator fill up the scale with intervals of 9/8, the tone. This leaves intervals of 256/243 as remainders, equal to the half tone.
Thus Plato has constructed the scale from arithmetic calculations alone, and not by experimenting with stretched strings to find out what sounded best, as did the Pythagoreans.
Project: Repeat Plato's calculations and see if you do indeed get a musical scale.
Forming the Celestial Circles
After marking the strip into these intervals, the creator then cut it lengthwise into two strips that are placed at an angle to each other and formed into circles. These correspond to the celestial
equator and the ecliptic, the start of an armillary sphere.
Slide 10-121: Armillary Sphere
Turner, Gerard. Antique Scientific Instruments. Dorset: Blandford, 1980. p. 61
Recall our quotation from Plato's Republic, where, in the Myth of Er he wrote,
". . . Upon each of its circles stood a siren who was carried round with its movements, uttering the concords of a single scale." [Republic p. 354]
This is the origin of the expression, Music of the Spheres.
The Elements
The idea tthat all things are composed of four primal elements: earth, air, fire, and water, is attributed to Empedocles (circa 493-433 BCE), Greek philosopher, statesman, and poet. He was born in
Agrigentum (now Agrigento), Sicily, and was a disciple of Pythagoras and Parmenides.
Remember the opposite forces, Yin and Yang, male and female, whose interaction created everything in the universe? Empedocles thought that active and opposing forces, love and hate, or affinity and
antipathy, act upon these elements, combining and separating them into infinitely varied forms.
He believed also that no change involving the creation of new matter is possible; only changes in the combinations of the four existing elements may occur.
Empedocles died about 6 years before Plato was born.
The Universe as a Geometric Progression
Plato deduces the need for the four elements. Timaeus, 31B-32C
1. First, we must have fire, to make the world visible, and earth to make it resistant to touch. These are the two extreme elements, fire belonging to heaven and earth to earth. He writes,
. . . it is necessary that nature should be visible and tangible ...
and nothing can be visible without fire or tangible without earth ...
2. But two cannot hold together without a third as a bond. [like glue]
. . . But it is impossible for two things to cohere without the intervention of a third ...
3. And the most perfect bond is the connued geometric proportion.
... [and] the most beautiful analogy is when in three numbers,
the middle is to the last as the first to the middle, . . . they become the same as to relation to each other.
4. But the primary bodies are solids, and must be represented by solid numbers (cubes).
To connect two plane numbers (squares) one mean is enough,
but to connect two solid numbers, two means are needed.
But if the universe were to have no depth, one medium would suffice to bind all the natures it contains. But . . . the world should be a solid, and solids are never harmonized by one, but always by
two mediums.
Hence the Divinity placed water and air in the middle of fire and earth, fabricating them in the same ratio to each other; so that fire might be to air as air is to water and that water is to earth.
fire/air = air/water = water/earth
Thus the ratio is constant between successive elements, giving a geometric progression.
The Platonic Solids
The Platonic Solids belong to the group of geometric figures called polyhedra.
A polyhedron is a solid bounded by plane polygons. The polygons are called faces; they intersect in edges, the points where three or more edges intersect are called vertices.
A regular polyhedron is one whose faces are identical regular polygons. Only five regular solids are possible
cube tetrahedron octahedron icosahedron dodecahedron
These have come to be known as the Platonic Solids
The Elements Linked to the Platonic Solids
Plato associates four of the Platonic Solid with the four elements. He writes,
We must proceed to distribute the figures [the solids] we have just described between fire, earth, water, and air. . .
Let us assign the cube to earth, for it is the most immobile of the four bodies and most retentive of shape
the least mobile of the remaining figures (icosahedron) to water
the most mobile (tetrahedron) to fire
the intermediate (octahedron) to air
Note that earth is associated with the cube, with its six square faces. This lent support to the notion of the foursquaredness of the earth.
The Cosmos
But there are five regular polyhedra and only four elements. Plato writes,
"There still remained a fifth construction, which the god used for embroidering the constellations on the whole heaven."
Plato's statement is vague, and he gives no further explanation. Later Greek philosophers assign the dodecahedron to the ether or heaven or the cosmos.
The dodecahedron has 12 faces, and our number symbolism associates 12 with the zodiac.
This might be Plato's meaning when he writes of "embroidering the constellations" on the dodecahedron.
Note that the 12 faces of the dodecahedron are pentagons. Recall that the pentagon contains the golden ratio. Perhaps this has something to do with equating this figure with the cosmos.
Other Polyhedra
The Archimedian Solids
Slide 6-4: Archimedian Solids
Wenniger, Magnus J. Polyhedron Models for the Classroom. NCTM 1966. p. 7
Other sets of solids can be obtained from the Platonic Solids. We can get a set by cutting off the corners of the Platonic solids and get truncated polyhedra.
They are no longer regular; they are called semi -regular; all faces are regular polygons, but there is more than one polygon in a particular solid, and all vertices are identical.
These are also called the Archimedian Solids, named for Archimedes, (287-212) the Greek mathematician who lived in Syracusa on the lower right corner of Sicily.
Mini-Project: Make some Archimedian Solids.
Star Polyhedra
Slide 6-5: The Four Kepler-Poinsot Solids
Wenniger, Magnus J. Polyhedron Models for the Classroom. NCTM 1966. p. 11
Slide 6-6: Engraving from Harmonices Mundi, 1619.
Emmer, Michele, Ed. The Visual Mind: Art and Mathematics. Cambridge: MIT Press, 1993. p. 218
The second obvious way to get another set of solids is to extend the faces of each to form a star, giving the so-called Star Polyhedra.
Two star polyhedra were discovered by Poinsot in 1809. The others were discovered about 200 years before that by Johannes Kepler (1571-1630), the German astronomer and natural philosopher noted for
formulating the three laws of planetary motion, now known as Kepler's laws, including the law that celestial bodies have elliptical, not circular orbits.
Mini-Project: Make some star polyhedra.
Polyhedra in Art & Architecture
Polyhedra are Nothing New
Polyhedra have served as art motifs from prehistoric times right up to the present.
Slide 6-7: Pyramids
Tompkins, Peter. Secrets of the Great Pyramid. NY: Harper, 1971. Cover
The Egyptians, of course, knew of the tetrahedron, but also the octahedron, and cube. And there are icosahedral dice from the Ptolomaic dynasty in the British Museum, London.
Slide 6-8: Etruscan Dodecahedron
Emmer, Michele, Ed. The Visual Mind: Art and Mathematics. Cambridge: MIT Press, 1993. p. 216
There were Neolithic solids found in Scotland, and excavations near Padova have unearthed an Etruscan dodecahedron, c. 500 BCE, probably used as a toy.
Slide 3-6: Kepler's Model of Universe
Lawlor, p. 106
In 1596 Kepler published a tract called The Cosmic Mystery in which he envisioned the universe as consisting of nested Platonic Solids whose inscribed spheres determine the orbits of the planets, all
enclosed in a sphere representing the outer heaven. Of course, his observations did not fit this scheme. We'll encounter Kepler again in our unit on Celestial Themes in Art.
Polyhedra and Plagiarism in the Renaissance
Slide 14-10 : JACOPO DE 'BARBERI: Luca Pacioli, c. 1499
This painting shows Fra Luca Pacioli and his student, Guidobaldo, Duke of Urbino. In the upper left is a rhombi-cuboctahedron, and on the table is a dodecahedron on top of a copy of Euclid's
Slide 15-11 : Leonardo's Illustrations for Luca's book.
Da Divina Proportione
Luca Pacioli wrote a book called Da Divina Proportione (1509) which contained a section on the Platonic Solids and other solids, which has 60 plates of solids by none other than his student Leonardo
da Vinci. We'll tell the whole story of how this material was stolen from Luca's teacher Piero della Francesca in our unit on Polyhedra and Plagiarism in the Renaissance.
Platonic Solids as Art Motifs
Slide 6-12: UCELLO: Mosaic from San Marco Cathedral, Venice,1425-1430 plate J2
Emmer, Michele, Ed. The Visual Mind: Art and Mathematics. Cambridge: MIT Press, 1993.
Slide 16-08: DURER: Melancolia I, 1514
Albrech Durer (1471-1528) had a keen interest in geometry, as we'll see in a later unit. This famous engraving shows an irregular polyhedron, as well as a sphere, a magic square, and compasses.
People who have analyzed this polyhedron have decided that its actually a cube with opposite corners cut off.
Slide 6-13: NEUFCHATEL, Nicolaus: Picture of Johannes Neudorfer and His Son,1561.
Kemp, Martin. Leonardo on Painting. New Haven: Yale U. Press, 1989. p. 63
Slide 6-14: Gold-plated lion from the front of the Gate of Heavenly Purity, Closeup of Ball.
Forbidden City, Beijing. From Qing Dynasty (1736-1796)
This ball has hexagons interspersed with pentagons.
Polyhedra in Art in the Twentieth Century
Slides 6-15, 6-16, 6-17: Giacometti's Works
Hohl, Reinhold. Alberto Giacometti. NY: Abrams, 1972.
The Swiss artist Alberto Giacometti (1901-1966) often included polyhedra in his earlier surrealist works such as these two drawings and a sculpture.
Slide 21-5: ESCHER:Stars
1948 (#123)
We'll talk about M.C. Escher (1902-1972) in detail when we get to the 20th Century, but lets just peek at his 1948 engraving, Stars. Note the similarity between this polyhedron and Leonardo's
illustrations for Pacioli's book.
Slide 21-06: Escher contemplating his nested set of Platonic Solids
Escher made a set of nested Platonic Solids. When he moved to a new studio he have away most of his belongings but took his beloved model.
Other Twentieth Century artists using polyhedra include Harriet Brisson, Paul Calter, and Lucio Saffaro.
Slide 6-18:Truncated Close-Packing Octahedra, Rhombidodecahedra, and Cubes. Plexiglass, aluminum tubes, and nylon cord, 1976
Emmer, Michele, Ed. The Visual Mind: Art and Mathematics. Cambridge: MIT Press, 1993. plate B3
Slides 6-19, 6-20, 6-21: Platonic Solids
Calter's Sorcerer's Circle
Slide 6-22: LUCIO SAFFARO: Platonic Forms. Computer graphic, 1989.
Emmer, Michele, Ed. The Visual Mind: Art and Mathematics. Cambridge: MIT Press, 1993. plate A3
Project: Make a work of art featuring polyhedra.
So we've seen the origins of the Platonic Solids, starting even before Plato, and have briefly traced the influence of the polygons in art right up to the present.
We've also had a first look at some subjects we'll look at in more detail later.
For mathematical topics, we've briefly looked at sequences and series and the geometry of the polyhedra.
Reading Assignment:
Plato, Timaeus, the selection in your reader.
Emmer, The Visual Mind, from your reader
Calter, selection from Technical Mathematics with Calculus, handout
Additional References from your Bibliography :
Wenninger Pedoe Kappraff Irma Richter Lawlor Euclid
Ivins Newman Ghyka Wittkower Critchlow
Project: Repeat Plato's calculations and see if you do indeed get a musical scale.
Make some Archimedian Solids
Make some star polyhedra.
Make a work of art featuring polyhedra
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were you allowed to use a graphing calculator? because if you were, you would be able to find the HA by looking at the graph
Find the areas of both the wall and the door and subtract the two 8 x 20 = 160 to find the width of the wall: 4+ 8+ 8= 20 Area of the wall: 25 x 20 = 500 Area to be painted: 500-160 = 340 ft^2
(7/8)-(3/4)= x ----equation (3/4) = (6/8) (7/8)- (6/8) = (1/8)
thank you!
Consider an amphoteric hydroxide, M(OH)2(s), where M is a generic metal. M(OH)2 (s) <--> M^2+ (aq) + 2OH- (aq) Ksp=4e-16 M(OH)2 (s) + 2OH- (aq) <--> M(OH)4 ^2- (aq) Kf=0.06 Estimate the solubility of
M(OH)2 in a solution buffered at pH= 7.0, 10.0, and 14.0 I'm ...
Chemistry help!!!
The class will react 50.00 mL samples of .200 mol/L potassium phosphate with an excess of 0.120 mol/L lead II nitrate solution. what is the minimum volume of lead II nitrate solution required? What
volume of lead II nitrate solution should the instructor tell the students to u...
Calculate the pH of the resulting solution if 30.0 mL of 0.300 M HCl(aq) is added to (a) 35.0 mL of 0.300 M NaOH(aq). (b) 40.0 mL of 0.350 M NaOH(aq).
Which of the following solution should be mixed with 50.0 mL of 0.050 M HF solution to make an effective buffer? A) 50.0 mL of 0.10 M NaOH B) 25.0 mL of 0.10 M NaOH C) 50.0 mL of 0.050 M NaOH D) 25.0
mL of 0.050 M NaOH
if the solutions of f(x)=0 are -1 and 2, then the solutions of f(x/2) = 0 are
set denominator equal to 0 and solve for x which would make the equation undefined v^2-4v-32=0 (v-8)(v+4)=0 v=8, v=-4 x is not equal to 8 or -4
Intermediate Algebra
=sqrt(24)x i = 2sqrt(6)i
Intermediate Algebra
[(7a-b)(7a+b)]/(7a-b) =7a+b
break them down into shapes that you know how to work with like squares and rectangles.
It should be 0.25. I don't really know how to get that from the equation (Amplitude=distance/frequency), but since the problem said that "the crests of the waves are about .5 meters above the
troughs." I divided that by two.
A certain form of albinism in humans is recessive and autosomal. Assume that 1% of the individuals in a given population are albino. Assuming that the population is in Hardy-Weinberg equilibrium,
what percentage of the individuals in this population is expected to be heterozyg...
The difference between fitness of a given genotype and another genotype considered optimal is called the selection coefficient (s). What is the selection coefficient for a genotype (aa) that produces
an average of 99 offspring when Aa individuals produce an average of 100 offs...
But how did you get the (Cl^-)?
A certain form of albinism in humans is recessive and autosomal. Assume that 1% of the individuals in a given population are albino. Assuming that the population is in Hardy-Weinberg equilibrium,
what percentage of the individuals in this population is expected to be heterozyg...
The difference between fitness of a given genotype and another genotype considered optimal is called the selection coefficient (s). What is the selection coefficient for a genotype (aa) that produces
an average of 99 offspring when Aa individuals produce an average of 100 offs...
The area of a square garden is 242m squared. How long is the diagonal?
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pulleys reduce the work needed
clarify this for me...
I know that pulleys reduce the work needed, depending on how its setup... But lets say at you add n number of pulleys to mass... And the work is reduced by 1/n... So you add infinite amount of
pulleys... and then you supply an infinity amount of force... inf/inf = ? What exactly are you getting here? Tug 5m... with 5 pulleys in this setup its just 5/5...hmmm confused.
Pulleys do NOT reduce "work needed"- not in the physics sense of the word "work". They reduce the
needed in "work= force times distance" by increasing the distance over which the force is applied.
If you attached an "infinite" number of pulleys you would use 0 force for an "infinite" distance- which means nothing, physically. As CookieMonster said, "Stay away from infinities"!
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Talk Tennis - View Single Post - Can someone identify this old racket?
Originally Posted by
Capt. Willie
Your 18x18 looks right to me...8 and 10 shared, skip 9. I think I would have tried finding pictures of Gonzalez with this racquet and attempted to count the patteren. But I'm pretty sure you have it
Let us know how it plays.
I did find some Gonzales pictures where I was able to count the pattern, as well as that 1969 advertisement for the Spalding Smasher. However, they both had a 16 mains x 18 crosses pattern. Looking
at my racket, 16 x 18 didn't look right for it, so I wondered why. So, using the 1969 ad, I counted the total number of string holes. It had 54. Well, like I said earlier, my racket has 64 string
holes. So I wondered if my racket had a larger head to account for the extra string holes, but it looked like a standard size head to me. But just to make sure, I did the math on it, along with the
math on the Davis wood racket (which I know for a fact is a standard size head), and they were the same -- and I don't mean "the same" in a rough sense, I mean
the same, out to the 5th decimal place. That surprised me. Here is the math for both of them, measuring just the string bed area on both of them (inside of the hoop):
TAD Davis Professional:
8 3/16" Width x 10 5/8" Length = 8.1875" x 10.625" =
average diameter =
69.49 in.^2
Spalding Smasher:
9" Width x 9 13/16" Length = 9" x 9.8125" =
average diameter =
69.49 in.^2
So apparently, Spalding went with a tighter pattern while leaving the head size the same, by the time they made the models like mine with the "S" in the throat piece.
The closest thing I could find for a picture to go by was
Chemold Rod Laver model, which has a similar shape to the Spalding Smasher, and 60 total string holes. It has an 18x18 pattern. And just for the hell of it, I checked out a Wilson T2000, which also
has a similar shape to the Spalding Smasher, and it was 18x18 as well. So that's what I went with.
I'll probably hit with it tonight, assuming it doesn't rain.
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Network ‘Small-World-Ness’: A Quantitative Method for Determining Canonical Network Equivalence
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More information
PLoS ONE. 2008; 3(4): e2051.
Network ‘Small-World-Ness’: A Quantitative Method for Determining Canonical Network Equivalence
Olaf Sporns, Editor^
Many technological, biological, social, and information networks fall into the broad class of ‘small-world’ networks: they have tightly interconnected clusters of nodes, and a shortest mean path
length that is similar to a matched random graph (same number of nodes and edges). This semi-quantitative definition leads to a categorical distinction (‘small/not-small’) rather than a quantitative,
continuous grading of networks, and can lead to uncertainty about a network's small-world status. Moreover, systems described by small-world networks are often studied using an equivalent canonical
network model – the Watts-Strogatz (WS) model. However, the process of establishing an equivalent WS model is imprecise and there is a pressing need to discover ways in which this equivalence may be
Methodology/Principal Findings
We defined a precise measure of ‘small-world-ness’ S based on the trade off between high local clustering and short path length. A network is now deemed a ‘small-world’ if S>1 - an assertion which
may be tested statistically. We then examined the behavior of S on a large data-set of real-world systems. We found that all these systems were linked by a linear relationship between their S values
and the network size n. Moreover, we show a method for assigning a unique Watts-Strogatz (WS) model to any real-world network, and show analytically that the WS models associated with our sample of
networks also show linearity between S and n. Linearity between S and n is not, however, inevitable, and neither is S maximal for an arbitrary network of given size. Linearity may, however, be
explained by a common limiting growth process.
We have shown how the notion of a small-world network may be quantified. Several key properties of the metric are described and the use of WS canonical models is placed on a more secure footing.
Networks are widely used to both represent real-world systems for topological study [1] and as a substrate for modeling their dynamics [2]. Many real technological, biological, social, and
information networks fall into the broad class of ‘small-world’ networks [3], a middle ground between regular and random networks: they have high local clustering of elements, like regular networks,
but also short path lengths between elements, like random networks. Membership of the ‘small-world’ network class also implies that the corresponding systems have dynamic properties different from
those of equivalent random or regular networks [3]–[7].
One popular method for studying small-world networks is to use an equivalent network model to generate other similar instances of the class of systems under study. Such generating models may also
possess analytic properties that, we assume, may be extrapolated to the target system. One canonical model used as a candidate for network equivalence is the original Watts-Strogatz (WS) model, which
has been used as a substrate for studying dynamics in the diverse fields of ecology [8], economics [9], [10], epidemiology [11], [12], and neuroscience [13].
However, the existing ‘small-world’ definition is a categorical one, and breaks the continuum of network topologies into the three classes of regular, random, and small-world networks, with the
latter being the broadest. It is unclear to what extent the real-world systems in the small-world class have common network properties and to what specific point in the “middle-ground” (between
random and regular) a network generating model must be tuned to genuinely capture the topology of such systems. Here we explore a continuous, quantitative, measure of ‘small-world-ness’, with the aim
of overcoming these inadequacies in the current theory of small-world networks.
Network formalism
When describing a real-world system as a network, each element of the system is represented by a vertex or node, and relationships or interactions between elements are represented by edges between
nodes. Two nodes are said to be neighbors if they are connected by an edge, and the degree k[i] of node i is the number of neighbors it has. The minimum path length between two nodes is the minimum
number of edges that must be traversed to get from one node to the other. The mean value of the minimum path length over all node pairs will be denoted by L.
A key concept in defining small-worlds networks is that of ‘clustering’ which measures the extent to which the neighbors of a node are also interconnected. Watts and Strogatz [3] defined the
clustering coefficient i by
where E[i] is the number of edges between the neighbors of i. The clustering coefficient of the network C^ws is then the mean of [14], based on transitivity, is expressed by
where a ‘triangle’ is a set of three nodes in which each contacts the other two. Both capture intuitive notions of clustering but, though often in good agreement, values for C^ws and C^Δ can differ
by an order of magnitude for some networks. We consider mainly C^Δ here, but report where using C^ws leads to different results.
A network G with n nodes and m edges is a small-world network [3] if it has a similar path length but greater clustering of nodes than an equivalent Erdös-Rényi (E–R) random graph [15] with the same
m and n (an E–R graph is constructed by uniquely assigning each edge to a node pair with uniform probability). More formally, let L[g] be the mean shortest path length of G and L[rand] and
semi-quantitative categorical definition of a small world network [3]
Definition 1
The network G is said to be a small-world network if L[g]≥L[rand] and
Here a similar definition applies if we use (1) to define clustering coefficients.
New measures of small-world-ness
We then define a quantitative metric of ‘small-world-ness’ S^Δ according to
In a similar way, putting
we define S^ws
The categorical definition of small-world network above implies λ[g]≥1 and S^Δ>1. We can, therefore, now make a quantitative categorical definition of a ‘small-world’ network
Definition 2
A network is said to be a small-world network if S^Δ>1
A similar definition may also be given with respect to S^ws.
However, notwithstanding the new categorical definition, we wish to emphasize here the utility of using a continuously graded notion of small-world-ness. We go on, therefore, to analyze the
properties of the new metrics, and apply them to real-world data for the first time (in [16] we originally proposed this metric as a tool for comparing theoretical neuroanatomy models; its subsequent
adoption by others [17], [18], [19] motivated us to consider its theoretical and empirical applications as a universal metric).
New metrics behave as required with the Watts-Strogatz model
We first checked that the metric S^Δ behaves as required on the canonical Watts-Strogatz (WS) model of small-world generation [3]. The WS model begins with a ring of n nodes, each node connected to
its nearest neighbors out to some range K. Each edge in turn is ‘re-wired’ to a new target node with probability p (Figure 1A). Values of ppp values resulting in ‘small-world’ networks that share
properties of both provided that the network is connected and sparse — densely connected networks trivially have small mean path lengths and high clustering coefficients.
Small-world-ness S behaves as required on the Watts-Strogatz (WS) [3] model of small-world networks.
Figure 1 shows that small-world-ness captures the topology changes: it has a unique maximum at intermediate values of the re-wiring parameter p, indicating the maximum trade-off between high
clustering and low path length (Figure 1B), and decays with increasing edge density for a fixed size of network, reflecting the requirement of sparseness (Figure 1C). We can see why this occurs for
increasing density. The edge density of a network is given by
As ξ→1 then both C^ws, C^Δ→1 and L→1 because all nodes become connected; and as this would apply for both a given real-world network and its E-R random graph equivalent, so S^Δ, S^ws→1 regardless of
n: high edge density results in low small-world-ness.
Small-world-ness scales linearly with n for real networks
We computed S^Δ and S^ws for a broad range of technological, biological, social, and information networks (33 networks in total; Table 1, see Materials and Methods). To our surprise, we found that
both forms of small-world-ness scale linearly with the size of the network across all systems falling into the small-world class (Figure 2A,B), irrespective of their originating domain or their other
topological properties (e.g. their degree distribution, degree correlation). For S^Δ, it was not possible to find or calculate C^Δ (and hence S^Δ) for 6 of the 33 networks. However, for the remaining
27, all had S^Δ>1 and were therefore deemed to be small-world in the new scheme (Definition 2). To ensure the robustness of the categorization, networks with borderline values 1≤S^Δ≤3 were tested for
significance using Monte Carlo sampling of 1000 equivalent E–R random graphs for each network, estimating 99% confidence intervals using standard methods (see Materials and Methods). All such
networks had small-world-ness scores significantly greater than an equivalent E–R random graph. For the 27 networks for which S^Δ>1, linear regression on log-transformed quantities (see Materials and
Methods) allowed an estimate of the best power law fit: S^Δn^0.96 (r^2p^−9). This is an essentially linear scaling of S^Δ with n.
Correlation of real-world network properties.
Table of small-world-ness values and other topological properties of real networks.
For S^ws, 3 networks in our data-set were not small-worlds: relationships amongst students [20] S^ws[21], [1] S^ws[22] S^wsS^ws≤3 were tested using Monte Carlo methods for significant membership of
the small-world category and were found to satisfy this criterion. For the 30 networks with S^ws>1, a similar regression to that used for S^Δ gave S^wsn^1.11 (r^2p^−11). Thus, there is also a robust
linear scaling of S^ws with n (see Text S1 for further details).
Linear scaling on the Watts-Strogatz model
Linear scaling of small-world-ness with network size was unexpectedly shown by the canonical Watts-Strogatz model [3] of small-world network generation. We now show that this result can be explained
analytically. In what follows, many of the relationships are held only approximately, but because these approximations are often very good we show them as equalities. Note that from here on we use
subscript names to identify analytic quantities that pertain to a particular network model only.
If L[ws] is the mean shortest path length in the WS model, then it is known [14] that
Similarly, it is known for E-R random graphs [23] that
where kkK, so that
Using (11), (9) and (4) the path length ratio λ[ws] for the WS model is
The function f(x) in (9) has an upper asymptote of ln(2x)/4x if
If [24] that
For E–R random graphs [23], to a good approximation, kK
Therefore, using, (14), (15) and (3)
From (5), (16) and (13),
where h(K, p) is a function of K and p only. The term in the square brackets tends to 1 as n→∞ and so, for large enough n, S^Δ for the WS model scales with n. To quantify this approximation, we
performed a linear regression on log-transformed quantities (just as for the real networks) over the typical range of n encountered in our sample of networks, 10^2≤n≤10^7, and found a linear fit,
with r^2 within 10^−5 of unity.
Establishing the precise WS model correlate of a real network
The WS network is often used as a generative model for real small-world networks [e.g. 8]–[13]. This is assumed to establish a ‘first-pass’ model of that system’s topology, which may be augmented by
considering other factors such as degree sequence [23], degree correlation [25], modularity [26] and other properties.
In matching the WS parameters K, p, n to the target system, we know n, can measure kKkp has, hitherto, remained problematic. However, using our new metric of small-world-ness, it is possible to
establish p in a principled way. Thus, if G is a real (target) network with measured small-world-ness S^Δ. That is, form e with respect to p, keeping K, n at their measured values. We did this for
our sample of real-world systems, omitting those for which kpp values for the equivalent WS model are listed in Table 1
Given that the real-world networks showed S^Δn, the WS networks derived from them under the procedure described here must do likewise (they have identical S^Δ values). However, the result in the
previous section would suggest that this implies K, p are roughly constant for this set of WS networks.
To investigate the constancy of K we used the result that km/n (where m is the number of edges in the network). So, using KkK is equivalent to establishing mn. Figure 2C shows the result of
regressing m against n (using log-transformed quantities) for the real world networks. For networks with S^Δ>1, the best fit model was mn^1.06 (27 networks, r^2p^−15), implying a mean node degree of
km/n≈5; for networks with S^ws>1, the best fit model was mn^1.03 (30 networks, r^2p^−15) implying kp values; we found that all testable real-world systems fall into a very limited range of p for the
equivalent WS model (0.64≤p≤0.95 and σ[p]
An alternative view of these results is as follows. We could start with the empirically observed approximate constancy of mean node degree kp for the real world networks, and deduce a linear scaling
of S^Δ for the WS models. Then, under the equivalence of S^Δ for both real-world networks and their WS counterparts, we could have predicted that S^Δ for the real-world networks would also scale
The linear scaling of small-world-ness with n is not inevitable
Is the relationship Sn inevitable for all systems? (The subsequent argument holds for S based on either definition of clustering coefficient and so superscripts Δ, ws are dropped). To investigate
this we note that it is always possible to write S[i][i]n[i] for the ith system, for some value α[i]; in the case of linear scaling, α[i] is constant. To proceed further, we now express α[i] in terms
of other system parameters. Using the definition of S and (10) for random graphs,
where C[i], L[i], k[i] are the clustering coefficient, mean shortest path length, and mean node degree of system i respectively. While we do not know exactly how L[i] depends on n, we note that the
mean shortest path length for small-world networks is usually assumed to scale logarithmically like random graphs: from (11), L[rand]kn); and for the WS model, using (9) with large n, L[ws]K^2p)ln(n
). Both relations are of the form Ln) where β is independent of n. We therefore write L[i][i]ln(n[i]), where β[i] is the factor that ensures the equality to be true (i.e it plays a similar role in
this respect as α[i]).
This gives
In general, there is no a priori reason to suppose that the variables C[i], k[i] and β[i] are either all constant, or co-vary in a way commensurate with constancy for α[i]. However, for the sample of
networks used here, as noted above, the mean node degree k[i] is approximately constant. It is now instructive to see how much co-variation is required between the remaining two variables in order to
ensure a significantly different power law holds between S and n.
Thus, suppose that we fit a model Sn^1.5 so that we expect n encountered here – approximately four orders of magnitude – α[i] would therefore have to range over 2 orders of magnitude. For this to
occur, there must be sufficient variation in C[i] and β[i], and these two quantities should correlate well with n. The ranges of the two variables are reasonably large in the data-set – using C^Δ,
0.209≤β[i]≤2.52 and 0.005≤C[i]≤0.72 – and could plausibly generate the required 100-fold variation. However, the correlation coefficients with n are very small: for β[i], r^2C[i], r^2S^Δ and n for
the networks studied here.
To study the effect of a lack of correlation between n and network parameters like C[i] on linear scaling between S^Δ and n, we ran a Monte Carlo simulation (see Materials and Methods). Each one of
1000 experiments consisted of sampling 27 randomly drawn C values for networks with constant β, and with a spread of n over 4 orders of magnitude. For each network its small-world-ness was computed
and a linear regression of S against n performed. This resulted in a mean of r^2
The linear relationship is, however, sensitive to deviations from the approximation that kmn model in turn deviate furthest from the linear Sn model (Figure 3). This was shown using a novel
regress-delete-regress procedure outlined in Materials and Methods (we were able to directly test the sensitivity to km with n provided a baseline from which we could quantify deviation of kFigure 3
shows that if we delete a random set of networks from the data-set, then the average effect is to not change the fit to the linear Sn model: the linear scaling is robust, and does not depend on a
specific network set.
Robustness of WS model prediction Sn.
The sensitivity of small-world-ness linearity with n to degree kkm/n
and see that mean degree scales linearly with edge density. Thus, a network with high edge density implies high mean degree, which in turn would fall far from the linear Sn model, as we have just
One exemplar of a real system with high edge density is the network of individual neurons within a single vertebrate brain region. Detailed network data for these are not available because of the
great technical difficulties in reliably reconstructing even small networks such as the 302 neuron C. Elegans nervous system [27]. Indeed, high edge density itself may be the primary cause of
technical problems in reconstructing complete systems from many domains, resulting in their absence from the network literature. Nonetheless, approximate reconstructions can be attempted.
Quantitative anatomical models of individual brain regions suggest that each of the hundreds of thousands or millions of neurons receive many thousands of connections, and each themselves connect to
similar numbers of target neurons [16], [28]. Such networks of neurons can have very low small-world-ness values for their size [16], and thus fall far from the linear Sn model discovered here.
We conclude here that the linear relationship between small-world-ness and system size does not hold for an arbitrary collection of networks, but is highly likely if all such networks have a similar
mean node degree.
Other scaling properties of small-world-ness
Having established that S scales linearly with n, it is also instructive to look at how its component ratios scale with n. We find, as expected, that most networks falling into the small-world class
have approximately the same mean shortest path length as their equivalent E–R random graphs, and so λ≈1. Given this, it is unsurprising that both γ^Δ and γ^ws then scale linearly with n (see Figure
S1). We did find that three networks in our data-set — email messages (#7), software packages (#21), and software classes (#22) — had λ≈0.1, indicating that their mean shortest path length was an
order of magnitude smaller than the equivalent E–R random graph. These networks are thus ultra-small [29], and indeed both email message (#7) and software package (#21) networks fall further from the
linear model than any others.
Given the existence of the linear scaling with n, the scaling of small-world-ness with some other topological properties is completely determined. We can directly determine from (20) how edge density
behaves in our data-set (values for ξ are given in Table 1). Taking our fitted linear model Sn, we can substitute nS/α in (20) and find that
Substituting our found values of kS^ws or S^Δ confirms that this is a good approximation. Therefore, because small-world-ness linearly scales with network size, and degree is approximately constant,
then S also has a simple inverse linear scaling with edge density.
Real-world systems do not maximize small-world-ness
We can show that the specific scaling coefficient α in the relationship Sn for the real-world networks studied here does not maximize small-world-ness for a particular size of network. First, we show
that the WS model predicts an approximately constant amount of rewiring p that maximizes S^Δ, independent of network size. To do this, given the above analytic expressions (13) for λ[ws] and (16) for
p would then give us the value of p that maximized Text S1 for details of the solution.
We did this over the range KkFigure 2C. If p^* is the value of p giving maximal n^3) p^*n→∞, p^*→0.246, so that the range of p^* is very small. The constant K and very small range of p^* imply that
the associated maximum n. It transpires that the theoretical maximum n with slope 0.181 (and plotted in Figure 2A). Thus, S^Δ is not maximized by the real-world networks.
A generative mechanism for a specific linear Sn relationship
We have established and explained many simple properties of real-world networks and of their equivalence class in the WS model. We now show how the specific, sub-maximal, linear scaling of Sn could
have been generated. The models we examine here are intended as informative examples of the generation and limits on S scaling, not an exhaustive list of those which could generate the specific
linear scaling we found — that remains the subject of future work.
Many of the real-world systems share common generative principals despite their widely differing origins. Most systems have a growth process, showing some form of preferential attachment [30] that is
limited by the cost of adding new edges and by the capacity to maintain them (as might be induced by aging) [31]. Simple models of this process result in ‘scale-free’ networks with power-law or
truncated power-law degree distributions [30], [31], a property that is also common to many real-world systems considered here [32] (but see [33] for an alternative view of some biological networks).
However, networks generated by these models are not ‘small-world’ by either Definition 1 or 2. Their clustering coefficient is inversely proportional to n, going to zero as n grows large [34]. Thus,
they cannot show linear scaling of S: it is at best constant and at worst goes to zero with increasing n.
A noisy, limited growth process can generate the specific linear Sn relationships we report. A generalized form of the Klemm-Eguiluz model (GKE)[34], [35] encapsulates this process, and has the
unique property of creating networks that are both small-world (short path length, high clustering) and ‘scale-free’ (having a truncated power-law degree distribution) as found for many real-world
systems considered here. (To the best of our knowledge, all known real-world systems with power-law-like degree distributions also fall into the broad ‘small-world’ class we discussed in the
Introduction; it is only the scale-free networks formed by the simple models that form a distinct set of ‘scale-free-only’ networks). By using the GKE model, we therefore also show that linear
scaling of S can occur whether or not the real-world systems have ‘scale-free’ properties.
The GKE model begins with an active set of M nodes. At every time-step a new node is added, connecting d edges: one edge added to a random inactive node with probability ρ, adding noise to the
process; all remaining edges connect to randomly chosen active nodes. One of the active nodes is deactivated with a probability proportional to the active nodes’ degrees; finally, the new node is
activated. The sequence repeats until the desired size of network is obtained.
We found that specific values for M and ρ could generate GKE networks with the same linear scaling relationships between network size and S^ws and S^Δ that we observed for the real-world networks (
Figure 4; see Materials and Methods, and Figure S2). Therefore, a possible general mechanism for particular linear scaling rates of small-world-ness is a common size of both active node set and
quantity of noise during creation of the real-world systems.
A limited growth process can generate the observed Sn relationships.
Small-world-ness is a topological property linking real-world systems across domains of research. Hitherto it has been defined only in semi-quantitative way (Definition 1). In this paper we propose
quantitative measures of small-world-ness – S^Δ and S^ws – and define a network to be in the small-world category with respect to either of them if the small-world-ness is greater than 1 (e.g
Definition 2). This quantification of small-world-ness allows for the statistical testing of its presence in any given network.
The Watts-Strogatz (WS) model plays a key role in the study of small-world networks. It uses a generative process to create classes of small-world network and is now widely used as a model for
studying dynamic systems [3]–[13]. However, until now, a precise parameterization of the WS model associated with a given kind of real-world network remained elusive. Our introduction of a
quantitative measure of small-world-ness remedies this by demanding that the WS counterpart to a specific network have the same value of S^Δ (or S^ws). For the WS models it is possible to show
analytically that, under certain circumstances (constant re-wiring parameter and range), the small-world-ness S^Δ will scale linearly with network size n. Intriguingly, a wide class of real-world
networks also shows this linear scaling. Given this similarity in behavior, the assumption of (limited) topological correspondence of the WS model with real networks implies certain constraints on
empirically measured parameters (like mean degree) of these networks. These constraints appear to hold, and so the ideas developed here provide further support for using the WS model in the study of
small-world systems.
We have shown that the linear scaling between S^Δ and n is not an inevitable property of networks; it would be possible, for example, to include networks with very large edge density that would
destroy any linear scaling. However, in the event of linear scaling, there is a variety of possible scaling constants and there is a noisy growth process that could give rise to the networks sharing
the same scale (slope) parameter. Finally, we have shown that the small-world networks used here do not maximize S^Δ (there are ‘steeper’ linear relationships between S^Δ and n).
We cannot, on the basis of the work presented here, answer the question of why small-world-ness was not maximised, but we can give some insights as to why this is the case. The possible explanations
split into two broad classes of structural and dynamical limitations. Our use of the GKE model showed that the limited capacity of a system's nodes to maintain edges (whether due to physical cost,
aging processes, or some other mechanism) is one structural limitation that could result in sub-maximal small-world-ness. Other structural limitations could include physical limits on node location
and length of edges, such as might occur for the sub-stations and transmission wires in the power grid network.
Even if structural limitations were not an issue, then the system may have dynamical requirements that prevent it from maximising small-world-ness. The constraints placed on a system's topology by
the dynamics required to fulfill its function are not well understood. Recent work has shown how the presence of particular network ‘motifs’ — repeating patterns of connections between a small number
of nodes — can guarantee, for example, a chaotic attractor for the network as a whole [36]. The functional requirements of some real-world system may then lead to the inclusion of particular motifs
to guarantee the necessary dynamics [37], and there is no necessary link between a system's motifs and its global topological properties (of which small-world-ness is but one). Nonetheless, given
that so many of the key motifs identified so far are either complete 3-node loops or contain them [37], [38], the global topology will have a high clustering coefficient, and will most likely be a
small-world network.
Other systems may have constraints placed directly on their global topology, and this too could prevent maximisation of small-world-ness. For example, in his original work on the small-world model,
Watts [39] explored the dynamics of Kuramoto oscillators on a WS model substrate, and showed that the fraction of synchronised oscillators had a phase transition that occurred for progressively
smaller p as the oscillators' symmetric coupling strength increased (for fixed n, K). Therefore, if a system's function required it to be at the phase transition, so that it could rapidly switch
between synchronised and desynchronised states with minimal perturbation, the required amount of (implied) rewiring may be far from that which maximised small-world-ness.
These are just a few of many possible explanations for why real-world systems do not maximise small-world-ness. Instead we might ask, when would small-world-ness be maximised? Maximum S essentially
identifies the point in the network's possible topologies where the highest clustering is achieved for the smallest deviation from the shortest mean path length. Such a network would be optimal for
message-passing, such that all the nodes receive a message in the shortest possible number of network steps [40]. On this basis, we expect that some form of dynamic phenomenon, whether based on
percolation (or, equivalently, epidemiological SIR models), oscillators, or some other general ordinary differential equation system, will have a strong correlation with small-world-ness. So, just as
we have used a continuously graded ‘small-world-ness’ to quantitatively examine the topologies of the broad class of small-world networks, we may use this as the starting point for quantifying the
continuum of dynamic properties that must also span this class.
Materials and Methods
Data-set of real-world systems
We collated a database of real-world networks' topological properties, combining published results with our own analyses of available data-sets. These are presented in Table 1, extending the previous
considerable effort of collating topological properties by Mark Newman [1]. All networks are treated as undirected. We list 33 real-world systems in total: we could compute S^ws for all systems and S
^Δ for 27 systems — C^Δ could not be found or computed for those systems.
We emphasise that the networks were not chosen for their ability to fit the linear model of Sn. The majority of the data-set (21 of 33) were obtained from a previous collation [1]: networks were only
omitted from that prior data-set if neither C^ws or C^Δ were available for them (and hence were of no practical use to us here). Many of the additional networks we added filled sub-domains missing
from the prior data-set, for example: the dolphin network [41] is an example of an animal social network; the cortical area connectivity map [42] is an example of large-scale neural connectivity. In
addition, the regress-delete-regress sequence we used in the main text (and see below) shows two properties. First, that we could have applied that method to the data-set in Table 1 before further
analysis, pruning the data-set down to those networks that showed the best fit to the linear model (by choosing the ‘most-deviant’ networks to omit), but did not. Second, that the linear scaling
property is robust across randomly chosen sub-sets of the network data-set: on average, randomly deleting networks from the data-set did not significantly reduce the fit to the linear model.
Testing significance of S scores
We assess the significance of borderline small-world-ness scores S1 using Monte Carlo methods. The null hypothesis for the Watts-Strogatz definition of small-world networks is that the system is an
Erdös-Rényi (E–R) random graph. We thus constructed Nn and edges m for each tested real-world system, computing ith E–R network. The 99% confidence limits for the null hypothesis were then defined
for each system. We first found the central 99% interval [a*, b*], that is [43]
and similarly for S^ws. The 99% confidence interval for the system is then
The upper 99% confidence limit is then CL^0.01 S^ws, S^ΔS>CL^0.01 was therefore considered to significantly differ from a random network. We note that adopting a quantitative definition (Definition
2) of small-world-ness has led us to a procedure for a general statistical test for the presence of small-world structure, as defined by Watts and Strogatz [3], which is particularly useful for
establishing meaningful departures from randomness in small networks.
Fits to linear scaling
Least-squares regressions on small-world-ness S and number of edges m against size of system n were performed on log[10]-transformed data to normalize magnitude of errors across range of n. Best fit
linear model log[10](x)a+blog[10](n) back-transformed to a linear basis, giving xn^β, where α^a and βb. MATLAB (Mathworks) function regress was used to perform the regressions. The validity of r^2
significance values was established by confirming that the residuals of each regression had a normal distribution at p[44].
A regress-delete-regress procedure for testing robustness of predictions
We test the effect of real-world networks deviating from the constant k
1. regress
for the data-set (as in
Figure 2B
2. select network to remove from data-set based on regression outcome (3 different selection criteria were used, detailed below);
3. regress n vs S for reduced data-set and record new goodness-of-fit (as r^2);
4. repeat from step 1 until 50% of networks removed.
We do this for 3 selection cases in step 2. First, we tested removing the network with the largest deviation from mn linear model in each iteration, hypothesizing that this should lead to an overall
increase in fit to a linear model (increased r^2) for Sf(n) if the WS model behaviour reflected that of real-world systems. Second, we tested removing the network with the smallest deviation from mn
linear model at each iteration, hypothesizing that this should lead to an overall decrease in fit to a linear model (decreased r^2) for Sf(n) if the WS model behaviour reflected that of real-world
systems. Third, we tested random deletion, where a random network was deleted at each iteration, irrespective of the regression outcome, to establish the baseline effect of removing systems from the
data-set. The first and second cases are unique sequences of deleted networks; the third case we repeated 1000 times.
Monte Carlo testing of linear scaling
We tested the dominance of linear Sn scaling given an approximately constant mean node degree kC values from a uniform distribution in [0,1], computing α[i] for each from Eq. (19) with constant β[i]k
[i]S[i][i]n[i] for each, using a logarithmic spread of 27 network sizes [10]-transform) was then performed on the set of simulated 27 S values and the r^2 recorded. We repeated this procedure 1000
Searching GKE model parameter space
We wished to determine if the generalized Klemm-Eguiluz model (GKE)[31], [32] model could explain the particular scaling relationships we found for the real-world systems:
We explored the (M, ρ) parameter space, searching over M is set by the number of edges added per new vertex, and here we set dkmn relationship for the real-world systems. For each (M, ρ) pair, we
constructed 5 GKE model networks for each value of S^ws and S^Δ scores. We took the mean of these 5 scores for each n, giving sets n. The parameters that minimised RMSE are given in Figure 4; the
error landscapes are shown in Figure S2.
Supporting Information
Text S1
Supporting information text
(0.22 MB PDF)
Figure S1
Correlation of real-world systems' clustering coefficient and path length ratios with system size. Clustering coefficient ratios (a) γ^ΔC^ΔC[rand] and (b) γ^wsC^wsC[rand] both scaled linearly with S.
Linear regressions found r^2L was approximately the same as that of an E-R random graph, and so λL/L[rand]1 for most networks (note that we show λ here for all 33 networks). All linear regressions
performed on log[10]-transformed data, as detailed in Materials and Methods of the main text.
(0.11 MB TIF)
Figure S2
The root mean square error (RMSE) distribution across tested values of the GKE model parameters. The RMSE is computed based on the difference between the mean values of small-world-ness for a set of
generated GKE networks and the corresponding small-world-ness values from the specific linear relationships found for the real-world systems. (a) RMSE error distribution for the fit to the S^wsn
relationship. RMSE plotted on log scale to emphasise valley of minimum values. Stick-and-ball indicates the parameter pair that minimised RMSE. (b) RMSE error distribution for the fit to the S^Δn
relationship. Stick-and-ball indicates the parameter pair that minimised RMSE.
(0.44 MB TIF)
We thank M. Newman and M. Kaiser for making their data-sets publicly available, and T. Stafford and T. Prescott for comments on drafts of this manuscript.
Competing Interests: The authors have declared that no competing interests exist.
Funding: MDH acknowledges support by the EU Framework 6 IST-027819-IP (ICEA) project. KG acknowledges the support of EPSRC grant EP/C516303/1. The funders had no role in study design, data collection
and analysis, decision to publish, or preparation of the manuscript.
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Higley Math Tutor
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step-by-step approach.In reading, I rebuild with phonics.
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college students with success.
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[Numpy-discussion] Evaluate bivariate polynomials
Nils Wagner nwagner@iam.uni-stuttgart...
Wed Oct 19 06:58:58 CDT 2011
Hi all,
how do I evaluate a bivariate polynomial
p(x,y)=c_0 + c_1 x + c_2 y +c_3 x**2 + c_4 x*y+ c_5 y**2 +
c_6 x**3 + c_7 x**2*y + c_8 x*y**2+c_9*y**3 + \dots
in numpy ?
In case of univariate polynomials I can use np.polyval.
Any pointer would be appreciated.
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Fifth class
Fifth class
Here is a list of all of the skills students learn in fifth class! These skills are organised into categories, and you can move your mouse over any skill name to view a sample question. To start
practising, just click on any link. IXL will track your score, and the questions will automatically increase in difficulty as you improve!
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How to Do Math Proofs
Edit Article
Edited by Stephen King, Mike, Sondra C, Tom Viren and 45 others
Performing mathematical proofs can be one of the hardest things for students to do. Students majoring in mathematics, computer science, or other related fields will likely encounter proofs at some
point. Simply following a few guidelines will help erase the doubt from the validity of your proof.
1. 1
Understand that math uses information that you already know, especially axioms or the results of other theorems.
2. 2
Write out what is given, as well as what is needed to be proven. It shows that you will start with what is given, use other axioms, theorems, or math that you already know to be true, and arrive
at what you want to prove. True understanding means you can repeat, and paraphrase the problem in at least 3 different ways: pure symbols, flowchart, and using words.
3. 3
Ask yourself questions as you move along. "Why is this so?" and "Is there any way this can be false?" are good questions for every statement, or claim. These questions will be asked by your
professor in every step, and as soon as he/she can't verify one of those questions, your grade will go down. Back up every statement with a reason! Justify your process.
4. 4
Make sure your proof is step-by-step. It needs to flow from one statement to the other, with support for each statement, so that there is no reason to doubt the validity of your proof. It should
be constructionist, like building a house: orderly, systematic, and with properly paced progress. There is a very graphic proof of the Pythagorean theorem which is found by a simple process [1].
5. 5
Ask your professor or classmate if you have questions. It's okay to ask questions every now and then—doing so is part of the learning process. Remember: There is no such thing as a silly
6. 6
Designate the end of your proof. There are several methods for doing this:
□ Q.E.D. (quod erat demonstrandum, which is Latin for "which was to be shown"). Technically, this is only appropriate when the last statement of the proof is itself the proposition to be
□ A filled-in square (a "bullet") at the end of the proof.
□ R.A.A. (reductio ad absurdum, translated as "a bringing back to absurdity") is for indirect proofs, or proofs by contradiction. If the proof is incorrect, however, these symbols are very bad
news for your grade.
□ If you're not sure if your proof is correct, just write a few sentences saying what your conclusion was and why it is significant. If you use one of the above symbols and you turned out to be
wrong, your grade will suffer.
7. 7
Remember the definitions you were given. Go through your notes and the book to see if the definition is correct.
8. 8
Take time to ponder about the proof. The goal wasn't the proof, it was the learning. If you only do the proof and then move on then, you are missing out on half of the learning experience. Think
about it. Will you be satisfied with this?
• Proofs are difficult to learn to write. One excellent way to learn proofs is to study related theorems, and how those were proved.
• Try to apply your proof to a case where it should fail, and see whether it actually does. For example, here's a possible proof that: The square root of a number (that means any number) tends to
infinity as that number tends to infinity.
□ "For all positive n, the square root of n+1 is greater than the square root of n.
□ So if that is true as n increases, then its square root also increases; and as n tends to infinity, its square root tends to infinity for all n." (That might sound okay at first.)
□ But, although the statement you are attempting to prove is true, the deduction is false. This proof should apply equally well to the arctan of n as it does to square root of n. Arctan of n+1
is always greater then arctan of n for all positive n. But arctan does not tend to infinity, it tends to pi/2.
□ Instead, we prove it as follows. To prove something tends to infinity, we need that for all numbers M there exists a number N such that for all n bigger than N, the square root of n is bigger
than M. There does exist such a number - it's M^2.
☆ This example also shows that you should carefully check the definition of the thing you are trying to prove.
• A good mathematical proof makes every step really obvious. Impressive-sounding statements might get marks in other subjects, but in mathematics they tend to hide holes in the reasoning.
• What looks like failure, but is more than you started with, is actually progress. It can inform the solution.
• The best thing about most proofs: they've already been proven, which means they are usually true! If you come to a conclusion that's different than what you were to prove, then you more than
likely messed up somewhere. Just go back and carefully review each step.
• There are thousands of "heuristics" or good ideas to try. Polya's book has two parts, a how to, and an encyclopedia of heuristics.
• Writing multiple drafts for your proofs is not uncommon. Considering some homework sets will comprise 10 pages or more, you will want to make sure you got it right.
• Realize that a proof is just a good argument with every step justified. You can see about 50 proofs online [2].
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Topic: Duplication of the Cube.
Replies: 0
Duplication of the Cube.
Posted: Oct 1, 1994 6:50 PM
An article entitled "New Elements For the Irrational Numbers" has been
published this month by :
"The Journal of Transfigural Mathematics". Berlin, Germany.
I think it could be of some interest to all the experts on the history
and the fundaments of mathematics.
Abstract of the article :
"It is often claimed that by agency of algebra a modern man could solve
the problems no ancient Greek could do. The present work shows that
early mathematicians certainly had at hand an extremely simple operation
for solving problems which would imply the solution of any algebraic
equation, as for example the well known problem on the duplication of
the cube. It is presented the 'Rational Process' an iterative procedure
for finding rational approximations to the N-th. root of any positive
number by agency of the 'Mediant' which will be called here as the
'Rational Mean'. This method could have been easily implemented since
ancient times, mainly, because it only involves sums. So it is directed
to all those mathematicians interested in the historical attempts for
solving algebraic equations and deciphering the irrational numbers, in
this way, a brief introduction on these subjects is also included. For
the sake of clarity the examples are focused to find the cube root of 2,
the Golden Section and the transcendental number e."
As all we know, the 'Mediant' is an extremely simple operation which
rules the generation of the 'Farey fractions' and the convergents in the
simple continued fractions.
It is astonishing this so simple method hasn't been used before the sway
of analytic geometry!!!.
Domingo Gà ³mez Morà Ân.
E-mail address : dgomezm@etheron.net
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Algebraic inequality. I couldn't resist one more.
September 13th 2006, 08:17 PM #1
Junior Member
Aug 2006
Algebraic inequality. I couldn't resist one more.
An algebraic inequality
The score on a test was betwen 80 and 70
My answer is 70< s < 83
Am I wrong???
p.s. I suffer from insomnia!!!!
The 83 should be 80.
Also is the range supposed to be inclusive or exclusive?
I would have expected it to be inclusive so I would have gone for:
70<= s <= 80
September 13th 2006, 10:28 PM #2
Grand Panjandrum
Nov 2005
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Just a matter of rigor
August 28th 2010, 02:59 AM #1
Aug 2010
Just a matter of rigor
My teacher really takes account about the rigor of a proof. The following is the proposition and a proof given by myself. Can someone please check it whether it is rigorous enough?
Proposition: Let $(s_n)$ be a sequence that converges to $s\in \mathbb{R}$. Show that if $s_n\ge a$ for all $n>N$, then $s\ge a$.
Proof: Let $\epsilon>0$, then there exists $N_0$ such that $n>N_0$ implies $|s_n-s|<\epsilon$. Now take $N_1=\text{max}\{N_0, N\}$ and $n>N_1$ implies
Since $x\le |x|$, we must have
Since the difference can be taken arbitrarily small, or even negative. This proves that $s\ge a$.
I really doubt the rigor of this proof. And after I type in the proof, I doubt the validity.
Last edited by Hinatico; August 28th 2010 at 03:16 AM.
Here is an easy proof.
If it were true that $s<a$ then $\varepsilon = a-s$.
From that we get $\left| {s_N-s} \right| < \varepsilon \, \Rightarrow \,s_N<a$.
What is wrong to that?
August 28th 2010, 05:05 AM #2
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[SciPy-User] am I using interpolate.PiecewisePolynomial correctly?
Evgeni Burovski evgeny.burovskiy@gmail....
Wed Feb 27 09:54:44 CST 2013
Dear All,
I'm trying to use the interpolate.PiecewisePolynomial for the first time,
and I'm wondering if the timings I see are not due to some simple
misunderstanding of mine. Specifically, given a simple example,
$ cat pp.py
import numpy as np
from scipy.interpolate import interp1d, PiecewisePolynomial
def f(x):
return np.tan(x)
def fprime(x):
return 1./np.cos(x)**2
Npts = 50
grid = np.array([(np.pi/2.-0.1)*j/Npts for j in xrange(Npts+1)])
interp = interp1d(grid, f(grid), kind='cubic')
piecewise = PiecewisePolynomial(grid, np.array([np.r_[f(x), fprime(x)] for
x in grid]), orders=3)
it looks like evaluation of a PiecewisePolynomial takes ages:
$ python -mtimeit -s"from numpy import random, pi; import pp; x_new =
random.rand(1000)*pi/3." "pp.piecewise(x_new)"
100 loops, best of 3: 3.2 msec per loop
$ python -mtimeit -s"from numpy import random, pi; import pp; x_new =
random.rand(1000)*pi/3." "pp.interp(x_new)"
10000 loops, best of 3: 143 usec per loop
(I understand the difference in functionality between the two).
I'm wondering if this sort of timings is an artifact of the generality of
PiecewisePolynomials, or am I just not using them properly?
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Merion, PA Precalculus Tutor
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...I am a professional physicist with over 20 years math experience, including calculus. Geometry has a lot of terms to remember but once you are past that, it can be engaging and a lot of fun --
like doing puzzles. I use a variety of different teaching techniques to help students master geometry.
10 Subjects: including precalculus, calculus, physics, geometry
...Standardized test prep is a huge industry. Many of my students have tried several centers and packaged systems before beginning with me, and I often hear them say that they've never learned
like this before and that my advice is perfect for their unique needs. When I work with you, I'm not following a scripted curriculum.
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[R-sig-Geo] Newbie Questions
GRETA G. GRAMIG gggramig at students.wisc.edu
Wed Dec 20 17:57:07 CET 2006
I have limited experience with spatial analysis, but I am trying to learn. Perhaps someone with patience for a newbie will field my questions? If this is not the appropriate forum for such questions perhaps someone could suggest a more appropriate forum.
I have a small dataset that consists of 128 x and y coordinates (latitude and longitude, on a small field scale). At each coordinate there are measurements for various soil properties along with a biomass yield measurement for each x,y.
Via spdep, using k=4 nearest neighbors for the weights, I have calculated Moran's I and Geary's C for the univariate relationships. For most of the measured variables there is statistically significant spatial autocorrelation.
I have also used Rgeo to plot some quintile scatterplots for each variable. Just from looking at these quintile plots I can see that, although the soil variables and yield are spatially autocorrelated, there does not seem to be a relationship between the patterns observed in soil variables vs. the patterns observed in the yield.
In other words, I would like to know if there is a relationship between yield and phosphorus, taking into account the spatial dependence. But how can I test this? I have used GeoDa to compute the multivariate Moran's but I don't really understand what the program is doing or if this is the correct approach.
Another thing I would like to be able to do is, for each variable, interpolate between measurements to create 2D maps of the response surface.
I've searched everywhere I could think of but can't find any clear (and simple) examples to follow. This site: http://leg.ufpr.br/geoR/geoRdoc/geoRintro.html was some help but still kind of beyond me.
Can someone suggest a starting point (a book, article, or website) that might point me in the right direction? It would be especially great to have some examples using R to follow.
I would really appreciate any advice on these matters!
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Classical Angular Momentum
Classically, L is not an operator, so you cannot define a commutator.
You can show that {L[i], L[j]}=ε[ijk]L[k]. I don't know if that's what you meant by saying "Classically". If so, just write out L[i] in terms of q[i] and p[i]. If you write the correct expression for
it using Levi-Civita symbol and apply definition of Poisson bracket, it should be a trivial matter.
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Here's the question you clicked on:
Ok. I have a very hard question. Hopefully someone here can solve it. Now for binary operations, well I have one here, and I need to know if there exists in R (the set of real numbers), an identity
element for the operation *. Here is the operation, x*y=abs(x-y). Now when I do it, I break up the absolute value into x-e for x-e greater than 0 and -(x-e) for x-e less than 0, where e is the
identity element. Please explain your steps to this problem.
• one year ago
• one year ago
Best Response
You've already chosen the best response.
The identity element is 0
Best Response
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No. Only if you restrict the domain to positive real numbers.
Best Response
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I dont get why you're not satisfied with 0. The absolute value of |x-0| as well as the absolute value of |x+0| is |x|. It doesn't matter whether x is positive or negative, this always holds.
Hence x*e = e*x = x for e = 0, and 0 clearly is the identity element.
Best Response
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This operation has identity zero if the domain is nonnegative R, and no identity in general if domain is all R. Because abs(x-0)=abs(0-x)=-x (if x<0) which doesnt equal x
Best Response
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Yes. Ryan got it right. That is correct.
Best Response
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Well, it's much easier to see that with a concrete counterexample, am I not right? Let's see, now, exempli gratia, take x=-5. Let us suppose that e=0. Then, x * e = -5 * 0 = | -5 - 0 | = |-5| = 5
=/= -5. To make things more clear, logicaly... You can break down absolute value into three parts, one for x>0, second for x=0 and third for x<0 (of course, we observe R as the set on which the
binary operation is defined). This is a disjunction, therefore if we were to conclude anything from it by deduction, we must obtain the same result for every branch, yet we do not: 1) x>0: x-e=x
-> e=0 2) x=0: 0-e=0 -> e=0 3) x<0: x-e=-x -> e=2x Now, we do not get the -same- conclusion for every member of disjunction, therefore we can only conclude that it is either e=0 or e=2x. But can
we talk about e=2x at all? No, because it is not the condition for a neutral element that "for every x exists e" (in this case we would also be able to restrict it to x<0), but that "exists e
such that for every x", and there is a big difference. Note that the condition for the inverse elements is similar to the first case "for every x there exists -x"... Conclusion, neutral element
would exist on the set of nonnegative real numbers.
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
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Stress Intensity Factor Solutions for Internal Longitudinal Semi-Elliptical Surface Flaws in a Cylinder Under Arbitrary Loadings
STP677: Stress Intensity Factor Solutions for Internal Longitudinal Semi-Elliptical Surface Flaws in a Cylinder Under Arbitrary Loadings
McGowan, JJ
Senior engineer and senior scientistassistant professor of Aerospace Engineering, Westinghouse Electric Corp., Nuclear Energy SystemsUniversity of Alabama, PittsburghTuscaloosa, Pa.Ala.
Raymund, M
Senior engineer and senior scientistassistant professor of Aerospace Engineering, Westinghouse Electric Corp., Nuclear Energy SystemsUniversity of Alabama, PittsburghTuscaloosa, Pa.Ala.
Pages: 16 Published: Jan 1979
The behavior of semi-elliptical surface flaws in cylinders is of interest in the technology of pressure vessels. The object of this study is to determine the stress intensity factor distribution
around the crack front under arbitrary loading conditions for a longitudinal semi-elliptical flaw with a/c = 1/3 and Ri/t = 10; where a is the semi-minor axis of the ellipse, c is the semi-major
axis, Ri is the inside radius of the cylinder, and t is the cylinder thickness. Three crack depths are studied under various loading conditions: a/t = 0.25, 0.50, and 0.80.
The finite element method is used to determine the displacement solution. Parks' stiffness derivative method is used to find the stress intensity factor distribution around the semi-ellipse. The
immediate crack tip geometry is modeled by use of a macroelement containing over 1600 degrees of freedom.
Four separate loadings are considered: (1) constant, (2) linear, (3) quadratic, and (4) cubic crack surface pressure. From these loadings nondimensional magnification factors are derived to represent
the resulting stress intensity factors. By the method of superposition, comparisons are made with other investigators for pressure loading of a cylinder; the results agree within 8 percent of
published results.
crack propagation, pressure vessels, stress intensity factors, weight functions, surface flaws, semi-elliptical flaws, structural analysis, fatigue (materials)
Paper ID: STP34923S
Committee/Subcommittee: E08.05
DOI: 10.1520/STP34923S
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l'Hospital proof problem
Hey guys, first year university math student here. I need some help explaining the proof used in the scripts I'm studying from - just part of the proof to be more precise. English isn't my first
language and I don't have much experience writing/rewriting down proofs and I don't know how to write those nice latex symbols, so sorry in advance if something doesn't make sense:
(1), a is element of R (|a| =/= +oo)
(2), f and g are real functions
(3), limit x->a_+ (f'(x) / g'(x)) exists (must be element of R, or +-oo)
(4), limit x->a_+ (f(x)) = limit x->a_+ (g(x)) = 0
limit x->a_+ (f(x))/(g(x)) = limit x->a_+ (f'(x))/(g'(x))
I think I understand most of the proof but there's something right at the start that I'm completely stuck at and still don't understand precisely enough:
Let L=limit x->a_+ (f'(x) / g'(x)).
There exists delta>0, such that for all x element of (a,a+delta), f and g are both defined on this interval,
- I think this can be proved easily from (4), correct? Also, |f| and |g| are both smaller than some Epsilon>0. The following however, I don't understand at all:
and both f' and g' have a finite (not = oo or -oo) derivation on this interval, and also g'=/=0.
Why is the derivation necessarily finite?
To explain where I see the problem a bit more precisely, let's say:
f(x)=0 for all x element R, and therefore f'(x)=0 for all x element R
Now, from limit x->a_+ (f'(x) / g'(x)) = 0 , it should be possible to somehow prove, that there exists a delta>0, such that for all x element (a,a+delta), g'(x) is finite and non zero. I really don't
see it though, why can g'(x) not be +oo somewhere in that interval?
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Algorithms on Linked Lists
Wayne Snyder
CS 112, Spring 2013
These notes collect together a number of important algorithms operating on linked lists. They are intended as a supplement to the material in the textbook on p.142 and following. We also include some
basic information about recursive algorithms. In particular, the topics we cover are as follows:
1. Basic (iterative) algorithms on singly-linked lists;
2. An introduction to recursion [Optional]; and
3. Recursive algorithms on SLLs.
The algorithms are generally presented as Java functions (methods) operating on lists of integers, however, note that these algorithms are only the minimal specification of what would be used in a
practical situation. Your text presents these as generic data structures, and this is of course the more appropriate way to write real applications of thes principles. You would need to embed these
in the appropriate classes to accomplish a specific purpose.
Iterative Algorithms on Linked Lists
Basic data declarations
All the code below assumes the following declaration:
public class Node {
public int item;
public Node next;
Node() { // this would be the default, put here for reference
item = 0;
next = null;
Node(int n) {
item = n;
next = null;
Node(int n, Node p) {
item = n;
next = p;
We will assume that there is a variable which points to the head of the list:
Node head;
Note that references in Java are initialized to null by default. These constructors will simplify a number of the algorithms below. For example, to create a list with one element containing the item
5, we could write:
head = new Node(5);
To add a node containing a 7 to the front of an existing list, we could simply write
head = new Node(7, list);
and if we have a reference p to a node in a list, we can add a new node containing 13 after p (i.e., between the node p and the node p.next) by writing:
p.next = new Node(13, p.next);
Note that it can also be used to create simple linked lists in a single Java statement. For example, to create a list containing the integers 7, 8, and 12, we could write:
head = new Node( 7, new Node( 8, new Node( 12 ) ) )
Basic Paradigms for Chaining down a List
Basic chaining (example: printing out a list)
The basic thing you do with a list is to "chain along" the list, setting a pointer p to each node in turn and performing some operation at each node (such as printing out the list). This is done with
a simple for loop that initializes a reference to the first node and stops when it reaches null. Let's take printing out all the member of a list as an example.
The basic pattern is that this will execute once for each node in the list, pointing p at each node in turn:
for(Node p = head; p != null; p = p.next ) {
// OR:
Node p = head;
while( p != null ) {
p = p.next;
Written as a self-contained method taking the list as a parameter (the usual case), this would be:
void printList(Node p) {
for( ; p!=null; p = p.next) {
The basic pattern is that this will will execute once for each node in the list, pointing p at each node in turn:
for(Node p = head; p != null; p = p.next ) {
// Do something at each node in the list
(Note we are guaranteed by the loop condition that p is not null, so we can refer to p.item or p.next anytime we want inside the loop without worrying about NullPointerExceptions.)
Another simple example would be finding the length of a list, which can be done by simply incrementing a counter inside the for loop:
int count = 0;
for(Node p = head; p != null; p = p.next ) {
// now count contains the number of items in the list
Chaining down the list and stopping at the end or upon some condition
Suppose we wish to find the first node in the list satisfying some condition C (e.g., we are looking for a particular item k, so C would be p.item == k). A simple modification of the previous
technique is to add an if statement to the loop, possibly jumping out of the loop after executing some statements:
for(Node p = head; p != null; p = p.next )
if( C ) {
// do something to the node that p refers to
break; // if you want to stop the chaining along; or one could set p = null
Note that we can refer to p.item and p.next in the loop body with no worry of a NullPointerException, because we have already checked that p != null in the for loop condition.
Basic Paradigms for Modifying a List (Example: deleting a node)
Chaining down a list is often combined with some kind of modification of the structure of the list, for example, you might want to insert or delete a node. We will consider node deletion as an
example to illustrate the basic problem with a singly-linked list: a list goes in one direction, and you can go backwards! Suppose you want to delete the node pointed to by p in the following list:
head -> 2 -> 6 -> 9 -> .
The problem is that to delete the 6, you need to "reroute" the pointer in the node containing 2 so that it points to 9. But you don't have a pointer to this node! The simplest solution is just to
keep a "trailing pointer" q that always points to the node right before p:
head -> 2 -> 6 -> 9 -> .
^ ^
| |
q p
Now, to delete the 6, we can simply perform the following to "reroute" the next pointer in q to point to the node after p:
q.next = p.next;
This general technique is sometimes called the "inchworm" since the movement of the two pointers is like the way an inchworm moves its body.
Node p = head;
Node q = null;
while( p != null ) {
// During first iteration, p points to first element and q has no value;
// thereafter, p points to a node and q points to the previous node
q = p;
p = p.next;
// OR, using a for loop:
for(Node p = head, Node q = null; p != null; q = p, p = p.next ) {
// During first iteration, p points to first element and q has no value;
// thereafter, p points to a node and q points to the previous node
But it is a little bit complicated! What if we want to delete the first node? What if the list is empty? These are the special cases that make iterative algorithms a bit messy (we'll introduce a
technique using "header nodes" to avoid this below). For example, to delete the first node in the list containing a negative number, we could do the following:
if( head == null) // Case 1: List is empty, do nothing
else if( head.item < 0 ) // Case 2: have to delete first node?
head = head.next; // Yes, so reroute around the first node
else { // Case 3: Don't have to delete first node, so can use inchworm
Node p = head.next; // p points to second node
Node q = head; // q points to first node
while( p != null ) {
if( p.item < 0 ) {
q.next = p.next;
q = p; // chain along, and keep q trailing p
p = p.next;
// OR, using a for loop:
if( head == null) // Case 1: List is empty, do nothing
else if( head.item < 0 ) // Case 2: have to delete first node?
head = head.next; // Yes, so reroute around the first node
else { // Case 3: Don't have to delete first node, so can use inchworm
for(Node p = head, Node q = null; p != null; q = p, p = p.next ) {
if( p.item < 0 ) {
q.next = p.next;
In general, the Inchworm technique is the easiest way to manage this problem, so we will use it in the rest of these notes. However, be careful about those special cases! In general, we have to worry
about the following cases when modifying a linked list:
1. The list is empty;
2. The modification takes place at the beginning of the list (e.g., you are inserting a node at the front, or deleting the first node);
3. The modification takes place at the end of the list; and
4. The modification takes place somewhere in the middle.
Deleting a node involves cases 1 - 3, as we just saw; when inserting we have to worry about all four, as we will see below.
Iterative Algorithms for Linked Lists
We now present a "cookbook" of a number of useful iterative algorithms. Other algorithms can usually be created by suitably modifying one of the ones found here.
Print the List
void printList(Node h) {
for(Node p = h ; p != null; p = p.next )
You would call the method like this:
Finding the length of a list
int length(Node h) {
int count = 0;
for(Node p = h ; p!=null; p = p.next)
return count;
// You would call the method like this:
int n = length(head);
Looking up a item
This next function is a standard one; it returns a reference to the node containing n if it exists, and null otherwise:
Node lookup(Node h, int n) {
for(Node p = h; p != null; p = p.next )
if( p.item == n )
return p;
return null; // return null if item not found
// You would call the method like this:
Node q = lookup(head, 23);
Deleting an item in a list
Here is a version of delete which removes the first instance of a specific number from the list; as observed above, the complication is that we have a number of special cases, depending on where the
node to be deleted is. We have to have access to the head variable inside the method, in case we are deleting the first node, so you would have to write a method like this for each list (i.e., you
can't just write a static method to be used on all such lists).
void delete(int n ) {
if(head == null) // case 1: list is empty, do nothing
else if(head.item == n) // case 2: n occurs in first node
head = head.next; // skip around first node
else { // case 3: find the node before n using the inchworm technique
for(Node p = head.next, q = head; p != null ; q = p, p = p.next ) {
if( p.item == n ) {
q.next = p.next;
return; // if you delete this line it will remove all instances of the number
// You would call the method like this:
Inserting an item into a sorted list
We have shown above how easy it is to insert an item into the first position using the Node() constructor; inserting into a sorted list (in ascending order) is quite similar to the delete algorithm
and again uses the two-pointer technique. Again, you would have to write a separate algorithm for each list, since you might have to modify the head pointer. In this algorithm we will use a while
loop just to show how that works, as compared with the for loop.
void insertInOrder( int n ) {
if(head == null) // case 1: list is empty
head = new Node(n);
else if(head.item >= n) // case 2: n should be before the first node
head = new Node(n, head); // push on front of list
else { // case 3: find the node before where n should be using the inchworm technique
Node p = head.next;
Node q = head;
while( p != null ) {
if(p.item >= n) { // found insertion point, between p and q
q.next = new Node(n, p);
q = p; // chain along!
p = p.next;
q.next = new Node(n, null); // case 4: Node must be added at end of list
// You would call this as follows:
Copying a list
This algorithm is an example of one that is fairly messy in the iterative case, but almost trivial in the recursive case; we include it here so that you can compare it with the recursive one given
later. It is also presented as an example of a technique that is very useful in maintaining a linked list, that is, maintaining a pointer to the last node in the list (e.g., if you want to
continually add nodes to the end of the list, as here).
Node copyList(Node p) {
if(p == null) // Case 1: list is empty
return null;
else { // Case 2: list has at least one node
Node c = new Node(p.item); // put first node in place in copy
Node last = c; // Maintain a pointer to the last node to facilitate adding to the end of the list
p = p.next;
while( p != null ) {
last.next = new Node(p.item);
last = last.next; // chain along original list and with the last node in new list
p = p.next;
return c;
// You would call this as follows:
Node q = copyList(head);
Reversing a list
This algorithm is probably one of the most difficult for linked lists; it uses three references which chain down the list together and rearrange the pointers so that node q, instead of pointing to p,
now points to r; doing this for each node reverses the entire list. This was a standard exam question on the PhD Qualifying Exam at UPenn when I was a student there..... I've also heard that it is a
common interview question at software companies!
void reverse() {
if(head == null)
Node p = head.next, q = head, r;
while ( p != null ) {
r = q; // r follows q
q = p; // q follows p
p = p.next; // p moves to next node
q.next = r; // link q to preceding node
head.next = null;
head = q;
Eliminating a Messy Special Case: Linked Lists with Header Nodes
The special cases 1 and 2 in these algorithms, when you need to worry about whether you need to change the head pointer, causes significant problems:
You have to write these special cases into the algorithms, as we have seen above; and
You must write a separate method for EACH linked list, as the pointer head must be explicitly mentioned in the algorithm.
This latter problem is really the most important, as it makes us write multiple versions of the same code, which, as we saw in the case of generics, is a real problem. We would like to write ONE
method which can be used for all such lists. This will be solved elegantly when we use recursion, but we can solve the problem in the iterative case by using a dummy first node that contains no value
(or a sentinel value, such as Integer.MIN_VALUE) as a item. In this case, you would initialize your list using:
Node head = new Node();
Alternatively, and sometimes very usefully, we can use the header node to store useful information such as the length of the list. The general outline of a for loop which chains down a list keeping a
training pointer is now as follows. (We will show the differences from the previous methods in red -- you will see the the differences are usually quite minimal!)
for(Node p = head.next, q = head; p != null; q = p, p = p.next ) {
// During first iteration, p points to first element and q to the header node;
// thereafter, p points to a node and q points to the previous node
For example, if we want to eliminate the first node in our list which contains a negative number, we would write the following:
for(Node p = head.next, q = head; p != null; q = p, p = p.next ) {
if( p.item < 0 ) {
q.next = p.next;
You do not HAVE to use such a dummy header node, especially if you are not modifying the lists; when you are inserting or deleting nodes, however, they make your life simpler. In any case, they are
unnecessary for recursive algorithms, as we shall see.
Iterative Algorithms for Linked Lists with Header Nodes
We now present the same algorithms in versions which assume a global head variable, and header node, as discussed above. They take the head pointer as a parameter, so that they can be used on
multiple lists.
Print the List
void printList(Node h) {
for(Node p = h.next ; p != null; p = p.next )
You would call the method like this:
Finding the length of a list
int length(Node h) {
int count = 0;
for(Node p = h.next ; p!=null; p = p.next) {
return count;
You would call the method like this:
int n = length(head);
Looking up a item
Node lookup(Node h, int n) {
for(Node p = h.next; p != null; p = p.next )
if( p.item == n )
return p;
return null; // return null if item not found
You would call the method like this:
Node q = lookup(head, 34);
Deleting an item in a list
Now we will see the big advantage of the header node technique: the header node removes cases 1 and 2, and only case 3 remains! The resulting algorithm is much simpler. In addition, you can now
delete all instances of a given node by simply removing one line from the code!
void delete(Node h, int n ) {
for(Node p = h.next, q = h; p != null ; q = p, p = p.next )
if( p.item == n ) {
q.next = p.next;
return; // if you delete this line it will remove all instances of the number
You would call the method like this:
Inserting an item into a sorted list
Again, the algorithm is simpler, since cases 1 and 2 are gone!
void insertInOrder( Node h, int n ) {
Node p = h.next; // p points to first real node in list
Node q = h; // q trails p
while( p != null ) {
if(p.item >= n) { // found insertion point, between p and q -- This is case 3
q.next = new Node(n, p);
q = p;
p = p.next;
q.next = new Node(n, null); // Node must be added at end of list --- This is case 4
You would call the method like this:
Reversing a list
void reverse(Node h) {
Node p = h.next, q = h, r;
while ( p != null ) {
r = q; // r follows q
q = p; // q follows p
p = p.next; // p moves to next node
q.next = r; // link q to preceding node
head.next.next = null;
head.next = q;
Recursive Algorithms
Recursively defined algorithms are a central part of any advanced programming course and occur in almost every aspect of computer science. Although they are difficult to understand initially, after
one gets the knack, they are easier to write, debug, and understand than their iterative counterparts. In many cases, the only realistic solution possible for a certain problem is recursive.
Let us examine the definition of the factorial function. We can define the factorial of a number n, notated n!, in two ways:
1. n! is the product of all the integers between 1 and n, inclusive;
2. if n = 1, then n! = 1, otherwise n! = n * ( n-1)!
The first definition gives us an explicit way to calculate n! which involves iterating through all the numbers from 1 to n and keeping a running sum; it could be expressed in Java as follows:
int factorial( int num ) {
int fact = 1;
for (int i = 1; i <= num; ++i)
fact = fact * i;
The second definition of n! is, at first glance, nonsense, because we are defining something in terms of itself. Its like asking someone what the food at a Thai restaurant is like and he tells you,
``Well, it's kind of like food from Thailand." Or you look up ``penultimate" in the dictionary and it says ``just after propenultimate;" but when you look up ``propenultimate" it's defined as ``just
before penultimate." Actually our example is not exactly this paradoxical, because we are defining our object, if you look closely, in terms of a slightly different object. That is, n! is defined in
terms of ( n-1)!, which has a smaller value before the ``!". Also, the definition has a condition: when the value of n is small enough, i.e., 1, the factorial is just given explicitly as 1. Since the
recursive part always defines the factorial in terms of the factorial of a smaller number, we must reach 1 eventually. This is the trick which allows us to define mathematical objects in this way. We
must define a mathematical function explicitly for some values, and then we can define other values in terms of the function itself, as long as the function will eventually reach one of the explicit
values. Let us look at the Java for this version of the function:
int factorial( int num ) {
if ( num == 1 )
return 1;
else return num * factorial( num - 1 );
This function has the following standard features of any recursively defined procedure or function:
1. it has an if or a switch statement;
2. this if statement tests whether the function input is one of the base cases, i.e., one for which a value is returned explicitly;
3. if the base case is found, an action is performed or a value is returned which does not involved calling the function again;
4. if the base case is not found, the function calls itself on an argument which is closer to the base case than the original argument.
When this program runs, the computer has to keep calling this function on increasingly smaller values of n until n equals 1. For example, to find the value of Factorial(4), the computer has to find
out the value of Factorial(3); to find this value, it has to know the value of Factorial(2); to get this value, it has to know the value of Factorial(1). But it knows the value of Factorial(1), since
we told it that this is 1. Now it can find the value of Factorial(2), etc. all the way back to Factorial(4). It is important to realize that the computation of the Java function for a given value has
to wait until it gets though with all the function calls it makes, even when it calls itself. Thus there will be many different invocations for the same piece of code even though only one of these
will actually be executing; the rest will be waiting for the function calls they made to finish. You should try tracing the factorial program above for, say, Factorial(5), to get a feel for the way
it works.
Let's look at another simple recursive algorithm similar to the factorial function:
int power( int num, int exponent ) {
if ( exponent == 1 )
return num;
else return num * power( num, exponent - 1 );
Here we are determining the value of an integer num raised to a power exponent. We could have written this explicitly by just creating a for loop to multiply num by itself exponent number of times,
i.e., 5^4 = 5 * 5 * 5 * 5. But the recursive algorithm says that, for example, 5^4\ is just 5 * (5^3), which is just 5 * (5 * (5^2) ), which is just 5 * (5 * (5 * (5^1) ) ), which is just 5 * 5 * 5 *
5. So they really do the same thing in different ways. Note again that the recursive call involves the function calling itself on arguments which get closer to the base case--if you keep subtracting
1 from exponent you will eventually reach 1. Try this algorithm on Power(2, 5).
Another recursive algorithm we could write would be for calculating the Nth Fibonacci number. Recall that the Fibonacci numbers form a series in which the first two values are both 1, and each
successive value is the sum of the previous two values:
1 1 2 3 5 8 13 21 34 55 89 ....
Thus the third Fibonacci number is 2, the seventh is 13, and so on. Note how the definition is phrased: ``the first two values are both 1" (an explicit answer is given), ``and each successive value
is the sum of the previous values"(the rest are defined in terms of previous values in the series). This is clearly translatable into a recursive algorithm with almost no effort:
int fibonacci( int n ) {
if ( n < 2 )
return 1;
else return fibonacci(n-2) + fibonacci(n-1);
This is obviously a Java version of the English definition above, but will it work? After all, it calls itself not once but twice! The base case assures us, however, that this must stop eventually,
since we call the function on smaller values each time. It must reach Fibonacci(1) or Fibonacci(2) eventually. In fact, this is not a very efficient way to calculate the Fibonacci numbers, since we
must cover the same ground twice to get each number. It does work, however, and is an exact translation of the English definition we started with. In other words, it is a more natural expression of
the original problem than an iterative algorithm, because the original definition is recursive. Again, try this on some small values to convince yourself that it works.
A slightly more difficult algorithm which can be written recursively is Euclid's Greatest Common Divisor algorithm, which has pride of place as the oldest recursive algorithm in existence. This
ancient Greek mathematician discovered that we can find the largest integer which divides two given integers evenly if we generate a series of values a follows:
1. write down the two integers;
2. divide the first by the second and write down the remainder from that division;
3. if the remainder is 0, then the greatest common divisor is the number immediately to the left of the 0, i.e., the number you divided into the previous number to get a remainder of 0;
4. if the remainder is not 0, then repeat from step 2 using the last two integers in the list.
For example, starting with the two integers 28 and 18, we would generate the series:
28 18 10 8 2 0.
Thus 2 is the greatest common divisor of 28 and 18. This method is essentially a recursive algorithm, although it may not be obvious at first. Notice that we perform the same action on each pair of
numbers: we divide the first by the second and write down the remainder, then continue with the second number and the remainder just obtained, etc., until we reach 0. The recursive algorithm looks
int gcd( int num1, int num2 ) {
if ( (num1 % num2) == 0 )
return num2;
return gcd( num2, (num1 % num2) );
This algorithm will implicitly create the list (except for the 0, which just indicates that the previous number divides the number before it evenly) that we showed above if you call gcd(28, 18). It's
tricky, but it does nothing really different than the recursive algorithms we have examined so far. It's worth tracing through on some simple input.
We have presented here a number of recursive functions returning values, but it is important to realize that void functions (not returning values) can be written recursively as well. For example,
many sorting algorithms are written as recursively.
Other recursive algorithms are presented below for singly-linked lists and for tree structures. These are important algorithms which show the advantages of recursion clearly. For some, like the
printing procedures for singly-linked lists, the iterative and recursive versions are of about equal complexity. For others, like the cell deletion algorithm, the difference is more pronounced in
favor of the recursive version. For another class of algorithms, such as the tree walk and insertion procedures, the recursive version is really the only reasonable solution. In general, when a data
structure is defined recursively(like a tree or a linked list) the most natural algorithms are recursive as well. To use these advanced data structures one must have a firm understanding of
recursion. Those interested in pursuing this topic should try writing the recursive algorithms suggested in the notes on singly-linked lists below and should look up other recursive algorithms, such
as Quicksort, Mergesort, the Tower of Hanoi, or practically any algorithm which involves trees.
Recursive Algorithms on Linked Lists
The recursive algorithms depend on a series of method calls to chain along the list, rather than an explicit for loop. The recursive versions of most linked-list algorithms are quite concise and
elegant, compared with their iterative counterparts. For example, we do not need dummy header nodes; we will therefore not use them in this section.
Print the List
This is a simple algorithm, and good place to start. Recursion allows us flexibility in printing out a list forwards or in reverse (by exchanging the order of the recursive call):
void printList( Node p ) {
if (p != null) {
printList( p.next );
void printReverseList( Node p ) {
if (p != null) {
printReverseList( p.next );
// Example of use:
printList( head ); // Note: No dummy header node!
Finding the length of a list
Another simple recursive function.
int length( Node p ) { // Example call: int len = length(head.next);
if (p == null)
return 0;
return 1 + length( p.next );
When changing the structure of a linked list by deleting to adding nodes, it is useful to think in terms of reconstructing the list. Suppose you wanted to trivially reconstruct a list by reassigning
all the references. You could do this:
Reconstruct a list
Node construct( Node p ) {
if( p == null )
return null;
else {
p.next = construct( p.next );
return p;
// Example of use:
head.next = construct( head );
Pretty silly, right? But if we use this as a basis for altering the list recursively, then it becomes a very useful paradigm. All you have to do is figure out when to interrupt the silliness and do
something useful. Here is a simple example, still kind of silly:
Node addOne( Node p ) {
if( p == null )
return null;
else {
p.next = addOne( p.next );
return p;
// Example of use:
head.next = AddOne( head );
This recursively traverses the list and adds one to every item in the list. Following are some definitely non-silly algorithms using the approach to traversing the list.
This one is extremely simple, useful and not at all silly. Instead of reconstructing the same list, reconstruct another list, thereby building a copy (compare with the complicated iterative version
Copy a list
Node copy( Node p ) {
if( p == null )
return null;
return new Node(p.item, copy( p.next ));
// Example of use
newList = copy( head );
I'll repeat again that when using this "reconstruct the list" paradigm, we do NOT need to use a dummy header node to avoid the special case of the first node in the list; this next algorithm shows
the advantage:
Inserting an item into a sorted list
Node insertInOrder( int k, Node p ) {
if( p == null || p.item >= k )
return new Node( k, p );
else {
p.next = insertInOrder( k, p.next );
return p;
// Example of use:
head = insertInOrder( 7, head );
Deleting an item from a list
This algorithm deletes the first occurrence of an item from a list. A simple change enables this algorithm to delete all occurrence of the item, by continuing to chain down the list after the item
has been found. We assume that the list is unordered; you can easily change this to stop after finding a item beyond the search item by changing the first if condition.
Node delete( int k, Node p ) {
if( p == null ) // if the list is ordered use: ( p == null || p.item > k )
return p;
else if ( p.item == k )
return p.next; // if you want to delete all instances, use: return deleteItem( k, p.next );
else {
p.next = delete( k, p.next );
return p;
Deleting the last element in the list
This is a rather messy process in the iterative case; the use of recursion makes it much simpler:
public static Node deleteLast( Node p ) {
System.out.println(" " + p.item);
if( p == null || p.next == null )
return null;
else {
p.next = deleteLast( p.next );
return p;
Appending two lists
Appending two lists is a simple way of creating a single list from two. This function adds the second list to the end of the first list:
Node append( Node p, Node q ) {
if ( p == null)
return q;
else {
p.next = append( p.next, q );
return p;
Example of use
head = append(head, anotherList);
Merging two lists
Here is a more complex function to combine two lists; it simply zips up two lists, taking a node from one, then from the other. The first list in the original call now points to the new list.
Node zip( Node p, Node q ) {
if ( p == null)
return q;
else if ( q == null)
return p;
else {
p.next = zip( q, p.next ); // Note how we exchange p and q here
return p;
Example of call:
head = zip( head, anotherlist );
If head points to 3 4 7 and anotherlist points to 2 5 6 8, then at the end of this call to zip, head will point to 3 2 4 5 7 6 8.
Merging two sorted lists
Here is another more complex function to combine two lists; this one merges nodes from two sorted lists, preserving their order:
Node merge( Node p, Node q ) {
if ( p == null)
return q;
else if ( q == null)
return p;
else if (p.item < q.item) {
p.next = merge( p.next, q );
return p;
else {
q.next = merge( p, q.next );
return q;
Example of call:
head = merge( head, anotherlist );
If head points to 3 4 7 and anotherlist points to 2 5 6 8, then at the end of this call to zip, head will point to 2 3 4 5 6 8.
Other linked-list algorithms to try.....
Some other recursive algorithms(in increasing order of difficulty) you might want to try writing along the lines of those above are:
• Summing all the elements in an integer list, or finding the largest element
• Checking if two lists are identical
• Unzip a list into two lists, so that zip(unzip(h)) returns the same list h
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Mathematics 2006 - Mathematical Modeling
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Mathematics 2006 - Mathematical Modeling
Mark Kot, University of Washington
Elements of Mathematical Ecology provides an introduction to classical and modern mathematical models, methods, and issues in population ecology. The first part of the book is devoted to simple,
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Edited by Ellis Cumberbatch, Claremont Graduate School, California
Edited by Alistair Fitt, University of Southampton
Industrial Mathematics is growing enormously in popularity around the world. This book deals with real industrial problems from real industries. Presented as a series of case studies by some of the
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different problems are then considered, ranging from the cooking of cereal to the analysis of epidemic waves in animal populations. Throughout the work the emphasis is on telling industry what they
really want to know. This book is suitable for all final year undergraduates, master's students, and Ph.D. students who are working on practical mathematical modeling.
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Glenn Fulford, University College, Australian Defence Force Academy, Canberra
Peter Forrester, La Trobe University, Victoria
Arthur Jones, La Trobe University, Victoria
The theme of this book is modeling the real world using mathematics. The authors concentrate on the techniques used to set up mathematical models and describe many systems in full detail, covering
both differential and difference equations in depth. Among the broad spectrum of topics studied in this book are: mechanics, genetics, thermal physics, economics and population studies.
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Elizabeth S. Allman, University of Southern Maine
John A. Rhodes, Bates College, Maine
Focusing on discrete models across a variety of biological subdisciplines, this introductory textbook includes linear and non-linear models of populations, Markov models of molecular evolution,
phylogenetic tree construction from DNA sequence data, genetics, and infectious disease models. Assuming no knowledge of calculus, the development of mathematical topics, such as matrix algebra and
basic probability, is motivated by the biological models. Computer research with MATLAB is incorporated throughout in exercises and more extensive projects to provide readers with actual experience
with the mathematical models.
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K. Chen, University of Liverpool
Peter J. Giblin, University of Liverpool
A. Irving, University of Liverpool
Mathematical Explorations with MATLAB examines the mathematics most frequently encountered in first-year university courses. A key feature of the book is its use of MATLAB, a popular and powerful
software package. The book's emphasis is on understanding and investigating the mathematics by putting the mathematical tools into practice in a wide variety of modeling situations. Even readers
who have no prior experience with MATLAB will gain fluency. The book covers a wide range of material: matrices, whole numbers, complex numbers, geometry of curves and families of lines, data
analysis, random numbers and simulations, and differential equations from the basic mathematics. These lessons are applied to a rich variety of investigations and modeling problems, from sequences
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This original text on the basics of option pricing is accessible to readers with limited mathematical training. It is for both professional traders and undergraduates studying the basics of
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option cost function and of the computational Black-Scholes formula; three different models of European call options with dividends; a new, easily implemented method for estimating the volatility
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|
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one and two sample propotion test
One Sample Proportion Test
To test the proportion of one certain outcome in a population which follows Bernoulli distribution, we can use the 1_proportion function in Minitab. This function produces a confidence interval and
hypothesis test of the proportion. The data format can be either raw( in the form of "failure" and " success") or summarized. Our example here is based on summarized data. Since we get summarized
data most of the time for this kind of experiments. If it is not the case, Tally command (see Generate Random Data) can give us summarized data right away.
In Exercise 8.3 (BPS, Chapter 8, page 431), the college president says "99% of the alumni support my firing of coach Boggs." However in the SRS of 200 out of the colleges 15,000 living alumni only 76
of them support firing coach. Now we need procedure a hypothesis test to address the question whether the proportion of alumni who support firing coach is equal to 99% as the president said. Hence,
we are going to test: Ho: p=0.99 Ha: p=/=0.99
Select Stat->Basic Statistics->1 proportion from the menu. Since we have only summarized data in this case, which is 200 trails and 76 successes, check the box "Summarized data", fill out the number
in appropriate location.
The Option subdialog allows us to specify the confidence level, test probability, alternative hypothesis, and whether Minitab should employ test statistic and interval based on Normal distribution.
If this box is checked, Minitab is going to calculation C.I. as z) asMinitab will use an exact method to calculate the test probability by default. There are slight difference between the two method,
which could become significant when n is small.
The 95% C.I. of the proportion who support firing coach is (0.313, 0.447), and the P-value shown in output is 0.000 which gives us strong evidence to reject null hypothesis.
With two binomial proportions in our hand, one frequently asked question is whether they are equal or not. In the word of statistics, the following hypothesis needs to be tested:
In Exercise 8.24 (BPS Chapter 8, page 451), 161 people who visited one hospital's emergency room in a 6-month study period with injuries from in-line skating were interviewed. The interviewer found
that 53 people were wearing wrist guards and 6 of them had wrist injuries. Of the 108 who did not wear wrist guards, 45 had wrist injuries. We are interested in the difference between the proportions
of wrist injuries in the population wearing wrist guard and the population without.
Select Stat->Basic Statistics->2 proportions from menu, since we got summarized data here, check the box of "Summarized Data" and fill it out as shown.
The Option subdialog box gives us a chance to specify the confidence level, test proportion, alternative hypothesis, and whether Minitab should use a pooled estimate of p for the test.
This 2-proportions function calculates the confidence interval as :
By default, Minitab uses separate estimates of p for each population and calculates z as :
Once we check the box of "Use pooled estimate of p for test", Minitab calculates z as :
The output shows that the P-value is 0.000, so there is strong evidence that wearing wrist guard reduce the rate of wrist injuries.
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Math Forum Discussions
Math Forum
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Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.
Topic: Fwd: Re: Saccheri quadrilaterals
Replies: 2 Last Post: Jun 13, 2007 11:29 AM
Messages: [ Previous | Next ]
Fwd: Re: Saccheri quadrilaterals
Posted: Jun 6, 2007 7:57 PM
I'm trying to picture your drawing... Does the quadrilateral inside
the other quadrilateral share a common bottom base (with the two
right angles) with the arms of the smaller lying on the arms of the
larger? In that case you have two quadrilaterals. The lower one is a
saccheri quadrilateral but the top one is not. It has two obtuse
angles and two acute angles.
I can see how case 2 is a contradiction, and cannot occur. Case 3 can
occur. However, they cannot fit one inside the other to make two
quadrilaterals. If they share a common base-line and perpendicular
bisectors to the bases, then the arms of one will cross the upper
base (summit) of the other.
>To: geometry-college@support1.mathforum.org
>I was trying to do this too. And this is what I think should work to
>prove the statement
>Assume the saccheri quadrilaterals are not congruent. That means for
>these two quadrilaterals,
>There are three cases where the quadrilaterals will not be equal:
>1. arms are not equal and bases are not equal
>2. arms are not equal and bases are equal
>3. arms are equal and bases are not equal
>If can be proven easily that when arms are equals base must be
>equal. Thus 3 is a contradiction.
>When arms are not equal, assume one of the quadrilateral arms are
>larger than the other. Then in the larger the quadrilateral you can
>construct another quadrilateral (inside) where arms are equal to the
>smaller quadrilateral's arms. You can show this inside quadrilateral
>is congruent to the smaller quadrilateral. But that is not possible
>as it would create a left-over quadrilateral with angle-sum larger than 360.
>Let me know if you got a different way to proving this.
Ben Saucer
e-mail: bsaucer2@comcast.net
web page: www.saucersdomain.com
ICQ: 20610314
Date Subject Author
6/6/07 Fwd: Re: Saccheri quadrilaterals Ben Saucer
6/12/07 Re: Fwd: Re: Saccheri quadrilaterals Mashrur Mia
6/13/07 Re: Fwd: Re: Saccheri quadrilaterals Ben Saucer
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Power cycle efficiency problem
1. The problem statement, all variables and given/known data
A power cycle has a thermal efficiency of 40% and generates electricity at a rate of 100 MW. The electricity is valued at $0.08 per kW * h. Based on the cost of fuel, the cost to supply Qin is $4.50
per GJ. For 8000 hours of operation annually, determine, in $.
a. The value of electricity generated per year.
b. The annual fuel cost.
c. Does the difference between the results of parts (a) and (b) represent profit? Discuss.
2. Relevant equations
Thermal Efficiency = W cycle/ Q in
3. The attempt at a solution
I don't even know where to begin.
Can someone please get me on the right track.
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DNS CFD-Simulation, Navier-Stokes Equations, and Turbulence by M. Kostic
DNS CFD-Simulation, Navier-Stokes Equations, and Turbulence
By M. Kostic , June 1, 2007. This page posted at: www.kostic.niu.edu/DNS-CFD-NS-Turbulence.htm
See the related Web Links at the end
The Direct Numerical Simulation (DNS) is a simulation in computational fluid dynamics (CFD) in which the Navier-Stokes equations are numerically solved without any turbulence model.
The Navier-Stokes equations are the basic governing equations for a continuum medium, a viscous fluid. It is a vector equation (or three scalar components) obtained by applying Newton's Law of Motion
to a fluid element and is also called the momentum equation. The solution is obtained by integrating the governing equations, including conservation of mass and energy equations, with boundary and
constitutive fluid-property equations (stress-strain fluid response, etc.) over fluid continuum domain.
Turbulence is a very complex stochastic phenomenon which is not (and may never be) well understood. Turbulence develops when the “loose” fluid-structure (constitutive property) responds to flow
instabilities (rapid variation of pressure and velocity in space and time with appearance of unsteady vortices and eddies on many scales which interact with each other) initiated by diverse
extraneous and internal “disturbances.” The large scale flow instability may develop by formation of eddies of many different length scales, which will be “resisted” by very complex fluid-structure
(visco-elastic molecular particulate structure), which in turn will "cascade" and break large scale flow instability into fine turbulent structure with additional viscous dissipation of energy.
Turbulence is “damping” (stabilizing) the flow instabilities by the small-scale energy dissipation and thus make “orderly disorder (how ironic!),” in similar (but much more complex) way as viscosity
(molecular momentum diffusion) does order laminar flow.
For example, a simple, well-ordered, fully-developed laminar flow in a smooth pipe may become unstable due to many reasons, like pipe surface small irregularities (roughness) or small vibration, or
fluid impurities (small foreign particles or bubbles). For a given pipe size and fluid property the flow will always be stabilized by viscosity if velocity (thus fluid inertia and Reynolds number,
Re) are billow certain value (Re<2000 for pipe flow). For higher velocity (inertia, i.e. Re) due to diverse flow disturbances (imperfections) always present in reality, the instability will develop
and turbulence will occur to “order” the flow by fluid “disorderly-stochastic” reaction. Careful experiments in laboratory were conducted by minimizing all disturbances (roughness, vibration,
impurities, initial and boundary conditions) and laminar flow in pipe was maintained up to 20,000, 40,000 and even 100,000 Reynolds number. There is no upper theoretical limit for laminar pipe flow
transition to turbulence, but only the lower practical limit (i.e., Re about 2000 for pipe flow) where all practical disturbances will be “damped” by fluid viscous forces.
Therefore, there is nothing in the Navier-Stokes equations to model real turbulence phenomena, since “external” disturbances and irregularities at the boundaries, which initiate flow instabilities,
along with very complex particulate fluid structure with impurities, which may promote and/or damp flow instabilities, are not modeled as such. For example, the usual direct numerical simulation
(DNS) will never predict pipe turbulent flow, but only laminar, regardless of Reynolds value, the same way the laminar pipe flow could be maintained in reality up to much higher Re number values (up
to 100,000 or more!) than the (minimum) critical number of about 2,000.
The results of direct numerical simulation (DNS) in more complex flow configurations will result in much more fine flow details including flow instabilities and “its own turbulence,” due to
instability and imperfections of continuum media simulation and numerical discretization methods used to achieve the solution. Such fine and transient flow fluctuations, as outcome of a very detailed
DNS simulation cannot be the same as the real turbulence (although it may look similar in some instances) since reality with all “extraneous” disturbances (including imperfect boundary conditions)
and discrete (sub- and molecular fluid structure) is not even modeled by the DNS governing and other equations.
Even under idealized simulation conditions the existence and uniqueness of classical solutions of the 3-D Navier-Stokes equations is still an open mathematical problem and is one of the Clay
Institute's Millennium Problems:
* http://www.claymath.org/millennium/Navier-Stokes_Equations/Official_Problem_Description.pdf
The above is only to highlight limits and uncertainties of CFD simulation which is in many ways similar to limits and uncertainties of experimental investigation. It is important to repeat here, that
computational simulation and experimentation engineering have their exclusive strengths and weaknesses and can not replace each-other, but if properly integrated, will strongly complement each-other,
resulting in a synergistic result which is much greater than the simple sum of the two constituents.
Some interesting (and useful) Web links:
*CFD-online/Wiki*CFD Resources Online*CFD Online Discussion Forums*DNS-Direct Numerical Simulation*Turbulence*Turbulence modeling* Turbulent Times for Fluids*Tackling Turbulence with Supercomputers*
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Physics Forums - Modal Logic & Topology
General Math
cronxeh Aug18-05 04:02 AM
Modal Logic & Topology
How do we unify modal logic with topology or perhaps complex analysis/probability&random variables?
Provided the principles of modal logic, is it possible to translate philosophy into computer language and create the real AI?
Is it possible to apply modal logic to Shor's computational algorithm? How about quantum encryption? Is anyone doing any research on this field?
Huh? What do you mean by modal logic? The logic that deals with necessity and possibility (modalities)? What does that have to do with complex analysis or topology?
All times are GMT -5. The time now is 11:02 AM.
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Shading a sphere in opengl Help please! [Archive] - OpenGL Discussion and Help Forums
03-19-2009, 08:18 PM
Hey I have a sphere that I created manually and am now trying to calculate the normals so that it shades correctly. I am shining a spotlight on the sphere from the camera's position. The light shines
but instead of the sphere looking like a sphere it looks like it has a grid on it. If anyone knows what I might be doing wrong it would be a life saver!
Here is a link to a picture of the light on the sphere:
And a pic of the light up close on the sphere:
Below is the code for creating the sphere, calculating the normals, and drawing the sphere.
// Calculates the sphere points.
void make_sphere( OBJECT *po, double ray, int rs, int vs ) {
double r,u,t,t_inc,u_inc;
int i,j, level_count;
r = ray;
u_inc = 2*M_PI/rs;
u = -M_PI/2;
for(i=0; i <=rs; i++)
t_inc = M_PI/vs;
for(j=0; j <=vs; j++)
po->vertices[i][j][0] = (r*cos(t)*cos(u));
po->vertices[i][j][1] = (r*sin(t));
po->vertices[i][j][2] = (-(r * cos(t)*sin(u)));
po->vertices[i][j][3] = 1;
t= t+t_inc;
Normalize(po, ray); // Calculates Normal vectors
// Calculates the normal vectors
void Normalize(OBJECT *po, GLfloat radius)
int i,j;
for(i=0; i <= crt_rs; i++)
for(j=0; j <= bcrt_vs; j++)
po->normals[i][j][0] = po->vertices[i][j][0] / radius;
po->normals[i][j][1] = po->vertices[i][j][1] / radius;
po->normals[i][j][2] = po->vertices[i][j][2] / radius;
po->normals[i][j][3] = po->vertices[i][j][3] / radius;
// Draws the sphere and normals.
void draw_sphere(OBJECT *po, int rs, int vs)
int i, j;
for(i=0; i < rs; i++)
for(j=0;j < vs;j++)
glVertex4f(po->vertices[i][j][0],po->vertices[i][j][1],po->vertices[i][j][2], po->vertices[i][j][3]);
glVertex4f(po->vertices[i+1][j][0],po->vertices[i+1][j][1],po->vertices[i+1][j][2], po->vertices[i+1][j][3]);
glVertex4f(po->vertices[i+1][j+1][0],po->vertices[i+1][j+1][1],po->vertices[i+1][j+1][2], po->vertices[i+1][j+1][3]);
glVertex4f(po->vertices[i][j+1][0],po->vertices[i][j+1][1],po->vertices[i][j+1][2], po->vertices[i][j+1][3]);
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Matches for:
Student Mathematical Library
2009; 234 pp; softcover
Volume: 47
ISBN-10: 0-8218-4699-X
ISBN-13: 978-0-8218-4699-5
List Price: US$39
Member Price: US$31.20
Order Code: STML/47
This book is based on notes from the course developed and taught for more than 30 years at the Department of Mathematics of Leningrad University. The goal of the course was to present the basics of
quantum mechanics and its mathematical content to students in mathematics. This book differs from the majority of other textbooks on the subject in that much more attention is paid to general
principles of quantum mechanics. In particular, the authors describe in detail the relation between classical and quantum mechanics. When selecting particular topics, the authors emphasize those that
are related to interesting mathematical theories. In particular, the book contains a discussion of problems related to group representation theory and to scattering theory.
This book is rather elementary and concise, and it does not require prerequisites beyond the standard undergraduate mathematical curriculum. It is aimed at giving a mathematically oriented student
the opportunity to grasp the main points of quantum theory in a mathematical framework.
Undergraduate and graduate students interested in learning the basics of quantum mechanics.
"The present volume has several desirable features. It speaks to mathematicians broadly, not merely practitioners of some narrow specialty. It faithfully explains physical ideas/concerns, rather than
addresses the mathematician eager only to glean from physics a purely mathematical problem to attack. This book accomplishes its task as quickly as one could hope but still achieves interesting
applications...Highly recommended."
-- D.V. Feldman, Choice
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Homework Help
Posted by Ryan on Wednesday, May 8, 2013 at 1:25am.
15. Given ensuing information, determine the least cost and the least cost mix of on-
and off-shore pipe.
The intake facility is in the water 2 miles vertically north of the shoreline
The water filtration plant is on land 1 mile vertically south of the shoreline
The intake and the plant are horizontally 3 miles apart
The cost to lay pipe off-shore is $60,000 per mile and the cost to lay pipe on-
shore is $30,000 per mile
The shoreline runs east and west in a horizontal straight line
• calc - Steve, Wednesday, May 8, 2013 at 10:55am
If the pipe comes ashore x miles from the point directly south of the intake, the cost (in $K) is
c(x) = 60√(x^2+4) + 30√((3-x)^2 + 1)
dc/dx = 60x/√(x^2+4) - 30(3-x)/√((3-x)^2 + 1)
dc/dx=0 when x=1
So, we have √5 miles of offshore pipe and √5 miles of onshore pipe.
Cost = 90√5 $K = $201,246.12
• calc - James, Wednesday, May 8, 2013 at 8:35pm
can you tell me how you got √(x^2+4) and √((3-x)^2 + 1)
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Cryptography FAQ (01/10: Overview)
Archive-name: cryptography-faq/part01
Last-modified: 1999/06/27
View all headers
This is the first of ten parts of the sci.crypt FAQ. The parts are
mostly independent, but you should read this part before the rest. We
don't have the time to send out missing parts by mail, so don't ask.
Notes such as ``[KAH67]'' refer to the reference list in the last part.
Disclaimer: This document is the product of the Crypt Cabal, a secret
society which serves the National Secu---uh, no. Seriously, we're the
good guys, and we've done what we can to ensure the completeness and
accuracy of this document, but in a field of military and commercial
importance like cryptography you have to expect that some people and
organizations consider their interests more important than open
scientific discussion. Trust only what you can verify firsthand.
And don't sue us.
Many people have contributed to this FAQ. In alphabetical order:
Eric Bach, Steve Bellovin, Dan Bernstein, Nelson Bolyard, Carl Ellison,
Jim Gillogly, Mike Gleason, Doug Gwyn, Luke O'Connor, Tony Patti,
William Setzer. We apologize for any omissions.
Archives: sci.crypt has been archived since October 1991 on
ripem.msu.edu, though these archives are available only to U.S. and
Canadian users. Another site is rpub.cl.msu.edu in /pub/crypt/sci.crypt/
from Jan 1992.
The sections of this FAQ are available via anonymous FTP to rtfm.mit.edu
as /pub/usenet/news.answers/cryptography-faq/part[xx]. The Cryptography
FAQ is posted to the newsgroups sci.crypt, talk.politics.crypto,
sci.answers, and news.answers every 21 days.
The fields `Last-modified' and `Version' at the top of each part track
1999: There is a project underway to reorganize, expand, and update the
sci.crypt FAQ, pending the resolution of some minor legal issues. The
new FAQ will have two pieces. The first piece will be a series of web
pages. The second piece will be a short posting, focusing on the
questions that really are frequently asked.
In the meantime, if you need to know something that isn't covered in the
current FAQ, you can probably find it starting from Ron Rivest's links
at <http://theory.lcs.mit.edu/~rivest/crypto-security.html>.
If you have comments on the current FAQ, please post them to sci.crypt
under the subject line Crypt FAQ Comments. (The crypt-comments email
address is out of date.)
Table of Contents
1. Overview
2. Net Etiquette
2.1. What groups are around? What's a FAQ? Who am I? Why am I here?
2.2. Do political discussions belong in sci.crypt?
2.3. How do I present a new encryption scheme in sci.crypt?
3. Basic Cryptology
3.1. What is cryptology? Cryptography? Plaintext? Ciphertext? Encryption? Key?
3.2. What references can I start with to learn cryptology?
3.3. How does one go about cryptanalysis?
3.4. What is a brute-force search and what is its cryptographic relevance?
3.5. What are some properties satisfied by every strong cryptosystem?
3.6. If a cryptosystem is theoretically unbreakable, then is it
guaranteed analysis-proof in practice?
3.7. Why are many people still using cryptosystems that are
relatively easy to break?
3.8. What are the basic types of cryptanalytic `attacks'?
4. Mathematical Cryptology
4.1. In mathematical terms, what is a private-key cryptosystem?
4.2. What is an attack?
4.3. What's the advantage of formulating all this mathematically?
4.4. Why is the one-time pad secure?
4.5. What's a ciphertext-only attack?
4.6. What's a known-plaintext attack?
4.7. What's a chosen-plaintext attack?
4.8. In mathematical terms, what can you say about brute-force attacks?
4.9. What's a key-guessing attack? What's entropy?
5. Product Ciphers
5.1. What is a product cipher?
5.2. What makes a product cipher secure?
5.3. What are some group-theoretic properties of product ciphers?
5.4. What can be proven about the security of a product cipher?
5.5. How are block ciphers used to encrypt data longer than the block size?
5.6. Can symmetric block ciphers be used for message authentication?
5.7. What exactly is DES?
5.8. What is triple DES?
5.9. What is differential cryptanalysis?
5.10. How was NSA involved in the design of DES?
5.11. Is DES available in software?
5.12. Is DES available in hardware?
5.13. Can DES be used to protect classified information?
5.14. What are ECB, CBC, CFB, and OFB encryption?
6. Public-Key Cryptography
6.1. What is public-key cryptography?
6.2. How does public-key cryptography solve cryptography's Catch-22?
6.3. What is the role of the `trapdoor function' in public key schemes?
6.4. What is the role of the `session key' in public key schemes?
6.5. What's RSA?
6.6. Is RSA secure?
6.7. What's the difference between the RSA and Diffie-Hellman schemes?
6.8. What is `authentication' and the `key distribution problem'?
6.9. How fast can people factor numbers?
6.10. What about other public-key cryptosystems?
6.11. What is the `RSA Factoring Challenge?'
7. Digital Signatures
7.1. What is a one-way hash function?
7.2. What is the difference between public, private, secret, shared, etc.?
7.3. What are MD4 and MD5?
7.4. What is Snefru?
8. Technical Miscellany
8.1. How do I recover from lost passwords in WordPerfect?
8.2. How do I break a Vigenere (repeated-key) cipher?
8.3. How do I send encrypted mail under UNIX? [PGP, RIPEM, PEM, ...]
8.4. Is the UNIX crypt command secure?
8.5. How do I use compression with encryption?
8.6. Is there an unbreakable cipher?
8.7. What does ``random'' mean in cryptography?
8.8. What is the unicity point (a.k.a. unicity distance)?
8.9. What is key management and why is it important?
8.10. Can I use pseudo-random or chaotic numbers as a key stream?
8.11. What is the correct frequency list for English letters?
8.12. What is the Enigma?
8.13. How do I shuffle cards?
8.14. Can I foil S/W pirates by encrypting my CD-ROM?
8.15. Can you do automatic cryptanalysis of simple ciphers?
8.16. What is the coding system used by VCR+?
9. Other Miscellany
9.1. What is the National Security Agency (NSA)?
9.2. What are the US export regulations?
9.3. What is TEMPEST?
9.4. What are the Beale Ciphers, and are they a hoax?
9.5. What is the American Cryptogram Association, and how do I get in touch?
9.6. Is RSA patented?
9.7. What about the Voynich manuscript?
10. References
10.1. Books on history and classical methods
10.2. Books on modern methods
10.3. Survey articles
10.4. Reference articles
10.5. Journals, conference proceedings
10.6. Other
10.7. How may one obtain copies of FIPS and ANSI standards cited herein?
10.8. Electronic sources
10.9. RFCs (available from [FTPRF])
10.10. Related newsgroups
User Contributions:
Comment about this article, ask questions, or add new information about this topic:
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Quincy, MA Algebra 1 Tutor
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MathGroup Archive: July 2004 [00647]
[Date Index] [Thread Index] [Author Index]
Follow up: Help wanted ... bounding function is pierced for n even > 10^7.
• To: mathgroup at smc.vnet.net
• Subject: [mg49733] Follow up: Help wanted ... bounding function is pierced for n even > 10^7.
• From: gilmar.rodriguez at nwfwmd.state.fl.us (Gilmar Rodr?guez Pierluissi)
• Date: Thu, 29 Jul 2004 07:45:42 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com
Thank you Mathematica Group for all your excellent feedback through e-mail!
Keep in mind that I welcome comments long after the date when this message
was posted.
(1.) Many of you pointed out that the first sentence in my original
presentation should read as:
"The function B[n] = (360.874) / (Exp[2 / Log[n]] - 1) is a bounding
functionfor all Minimal Goldbach Prime Partition Points ("MGPPP's"
for short),for all even integers n between (and including) 4 and 10^7"
i.e. not 10^8 and this was indeed a typing error. Sorry!
(2.) Others pointed out that one spike seems to pierce the bounding
function in the last plot (i.e. Show[Plt3,Plt1]) near the (vertical)
line N = 2*10^6. This is only an optical illusion caused by the way
that we built Plt1 and Plt3, and put them together, without using
a lot of computer memory.
To convince yourself that the data is well separated from its bounding
function (for large even N) please, either:
(1.) download and evaluate the following notebook
found in:
(2.) add the following input lines to the original notebook:
In[14]: B[n_] = (360.874) / (Exp[2 / Log[n]] - 1)
In[15]: bfdata = Table[BF[data[[i]][[1]]], {i, 1, Length[data]}];
In[16]: Plt4=ListPlot[bfdata, AspectRatio -> 1\/GoldenRatio,
PlotJoined -> True, PlotLabel ->"MINIMAL GOLDBACH PRIME
PARTITION POINTS FOR EVEN INTEGERS N BETWEEN 4 and 10^7
(DEPICTED IN RED).", FrameLabel -> {"BOUNDING FUNCTION BF[N]
=360.874/(Exp[2/Log[N]] - 1)","BOUNDING FUNCTION (DEPICTED IN BLUE).",
"PLANE ROTATED CLOCKWISE BY AN ANGLE THETA = PI/4 RADIANS ABOUT
THE ORIGIN."}, Frame -> True, PlotStyle -> {Hue[.7],{RGBColor[1,1,0]},
Thickness[.001]},Background -> RGBColor[1,1,0], ImageSize -> 800,
PlotRange -> All]
In[17]: Show[Plt1,Plt4]
If you just want to take a quick look at the plot without
evaluating neither notebook, please visit:
Notice that the the data set (in red) is well separated from
its bounding function(in blue) for large (even) N. Building
Plt4 increased the size of my Mathematica notebook to a whopper
332209 KB's, and I this is why I didn't include inputs[14]
to [17] in my original presentation. Thanks again!
P.S. I still have not received an e-mail from anyone claiming
that the bounding function was pierced for N > 10^7.
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A Large Software Company Gives Job Applicants A ... | Chegg.com
a large software company gives job applicants a test of programming ability and the mean for that test has been 160 in the past. twenty-five job applicants are randomly selected from one large
university and they produce a mean score and standard deviation of 183 and 12, respectively. use a 0.05 level og significance to test the claim that this sample comes from a population with a mean
score greater than 160. use the P-value method of testing hypotheses.
Statistics and Probability
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I was so interested in getting an answer to the Stomachion that I resorted to old-fashioned capitalism: I posted a "bounty" for the solution to my programmer friends. Here is the exact text of the
e-mail message I sent them. Kate Jones passed it on to Bill Cutler, who was already working on the problem at her request.
There is this geometrical puzzle called the Stomachion that dates from the time of Archimedes. Here is the puzzle: The numbers just refer to the area of each piece relative to the total area of 144
units. For more details, see: http://www.barbecuejoe.com/stomachion.htm and http://www.gamepuzzles.com/archsqu.htm.
It would appear that there is an interesting UNSOLVED problem relative to this puzzle. How many unique solutions are there?
Clearly, there are many. To start with, note the left and right symmetry caused by the vertical line going down the middle. Further, note that there are essentially four right triangles, each
composed of either three or four pieces, that make up the solution (for example, the pieces marked 24, 3, and 9 make up one of them; the two 12's and two 6's make up another.)
There are some "counting" issues. For example, note that there are two 12's that are identical, so merely interchanging them should not constitute a "different" solution, unless you want to define it
Also to be considered is the issue of turning the puzzle over, as well as taking any solution such as this one and creating three "new" ones by successive 90 degree rotations around one corner.
I leave it up to you to define what constitutes a "unique solution." Pick your own rules, and state them clearly.
Then write a program that tells us how many solutions there are. Should be simple enough.
I have it on good authority that this problem remains UNSOLVED and that it is at least 2200 years old. If you find a solution, I can get you credit in a forthcoming paper by professor Reviel Netz of
Stanford, a classics professor who is writing on the subject of the Stomachion and would like to know the answer.
It's not exactly Fermat's Last Theorem, but it's a worthy problem. And I think it is non-trivial, although do-able by the right group of people.
To sweeten the pot, I will offer a BOUNTY of $100 for a solution before midnight November 15, 2003. To qualify for the bounty, you must be willing to demonstrate the program and defend your solution.
Any takers?
Clarification on the bounty:
I will pay one (1) prize of $100 to the first correct solution received before midnight November 15, 2003. No multiple awards, and no awards for incorrect solutions.
If you just give me a number, such as "537," then you will have to wait for either confirmation or refutation to earn (or not earn) the reward.
If you can convince me that the solution must be correct, the award will be accelerated accordingly.
The bounty can of course be awarded to a team of solvers, to be shared as they see fit.
I want to reward solution finders, not language lawyers or loophole finders. Decisions of the judges will be final.
Bill Cutler won the bounty with his correct answer of 536 solutions on October 31, 2003. He was the only programmer to submit a solution.
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Summary: Math 310: Midterm 1
October 10, 2006
Name: ID number:
There are 5 questions worth a total of 100 points, plus one small bonus question
worth 10 points. Please justify all your statements, and write neatly so that we can
read and follow your answers. Continue your answers on the back of the pages. Also,
please turn off cell phones.
Question 1. (20 points) (i) Define a subspace of a vector space.
(ii) Suppose that V is a finite dimensional vector space and that W V is a subspace
such that dim W = dim V . Prove carefully that W = V .
1 20pt
2 20pt
3 20pt
4 30pt
5 10pt
Total 100pt
bonus 10pt
Question 2. (20 points) (i) Let (v1, . . . , vn) be a list of vectors in V . What does it
mean to say that this list is linearly independent? Give the formal definition.
(ii) Give an example of a list of three vectors in R4
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Fitting Continuous Piecewise Linear Functions
Continuous piecewise linear functions (CPWL) are an interesting class of functions. The attractive features include their efficiency and continuity. The CPWL function regression can be better
behaving than the polynomial regression, and is often used for approximation of complex functions. The CPWL functions are quite efficient in terms of the computational and memory requirements, which
allows demanding applications, such as resource-constrained microcontrollers or graphics processing.
However, there seems to be a lack of methods of fitting the CPWL functions to other functions or experimental sets of data. In particular, when the x coordinates of the CPWL function segments are
fixed and only the y coordinates are unknown. The following paper offers a solution to this problem.
Least-squares Fit of a Continuous Piecewise Linear Function
The paper describes an application of the least-squares method to fitting a continuous piecewise linear function. It shows that the solution is unique and the best fit can be found without resorting
to iterative optimization techniques.
MATLAB function(s)
Page created: 3-Sep-2004. Last updated: 3-Sep-2004
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Change Number format in MS Word
up vote 1 down vote favorite
My country uses this numbering system so 100000 is One Lakh and not One Hundered Thousand.
Now I'm trying to convert a numeric value to words. Pretty simple using this technique. The issue is, it uses a normal western style system and I want the result in the south asian system.
Is there a way to do this?
you might take a look here: office.microsoft.com/en-us/word-help/… – kmote Jul 11 '13 at 21:33
I'm reaching here, because I really doubt this feature is available for foreign languages, but if it is, I would assume it would require a foreign version of Word. You might check out these Office
Language Packs: office.microsoft.com/en-us/downloads/… – kmote Jul 11 '13 at 21:37
this one is even a longer stretch, but you might look into the Translation functionality: library.vicu.utoronto.ca/library_services/technology/… – kmote Jul 11 '13 at 21:38
add comment
1 Answer
active oldest votes
You can extend the CardText switch using a variety of fields so that it follows the numbering system you want. The following field code will handle numbers up to 99 lakh correctly, if you
want to go higher such as to crore you just need to add an extra level of IF fields. Note that the CardText switch by itself can only handle numbers up to a million anyway.
Input field code
{ QUOTE { SET n 1099999 } { IF n < 100000 “{ = n \*cardtext }” “{ = int(n/100000) \* cardtext } lakh { SET r { = MOD(n, 100000) } *}*{ IF r = 0 “” “{ = r \* cardtext }” }” } \* caps \*
CharFormat }
up vote 1
down vote Output
Ten Lakh Ninety-Nine Thousand Nine Hundred Ninety-Nine
I have uploaded a demo document which includes the above field (click the link and then choose File -> Download). You may wish to add this as an autotext entry to make it easier to
add comment
Not the answer you're looking for? Browse other questions tagged microsoft-word or ask your own question.
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Feature selection: Using the caret package
November 16, 2010
By Allan Engelhardt
Feature selection is an important step for practical commercial data mining which is often characterised by data sets with far too many variables for model building. In a previous post we looked at
all-relevant feature selection using the Boruta package while in this post we consider the same (artificial, toy) examples using the caret package. Max Kuhn kindly listed me as a contributor for some
performance enhancements I submitted, but the genius behind the package is all his.
The caret package provides a very flexible framework for the analysis as we shall see, but first we set up the artificial test data set as in the previous article.
## Feature-bc.R - Compare Boruta and caret feature selection
## Copyright © 2010 Allan Engelhardt (http://www.cybaea.net/)
run.name <- "feature-bc"
## Load early to get the warnings out of the way:
## Set up artificial test data for our analysis
n.var <- 20
n.obs <- 200
x <- data.frame(V = matrix(rnorm(n.var*n.obs), n.obs, n.var))
n.dep <- floor(n.var/5)
cat( "Number of dependent variables is", n.dep, "\n")
m <- diag(n.dep:1)
## These are our four test targets
y.1 <- factor( ifelse( x$V.1 >= 0, 'A', 'B' ) )
y.2 <- ifelse( rowSums(as.matrix(x[, 1:n.dep]) %*% m) >= 0, "A", "B" )
y.2 <- factor(y.2)
y.3 <- factor(rowSums(x[, 1:n.dep] >= 0))
y.4 <- factor(rowSums(x[, 1:n.dep] >= 0) %% 2)
The flexibility of the caret package is to a large extent implemented by using control objects. Here we specify to use the randomForest classification algorithm (which is also what Boruta uses) and
if the multicore package is available then we use that for extra perfomance (you can also use MPI etc – see the documentation):
control <- rfeControl(functions = rfFuncs, method = "boot", verbose = FALSE,
returnResamp = "final", number = 50)
if ( require("multicore", quietly = TRUE, warn.conflicts = FALSE) ) {
control$workers <- multicore:::detectCores()
control$computeFunction <- mclapply
control$computeArgs <- list(mc.preschedule = FALSE, mc.set.seed = FALSE)
We will consider from one to six features (using the sizes variable) and then we simply let it lose:
sizes <- 1:6
## Use randomForest for prediction
profile.1 <- rfe(x, y.1, sizes = sizes, rfeControl = control)
cat( "rf : Profile 1 predictors:", predictors(profile.1), fill = TRUE )
profile.2 <- rfe(x, y.2, sizes = sizes, rfeControl = control)
cat( "rf : Profile 2 predictors:", predictors(profile.2), fill = TRUE )
profile.3 <- rfe(x, y.3, sizes = sizes, rfeControl = control)
cat( "rf : Profile 3 predictors:", predictors(profile.3), fill = TRUE )
profile.4 <- rfe(x, y.4, sizes = sizes, rfeControl = control)
cat( "rf : Profile 4 predictors:", predictors(profile.4), fill = TRUE )
The results are:
rf : Profile 1 predictors: V.1 V.16 V.6
rf : Profile 2 predictors: V.1 V.2
rf : Profile 3 predictors: V.4 V.1 V.2
rf : Profile 4 predictors: V.10 V.11 V.7
If you recall the feature selection with Boruta article, then the results there were
1. Profile 1: V.1, (V.16, V.17)
2. Profile 2: V.1, V.2, V,3, (V.8, V.9, V.4)
3. Profile 3: V.1, V.4, V.3, V.2, (V.7, V.6)
4. Profile 4: V.10, (V.11, V.13)
To show the flexibility of caret, we can run the analysis with another of the built-in classifiers:
## Use ipred::ipredbag for prediction
control$functions <- treebagFuncs
profile.1 <- rfe(x, y.1, sizes = sizes, rfeControl = control)
cat( "treebag: Profile 1 predictors:", predictors(profile.1), fill = TRUE )
profile.2 <- rfe(x, y.2, sizes = sizes, rfeControl = control)
cat( "treebag: Profile 2 predictors:", predictors(profile.2), fill = TRUE )
profile.3 <- rfe(x, y.3, sizes = sizes, rfeControl = control)
cat( "treebag: Profile 3 predictors:", predictors(profile.3), fill = TRUE )
profile.4 <- rfe(x, y.4, sizes = sizes, rfeControl = control)
cat( "treebag: Profile 4 predictors:", predictors(profile.4), fill = TRUE )
This gives:
treebag: Profile 1 predictors: V.1 V.16
treebag: Profile 2 predictors: V.2 V.1
treebag: Profile 3 predictors: V.1 V.3 V.2
treebag: Profile 4 predictors: V.10 V.11 V.1 V.7 V.13
And of course, if you have your own favourite model class that is not already implemented, then you can easily do that yourself. We like gbm from the package of the same name, which is kind of silly
to use here because it provides variable importance automatically as part of the fitting process, but may still be useful. It needs numeric predictors so we do:
## Use gbm for prediction
y.1 <- as.numeric(y.1)-1
y.2 <- as.numeric(y.2)-1
y.3 <- as.numeric(y.3)-1
y.4 <- as.numeric(y.4)-1
gbmFuncs <- treebagFuncs
gbmFuncs$fit <- function (x, y, first, last, ...) {
n.levels <- length(unique(y))
if ( n.levels == 2 ) {
distribution = "bernoulli"
} else {
distribution = "gaussian"
gbm.fit(x, y, distribution = distribution, ...)
gbmFuncs$pred <- function (object, x) {
n.trees <- suppressWarnings(gbm.perf(object,
plot.it = FALSE,
method = "OOB"))
if ( n.trees <= 0 ) n.trees <- object$n.trees
predict(object, x, n.trees = n.trees, type = "link")
control$functions <- gbmFuncs
n.trees <- 1e2 # Default value for gbm is 100
profile.1 <- rfe(x, y.1, sizes = sizes, rfeControl = control, verbose = FALSE,
n.trees = n.trees)
cat( "gbm : Profile 1 predictors:", predictors(profile.1), fill = TRUE )
profile.2 <- rfe(x, y.2, sizes = sizes, rfeControl = control, verbose = FALSE,
n.trees = n.trees)
cat( "gbm : Profile 2 predictors:", predictors(profile.2), fill = TRUE )
profile.3 <- rfe(x, y.3, sizes = sizes, rfeControl = control, verbose = FALSE,
n.trees = n.trees)
cat( "gbm : Profile 3 predictors:", predictors(profile.3), fill = TRUE )
profile.4 <- rfe(x, y.4, sizes = sizes, rfeControl = control, verbose = FALSE,
n.trees = n.trees)
cat( "gbm : Profile 4 predictors:", predictors(profile.4), fill = TRUE )
And we get the results below:
gbm : Profile 1 predictors: V.1 V.10 V.11 V.12 V.13
gbm : Profile 2 predictors: V.1 V.2
gbm : Profile 3 predictors: V.4 V.1 V.2 V.3 V.7
gbm : Profile 4 predictors: V.11 V.10 V.1 V.6 V.7 V.18
It is all good and very flexible, for sure, but I can’t really say it is better than the Boruta approach for these simple examples.
Jump to comments.
You may also like these posts:
1. Benchmarking feature selection with Boruta and caret
Feature selection is the data mining process of selecting the variables from our data set that may have an impact on the outcome we are considering. For commercial data mining, which is often
characterised by having too many variables for model building, this is an important step in the analysis process. And since we often work on very large data sets the performance of our process is
very important to us. Having looked at feature selection using the Boruta package and feature selection using the caret package separately, we now consider the performance of the two approaches.
Neither approach is suitable out of the box for the sizes of data sets that we normally work with.
2. Feature selection: All-relevant selection with the Boruta package
Feature selection is an important step for practical commercial data mining which is often characterised by data sets with far too many variables for model building. There are two main approaches
to selecting the features (variables) we will use for the analysis: the minimal-optimal feature selection which identifies a small (ideally minimal) set of variables that gives the best possible
classification result (for a class of classification models) and the all-relevant feature selection which identifies all variables that are in some circumstances relevant for the classification.
In this article we take a first look at the problem of all-relevant feature selection using the Boruta package by Miron B. Kursa and Witold R. Rudnicki. This package is developed fo…
3. R code for Chapter 1 of Non-Life Insurance Pricing with GLM
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by Esbjorn Ohlsson and Born Johansson. We have been doing some work in this area recently. Needing a robust internal training course and documented methodology, we have been working our way
through the book again and converting the examples and exercises to R , the statistical computing and analysis platform. This is part of a series of posts containing elements of the R code.
4. R code for Chapter 2 of Non-Life Insurance Pricing with GLM
We continue working our way through the examples, case studies, and exercises of what is affectionately known here as “the two bears book” (Swedish björn = bear) and more formally as Non-Life
Insurance Pricing with Generalized Linear Models by Esbjörn Ohlsson and Börn Johansson (Amazon UK | US ). At this stage, our purpose is to reproduce the analysis from the book using the R
statistical computing and analysis platform, and to answer the data analysis elements of the exercises and case studies. Any critique of the approach and of pricing and modeling in the Insurance
industry in general will wait for a later article.
5. Area Plots with Intensity Coloring
I am not sure apeescape’s ggplot2 area plot with intensity colouring is really the best way of presenting the information, but it had me intrigued enough to replicate it using base R graphics.
The key technique is to draw a gradient line which R does not support natively so we have to roll our own code for that. Unfortunately, lines(..., type=l) does not recycle the colour col=
argument, so we end up with rather more loops than I thought would be necessary. We also get a nice opportunity to use the under-appreciated read.fwf function.
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Syllabus Entrance
MA 120 Basic Concepts of Statistics
Milner, Thomas S.
Mission Statement: The mission of Park University, an entrepreneurial institution of learning, is to provide access to academic excellence, which will prepare learners to think critically,
communicate effectively and engage in lifelong learning while serving a global community.
Vision Statement: Park University will be a renowned international leader in providing innovative educational opportunities for learners within the global society.
Course MA 120 Basic Concepts of Statistics
Semester F1EE 2008 MO
Faculty Milner, Thomas S.
Title Senior Adjunct Faculty
Degrees/Certificates M.Ed. Math Education, Valdosta State U.
B.S. Math, Valdosta State U.
Office Hours I will be available after class in the classroom to answer questions.
Daytime Phone 229 896-7255
E-Mail tom.milner@pirate.park.edu
tmilner1@alltel.net tmilner@gmc.cc.ga.us
Semester Dates 11 Aug - 3 Oct 2008
Class Days T/Th
Class Time 1700
Prerequisites None
Credit Hours 3
Triola, Mario Elementary Statistics (10th) Addison-Wesley
Additional Resources:
Graphing calculator is required. TI-83/84 recommended; TI83/84 will be supported in class. BRING YOUR CALCULATOR TO EVERY CLASS INCLUDING THE FIRST MEETING.
McAfee Memorial Library - Online information, links, electronic databases and the Online catalog. Contact the library for further assistance via email or at 800-270-4347.
Career Counseling - The Career Development Center (CDC) provides services for all stages of career development. The mission of the CDC is to provide the career planning tools to ensure a lifetime of
career success.
Park Helpdesk - If you have forgotten your OPEN ID or Password, or need assistance with your PirateMail account, please email helpdesk@park.edu or call 800-927-3024
Resources for Current Students - A great place to look for all kinds of information http://www.park.edu/Current/.
Course Description:
A development of certain basic concepts in probability and statistics that are pertinent to most disciplines. Topics include: probability models, parameters, statistics and sampling procedures,
hypothesis testing, correlation, and regression. 3:0:3
Educational Philosophy:
My educational philosophy is one of interactiveness based on lectures, readings, quizzes, dialogues, examinations, internet, web sites and writing. I engage each learner to encourage the lively
exploration of ideas, issues and contradictions.
Learning Outcomes:
Core Learning Outcomes
1. Compute descriptive statistics for raw data as well as grouped data
2. Determine appropriate features of a frequency distribution
3. Apply Chebyshev's Theorem
4. Distinguish between and provide relevant descriptions of a sample and a population
5. Apply the rules of combinatorics
6. Differentiate between classical and frequency approaches to probability
7. Apply set-theoretic ideas to events
8. Apply basic rules of probability
9. Apply the concepts of specific discrete random variables and probability distributions
10. Compute probabilities of a normal distribution
Core Assessment:
Description of MA 120 Core Assessment
One problem with multiple parts for each numbered item, except for item #3, which contains four separate problems.
1. Compute the mean, median, mode, and standard deviation for a sample of 8 to 12 data.
2. Compute the mean and standard deviation of a grouped frequency distribution with 4 classes.
3. Compute the probability of four problems from among these kinds or combinations there of:
a. the probability of an event based upon a two-dimensional table;
b. the probability of an event that involves using the addition rule;
c. the probability of an event that involves conditional probability;
d. the probability of an event that involves the use of independence of events;
e. the probability of an event based upon permutations and/or combinations;
f. the probability of an event using the multiplication rule; or
g. the probability of an event found by finding the probability of the complementary event.
4. Compute probabilities associated with a binomial random variable associated with a practical situation.
5. Compute probabilities associated with either a standard normal probability distribution or with a non-standard normal probability distribution.
6. Compute and interpret a confidence interval for a mean and/ or for a proportion.
Link to Class Rubric
Class Assessment:
Examinations: there will be three tests plus a final exam in the course. The second test score will be your midterm exam. There will be three take home quizzes for grade, and several projects to
complete. These assignments will be given at least one week prior to the due date.
Each assignment will be graded by the percentage of correct answers. The average on the three tests will count 40% of the final grade. The final exam will count 20% of the final grade. Quizzes and
projects will count 30% of the final grade. Class participation will count 10% of the final grade.
NOTE: The final (20% of your final grade) is part of the core assessment. The final is a departmental exam and it will be provided to the instructor by the department of mathematics. The final is
2hrs and is open book, open notes, use any handouts provided throughout the course, and the use of a calculator is essential. Students are NOT allowed to share any textbooks, notes, handouts or
calculators during the final exam.
Late Submission of Course Materials:
Work will be submitted on time. If a student is unable to finish an assignment on time, arrangements must be made with the instructor beforehand if an extension is requested. Other late work will not
be accepted. If a student misses a test unexcused, the test will be accomplished on the day of the final examination.
Classroom Rules of Conduct:
Come on time. If you are late and unexcused, you will lose class participation points for the week. If you cannot make class, let me know beforehand. If work is your excuse for absence, a letter from
your supervisor to me will be required. Turn off all electronic devices before class starts; if you are on call let me know and have your cellphone set to a silent setting. If you emit an electronic
beep or have your cell phone / personal communication device out during the class, you will lose 50% of your class participation points for the term. Be an active participant in class. Ask questions
and join in the discussion.
Course Topic/Dates/Assignments:
All due dates will be announced one week in advance.
Week 1- Administrative and Chapters 1 and 2. Calculator practice. Understand uses and misuses of statistics. Introduction to sampling, including takehome sampling exercise. Summarizing and graphing
Week 2- Chapter 3, Statistical Description. Takehome quiz covering data collection, grouping, graphs, and measures of central tendency and dispersion.
Weeks 3and 4- Test on Ch. 3. Chapter 4,5, and 6.1-6.3, probability concepts and probability distributions. Takehome quiz covering probability and probability distrubutions (Binomial, Poisson, and
Normal distributions).
Week 5-- Midterm exam. Chapter 6.4-6.5, sampling concepts and the Central Limit Theorem.
Week 6- Chapter 7, Estimating means and proportions. Takehome quiz covering interval estimates and finding the sample size required.
Week 7- Test on Chapter 7. Chapter 8, One sample hypothesis testing.
Week 8- Final exam.
Academic Honesty:
Academic integrity is the foundation of the academic community. Because each student has the primary responsibility for being academically honest, students are advised to read and understand all
sections of this policy relating to standards of conduct and academic life. Park University 2008-2009 Undergraduate Catalog Page 87
Academic integrity is the foundation of the academic community. Because each student has the primary responsibility for being academically honest, students are advised to read and understand all
sections of this policy relating to standards of conduct and academic life. Park University 2006-2007 Undergraduate Catalog Page 87-89.
Plagiarism involves the use of quotations without quotation marks, the use of quotations without indication of the source, the use of another's idea without acknowledging the source, the submission
of a paper, laboratory report, project, or class assignment (any portion of such) prepared by another person, or incorrect paraphrasing. Park University 2008-2009 Undergraduate Catalog Page 87
Plagiarism involves the use of quotations without quotation marks, the use of quotations without indication of the source, the use of another's idea without acknowledging the source, the submission
of a paper, laboratory report, project, or class assignment (any portion of such) prepared by another person, or incorrect paraphrasing.
Park University 2006-2007 Undergraduate Catalog
Attendance Policy:
Instructors are required to maintain attendance records and to report absences via the online attendance reporting system.
1. The instructor may excuse absences for valid reasons, but missed work must be made up within the semester/term of enrollment.
2. Work missed through unexcused absences must also be made up within the semester/term of enrollment, but unexcused absences may carry further penalties.
3. In the event of two consecutive weeks of unexcused absences in a semester/term of enrollment, the student will be administratively withdrawn, resulting in a grade of "F".
4. A "Contract for Incomplete" will not be issued to a student who has unexcused or excessive absences recorded for a course.
5. Students receiving Military Tuition Assistance or Veterans Administration educational benefits must not exceed three unexcused absences in the semester/term of enrollment. Excessive absences will
be reported to the appropriate agency and may result in a monetary penalty to the student.
6. Report of a "F" grade (attendance or academic) resulting from excessive absence for those students who are receiving financial assistance from agencies not mentioned in item 5 above will be
reported to the appropriate agency.
Park University 2008-2009 Undergraduate Catalog Page 89-90
Any unexcused absence, including the first class meeting, will result in a class participation grade of Zero for that class.
Disability Guidelines:
Park University is committed to meeting the needs of all students that meet the criteria for special assistance. These guidelines are designed to supply directions to students concerning the
information necessary to accomplish this goal. It is Park University's policy to comply fully with federal and state law, including Section 504 of the Rehabilitation Act of 1973 and the Americans
with Disabilities Act of 1990, regarding students with disabilities. In the case of any inconsistency between these guidelines and federal and/or state law, the provisions of the law will apply.
Additional information concerning Park University's policies and procedures related to disability can be found on the Park University web page: http://www.park.edu/disability .
┃Competency │ Exceeds Expectation (3) │ Meets Expectation (2) │ Does Not Meet Expectation (1) │ No Evidence (0) ┃
┃Evaluation │Can perform and interpret a hypothesis test with 100% │Can perform and interpret a hypothesis test with at │Can perform and interpret a hypothesis test with less │Makes no attempt to┃
┃Outcomes │accuracy. │least 80% accuracy. │than 80% accuracy. │perform a test of ┃
┃10 │ │ │ │hypothesis. ┃
┃ │ │ │ │Makes no attempt to┃
┃Synthesis │Can compute and interpret a confidence interval for a │Can compute and interpret a confidence interval for a │Can compute and interpret a confidence interval for a │compute or ┃
┃Outcomes │sample mean for small and large samples, and for a │sample mean for small and large samples, and for a │sample mean for small and large samples, and for a │interpret a ┃
┃10 │proportion with 100% accuracy. │proportion with at least 80% accuracy. │proportion with less than 80% accuracy. │confidence ┃
┃ │ │ │ │interval. ┃
┃ │ │ │ │Makes no attempt to┃
┃ │ │ │ │apply the normal ┃
┃Analysis │Can apply the normal distribution, Central limit │Can apply the normal distribution, Central limit │Can apply the normal distribution, Central limit │distribution, ┃
┃Outcomes │theorem, and binomial distribution to practical │theorem, and binomial distribution to practical │theorem, and binomial distribution to practical │Central Limit ┃
┃10 │problems with 100% accuracy. │problems with at least 80% accuracy. │problems with less than 80% accuracy. │Theorem, or ┃
┃ │ │ │ │binomial ┃
┃ │ │ │ │distribution. ┃
┃Terminology│Can explain event, simple event, mutually exclusive │Can explain event, simple event, mutually exclusive │Can explain event, simple event, mutually exclusive │Makes no attempt to┃
┃Outcomes │events, independent events, discrete random variable, │events, independent events, discrete random variable, │events, independent events, discrete random variable, │explain any of the ┃
┃4,5,7 │continuous random variable, sample, and population │continuous random variable, sample, and population with│continuous random variable, sample, and population │terms listed. ┃
┃ │with 100% accuracy. │at least 80% accuracy. │with less than 80% accuracy. │ ┃
┃Concepts │Can explain mean, median, mode, standard deviation, │Can explain mean, median, mode, standard deviation, │Can explain mean, median, mode, standard deviation, │Makes no attempt to┃
┃Outcomes │simple probability, and measures of location with 100%│simple probability, and measures of location with at │simple probability, and measures of location with less│define any concept.┃
┃1,6 │accuracy. │least 80% accuracy. │than 80% accuracy. │ ┃
┃ │Compute probabilities using addition multiplication, │Compute probabilities using addition multiplication, │Compute probabilities using addition multiplication, │ ┃
┃Application│and complement rules and conditional probabilities. │and complement rules and conditional probabilities. │and complement rules and conditional probabilities. │Makes no attempt to┃
┃Outcomes │Compute statistical quantities for raw and grouped │Compute statistical quantities for raw and grouped │Compute statistical quantities for raw and grouped │compute any of the ┃
┃1,2,3,8,9 │data. Compute probabilities using combinatorics, │data. Compute probabilities using combinatorics, │data. Compute probabilities using combinatorics, │probabilities or ┃
┃ │discrete random variables, and continuous random │discrete random variables, and continuous random │discrete random variables, and continuous random │statistics listed. ┃
┃ │variables. All must be done with 100% accuracy. │variables. All must be done with at least 80% accuracy.│variables. All are done with less than 80% accuracy. │ ┃
┃Whole │Can apply the concepts of probability and statistics │Can apply the concepts of probability and statistics to│Can apply the concepts of probability and statistics │Makes no attempt to┃
┃Artifact │to real-world problems in other disciplines with 100 %│real-world problems in other disciplines with at least │to real-world problems in other disciplines with less │apply the concepts ┃
┃Outcomes │accuracy. │80 % accuracy. │than 80% accuracy. │to real-world ┃
┃7,8 │ │ │ │problems. ┃
┃Components │ │ │ │Makes no attempt to┃
┃Outcomes │Can use a calculator or other computing device to │Can use a calculator or other computing device to │Can use a calculator or other computing device to │use any computing ┃
┃1 │compute statistics with 100% accuracy. │compute statistics with at least 80% accuracy. │compute statistics with less 80% accuracy. │device to compute ┃
┃ │ │ │ │statistics. ┃
This material is copyright and can not be reused without author permission.
Last Updated:7/11/2008 3:23:47 PM
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Post a reply
Found this:
The earliest known use of "m" for slope is from an 1844 British text by Matthew O'Brien entitled "A Treatise on Plane Co-Ordinate Geometry". Later in 1848 George Salmon (1819-1904) referred to
O'Brien's 1844 article within his "A Treatise on Conic Sections" and used the slope-intercept formula "y = mx + b", where "b" is the ordinate (vertical component) of the point where the line
intersects the y-axis. It is also known that the four authors Isaac Todhunter in 1855 (Treatise on Plane Co-Ordinate Geometry), George A. Osborne in 1891 (Differential and Integral Calculus), and
Arthur M. Harding and George W. Mullins in 1924 (Analytic Geometry) each used "m" to refer to slope in their mathematical writings.
(Source: http://www.bookrags.com/sciences/mathematics/slope-wom.html)
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An analysis of first-order logics of probability
Results 1 - 10 of 204
- Machine Learning , 2006
"... Abstract. We propose a simple approach to combining first-order logic and probabilistic graphical models in a single representation. A Markov logic network (MLN) is a first-order knowledge base
with a weight attached to each formula (or clause). Together with a set of constants representing objects ..."
Cited by 569 (34 self)
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Abstract. We propose a simple approach to combining first-order logic and probabilistic graphical models in a single representation. A Markov logic network (MLN) is a first-order knowledge base with
a weight attached to each formula (or clause). Together with a set of constants representing objects in the domain, it specifies a ground Markov network containing one feature for each possible
grounding of a first-order formula in the KB, with the corresponding weight. Inference in MLNs is performed by MCMC over the minimal subset of the ground network required for answering the query.
Weights are efficiently learned from relational databases by iteratively optimizing a pseudo-likelihood measure. Optionally, additional clauses are learned using inductive logic programming
techniques. Experiments with a real-world database and knowledge base in a university domain illustrate the promise of this approach.
- Information and Computation , 1990
"... We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the
probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable ( ..."
Cited by 214 (19 self)
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We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability
of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice,
where they may not be measurable. The measurable case is essentially a formalization of (the proposi-tional fragment of) Nilsson’s probabilistic logic. As we show elsewhere, the general
(nonmeasurable) case corresponds precisely to replacing probability measures by Dempster-Shafer belief functions. In both cases, we provide a complete axiomatiza-tion and show that the problem of
deciding satistiability is NP-complete, no worse than that of propositional logic. As a tool for proving our complete axiomatiza-tions, we give a complete axiomatization for reasoning about Boolean
combina-tions of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow
reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields. ( 1990 Academic Press. Inc 1.
- FUZZY SETS AND SYSTEMS , 2003
"... ..."
- AI Magazine vol
"... This article describes a methodology for programming robots known as probabilistic robotics. The probabilistic paradigm pays tribute to the inherent uncertainty in robot perception, relying on
explicit representations of uncertainty when determining what to do. This article surveys some of the progr ..."
Cited by 166 (9 self)
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This article describes a methodology for programming robots known as probabilistic robotics. The probabilistic paradigm pays tribute to the inherent uncertainty in robot perception, relying on
explicit representations of uncertainty when determining what to do. This article surveys some of the progress in the field, using in-depth examples to illustrate some of the nuts and bolts of the
basic approach. Our central conjecture is that the probabilistic approach to robotics scales better to complex real-world applications than approaches that ignore a robot’s uncertainty. 1
- Journal of Artificial Intelligence Research , 2001
"... Description Logics (DLs) are suitable, well-known, logics for managing structured knowledge. They allow reasoning about individuals and well defined concepts, i.e. set of individuals with common
properties. The experience in using DLs in applications has shown that in many cases we would like to ext ..."
Cited by 151 (21 self)
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Description Logics (DLs) are suitable, well-known, logics for managing structured knowledge. They allow reasoning about individuals and well defined concepts, i.e. set of individuals with common
properties. The experience in using DLs in applications has shown that in many cases we would like to extend their capabilities. In particular, their use in the context of Multimedia Information
Retrieval (MIR) leads to the convincement that such DLs should allow the treatment of the inherent imprecision in multimedia object content representation and retrieval. In this paper we will present
a fuzzy extension of ALC, combining...
- Artificial Intelligence , 1997
"... Inspired by game theory representations, Bayesian networks, influence diagrams, structured Markov decision process models, logic programming, and work in dynamical systems, the independent
choice logic (ICL) is a semantic framework that allows for independent choices (made by various agents, includi ..."
Cited by 150 (9 self)
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Inspired by game theory representations, Bayesian networks, influence diagrams, structured Markov decision process models, logic programming, and work in dynamical systems, the independent choice
logic (ICL) is a semantic framework that allows for independent choices (made by various agents, including nature) and a logic program that gives the consequence of choices. This representation can
be used as a specification for agents that act in a world, make observations of that world and have memory, as well as a modelling tool for dynamic environments with uncertainty. The rules specify
the consequences of an action, what can be sensed and the utility of outcomes. This paper presents a possible-worlds semantics for ICL, and shows how to embed influence diagrams, structured Markov
decision processes, and both the strategic (normal) form and extensive (game-tree) form of games within the Thanks to Craig Boutilier and Holger Hoos for detailed comments on this paper. This work
was supporte...
, 1991
"... We argue that rather than representing an agent's knowledge as a collection of formulas, and then doing theorem proving to see if a given formula follows from an agent's knowledge base, it may
be more useful to represent this knowledge by a semantic model, and then do model checking to see if the g ..."
Cited by 117 (5 self)
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We argue that rather than representing an agent's knowledge as a collection of formulas, and then doing theorem proving to see if a given formula follows from an agent's knowledge base, it may be
more useful to represent this knowledge by a semantic model, and then do model checking to see if the given formula is true in that model. We discuss how to construct a model that represents an
agent's knowledge in a number of different contexts, and then consider how to approach the model-checking problem.
- PROCEEDINGS OF THE WORK-IN-PROGRESS TRACK AT THE 10TH INTERNATIONAL CONFERENCE ON INDUCTIVE LOGIC PROGRAMMING , 2001
"... Various proposals for combining first order logic with Bayesian nets exist. We introduce the formalism of Bayesian logic programs, which is basically a simplification and reformulation of Ngo
and Haddawys probabilistic logic programs. However, Bayesian logic programs are sufficiently powerful to ..."
Cited by 109 (7 self)
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Various proposals for combining first order logic with Bayesian nets exist. We introduce the formalism of Bayesian logic programs, which is basically a simplification and reformulation of Ngo and
Haddawys probabilistic logic programs. However, Bayesian logic programs are sufficiently powerful to represent essentially the same knowledge in a more elegant manner. The elegance is illustrated by
the fact that they can represent both Bayesian nets and definite clause programs (as in "pure" Prolog) and that their kernel in Prolog is actually an adaptation of an usual Prolog meta-interpreter.
- Proceedings of the 13th Conference of Uncertainty in Artificial Intelligence (UAI-13 , 1997
"... A new method is developed to represent probabilistic relations on multiple random events. Where previously knowledge bases containing probabilistic rules were used for this purpose, here a
probabilitydistributionover the relations is directly represented by a Bayesian network. By using a powerful wa ..."
Cited by 104 (10 self)
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A new method is developed to represent probabilistic relations on multiple random events. Where previously knowledge bases containing probabilistic rules were used for this purpose, here a
probabilitydistributionover the relations is directly represented by a Bayesian network. By using a powerful way of specifying conditional probability distributions in these networks, the resulting
formalism is more expressive than the previous ones. Particularly, it provides for constraints on equalities of events, and it allows to define complex, nested combination functions. 1
, 1993
"... According to the logical model of Information Retrieval (IR), the task of IR can be described as the extraction, from a given document base, of those documents d that, given a query q, make the
formula d → q valid, where d and q are formulae of the chosen logic and “→ ” denotes the brand of logical ..."
Cited by 93 (19 self)
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According to the logical model of Information Retrieval (IR), the task of IR can be described as the extraction, from a given document base, of those documents d that, given a query q, make the
formula d → q valid, where d and q are formulae of the chosen logic and “→ ” denotes the brand of logical implication formalized by the logic in question. In this paper, although essentially
subscribing to this view, we propose that the logic to be chosen for this endeavour be a Terminological Logic (TL): accordingly, the IR task becomes that of singling out those documents d such that d
� q, where d and q are terms of the chosen TL and “�” denotes subsumption between terms. We call this the terminological model of IR. TLs are particularly suitable for modelling IR; in fact, they can
be employed: 1) in representing documents under a variety of aspects (e.g. structural, layout, semantic content); 2) in representing queries; 3) in representing lexical, “thesaural ” knowledge. The
fact that a single logical language can be used for all these representational endeavours ensures that all these sources of knowledge will participate in the retrieval process in a uniform and
principled way. In this paper we introduce Mirtl, a TL for modelling IR according to the above guidelines; its syntax, formal semantics and inferential algorithm are described. 1
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affine scheme
An affine scheme is a scheme that as a sheaf on the opposite category CRing${}^{op}$ of commutative rings (or equivalently as a sheaf on the subcategory of finitely presented rings) is representable.
In a ringed space picture an affine scheme is a locally ringed space which is locally isomorphic to the prime spectrum of a commutative ring. Affine schemes form a full subcategory $Aff\
hookrightarrow Scheme$ of the category of schemes.
The correspondence $Y\mapsto Spec(\Gamma_Y \mathcal{O}_Y)$ extends to a functor $Scheme\to Aff$. The fundamental theorem on morphisms of schemes says that there is a bijection
$CRing(\Gamma_Y\mathcal{O}_Y, R) \cong Scheme(Spec R, Y).$
In other words, for fixed $Y$, and for varying $R$ there is a restricted functor
$Scheme(-,Y)|_{Aff^{op}} = h_Y|_{Aff^{op}} = h_Y|_{CRing} : CRing\to Set,$
and the functor $Y\mapsto h_Y|_{CRing}$ from schemes to presheaves on $Aff$ is fully faithful. Thus the general schemes if defined as ringed spaces, indeed form a full subcategory of the category of
presheaves on $Aff$.
There is an analogue of this theorem for relative noncommutative schemes in the sense of Rosenberg.
Relative affine schemes
A relative affine scheme over a scheme $Y$ is a relative scheme $f:X\to Y$ isomorphic to the spectrum of a (commutative unital) algebra $A$ in the category of quasicoherent $\mathcal{O}_Y$-modules;
such a β relativeβ spectrum has been introduced by Grothendieck. It is characterized by the property that for every open $V\subset Y$ the inverse image $f^{-1}V\subset X$ is an open affine
subscheme of $X$ isomorphic to $Spec(A(V))$ and such open affines glue in such a way that $f^{-1}V\hookrightarrow f^{-1}W$ corresponds to the restriction morphism $A(W)\to A(V)$ of algebras.
Relative affine scheme is a concrete way to represent an affine morphism of schemes.
Affine Serreβ s theorem
Given a commutative unital ring $R$ there is an equivalence of categories ${}_R Mod\to Qcoh(Spec R)$ between the category of $R$-modules and the category of quasicoherent sheaves of $\mathcal{O}_
{Spec R}$-modules given on objects by $M\mapsto \tilde{M}$ where $\tilde{M}$ is the unique sheaf such that the restriction on the principal Zariski open subsets is given by the localization $\tilde
{M}(D_f) = R[f^{-1}]\otimes_R M$ where $D_f$ is the principal Zariski open set underlying $Spec R[f^{-1}]\subset Spec R$, and the restrictions are given by the canonical maps among the localizations.
The action of $\mathcal{O}_{Spec R}$ is defined using a similar description of $\mathcal{O}_{Spec R} = \tilde{R}$. Its right adjoint (quasi)inverse functor is given by the global sections functor $\
mathcal{F}\mapsto\mathcal{F}(Spec R)$.
• Robin Hartshorne, Algebraic geometry
• Demazure, Gabriel, Algebraic groups
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MATHEMATICA BOHEMICA, Vol. 133, No. 3, pp. 259-266 (2008)
On $\Cal C$-starcompact spaces
Yan-Kui Song
Yan-Kui Song, Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing 210097, P. R China, e-mail: songyankui@njnu.edu.cn
Abstract: A space $X$ is $\Cal C$-starcompact if for every open cover $\Cal U$ of $X,$ there exists a countably compact subset $C$ of $X$ such that $\St(C,{\Cal U})=X.$ In this paper we investigate
the relations between $\Cal C$-starcompact spaces and other related spaces, and also study topological properties of $\Cal C$-starcompact spaces.
Keywords: compact space, countably compact space, Lindel{ö}f space, $\Cal K$-starcompact space, $\Cal C$-starcompact space, $\Cal L$-starcompact space
Classification (MSC2000): 54D20, 54D55
Full text of the article:
[Previous Article] [Next Article] [Contents of this Number] [Journals Homepage] © 2008–2010 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition
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Howard Hathaway Aiken
Born: 9 March 1900 in Hoboken, New Jersey, USA
Died: 14 March 1973 in St Louis, Missouri, USA
Click the picture above
to see two larger pictures
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Howard Aiken studied at the University of Wisconsin, Madison obtaining a doctorate from Harvard in 1939. While he was a graduate student and an instructor in the Department of Physics at Harvard
Aiken began to make plans to build a large computer. These plans were made for a very specific purpose, for Aiken's research had led to a system of differential equations which had no exact solution
and which could only be solved using numerical techniques. However, the amount of hand calculation involved would have been almost prohibitive, so Aiken's idea was to use an adaptation of the punched
card machines which had been developed by Hollerith.
Aiken wrote a report on how he envisaged the machine, and in particular how such a machine designed to be used in scientific research would differ from a punched card machine. He listed four main
points [2]:-
... whereas accounting machines handle only positive numbers, scientific machines must be able to handle negative ones as well; that scientific machines must be able to handle such functions as
logarithms, sines, cosines and a whole lot of other functions; the computer would be most useful for scientists if, once it was set in motion, it would work through the problem frequently for
numerous numerical values without intervention until the calculation was finished; and that the machine should compute lines instead of columns, which is more in keeping with the sequence of
mathematical events.
The report was sufficient to prompt senior staff at Harvard to contact IBM and an agreement was made that Aiken would build his computer at the IBM laboratories at Endicott, helped by IBM engineers.
Working with three engineers, Aiken developed the ASCC computer (Automatic Sequence Controlled Calculator) which could carry out five operations, addition, subtraction, multiplication, division and
reference to previous results. Aiken was much influenced in his ideas by Babbage's writings and he saw the project to build the ASCC computer as completing the task which Babbage had set out on but
failed to complete.
The ASCC had more in common with Babbage's analytical engine that one might imagine. Although it was powered by electricity, the major components were electromechanical in the form of magnetically
operated switches. It weighed 35 tons, had 500 miles of wire and could compute to 23 significant figures. There were 72 storage registers and central units to perform multiplication and division. The
gain an idea of the performance of the machine, a single addition took about 6 seconds while a division took about 12 seconds. ASCC was controlled by a sequence of instructions on punched paper
tapes. Punched cards were used to enter data and the output from the machine was either on punched cards or by an electric typewriter.
Having completed construction of ASCC in 1943 it was decided to move the computer to Harvard University where it began to be used from May 1944. Grace Hopper worked with Aiken from 1944 on the ASCC
computer which had been renamed the Harvard Mark I and given by IBM to Harvard University. The computer figured highly in the Bureau of Ordnance's Computation Project at Harvard University, to which
Hopper had been assigned, being used by the US navy for gunnery and ballistics calculations.
Aiken completed the Harvard Mark II, a completely electronic computer, in 1947. He continued to work at Harvard on this series of machines, working next on the Mark III and finally the Mark IV up to
1952. He not only worked on computer construction, but he also published on electronics and switching theory.
In 1964 Aiken received the Harry M Goode Memorial Award, a medal and $2,000 awarded by the Computer Society:-
For his original contribution to the development of the automatic computer, leading to the first large-scale general purpose automatic digital computer.
This was one of many honours which Aiken received for his pioneering work with the development of computers. These awards were from many countries including the United States, France, The
Netherlands, Belgium, and Germany.
Article by: J J O'Connor and E F Robertson
Click on this link to see a list of the Glossary entries for this page
List of References (6 books/articles) A Quotation
A Poster of Howard Aiken Mathematicians born in the same country
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JOC/EFR © July 1999 School of Mathematics and Statistics
Copyright information University of St Andrews, Scotland
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Items where Department is "Faculty of Science > Statistics"
The Library
Items where Department is "Faculty of Science > Statistics"
Number of items: 54.
Journal Article
Borodin, Alexei, Ferrari, Patrik L., Prähofer, Michael, Sasamoto, Tomohiro and Warren, Jon. (2009) Maximum of Dyson Brownian motion and non-colliding systems with a boundary. Electronic
communications in probability, Vol.14 (No.47). pp. 486-494. ISSN 1083-589X
Copas, John B. and Lozada-Can, Claudia. (2009) The radial plot in meta-analysis : approximations and applications. Journal of the Royal Statistical Society Series C: Applied Statistics, Vol.58
(No.3). pp. 329-344. ISSN 0035-9254
Croydon, David A.. (2009) Hausdorff measure of arcs and Brownian motion on Brownian spatial trees. Annals of Probability, Vol.37 (No.3). pp. 946-978. ISSN 0091-1798
Croydon, David A.. (2009) Random walk on the range of random walk. Journal of Statistical Physics, Vol.136 (No.2). pp. 349-372. ISSN 0022-4715
Didelot, Xavier, Urwin, Rachel, Maiden, Martin C. J. and Falush, Daniel. (2009) Genealogical typing of Neisseria meningitidis. Microbiology, Vol.15 (No.10). pp. 3176-3186. ISSN 1350-0872
Howitt, Chris and Warren, Jon. (2009) Consistent families of Brownian motions and stochastic flows of kernels. The Annals of Probability, Vol.37 (No.4). pp. 1237-1272. ISSN 0091-1798
Jacka, Saul D.. (2009) Markov chains conditioned never to wait too long at the origin. Journal of Applied Probability, Vol.46 (No.3). pp. 812-826. ISSN 0021-9002
Jewell, Chris P., Kypraios, Theodore, Neal, Peter and Roberts, Gareth O.. (2009) Bayesian analysis for emerging infectious diseases. Bayesian analysis, Vol.4 (No.3). pp. 465-496. ISSN 1931-6690
Johansen, Adam M.. (2009) SMCTC : sequential Monte Carlo in C++. Journal of Statistical Software, Vol.30 (No.6). pp. 1-41. ISSN 1548-7660
Kendall, Wilfrid S.. (2009) Brownian couplings, convexity, and shy-ness. Electronic communications in probability, Vol.14 (No.7). pp. 66-80. ISSN 1083-589X
Kiddle, Steven J., Windram, Oliver P., McHattie, Stuart, Mead, A. (Andrew), Beynon, Jim, 1956-, Buchanan-Wollaston, Vicky, Denby, Katherine J. and Mukherjee, Sach. (2009) Temporal clustering by
affinity propagation reveals transcriptional modules in Arabidopsis thaliana. Bioinformatics, Vol.26 (No.3). pp. 355-362. ISSN 1367-4811
Kilkenny, Carol, Parsons, Nicholas R., Kadyszewski, Ed, Festing, Michael F. W., Cuthill, Innes C., Fry, Derek, Hutton, Jane L. and Altman, Douglas G.. (2009) Survey of the quality of experimental
design, statistical analysis and reporting of research using animals. PL o S One, Vol.4 (No.11). Article no. e7824. ISSN 1932-6203
Kolokoltsov, V. N. (Vasiliĭ Nikitich). (2009) Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics. Theory of Probability and its Applications, Vol.53
(No.4). pp. 594-609. ISSN 0040-585X
Komorowski, Michal, Finkenstädt, Bärbel, Harper, Claire V. and Rand, D. A. (David A.). (2009) Bayesian inference of biochemical kinetic parameters using the linear noise approximation. BMC
Bioinformatics, Vol.10 (No.343). ISSN 1471-2105
Liverani, Silvia, Anderson, Paul E., Edwards, Kieron D., Millar, A. J. (Andrew J.) and Smith, J. Q., 1953-. (2009) Efficient utility-based clustering over high dimensional partition spaces. Bayesian
analysis, Vol.4 (No.3). pp. 539-572. ISSN 1931-6690
Quintana, Fernando A., Steel, Mark F. J. and Ferreira, Jose T. A. S.. (2009) Flexible univariate continuous distributions. Bayesian analysis, Vol.4 (No.3). pp. 497-522. ISSN 1931-6690
Sherlock, Chris and Roberts, Gareth O.. (2009) Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets. Bernoulli, Vol.15 (No.3). pp. 774-798. ISSN 1350-7265
Warren, Jon and Windridge, Peter. (2009) Some examples of dynamics for Gelfand-Tsetlin patterns. Electronic Journal of Probability, Vol.14 . pp. 1745-1769. ISSN 1083-6489
Zhang, Junlong, Zhang, F. (Fang), Didelot, Xavier, Bruce, Kimberley D., Cagampang, Felino R., Vatish, Manu, Hanson, Mark A., Lehnert, Hendrik, Ceriello, Antonio and Byrne, Christopher D.. (2009)
Maternal high fat diet during pregnancy and lactation alters hepatic expression of insulin like growth factor-2 and key microRNAs in the adult offspring. BMC Genomics, Vol.10 (No.478). ISSN 1471-2164
Working or Discussion Paper
Akacha, Mouna, Fonseca, Thaís C. O. and Liverani, Silvia (2009) First CLADAG data mining prize : data mining for longitudinal data with different marketing campaigns. Working Paper. Coventry:
University of Warwick. Centre for Research in Statistical Methodology. (Working papers).
Akacha, Mouna and Hutton, Jane L. (2009) Analysing the rate of change in a longitudinal study with missing data, taking into account the number of contact attempts. Working Paper. Coventry:
University of Warwick. Centre for Research in Statistical Methodology. (Working papers).
Alexander, Kenneth S. and Zygouras, Nikos (2009) Equality of critical points for polymer depinning transitions with loop exponent one. Working Paper. Coventry: University of Warwick. Centre for
Research in Statistical Methodology. (Working papers).
Alexander, Kenneth S. and Zygouras, Nikos (2009) Quenched and annealed critical points in polymer pinning models. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical
Methodology. (Working papers).
Bai, Yan, Roberts, Gareth O. and Rosenthal, Jeffrey S. (Jeffrey Seth) (2009) On the containment condition for adaptive Markov Chain Monte Carlo algorithms. Working Paper. Coventry: University of
Warwick. Centre for Research in Statistical Methodology. (Working papers).
Carta, Alessandro and Steel, Mark F. J. (2009) Modelling multi-output stochastic frontiers using copulas. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical
Methodology. (Working papers).
Chakrabarty, Dalia (2009) Different traces give different gravitational mass distributions. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working
Copas, John B. and Eguchi, Shinto (2009) Likelihood for statistically equivalent models. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working
Freeman, Guy and Smith, J. Q., 1953- (2009) Bayesian MAP model selection of chain event graphs. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology.
(Working papers).
Griffin, Jim E. and Steel, Mark F. J. (2009) Time-dependent stick-breaking processes. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working papers).
Griffiths, Robert C. and Spanò, Dario (2009) Diffusion processes and coalescent trees. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working
Guan, Shiqin Helen, Bonnett, Laura and Brettschneider, Julia (2009) Using gene subsets in the assessment of microarray data quality for time course experiments. Working Paper. Coventry: University of
Warwick. Centre for Research in Statistical Methodology. (Working papers).
Hairer, Martin and Pillai, Natesh S., 1981- (2009) Ergodicity of hypoeliptic SDEs driven by fractional Brownian motion. Working Paper. Coventry: University of Warwick. Centre for Research in
Statistical Methodology. (Working papers).
Hutton, Jane L. and Stanghellini, E. (2009) Modelling health scores with the skew-normal distribution. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology.
(Working papers).
Jiang, Ci-Ren, Aston, John A. D. and Wang, Jane-Ling (2009) Smoothing dynamic positron emission tomography time courses using functional principal components. Working Paper. Coventry: University of
Warwick. Centre for Research in Statistical Methodology. (Working papers).
Kendall, Wilfrid S. (2009) Geodesics and flows in a Poissonian city. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working papers).
Kosmidis, Ioannis (2009) On iterative adjustment of responses for the reduction of bias in binary regression models. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical
Methodology. (Working papers).
Lamnisos, Demetris, Griffin, Jim E. and Steel, Mark F. J. (2009) Adaptive Monte Carlo for binary regression with many regressors. Working Paper. Coventry: University of Warwick. Centre for Research
in Statistical Methodology. (Working papers).
Li, Wenbo V., Pillai, Natesh S., 1981- and Wolpert, Robert L. (2009) On the supremum of certain families of stochastic processes. Working Paper. Coventry: University of Warwick. Centre for Research
in Statistical Methodology. (Working papers).
Liverani, Silvia, Cussens, James and Smith, J. Q., 1953- (2009) Searching a multivariate partition space using weighted MAX-SAT. UNSPECIFIED. Coventry: University of Warwick. Centre for Research in
Statistical Methodology. (Working papers).
Marshall, Tristan and Roberts, Gareth O. (2009) An ergodicity result for adaptive Langevin algorithms. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology.
(Working papers).
Mattingly, Jonathan C., McKinley, Scott A. and Pillai, Natesh S., 1981- (2009) Geometric erogdicity of a bead-spring pair with stochastic Stokes forcing. Working Paper. Coventry: University of
Warwick. Centre for Research in Statistical Methodology. (Working papers).
Mattingly, Jonathan C., Pillai, Natesh S., 1981- and Stuart, A. M. (2009) SPDE limits of the random walk Metropolis algorithm in high dimensions. Working Paper. Coventry: University of Warwick.
Centre for Research in Statistical Methodology. (Working papers).
Papaspiliopoulos, Omiros (2009) A methodological framework for Monte Carlo probabilistic inference for diffusion processes. Working Paper. Coventry: University of Warwick. Centre for Research in
Statistical Methodology. (Working papers).
Papaspiliopoulos, Omiros and Roberts, Gareth O. (2009) Importance sampling techniques for estimation of diffusions models. Working Paper. Coventry: University of Warwick. Centre for Research in
Statistical Methodology. (Working papers).
Papavasiliou, Anastasia and Ladroue, Christophe (2009) Parameter estimation for rough differential equations. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical
Methodology. (Working papers).
Pokern, Yvo, Papaspiliopoulos, Omiros, Roberts, Gareth O. and Stuart, A. M. (2009) Non parametric Bayesian drift estimation for one-dimensional diffusion processes. Working Paper. Coventry:
University of Warwick. Centre for Research in Statistical Methodology. (Working papers).
Rigat, Fabio, 1975- and Mira, Antonietta (2009) Parallel hierarchical sampling : a general-purpose class of multiple-chains MCMC algorithms. Working Paper. Coventry: University of Warwick. Centre for
Research in Statistical Methodology. (Working papers).
Roberts, Gareth O. and Rosenthal, Jeffrey S. (Jeffrey Seth) (2009) Quantitative non-geometric convergence bounds for independence samplers. Working Paper. Coventry: University of Warwick. Centre for
Research in Statistical Methodology. (Working papers).
Rogers, Jennifer Kaye, Hutton, Jane L. and Hemming, Karla (2009) Joint modelling of event counts and survival times. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical
Methodology. (Working papers).
Sherlock, Chris, Fearnhead, Paul and Roberts, Gareth O. (2009) The random walk Metropolis : linking theory and practice through a case study. Working Paper. Coventry: University of Warwick. Centre
for Research in Statistical Methodology. (Working papers).
Thwaites, Peter, Freeman, Guy and Smith, J. Q., 1953- (2009) Chain event graph MAP model selection. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology.
(Working papers).
Yau, C., Papaspiliopoulos, Omiros, Roberts, Gareth O. and Holmes, Christopher (2009) Bayesian nonparametric hidden Markov models with application to the analysis of copy-number-variation in mammalian
genomes. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working papers).
Łatuszyński, Krzysztof, Kosmidis, Ioannis, Papaspiliopoulos, Omiros and Roberts, Gareth O. (2009) Simulating events of unknown probabilities via reverse time martingales. Working Paper. Coventry:
University of Warwick. Centre for Research in Statistical Methodology. (Working papers).
Journal Item
Kendall, Wilfrid S. (2009) Academy for PhD Training in Statistics (APTS). MSOR Connections, Vol.9 (No.4).
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Posts from September 28, 2008 on The Gauge Connection
A shocking reality
I am currently a member of American Mathematical Society and when I have the chance I support the activity of this venerable and fundamental association by working as a reviewer for Mathematical
Reviews. So, I have the opportunity to read Notices that is a beatiful journal published by AMS and send freely to all members. It can be found here. In the August issue I have read an Opinion by
Melvyn B. Nathanson of CUNY that shocked me and mostly my fundations on mathematical truth. The article is here. I think that the core of this article resides on the idea that bosses of the community
make truth, mostly when proof implies thousands pages of published material very difficult to be checked carefully. We all know that, for the Wiles’s proof of Fermat theorem, the community was lucky
enough to catch such a bug in a long proof. This was conveniently corrected and all agreed that such a demonstration indeed was given. But when, for any reason, we have to rely on authorities to get
the truth we are in serious troubles. This means that whatever they said should be taken as granted but in this case does seem that no protestation is indeed possible. As scientists we cannot accept
“ipse dixit” position and if mathematics is in such a situation something must be done to correct it.
The next question to be answered is: What is the situation in physics? Physics, differently from mathematics, is an experimental science. This means that experiments should grant our ability to tell
where the truth is. Today things do not seem to be that easy for a lot of reasons but I think other sources can give judgments better than mine.
Physics laws and strong coupling
It is a well acquired fact that all the laws of physics are expressed through differential equations and our ability as physicists is to unveil their solutions in a way or another. Indeed, almost all
these equations are really difficult to solve in a straightforward way and are very far from the exercises at undergraduate courses. During the centuries people invented several techniques to manage
such equations and the most generally known is surely perturbation theory. Perturbation theory applies when a small parameter enters into the equations and a series solution is so allowed. I remember
I have seen this method the first time at the third year of my “laurea” course and was Giovanni Jona-Lasinio that showed it to me and other fellow students.
Presently, we see as small perturbation theory has become so pervasive that conclusions derived just at a perturbation level are sometime believed always true. An example of this is the Landau pole
or, generally, what implies the renormalization program. It is not generally stated but it is quite common the prejudice that when a large parameter enters into a differential equation we are stuck
and nothing can be done than using our physical intuition or numerical computation. This is true despite the fact that the inverse of such a large parameter is indeed a small parameter and most known
functions have both a small parameter and a large parameter series as well.
As I said elsewhere this is just a prejudice and I have proved it wrong in a series of papers on Physical Review A (see here, here and here). I have given an overview in a recent paper. With such a
great innovation to solve differential equations at hand is really tempting to try to apply it to all fields of physics. Indeed, I have worked for a lot of years in quantum optics testing the
approach in a lot of successful ways and I have also found applications to condensed matter physics appeared on Physical Review B and Physica E.
The point is quite clear. How to apply all this to partial differential equations? What is the effect of a large perturbation on such equations? Indeed, I have had this understanding under my nose
since the start but I have not been so able to catch it immediately. The reason is that the result is really counterintuitive. When a physical system is strongly perturbed all the terms that imply
spatial variation can be neglected. So, a strong perturbation series is a gradient expansion and the converse is true as well. I have proved it numerically in a quite easy way using two or three
lines of Maple. These results can be found in my very recent paper on quantum field theory (see here and here). Other results can be found by yourself with similar simple means and are very easy to
As strange as may seem this conclusion, it has obtained a striking confirmation through numerical computations in general relativity. Indeed, I have applied this method also to general relativity
(see here and here). Indeed this paper gives a sensible proof of the Belinski-Khalatnikov-Lifshitz or BKL conjecture on the behavior of space-time approaching a singularity. Indeed, BKL conjecture
has been analyzed numerically by David Garfinkle with a very beatiful paper published on Physical Review Letters (see here and here). It is seen in a striking way how all the gradient contributions
from Einstein equations become increasingly irrelevant as the singularity is approached. This is a clear proof of BKL conjecture and our approach of strong perturbations at work. Since then Prof.
Garfinkle has done a lot of other very good work on general relativity (see here).
We hope to show in future posts how this machinery works for pdes. In case of odes we have already posted about (see here).
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13.1 Newton's Method
Home | 18.013A | Chapter 13 Tools Glossary Index Up Previous Next
13.1 Newton's Method
This method, as just noted, consists of iterating the process of setting the linear approximation of f to 0 to improve guesses at the solution to f = 0.
The linear approximation to f(x) at argument x[0] which we will call fLx[0](x) can be described by the equation
if we set fLx[0](x) = 0, and solve for x, we obtain
and so we obtain
In the applet that follows you can enter a standard function, choose the number (nb points) of iterations that can be shown, adjust x[0] with the second slider, and look at each iteration with the
first slider. You will see the function and the effects of the iterations.
You will notice that with this method you may arrive at a nearby 0, or a far away one, depending a bit on luck.
In the old days the tedium of performing the steps of finding the x[j]'s was so formidable that it could not be safely inflicted on students.
Now, with a spreadsheet, we can set this up and do it with even a messy function f, in approximately a minute.
To do so put your initial guess, x[0] in one box say a2, put "= f(a2)" in b2, and "= f '(a2)" in c2. Then put "= a2-b2/c2" in a3 and copy a3, b2 and c2 down as far as you like. (Of course you have to
spell out what f and f ' are in doing this.)
That's all there is to it.
If column b goes to zero, the entries in column a will have converged to an answer.
If you want to change your initial guess you need only enter something else in a2; to solve a different equation you need only change b2 and c2 and copy them down.
This raises some interesting questions; namely, can we say anything about when this method will work and when it will not?
First you should realize that many functions have more than one argument for which their values are 0. Thus you may not get the solution you want.
Also, if the function f has no zero, like x^2 + 1, you will never get anywhere.
Here is another problem: if your initial guess,
And if f is implicitly defined you may find that some new guess x[j] in which f is not even defined, and the iteration will dead end.
Can we say anything positive about the use of the method?
Yes! If f goes from negative to positive at the true solution x, and f ' is increasing between x and your guess x[0], which is greater than x, then the method will always converge. Similar statements
hold when f goes from positive to negative, and under many other circumstances.
Why is this?
If f ' is increasing, then the tangent line to f at x[0] will go under the f curve at between x and x[0], so that the linear approximation, whose curve it is, will hit zero between x and x[0], and
the same thing will be true in each iteration. Thus the x's will march off toward the true solution without hope of escape and will eventually get there.
Another virtue of the method is that as one gets closer to the solution, the differentiable function will tend to look more and more like its linear approximation between the current guess and the
true solution. Thus the method tends to converge very rapidly once that current guess is near a solution.
Set up a spread sheet to apply it to the following functions:
13.1 exp(x) - 27
13.2 sin (x) - 0.1
13.3 x^2
13.4 tan x
13.5 x^1/3
13.6 x^1/3 - 1
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PV Circuit Sizing & Current Calculations
Section 690.8 of the National Electrical Code (NEC) deals with PV circuit sizing and current calculations, and defines how to calculate four maximum circuit current values. These maximum circuit
currents are used in additional calculations in sections 690.8(B). But before jumping into calculations, a few NEC definitions will be helpful, since the rules for correction factors and overcurrent
requirements can change based on the specific circuit. Working from the array to the inverter, we have:
PV source circuits are conductors between the modules, and from modules to a common point of connection, typically a junction box or combiner box. In industry terms, these are often called the “home
runs” from the individual strings.
PV output circuits are conductors between the PV source circuits and the inverter or DC utilization equipment. These are the circuit conductors after a combiner box to the inverter or charge
Inverter input circuits, in a battery-based system, are the conductors between the inverter and the battery bank. In a grid-tied system, they are the conductors between the inverter and PV output
circuits. Typically, these are the conductors between the inverter’s integrated DC disconnect and the inverter’s DC input connection.
Inverter output circuits are the AC conductors from the inverter to the ultimate connection to the AC distribution system for either stand-alone or utility-interactive systems.
The first calculation, from 690.8(A)(1), results in the maximum PV source-circuit current. The rated short-circuit current (Isc) is multiplied by 125%. For example, if a PV module has an Isc of 8.8
amps, this calculation is: 8.8 A × 1.25 = 11 A.
Section 690.8(A)(2) covers the maximum current for PV output circuits. For output circuits, multiply the Isc by the number of circuits in parallel, and then by 125%. A common installation method is
to keep the source circuits separate until they reach the inverter’s integrated DC combiner and disconnect. In that case, there are no output circuits to consider because the source circuits are not
placed in parallel outside of the inverter.
Section 690.8(A)(3) defines the maximum current for the inverter’s output circuit. For utility-interactive inverters, there isn’t a calculation required, since the maximum current is defined as the
inverter’s continuous output rating.
Section 690.8(A)(4) shows the calculation for the highest input current of a stand-alone inverter. This value helps determine the conductor size and overcurrent protection device (OCPD) rating
between the batteries and the inverter. Divide the inverter’s continuous power output rating by its lowest DC operating voltage, and then multiply by the inverter’s rated efficiency under those
Part 5 of 690.8(A), added to the 2014 Code, defines the maximum output current of DC-to-DC converters as the rated output per the manufacturer’s specifications. No additional calculations are
In the 2014 NEC, 690.8(B), which outlines the rules for calculating minimum conductor sizes in PV circuits, is titled “Conductor Ampacity.” The OCPD section has been relocated to 690.9. The method
for conductor sizing has not changed, although the 2014 sections incorporate some clarifications.
In 690.8(B)—690.8(B)(2) in the 2011 edition—two calculations must be run; the circuit conductor size must be based on the larger of the two values calculated. The first calculation is in 690.8(B)(1)
—690.8(B)(2)(a) in the 2011 edition. Because PV system currents are considered continuous, the maximum currents calculated in 690.8(A) must be multiplied by 125% to calculate the minimum conductor
size. This calculation ensures that the conductors do not carry more than 80% of the continuous current value (0.8 is the inverse of 1.25), a standard procedure in earlier Code articles. In the PV
industry, the result of this calculation is commonly referred to as the “156% factor.” When this rule is applied, the module’s rated Isc has been multiplied by 156% (125% × 125% = 156%). However,
don’t just multiply everything by 156%. Inverter output circuits were not multiplied by 125% originally, so the 156% factor doesn’t apply to them. This calculation is done before applying any
adjustment and correction factors, commonly referred to as “conditions of use,” which include corrections for conductors exposed to temperatures in excess of 30°C or more than three current-carrying
conductors within a conduit. The ampacity of the conductor, at a minimum, then, needs to be greater than or equal to the maximum current in 690.8(A) × 1.25.
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Re: Electronic Paper Request (Tamas Harczos )
Subject: Re: Electronic Paper Request
From: Tamas Harczos <tamas.harczos@xxxxxxxx>
Date: Fri, 23 Mar 2012 15:53:54 +0100
Dear List,
thank you for your replies! Special thanks to Dick for sending me not
only papers but also useful hints on reviewing the history of BM-models.
So far I have the Lumer and Johannesma papers. I would still be very
thankful for the following two:
D. Schofield, "Visualizations of speech based on a model of the
peripheral auditory system," NPL Report DITC 62/85, 1985.
J. L. Eriksson and A. Robert, "A simple nonlinear active cochlear model
with distributed feedback," Proc. International Symposium on Nonlinear
Theory and its Applications (NOLTA), Le Regent, Crans-Montana,
Switzerland, 1998.
Thank you!
"Richard F. Lyon" schrieb:
> Tamas,
> I have sent you the Johannesma 1972 paper, the origin of the term
> "gamma-tone".
> I may have Schofield 1985 on paper some place, since I quoted from it
> back in '96; I'll try to find it if someone else doesn't.
> I haven't heard of the Lumer 1987 papers before, but they look good,
> and are available online for money. I might get...
> Eriksson & Robert 1998 is just an early version of Robert & Eriksson
> 1999 JASA, I think. All they say about it is "Previous versions of
> the model were pre- sented in Eriksson and Robert, 1998." They were
> among the few people who picked up on my 1996 all-pole gammatone
> filter idea and used it to make a good model. I'd be interested if
> you find it.
> If you're interested in the history of gammatones, a relevant
> often-overlooked paper is Dirk Van Compernolle's 1991 IPO report
> "Development of a Computational Auditory Model". It turns out he did
> the pole-zero decomposition of the gammatone, and the all-pole version
> as an approximation, a bit before Slaney did. From the Laplace
> transforms that he tabulated, he concluded, "From the above table it
> can be seen that an all-pole filter approximation will be excellent as
> long as alpha is small, i.e. for sharp filters, which is the case for
> a cochlear filterbank." It's easy to find online.
> Dick
This message came from the mail archive
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Algebra - Math Learning Guides
Equations and Inequalities
The word algebra comes from the Arabic phrase al-jebr, meaning "the reunion of broken parts." Like your high school reunion will be if you go on to become a linebacker.
Besides being a mathy sort of word, around the 1500s or so, algebra was also used to mean "bonesetting." (The linebacker analogy still applies). Spanish still uses algebra that way, and algebrista in
Spanish means "bonesetter." You thought this was a math class. Mwa-ha-ha-ha. That was our evil laugh. Yes, we've been practicing.
Algebra, as taught in schools today, has two main parts:
1. Translating real-world problems into symbols, and
2. Manipulating the symbols to find answers.
This second one is nice, because it means a lot of problems can be broken down into two smaller ones that are easier to solve individually. Hey, we like "easier." First, we translate the problem into
the language of math, and then we do some arithmetic and manipulation of symbols to find an answer. In other words, we manhandle those symbols. We show them no mercy. Oh yeah, we also need to
remember to answer the right question while we're at it. Test-graders are real sticklers for accuracy, for some reason.
In this section we deal mostly with manipulating/manhandling symbols, but we also start tossing in problems that let us practice translating questions from English into math. (Just be grateful you
don't need to translate War and Peace into math. Can you imagine how complicated that formula would be?) The translation stage is where we turn words into equations or inequalities. Then we
manipulate symbols to solve the equations or inequalities, and find an answer.
We prefer answers because answers are the cat's pajamas. Equations and inequalities are the cat's hiking boots, at best.
But hold on—sometimes we don't get any answers, and sometimes we get more than one. The nice thing is that no matter how many answers there are, we will know when we are done. Often, we can even
check our answer(s) to make sure we're right. Might not be a bad thing to do before sending texts, too. It's time to stop blaming everything on AutoCorrect.
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Can anyone tell some of the chemical engg softwares ....... - Rediff Questions & Answers
Can anyone tell some of the chemical engg softwares .......
Asked by
, 01 Apr '08 09:27 pm
Earn 10 points for answering
Answers (1)
1............ ChemMaths is a chemical engineering,mathematical and chemistry program. Software suitable for chemistry,chemical engineering students and professionals.Contains information on 3000+
chemical compounds,allows predition of chemical compound properties,critical constants, thermodynamic properties,gas & liquid diffusivitiy ,surface tension,viscosity calculations etc,periodic table,
solves 500+ chemical/electrical/civil/mechcanical engineering,design,distillation, phyics, and mathematical equations. Contains 200+ unit conversions. Contains a graphical program to draw 2D/3D
graphs,general chemical sturcture drawing program,process simulation program. Solve for matrices, triangles, finance, geometry,area/surface/volume,statistics and many other mathematical problems and
2................Molecular Weight Calculator v1.1
A molecular weight calculator for download. MS Windows application.
3............... UNIFAC Activity Coefficient Calculator
A Win 95/NT pro
Answered by Virendra K, 02 Apr '08 01:10 am
Report abuse
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Reference for complex analysis jargon
up vote 2 down vote favorite
I am not a (complex) analyst but it seems that some of the questions I am working on are related to the following concepts:
1. logarithmic capacity
2. transfinite diameter
3. Green's function of a compact set
4. System of Fekete points
5. upper regularization
I have looked at various books but some of them are very old (e.g., Hille's Analytic function theory which was published in 1962), and most of them just explain the subject briefly. I am looking for
references which are more on the advanced side rather than elementary, to be able to find in them the results I need.
Any suggestions?
Thanks everyone for the helpful references. – Hadi Feb 17 '11 at 23:52
The question is not the resources of the Book you want to read, Do you have patience to read the book? – Jame Ake Oct 8 '13 at 11:38
add comment
6 Answers
active oldest votes
I would say these concepts rather belong to the field of potential theory. You will find most of the definitions and a fairly advanced treatment of the subject in Logarithmic
up vote 7 down Potentials with External Fields by Saff and Totik.
add comment
I really recommend the book Potential Theory in the complex plane" by Thomas Ransford.
up vote 5 down It's a very nice book with exercises and it covers each of the 5 points you mentioned.
+1. I had intended, while in Quebec, to work my way through the book properly... ;) – Yemon Choi Dec 2 '11 at 2:14
Although to be fair, Tom never intended the book as a reference text. It might still be useful to the OP as an indication of where to look next and what one might expect – Yemon
Choi Dec 2 '11 at 7:55
add comment
For the first two (maybe 3): L. Ahlfors: Conformal Invariants: Topics in Geometric Function Theory
up vote 2 down vote
Yes, a good book. – Gerald Edgar Oct 8 '13 at 12:11
add comment
Conformal radius of a domain and Transfinite diameter seem to have most of these terms; see also http://en.wikipedia.org/wiki/Conformal_radius .
up vote 2 down vote
add comment
Goluzin, Geometric theory of functions of a complex variable, contains a very comprehensive discussion of transfinite diameter, Fekete points etc. Ransford's book mentioned above is
up vote 2 down also very nice.
add comment
J. B. Garnett, D. E. Marshall: Harmonic Measure And also Carleson's (I don't remember the name of the book) book contains at least first two.
up vote 0 down vote
add comment
Not the answer you're looking for? Browse other questions tagged ca.analysis-and-odes cv.complex-variables fa.functional-analysis harmonic-analysis or ask your own question.
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The main idea in the proof of Artin's vanishing
up vote 5 down vote favorite
Does anybody know an easy explanation of the proof of Artin's vanishing theorem (that the etale cohomology of an affine variety of dimension $n$ over an algebraically closed field vanishes in degrees
$>n$, or of any other version of this statement)? I have found some proofs; all of them are step by step, and it is not clear to me which of these steps are the most important ones. So, what is the
central idea here?
ag.algebraic-geometry etale-cohomology reference-request
add comment
1 Answer
active oldest votes
I was curious myself after learning this result sometime ago from Lazarsfeld's book on positivity (he calls it the Artin-Grothendieck theorem). The corresponding statement for smooth
varieties over the complex numbers and singular cohomology (theorem of Andreotti-Frankel) follows from the fact that Morse theory shows that the variety is homotopy equivalent to a
CW-complex with no cells in dimensions $>n$.
The etale cohomology counterpart works more generally for constructible sheaves. This is probably not very helpful, but here is a sketch of the argument from Lazarsfeld (he uses
constructible sheaves in the complex topology, but it should adapt to etale sheaves):
1. Reduce to the affine space by choosing a finite map $X\to \mathbb{A}^n$ using Noether normalization (already here it is crucial to work with constructible sheaves, not constant
up vote 9 down sheaves, so we clearly gain something from generalization),
vote accepted 2. Prove the result for $\mathbb{A}^1$,
3. Prove the result for $\mathbb{A}^n$ by induction on $n$ using the Leray spectral sequence. Here the crucial observation is that if we choose a sufficiently generic linear
projection $\mathbb{A}^n\to \mathbb{A}^{n-1}$, then the stalks of the higher direct images will compute cohomology on the fibers.
So all in all it is a typical example of devissage, which I usually to dislike but slowly learn to appreciate. I think from the outline it is clear which are the key ideas, but I
would still really like to see a conceptual proof.
Thank you! Part 3 of this argument seems to be the most difficult one; I should think about it. – Mikhail Bondarko Oct 8 '12 at 9:11
3 Devissage is a bit like induction, which can also be confusing. Most of the work goes into the induction step (step 3). – Donu Arapura Oct 8 '12 at 11:12
1 The other thing that I should point out is that for this devissage argument to work, one has to prove it for general coefficients. – Donu Arapura Oct 8 '12 at 11:42
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Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry etale-cohomology reference-request or ask your own question.
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Using Power Series to solve Non-Homog. DE
March 23rd 2010, 04:45 PM #1
Oct 2009
Using Power Series to solve Non-Homog. DE
The equation is y''-4xy'-4y= e^x, assuming that y=(sigma from 0 to infinity) c_nx^n, a=0
So I did the substitution and found the recurrence term to be
c_n+2 = [(8nc_n)/(n+2)(n+1)]
I just have to find the first six terms of the solution but how do I get the first two terms? (since when n=0 I get c_2)
The first two are arbitrary and are due to the initial conditions so if y(0)=0 and y'(0)=1 then c0=0 and c1=1. Here's some Mathematica code to check the first 25 terms of the power series with
those initial conditions against a numerically computed solution. It's a little messy. See if you can interpret it if you want. Note how I set c0 to 0 and c1 to 1 then created a table using your
recurrence relation to compute the next 25 terms then used NDSolve to solve it numerically then superimposed the two plots. The agreement is not as good as I would expect.
Subscript[c, 0] = 0;
Subscript[c, 1] = 1;
myclist = Table[Subscript[c, n + 2] =
(8*n*Subscript[c, n])/((n + 2)*
(n + 1)), {n, 0, 25}]
myf[x_] := Sum[Subscript[c, n]*x^n,
{n, 0, 25}];
p1 = Plot[myf[x], {x, 0, 1}]
mysol = NDSolve[{Derivative[2][y][x] -
4*x*Derivative[1][y][x] -
4*y[x] == Exp[x], y[0] == 0,
Derivative[1][y][0] == 1}, y,
{x, 0, 1}];
p2 = Plot[Evaluate[y[x] /. mysol],
{x, 0, 1}, PlotStyle -> Red]
Show[{p1, p2}]
Last edited by shawsend; March 24th 2010 at 03:30 AM.
March 24th 2010, 03:16 AM #2
Super Member
Aug 2008
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Distinguished Lecture Series in Statistical Science
October 5 - 3:30 p.m.
Statistical Learning with Large Numbers of Predictor Variables
Many present day applications of statistical learning involve large numbers of predictor variables. Ofter that number is much larger than the number of cases or observations available to train the
learning algorithm. In such situations traditional methods fail. Recently new techniques based on regularization have been developed that can often produce accurate learning models in these settings.
This talk will describe the basic principles underlying the method of regularizationand then focus on those methods exploiting the sparsity of the predicting model. The potential merits of these
methods are then explored by example.
October 6 - 11:00 a.m.
Predictive Learning via Rule Ensembles
General regression and classification models are constructed as linear combinations of simple rules derived from the data. Each rule consists of a conjunction of a small number of simple statements
concerning the values of individual input variables. These rule ensembles are shown to produce predictive accuracy comparable to the best methods. However their principal advantage lies in
interpretation. Because of its simple form, each rule is easy to understand, as is its influence on individual predictions, selected subsets of predictions, or globally over the entire space of joint
input variable values. Similarly, the
degree of relevance of the respective input variables can be assessed globally, locally in different regions of the input space, or at individual prediction points. Techniques are presented for
automatically identifying those variables that are involved in interactions with other variables, the strength and degree of those interactions, as well as the identities of the other variables with
which they interact. Graphical representations are used to visualize both main and interaction effects.
Dr. Friedman is one of the world's leading researchers in statistics and data mining. He has been a Professor of Statistics at Stanford University for nearly 20 years and has published on a wide
range of data-mining topics including nearest neighbor classification, logistical regressions, and high dimensional data analysis. His primary research interest is in the area of machine learning.
The Distinguished Lecture Series in Statistical Science series was established in 2000 and takes place annually. It consists of two lectures by a prominent statistical scientist. The first lecture is
intended for a broad mathematical sciences audience. The series occasionally takes place at a member university and is tied to any current thematic program related to statistical science; in the
absence of such a program the speaker is chosen independently of current activity at the Institute. A nominating committee of representatives from the member universities solicits nominations from
the Canadian statistical community and makes a recommendation to the Fields Scientific Advisory Panel, which is responsible for the selection of speakers.
Distinguished Lecture Series in Statistical Science Index
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