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Here's the question you clicked on:
Where is the median in this stem-and-leaf plot?
• one year ago
• one year ago
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did u googled it :)
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Tried to.
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All the median is, is just the middle number. The middle umber of this stem-and-leaf plot is the median. Do you know what that number is? :)
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Hmm, do I know the number thats the median of the stem-and-leaf-plot??
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3 is. rightt???
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Not necessarily. For instance the first number on this plot would be 11, the 1 on the stem side =10 and the leaf side =1 , then the next number woudl be 13, and then 14 would be on the first row.
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Oh, okay. I get it now! ((:
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K do you know what the median is of this plot? :)
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Let me see.. brb
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K :)
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35! :D
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Yes :D Great job @iiCourtney !!
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Haha, thanks soo much. Couldn't have done it with out ya! ;DD
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No problem :) If you need any more help with anything else, just ask me and I'll try my best to help :)
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Sweet, I will. Thanks a billion♥♥
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Vibration and Sound
Philip M. Morse
Originally published in 1936; Reprinted in 1981
(Note: This is a very long table of contents with 8 chapters and 34 sections)
1. Definitions and Methods
Units. Energy
2. A Little Mathematics
The Trigonometric Functions, Bessel Functions. The Exponential. Conventions as to Sign. Other Solutions. Contour Integrals. Infinite Integrals. Fourier Transforms
3. Free Oscillations
The General Solution. Initial Conditions. Energy of Vibration
4. Damped Oscillations
The General Solution. Energy Relations
5. Forced Oscillations
The General Solution. Transient and Steady State. Impedance and Phase Angle. Energy Relations. Electromechanical Driving Force. Motional Impedance. Piezoelectric Crystals.
6. Response to Transient Forces
Representation by Contour Integrals. Transients in a Simple System. Complex Frequencies. Calculating the Transients. Examples of the Method. The Unit Function. General Transient. Some
Generalizations. Laplace Transfoms.
7. Coupled Oscillations
The General Equation. Simple Harmonic Motion. Normal Modes of Vibration. Energy Relations. The Case of Small Coupling. The Case of Resonance. Transfer of Energy. Forced Vibrations. Resonance and
Normal Modes. Transient Response.
8. Waves on a String
The Wave Velocity. The General Solution for Wave Motion. Initial Conditions. Boundary Conditions. Reflection at a Boundary. Strings of Finite Length.
9. Simple Harmonic Oscillations
The Wave Equation. Standing Waves. Normal Modes. Fourier Series. Initial Conditions. The Series Coefficients. Plucked String, Struck String, Energy of Vibration
10. Forced Vibrations
Wave Impedance and Admittance. General Driving Force. String of Finite Length. Driving Force Applied Anywhere. Alternative Series Form. Distributed Driving Force. Transient Driving Force. The Piano
String. The Effect of Friction. Characteristic Impedances and Admittances.
11. Strings of Variable Density and Tension
General Equation of Motion. Orthogonality of Characteristic Functions. Driven Motion. Nonuniform Mass. The Sequence of Characteristic Functions. The Allowed Frequencies. Vibrations of a Whirling
String. The Allowed Frequencies. The Shape of the String. Driven Motion of the Whirling String.
12. Perturbation Calculations
The Equation of Motion. First-order Corrections. Examples of the Method. Characteristic Impedances. Forced Oscillation. Transient Motion
13. Effect of Motion of the End Supports
Impedance of the Support. Reflection of Waves. Hyperbolic Functions. String Driven from One End. Shape of the String. Standing Wave and Position of Minima. Characteristic Functions. Transient
Response. Recapitulation.
CHAPTER IV - THE VIBRATION OF BARS
14. The Equation of Motion
Stresses in a Bar. Bending Moments and Shearing Forces. Properties of the Motion of the Bar. Wave Motion in an Infinite Bar.
15. Simple Harmonic Motion
Bar Clamped at One End. The Allowed Frequencies. The Characteristic Functions. Plucked and Struck Bar. Clamped-clamped and Free-free Bars. Energy of Vibration. Nonuniform Bar. Forced Motion.
16. Vibrations of a Stiff String
Wave Motion on a Wire. The Boundary Conditions. The Allowed Frequencies.
17. The Equation of Motion
Forces on a Membrane. The Laplacian Operator. Boundary Conditions and Coordinate Systems. Reaction to a Concentrated Applied Force
18. The Rectangular Membrane
Combinations of Parallel Waves. Separating the Wave Equation. The Normal Modes. The Allowed Frequencies. The Degenerate Case. The Characteristic Functions
19. The Circular Membrane
Wave Motion on an Infinite Membrane. Impermanence of the Waves. Simple Harmonic Waves. Bessel Functions. The Allowed Frequencies. The Characteristic Functions. Relation between Parallel and Circular
Waves. The Kettledrum. The Allowed Frequencies
20. Forced Motion. The Condenser Microphone
Neumann Functions. Unloaded Membrane, Any Force. Localized Loading, Any Force. Uniform Loading, Uniform Force. The Condenser Microphone. Electrical Connections. Transient Response of Microphone
21. The Vibration of Plates
The Equation of Motion. Simple Harmonic Vibrations. The Normal Modes. Forced Vibration
CHAPTER VI - PLANE WAVES OF SOUND
22. The Equation of Motion
Wave along a Tube. The Equation of Continuity. Compressibility of the Gas. The Wave Equation. Energy in a Plane Wave. Intensity. The Decibel Scale. Intensity and Pressure Level. Sound Power.
Frequency Distribution of Sounds. The Vowel Sounds.
23. The Propagation of Sounds in Tubes
Analogous Circuit Elements. Constriction. Tank. Examples. Characteristic Acoustic Resistance. Incident and Reflected Waves. Specific Acoustic Impedance. Standing Waves. Measurement of Acoustic
Impedance. Damped Waves. Closed Tube. Open Tube. Small-diameter Open Tube. Reed Instruments. Motion of the Reed. Pressure and Velocity at the Reed.
Even Harmonics. Other Wind Instruments. Tube as an Analogous Transmission Line. Open Tube, Any Diameter. Cavity Resonance. Transient Effect, Flutter Echo.
24. Propagation of Sound in Horns
One-parameter Wave. An Approximate Wave Equation. Possible Horn Shapes. The Conical Horn. Transmission Coefficient. A Horn Loud-speaker. The Exponential Horn. The Catenoidal Horn. Reflection from the
Open End, Resonance. Wood-wind Instruments. Transient Effects
25. The Wave Equation
The Equation for the Pressure Wave. Curvilinear Coordinates
26. Radiation from Cylinders
The General Solution. Uniform Radiation. Radiation from a Vibrating Wire. Radiation from an Element of a Cylinder. Long- and Short-wave Limits. Radiation from a Cylindrical Source of General Type.
Transmission inside Cylinders. Wave Velocities and Characteristic Impedances. Generation of Wave by Piston.
27. Radiation from Spheres
Uniform Radiation. The Simple Source. Spherical Waves of General Form. Legendre Functions. Bessel Functions for Spherical Coordinates. The Dipole Source. Radiation from a General Spherical Source.
Radiation from a Point Source on a Sphere. Radiation from a Piston Set in a Sphere
28. Radiation from a Piston in a Plane Wall
Calculation of the Pressure Wave. Distribution of Intensity. Effect of Piston Flexure on Directionality. Radiation Impedance, Rigid Piston. Distribution of Pressure over the Piston. Nonuniform Motion
of the Piston. Radiation out of a Circular Tube. Transmission Coefficient for a Dynamic Speaker. Design Problems for Dynamic Speakers. Behavior of the Loud-speaker. Transient Radiation from a Piston.
29. The Scattering of Sound
Scattering from a Cylinder. Short Wavelength Limit. Total Scattered Power. The Force on the Cylinder. Scattering from a Sphere. The Force on the Sphere. Design of a Condenser Microphone. Behavior of
the Microphone.
30. The Absorption of Sound at a Surface
Surface Impedance. Unsupported Panel. Supported Panel. Porous Material. Equivalent Circuits for Thin Structures. Formulas for Thick Panels. Reflection of Plane Wave from Absorbing Wall.
31. Sound Transmission through Ducts
Boundary Conditions. Approximate Solutions. Principal Wave. Transient Waves. The Exact Solution. An Example
32. Normal Modes of Vibration
Room Resonance. Statistical Analysis for High Frequencies. Limiting Case of Uniform Distribution. Absorption Coefficient. Reverberation. Reverberation Time. Absorption Coefficient and Acoustic
Impedance. Standing Waves in a Rectangular Room. Distribution in Frequency of the Normal Modes. Axial, Tangential, and Oblique Waves. Average Formulas for Numbers of Allowed Frequencies. Average
Number of Frequencies in Band. The Effect of Room Symmetry. Nonrectangular Rooms. Frequency Distribution for Cylindrical Room
33. Damped Vibrations, Reverberation
Rectangular Room, Approximate Solution. Wall Coefficients and Wall Absorption. Reverberation Times for Oblique, Tangential, and Axial Waves. Decay Curve for Rectangular Room. Cylindrical Room.
Second-order Approximation. Scattering Effect of Absorbing Patches.
34. Forced Vibrations
Simple Analysis for High Frequencies. Intensity and Mean-square Pressure. Solution in Series of Characteristic Functions. Steady-state Response of a Room. Rectangular Room. Transmission Response. The
Limiting Case of High Frequencies. Approximate Formula for Response. Exact Solution. The Wall Coefficients. Transient Calculations, Impulse Excitation. Exact Solution for Reverberation.
I and II, Trigonometric and Hyperbolic Functions. III and IV, Hyperbolic Tangent of Complex Quantity. V, VI and VII, Bessel Functions. VIII, Impedance Functions for Piston. IX, Legendre Functions.
XII, General Impedance Functions for Piston. XIII, Absorption Coefficients
I and II, Hyperbolic Tangent Transformation. III, Magnitude and Phase Angles of sinh and cosh. IV, Standing Wave Ratio and Phase vs. Acoustic Impedance. V, Exact Solutions for Wave Modes in
Rectangular Ducts and Rooms. VI, Absorption Coefficient vs. Acoustic Impedance
Return to List of Publications | {"url":"http://asa.aip.org/books/vib.html","timestamp":"2014-04-21T04:38:29Z","content_type":null,"content_length":"20217","record_id":"<urn:uuid:73bcf321-1dfa-4c18-86f6-70fa76271d05>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00224-ip-10-147-4-33.ec2.internal.warc.gz"} |
Need help sketching
April 2nd 2009, 06:31 PM #1
Junior Member
Mar 2009
Need help sketching
y = x(ln x)^3
How do you sketch this? is there any asymptotes? intercepts? min/max value? Any help will be greatly appreciated!
Last edited by proski117; April 2nd 2009 at 06:44 PM.
Well, the graph of just y=ln(x) just keeps growing on for infinity, and so does the graph of y=x, so combining those two together, you can pretty much guarantee that the graph is going to still
grow on to infinity.
You also know that the ln(1) = 0, so the function will equal 0 at x=1.
and because you can't take the ln of a negative number, you know there's a vertical asymptote at x=0
April 2nd 2009, 07:25 PM #2
Junior Member
Mar 2009
April 2nd 2009, 07:52 PM #3 | {"url":"http://mathhelpforum.com/calculus/82033-need-help-sketching.html","timestamp":"2014-04-24T16:53:29Z","content_type":null,"content_length":"35641","record_id":"<urn:uuid:ea91509a-db48-4a92-a61f-b28c19a6572e>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00383-ip-10-147-4-33.ec2.internal.warc.gz"} |
Pete L. Clark's Math 3200 Course, Fall 2009
Instructor: Assistant Professor Pete L. Clark, Ph.D., pete (at) math (dot) uga (dot) edu
Course webpage: http://www.math.uga.edu/~pete/MATH3200F2009.html (i.e., right here)
Office Hours: Boyd 502, TuTh 2-3 pm, and by appointment
Course Text (required): Mathematical Proofs: A Transition to Advanced Mathematics, by Gary Chartrand, Albert D. Polimeni and Ping Zhang, 2nd edition.
For information on grades, exams and other procedural matters, please consult the course syllabus.
Lecture Notes on Mathematical Induction: click here
These notes are provided to you for your enlightenment and edification. In some places they go further than what was presented in the lectures. You are not responsible for any of this additional
material. Moreover, the notes contain some "exercises", which are not to be turned in, and some of them are quite challenging, but those of you who are seeking deeper understanding of the material
may enjoy thinking about them.
Lecture Notes on Relations and Functions: click here
Assignment 1: Due in class, Thursday August 27th
To solve: Chapter 1, exercises 1-45, 47
To be turned in: Chapter 1, exercises 3, 6, 8, 9, 10, 16, 17, 20, 24, 26, 31, 32, 41
Optional Typed Problems:
OT1.1) Draw Venn diagrams for 4 and for 5 sets. (It is not possible to use circles. You might think about trying to prove this if that sounds interesting to you.)
OT1.2) Believe it or not, there has been some recent work done on Venn diagrams with certain nice properties. For instance, it is possible to make a very pretty Venn diagram for 5 sets using
congruent ellipses. Research this on the internet and write a short essay (approximately two pages) detailing some of the interesting results.
Assignment 2: Due in Class, Tuesday September 8th To solve: Chapter 2, exercises: 1-6, 8-12, 14-20, 22, 23, 25, 26, 28, 30-35, 37-41, 45, 46
To be turned in: 2, 4, 8, 10, 14, 16, 18, 20, 22, 30, 32, 34, 38, 40, 46
Typed Problems: Please do at least two of the following problems.
T2.1) Is there a partition of the empty set? (Comment: The definition in your text explicitly excludes the empty set. I am asking you to nevertheless consider whether there exists a family of sets
satisfying the three properties of the partition of X when X is the empty set.)
T2.2) a) Let a,b,a',b' be objects. Show that { {a}, {a,b} } = { {a'}, {a',b'} } if and only if a = a' and b = b'.
b) Explain why the result of part a) would allow us to define the ordered pair (a,b) as { {a}, {a,b} }.
c) Do you have any reservations about this definition? (For example, is it the only possible definition? Is it helpful to explicitly define ordered pairs in this way?) Discuss.
T2.3) a) Let X,Y,Z be sets. Prove that (X union Y) intersect Z = (X intersect Z) union (Y intersect Z).
b) Let P,Q,R be statements. Prove that (P or Q) and R = (P and R) or (Q and R).
c) Can you give an argument which proves both a) and b) at the same time?
T2.4) a) Show that there exists a binary logical operator, P*Q, such that not P, (P or Q) and (P and Q), can all be constructed in terms of the operator *.
b) Of the 16 binary logical operators, how many have the property in part a)?
Assignment 3: Due in Class, Tuesday September 15th
To solve: chapter 3, all exercises (but none of the additional exercises)
To be turned in: the even-numbered problems 3.2-3.30 except 3.26. (N.B.: Feel free to disregard 3.26 until further notice.)
Typed Probems:
T3.1) a) You are shown a selection of cards, each of which has a single letter printed on one side and a single number printed on the other side. Then four cards are placed on the table. On the up
side of these cards you can see, respectively, D, K, 3 and 7. Here is a rule: "Every card that has a D on one side has a 3 on the other." Your task is to select all those cards, but only those cards,
which you would have to turn over in order to discover whether or not the rule has been violated.
b) You have been hired to watch, via closed-circuit camera, the bouncer at a certain 18-and-over club. In order to be allowed to drink once inside the club, a patron must display valid 21-and-over ID
to the bouncer, who then gives him/her a special bracelet. In theory the bouncer should check everyone's ID, but (assume for the purposes of this problem, at least!) it is not illegal for someone who
is under 18 to enter the club, so you are not concerned about who the bouncer lets in or turns away, but only who gets a bracelet. You watch four people walk into the club, but because the bouncer is
so large, sometimes he obscures the camera. Here is what you can see:
The first person gets a bracelet.
The second person does not get a bracelet.
The third person displays ID indicating they are 21.
The fourth person does not display any ID.
You realize that you need to go down to the club to check some IDs. Precisely whose ID's do you need to check to verify that the bouncer is obeying the law?
c) Any comments?
Optional Typed Problems (Turn in Before 9/22/09)
OT3.2) a) Write down clearly the parity rules for addition and multiplication that were discussed and used in class, and verify (i.e., prove!) all of them using the 2k / 2l+1 technique seen in class
and in the textbook.
b) Are there similar parity rules for exponentiation? E.g., is an even/odd number raised to an even/odd power always even/odd? Discuss.
Assignment 4: Due in Class, Thursday September 24th
To solve: Chapter 4, all even exercises except 4.10, 4.12, 4.14, 4.16.
To be handed in: Problems 4.2, 4.4, 4.6, 4.8, 4.28, 4.30, 4.38, 4.40, 4.44
Typed problems:
T4.1) Write a one page essay describing your experience with the course so far, especially any concerns or suggestions that you may have. For instance, the pace of the course, the difficulty of
specific topics, and/or the amount of homework are all good topics. How would you feel about being asked to present solutions to problems on the board during class time?
Assignment 5: Due in Class, Tuesday October 6th
To solve: Chapter 5, all even exercises (none of the additional exercises).
To be handed in: Problems 5.2, 5.4, 5.8, 5.14, 5.18, 5.20, 5.22, 5.24, 5.26, 5.28, 5.29, 5.30, 5.31, 5.32, 5.36
Typed problems: Chapter 5, Exercises 5.38, 5.45, 5.46
Assignment 6: Due in class, Thursday October 15th
To be read: Chapter 7 Quiz, pp. 167-168
To solve but not hand in: Problems 7.1, 7.3, 7.4, 7.5, 7.7, 7.8, 7.9, 7.11, 7.12, 7.13, 7.15
To be handed in: Problems 6.6, 6.10, 6.11, 6.14, 6.15, 6.17, 7.2, 7.6, 7.10, 7.14, 7.16
Typed Problems:
T6.1) Exercise 6 on p. 10 of the induction handout.
T6.2) Exercise 8 on p. 12 of the induction handout.
Optional Typed Problems (turn in any time before the end of the semester):
OT6.1) Exercise 2 on p.6 of the induction handout.
OT6.2) Exercise 3 on p. 6 of the induction handout.
OT6.3) Exercise 4 on p.7 of the induction handout.
OT 6.4) Find the smallest real number a such that for all real numbers x, a^x >= x.
OT 6.5) Explore variants of Proposition 7 (p. 8 of the induction handout) to partial sums of other p-series.
Assignment 7: Due in class, Tuesday, November 17th
To solve: Chapter 6, Exercises 6.18, 6.19, 6.20, 6.21, 6.22, 6.23, 6.26, 6.28, 6.30, 6.31, 6.32, 6.34, 6.35, 7.(18+3n) for 0 <= n <= 16, 7.67
To be handed in: 6.18, 6.20, 6.22, 6.26, 6.28, 6.30, 6.32, 6.34, 7.18, 7.24, 7.30, 7.36, 7.42, 7.48, 7.54, 7.60, 7.66
Typed problems: none due this week
Optional Typed Problems:
OT7.1: Let x,y,z be any integers such that x^2 + y^2 = z^2. Show that 60 | xyz. Show also that 60 cannot be replaced by any larger integer.
Assignment 8: Due in class, Thursday, November 19th
To solve: Chapter 8: Exercises 1-27 To be handed in: Chapter 8: Even-numbered exercises from 2-26
Optinonal Typed Problems: click here
Spring 2009 Review Materials:
I will let you know if we cover anything differently this time around so as to require less, more, or different review materials.
Review problems for first midterm exam
Commment: Please disregard Problem 12 on congruences. We have not yet covered this material, and you are not responsible for it on the first midterm.
click here for solutions
Review problems for second midterm exam
Review problems for third midterm exam
First midterm exam
First midterm exam, Fall 2009
Second midterm exam
Second midterm exam, Fall 2009
Third midterm exam
Third midterm exam, Fall 2009
Some interesting -- and perhaps relevant -- links:
Why You Should Be A Math Major
Why Major in Mathematics?
What Can a Math Degree Do For You?, Jaqueline Jensen, Sam Houston State University.
Mathematics of Venn Diagrams
Wikipedia article on Venn diagrams.
Are Venn Diagrams Limited to Three or Fewer Sets?, by Amy N. Myers.
A Survey of Venn Diagrams, by Frank Ruskey and Mark Weston.
Wikipedia article on ordered pairs.
Some logical weaponry: the Sheffer stroke, the Peirce arrow, the Quine dagger
A Great Answer to T2.2c)
Does Mathematics Need a Philosophy?, by Timothy Gowers. The essay as a whole is highly recommended, but see especially Section 5 on ordered pairs.
On the Psychology Behind T3.1
For a very quick introduction, try Wason Selection Task (Wikipedia)
But wikipedia does not do justice to this unsolved problem in the psychology of logical reasoning. A google search reveals a rich literature on the subject. | {"url":"http://www.math.uga.edu/~pete/MATH3200F09.html","timestamp":"2014-04-19T01:47:37Z","content_type":null,"content_length":"13744","record_id":"<urn:uuid:ac4eab68-c25b-42df-9c09-896126cc2fb2>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00618-ip-10-147-4-33.ec2.internal.warc.gz"} |
Haskell Lesson
<asdftimo> hey roconnor, what do you think about this function for determining if a given number is prime:
<asdftimo> isprime n=not (elem 0 $ map(\x -> mod n x) $ takeWhile(\x -> x^2 <= n)(2:[3,5..])
<roconnor> asdftimo: that's not a bad start
<roconnor> asdftimo: it would be better if you test only prime numbers instead of 2 and the odd numbers
<asdftimo> im using isprime to generate
<asdftimo> primelist = [x | x <- 2:[3,5..], isprime x]
<asdftimo> yeah, im going to then use this list to find prime factors
<roconnor> how about primelist = 2:[x | x<-[3,5..], isprime x]
<asdftimo> i don't understand why that is an improvement
<roconnor> and then use takeWhile (\x -> x^2 <= n) primelist in your isprime
<asdftimo> roconnor: that's kind of strange, since they both point to each other, but i guess i'll go try that...
<roconnor> :)
<roconnor> you need to pull the 2: out in front in order for the recursive definition to get going.
<roconnor> kinda like a starter motor on an engine
<asdftimo> roconnor: that is fucking cool
<asdftimo> it works
<asdftimo> i love haskell
<asdftimo> it is going to take me another good 5 mins to comprehend this, but i can tell it is pretty amazing
— #haskell | {"url":"http://r6.ca/blog/20081116T213644Z.html","timestamp":"2014-04-19T09:24:52Z","content_type":null,"content_length":"4534","record_id":"<urn:uuid:373e6fb4-513f-45e4-bcab-7e5040340280>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00205-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Math Mama Writes...
I was doing a workshop today for a few of my calculus students on using numerical integration techniques. I started to show them how to use Excel, and it had changed!
1. The columns are no longer lettered, they're numbered, just like the rows. I don't even know how to refer to a particular cell any more!
2. The formulas no longer refer to cell labels, but to the distance from the formula cell. (This is relative addressing, which was implied before, but not obvious in the formula.)
3. The way you click to select has changed too, and I had real trouble typing in my formulas.
Her birthday is January 15, but today gets to be her day. Cathy O'Neil (aka Mathbabe) started Sonia Kovalevsky Day at Barnard College in 2006, and it sounds like it's been going strong ever since.
Sonia is one of my heroes. Born in 1850 in Russia, she loved mathematics, fought her parents to study it, and entered into a marriage of convenience so that she could study abroad. She earned a PhD
studying under Weierstrass, and still couldn't find work doing mathematics.
Eventually she got a position at the University of Stockholm, and later won the prestigious Prix Bordin. She also had a daughter, who she raised alone. She died at age 41, when her daughter was only
Mathematician, activist, and mother, Sonia was also a writer, whose novels were published to some acclaim. My kind of woman.
(Read more here and here.)
by Cristóbal Vila
His explanations of the math behind this sweet movie are even more exciting, for me.
(Thanks, Murray.)
I've been enjoying teaching this course, and I was liking the text, by David C. Lay ... until we got to chapter 3 on determinants. I worked through all the problems last weekend, verifying that
everything was easy for me, and read through the proofs. Unfortunately, I thought I got it all, but I was moving too fast, and ended up in front of my class on Monday unable to really do the proofs.
With my house being broken into later that day (3rd time in 5 months; yes, it's hideous, but nothing was taken this time), I never caught up this week. Yesterday and today, I've spent about 5 hours
writing up the proofs for 3 theorems. I still see a few holes, but I'm pretty proud of what I've put together.
My text defines the determinant by expanding on the first row. Looking around online, that doesn't look like a standard definition, but it seems like a fine starting point. From there we want to
prove that you can get the determinant by expanding on any row or column. (My text says "We omit the proof to avoid a lengthy digression." Bah! It's not math if you don't know why it's true!) My
proof may still have a bit of a hole (regarding which terms are negative), but I think it's more helpful than what I found online.
My proof starts with the definition, expands completely so there are n! terms, each having n factors (which come one from each row, and simultaneously one from each column), observing the symmetry
shows that we'd get the same terms no matter what row or column we expand on. The one sticky point is showing that the signs of each term stay the same. I don't think I've quite got that properly
proven. Tell me what you think.
det(AB)=det(A)^.det(B). This proof is done in my text, but I felt it was done badly. I'm following his outline, but writing it up in my own words. I think there's a bit of a hole where I use L*. (The
author does this step a bit differently, and I don't like his explanation.) To outline the proof:
• First, we prove it's true for any elementary matrix times a 2x2 matrix (EA),
• Then we do induction on the size of the matrix,
• Last, we show that (almost) any AB can be seen as a series of multiplications by elementary matrices (EB).
Here's my proof. What do you think? Is there a clear way to clean up the induction step?
My 3rd proof was on area of a parallelogram = absolute value of determinant (with column vectors representing adjacent sides of the parallelogram). Not particularly impressive, and I don't have the
energy to do the volume proof too. Anyone want to show me a good proof of that? (We have not yet covered dot product or cross product, so it can't reference those notions.) I got this version from a
mathematician I spoke with at my math circle a few days ago. I had fun using geogebra to illustrate.
Now, back to my regularly scheduled grading...
Rules of the game:
Each player gets one card. The deck goes in the middle. When you see a match between your card and the top card on the deck, you call it and collect that card, putting it on top of your pile. Your
card will always have a match with the center card, so it's just a question of who can spot their match first.
(There are 3 or 4 other games, but they're all pretty similar.)
After spending lots of time analyzing Spot It over the winter holidays, I thought it might make a good topic for the Oakland Math Circle. Two weeks ago, when I found out just a few hours ahead that I
was scheduled to lead a circle, I jumped on BART, got off in downtown Berkeley, ran across the street to Games of Berkeley, bought 3 more tins of Spot It, ran back to BART, realized I'd left the tins
on the counter at the store, ran back and forth another time, and made it to the circle just in time.
I had the students play a few rounds, and then we explored. The kids counted and found that most pictures appeared on 8 cards, but many appeared on only 7 cards. We eventually used Michelle's
technique to figure out there were at least 57 pictures. (Pick a particular picture, the heart, for example. Pull out all the cards with a heart - there are 8 - and think about what we now know: 7
other pictures/card*8 cards + the heart = 57 pictures).
After a while, we focused on the question:
How did the makers of this game make sure that every pair of cards has exactly one match?
We didn't get much farther the first week. During the second week we had mostly new kids, so we started in the same place. But we got a bit farther, and tried to make our own cards with 4 pictures
per card. We also had one group with a kid and an adult who had both worked on the problem before. They found a way to make every card in their deck match every other card. But their deck only had 5
cards. They figured out that they could make a similar deck of 9 cards with 8 pictures each. It wouldn't make a very satisfying Spot It game, though.
Yesterday was my third week of Spot It analysis with the Oakland Math Circle. We got back most of the kids from the first week, and played a few rounds again to get warmed up. (My first week with
them, they seemed uninterested in math circle, this time they were really engaged, and much more fun to work with.) Then I let them each pick their color and gave them stacks of half-size index cards
(3"x2.5"). They could choose to create cards using numbers or pictures, and were trying to make decks with 4 pictures per card, with one match between each pair of cards.
Lots of folks got the 5 card deck. We started calling that the minimal solution. I realized that was an easier solution to find than the solution that makes a bigger deck. (Although Chris and I never
stumbled on it while we were creating our decks over the holidays.) One person pointed out that if they all matched on the same picture, you could have as many cards as you wanted. We called that the
infinite solution. Since it would make a super-boring game, we added the condition that you have to use more than one picture for your matches, overall. People were so stuck on the minimal solution,
I suggested starting with the infinite solution, and making a bunch of cards, trying to figure out when that would get you in trouble.
It turns out there are 13 cards in what I'll call the maximal solution. I realized that this brought up another question: Are there any symmetrical solutions with more than 5 cards, but less than 13?
One of my questions is whether the cards could use any number of pictures, or if there might be a constraint. (I'm thinking that the number of pictures per card might need to be 1 more than a prime,
but I'm not at all sure. Yet.) Another cool aspect of this problem is that it illustrates the mathematical concept of duality. (I can't quite explain that yet, beyond saying there are 8 pictures per
card, and we could have 8 of each picture in the deck.)
As I got the students into small groups yesterday, I told them, "I don't do math circles, I do math clusters." I find it's much easier to get lots of participation from the students if they're in
small groups. Then my job is cross-fertilization. These last two weeks have been my favorite math circles yet. I think I'm finally getting the hang of it. My eternal thanks to Bob and Ellen Kaplan
for helping me get started. I highly recommend their Summer Math Circle Institute, on July 8 to 14, in South Bend, Indiana. [Rodi Steinig, who went last summer, is doing great work in Pennsylvania,
and blogs about her circles here.]
Each math circle leader has their own style, so if you're thinking about leading math circles, you need to find cool problems that work for you. My own favorites are: this problem (!), the magic
pancake, playing with base 3 and base 8, and Pythagorean triples. You can find lots of math-tested-problems at the National Association of Math Circles site. | {"url":"http://mathmamawrites.blogspot.com/2012_03_01_archive.html","timestamp":"2014-04-17T01:08:42Z","content_type":null,"content_length":"110784","record_id":"<urn:uuid:a5abce7d-3aae-4939-a9db-1e297fcd4196>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00034-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calculating skip distance
October 13th 2013, 02:25 AM
Calculating skip distance
Hi there,
I am struggling a great deal with this problem, the main issue being where/how to start.
For failsafe long distance communications, the military still uses short-wave radio signals. Thesesignals are bounced off the atmosphere and can skip even the distance between continents.
For a given inclination α and for a givenatmospheric setting (specified below),your task is to find the the skip distance.
1. Physics Department: The radio signal propagates along a curve C that minimises thetime of flight for fixed start and end points (Fermat’s principle).The speed of the signal is given by c = c0/
n(x, y), where c0 = 3 × 108 m/s is the vacuumspeed of light and n(x, y) is the position dependent refractive index.
1. Met Department: The refractive index can be modeled by n(x, y) = 1 + ∆n exp{−y/h}with ∆n = 0.0027 and h = 25km.
I had an idea that if c = c0/ 1 + ∆n exp{−y/h} was used as Lagrange function I could then use the Beltrami Identity. However, this is where I'm stuck, I'm not sure how to get it in that form. Any
advice would be much appreciated.
October 16th 2013, 07:35 AM
Re: Calculating skip distance
Solved this!
October 23rd 2013, 11:28 AM
Re: Calculating skip distance
Heya i have pretty much the same problem and i am also stuck could you give me a push in the right direction | {"url":"http://mathhelpforum.com/calculus/222982-calculating-skip-distance-print.html","timestamp":"2014-04-20T16:44:48Z","content_type":null,"content_length":"7623","record_id":"<urn:uuid:5b390622-eed5-43b3-9762-f7b7c86874af>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00660-ip-10-147-4-33.ec2.internal.warc.gz"} |
Connections on Nonabelian Gerbes and their Holonomy
Posted by Urs Schreiber
Finally this sees the light of day:
U.S. and Konrad Waldorf
Connections on non-abelian Gerbes and their Holonomy
Abstract: We introduce transport 2-functors as a new way to describe connections on gerbes with arbitrary strict structure 2-groups. On the one hand, transport 2-functors provide a manifest notion of
parallel transport and holonomy along surfaces. On the other hand, they have a concrete local description in terms of differential forms and smooth functions.
We prove that Breen-Messing gerbes, abelian and non-abelian bundles gerbes with connection, as well as further concepts arise as particular cases of transport 2-functors for appropriate choices of
structure 2-group. Via such identifications transport 2-functors induce well-defined notions of parallel transport and holonomy for all these gerbes. For abelian bundle gerbes with connection, this
induced holonomy coincides with the existing definition. In all other cases, finding an appropriate definition of holonomy is an interesting open problem to which our induced notion offers a
systematical solution.
This builds on
Smooth functors vs. differential forms - which establishes the relation between smooth 2-functors with values in Lie 2-groups and differential $L_\infty$-algebraic connection data;
Parallel transport and functors - which establishes the relation between transport $n$-functors and $n$-bundles/($n-1$)-gerbes with connection for $n=1$
and realizes the construction I did with John in Higher gauge theory as a cocycle in second nonabelian differential cohomology which represents a globally defined transport 2-functor.
$n$-Café regulars will remember some discussion of the development of the notion of locally trivializable transport $n$-functors and their classification in nonabelian differential cohomology in
early Café entries such as
On Transport, Part I
On Transport, Part II
and many other ones, to some extent summarized in this big set of slides. You might enjoy Konrad’s more readable slides .
For instance you can read about 2-vector transport for associated String 2-bundles (discussed for instance here), twisted vector bundles with connection as quasi-trivializations of 2-vector transport
(which I talked about at the Fields institute), local formulas for abelian and nonabelian surface holonomy as vaguely conceived before here, see all this now related to Street’s theory of codescent
in the context of differential cohomology which is the right formalization of those 2-paths in the Cech 2-groupoid which I kept talking about once upon a time.
Alas, a couple of things didn’t quite make it into this file or are only indicated. But if you could wait that long you probably can also wait a little longer still…
Posted at August 15, 2008 6:59 PM UTC
Re: Connections on Nonabelian Gerbes and their Holonomy
Would it be asking too much if someone can open an entry on Gerbes on ncatlab? I would like to learn about them…
Posted by: Daniel de França MTd2 on June 24, 2009 6:36 AM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
There does seem to be a lack of GERBE in the nLab. Until something is added on this, you can find an introduction to gerbes (very much derived from Larry Breen’s notes) in the Menagerie notes that
you can access from my nLab page.
Breen’s notes are on the archive as math.CT/0611317. They are good and clear.
Posted by: Tim Porter on June 24, 2009 2:10 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
I don’t think Breen’s notes are clear. They sure aren’t clear to me. They may be clear if you know what he means. But what if you don’t?
Posted by: Eugene Lerman on June 24, 2009 4:24 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
I suppose that, as always, it depends from where you are coming! If we were to launch into Gerbes for nLab what would be the starting point that would be useful. What slant should be prioritised,
e.g. gerbes from non-abelian cohomology, which is what Breen is using (is that the problem?) Should things be on a space with an open cover or is a topos and an approach from that angle better. I
know which I like since my background was in homotopy theory (spatial with open covers) but that is just me.
Of course one way to do this is to start an entry in the nLab yourself. You need not worry, it will quickly fill up and you will be able to complain when it does not explain! (and will be encouraged
to do so).
Posted by: Tim Porter on June 24, 2009 4:48 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
I have created a stub on the nLab and will start writing something later on today if possible… but feel free to start without me!
Posted by: Tim Porter on June 24, 2009 4:57 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
I am afraid that I will be cheering from the sidelines for the next few weeks.
If I were to write such an entry I would start by writing out the example of principal G-bundles (G fixed) over the category of manifolds (big site). Follow up by principal G-bundles with
Mention why this is a different piece of an elephant than the one Hitchin described in “What is a gerbe?”…
Posted by: Eugene Lerman on June 24, 2009 7:57 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
Ordinary bundles in their Cech incarnation I find the most accessible route into gerbish.
Posted by: jim stasheff on June 25, 2009 2:11 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
Do you know if anyone has done it this way?
I mean by this: took Cech cocycles on some space with coefficients in a fixed group and checked the axioms for a gerbe?
Posted by: Eugene Lerman on June 25, 2009 9:34 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
Do you know if anyone has done it this way? I mean by this: took Cech cocycles on some space with coefficients in a fixed group and checked the axioms for a gerbe?
Hm, what do you mean? The only way I currently see how to interpret this question it has the obvious answer:
the basic theorem of $G$-gerbes is that they are classified by nonabelian cohomology with coefficients in the automorphism 2-group $AUT(G)$.
This says that they are given by nonabelian Cech cocycles given by “Morita generalized” morphisms or anafunctor (or whatever you like to call them)
$X \stackrel{\simeq}{\leftarrow} C(Y) \to \mathbf{B}AUT(G)$
with $C(Y)$ the Cech 2-groupoid of a cover $Y \to X$.
So, what Jim Stasheff is asking for is indeed the starting pivotal point of [[gerbe (general idea)]]:
a $G$-gerbe on $X$ is whatever is classified by a cocycle
$X \to \mathbf{B}AUT(G)$
just like a $G$-principal bundle is whatever is classified by a cocycle
$X \to \mathbf{B}G \,.$
And what is it that is classified by cocycles? Their [[homtopy fiber]]s $P \to X$.
Regarded in the world of stacks, that gives gerbes.
(To Jim: I didn’t forget Stasheff-Wirth at [[gerbe (general idea)]]. I thought that would be the perspective you’d like most… :-)
Posted by: Urs Schreiber on June 25, 2009 9:53 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
I meant my question literally: is there a textbook/paper (preferably in English, but I suppose I can manage Russian too) where this is spelled out?
Posted by: Eugene Lerman on June 26, 2009 2:30 AM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
I meant my question literally: is there a textbook/paper (preferably in English, but I suppose I can manage Russian too) where this is spelled out?
But let me know what “this” is. My impression so far is that you are asking for the nonabelian Cech-cocycle description of gerbes. If so, this is in every standard text on gerbes, the first theorem
to be proven about them.
For instance in Breen’s Notes on 1- and 2-gerbes it is section 5 cocycles and coboundaries for gerbes.
The cocycle description itself is equation (5.1.10), the classification theorem is mentioned and referenced on the bottom of page 14.
Or Ieke Moerdijk’s Introduction to the language of gerbes and stacks discusses the Cech-cocycle description of gerbes from page 16 on, and the classification theorem appears as theorem 3.1 on p. 21.
The statement is originally due to Giraud’s work Cohomologie non-abélienne.
In Brylinski’s book Loop spaces, characteristic classes and geometric quantization the cocycle description of gerbes is extracted in chapter 5.2 Sheaves of groupoids and gerbes and the classification
theorem is theorem 5.2.8 on p. 200, 201.
The discussion of cocycles for gerbes is traditionally complicated by the fact that general sheaves of groups are used, instead of just a group, then there is the discussion of band, etc., all of
which somewhat contributes to tending to hide a simple idea behind non-essential technical details.
Another thing that gerby tradition has is to express in linear formulas or rectangular diagrams what is intrinsically a nice geometric higher dimensional structure. The funny-looking nonabelian
cocycle for a gerbe is really just a tetrahedron (the 3-simplex, since we are talking about a 2-cocycle) in $\mathbf{B} AUT(G)$.
I find this helpful, since it makes at once clear a lot of structure, such as for instance the nature of coboundaries. You can find these tetrahedra drawn in my work with John Baez, for instance the
gerbe 2-cocycle tetrahedron is the title piece of John’s Namboodiri lecture slides Higher Categories, Higher Gauge Theory.
John recalls the theorem in question there on slide 10.
Finally, gerbes, in as far as they are nonabelian, are really objects associated to principal 2-bundles. The cocycle description of principal 2-bundles is more transparent, conceptually, as it is the
2-bundle that is associated by abstract nonsense to the 2-cocycle, whereas the gerbe comes from that only after some fiddling.
Accordingly, the nonabelian Cech cocycles in question here are discussed at length and in detail in the literature on 2-bundles by Toby Bartels, Igor Baković and Christoph Wockel. The relevant links
are collected at [[principal 2-bundle]].
Finally, in case I am misunderstanding your question, maybe you are asking for literature that describes nonabelian Cech cocycles as $n$-functors out of Cech $n$-groupoids?
This is described in some detail for instance in the article that this thread here is about. Another discussion more in the style of the Lie-groupoid community is in section 2 of Ginot, Stiénon, $G$
-gerbes, principal 2-group bundles and characteristic classes.
Please let me know if that serves to answer the question.
Posted by: Urs Schreiber on June 26, 2009 7:05 AM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
This answers my question.
Thank you.
Posted by: Eugene Lerman on June 26, 2009 4:41 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
This answers my question. Thank you.
Okay, thanks for letting me know. So this shows that apparently one crucial point of our $n$Lab entries on gerbes had been missing.
To remedy this, I have now created [[gerbe (in nonabelian cohomology)]] and started filling in the above material.
Posted by: Urs Schreiber on June 26, 2009 5:02 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
Maybe to amplify:
[[Cohomology]] (Cech-, sheaf-, nonabelian-, whatever) is just homs $H(X,A) = Hom(X,A)$ in the right context.
A cocycle is just a morphism $c : X \to A$ in the right context.
A coboundary just a 2-morphism $c \Rightarrow c'$.
A cohomology class just an equivalence class $[c]$.
The thing classified by a cocycle $c : X \to P$ just its [[homotopy fiber]].
Where throughout:
the “right context” where these statements are true/ make sense is not just the $(\infty,1)$-catgeory Top, but any $(\infty,1)$-category that “looks like Top”.
This means: any [[(infinity,1)-topos]].
Which in turn means: any [[$(\infty,1)$-category of $(\infty,1)$-sheaves]].
Which in plain English means: any context of [[$\infty$-stacks]].
Which in practice means: [[simplicial sheaves]] with weak equivalence remembered.
Which in the language favored around here means (for the case that the site is $Diff$):
$\infty$-groupoids internal to [[diffeological spaces]] with $\infty$-[[anafunctors]] between them.
Posted by: Urs Schreiber on June 25, 2009 10:29 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
Well, the reason for my request is to study non cohomological homology, because I’d like to understand the stringy stuff Urs writes.
Posted by: Daniel de França MTd2 on June 24, 2009 5:19 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
the reason for my request is to study non cohomological homology
I guess you mean [[nonabelian cohomology]]. (?)
Would it be asking too much if someone can open an entry on Gerbes on ncatlab?
I have now added to the entry [[gerbe]] that Timothy Porter kindly created a chunk of material which I consider as the “general idea” of gerbes.
Please have a look and let me know about whatever questions arise, so that we can proceed with working the answers in.
Tim Porter and others planning to work on this entry I’d kindly ask to see my log about my changes at [[latest changes]].
Posted by: Urs Schreiber on June 25, 2009 10:45 AM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
“I guess you mean nonabelian cohomology”.
Sure, this is why I chose this blog post. But as you can see all the time in my posts, I make silly mistakes in writting, even in my mother language. Since I was a little child, It seems that
sometimes words scramble in my mind. I’m thankful you took the effort to understand me.
Posted by: Daniel de França MTd2 on June 25, 2009 1:06 PM | Permalink | Reply to this
Re: Connections on Nonabelian Gerbes and their Holonomy
We have split [[gerbe]] apart into
[[gerbe (as a stack)]],
[[gerbe (general idea)]]
[[bundle gerbe]].
I also started
[[principal 2-bundle]],
[[principal infinity-bundle]]
[[fibration sequence]].
Everything here boils down to the discussion of fibration sequences in $(\infty,1)$-toposes of $\infty$-sheaves, really.
Unfortunately the server is really not responsive today. It must be a problem with the server, because also ssh-ing into the machine is a pain.
So I need to migrate the Lab to a different hosting company. Does anyone have any experience/suggestions/advice?
Posted by: Urs Schreiber on June 25, 2009 3:23 PM | Permalink | Reply to this
nLab responsiveness
I wrote:
Unfortunately the server is really not responsive today.
It’s working consistently fine now for a few hours already, unless I am dreaming.
And before that I did something that Jacques Distler urged me to do anyway, but which I hadn’t done in a while:
I removed the $\gt$ 70 MB backup copy of the total $n$Lab database file that was still sitting in “my web directory” waiting for being downloaded to my local machine.
I am not sure if this is just a coincidence, but maybe the wiki software is being slowed down by such a huge file just sitting around?
Posted by: Urs Schreiber on June 25, 2009 8:50 PM | Permalink | Reply to this | {"url":"http://golem.ph.utexas.edu/category/2008/08/connections_on_nonabelian_gerb.html","timestamp":"2014-04-20T19:16:53Z","content_type":null,"content_length":"54483","record_id":"<urn:uuid:fa2902d7-d643-4bb4-bee4-3928fed40aae>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00140-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mathematics Classroom Assessments
Illinois Learning Standards
Stage H - Mathematics
Mathematics Classroom Assessments Aligned to the Illinois Learning Standards
Note: All documents are in http://www.adobe.com to download the most current version of Adobe Reader
The assessments are coded according to learning standard and stage. Example: 6A.H is aligned to standard 6A, stage H (eigth grade). Sample student work, when available, will follow the assessment. | {"url":"http://isbe.net/ils/math/stage_H/assessment.htm","timestamp":"2014-04-18T19:01:44Z","content_type":null,"content_length":"17594","record_id":"<urn:uuid:c38eb447-a116-4dc3-b5c5-3b2f1a94f574>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00075-ip-10-147-4-33.ec2.internal.warc.gz"} |
Translate the following English statement into a differential equation:
Jenna's savings account grows at a rate of $300 per year.
The two quantities described are savings (in $) and time (in years). Let S stand for savings and t for time in years. Then the "rate" at which Jenna's savings account grows is the same thing as the
"derivative" of savings with respect to time, or
The statement says the rate of change of savings is $300 dollars per year. As a differential equation, | {"url":"http://www.shmoop.com/differential-equations/word-problems-examples.html","timestamp":"2014-04-16T04:19:37Z","content_type":null,"content_length":"26626","record_id":"<urn:uuid:b0413aca-26a9-4f7a-911f-fa8b67680698>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00308-ip-10-147-4-33.ec2.internal.warc.gz"} |
Pathways to CyberInfrastructure
Shodor > CyberPathways > Workshops 2007 > WDHill
• Week 7
• Week 12
• Week 13
• Week 17
• Week 19
• Week 20
• Week 22
• Week 23
• Week 24
• Week 26
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• Week 29
• Week 30
• Week 31
• Week 33
• Week 34
Scavenger Hunt
Topic: Scavenger Hunt
Time Duration: 1 hour activity and 20 minute lesson discussion
Grades: K-7th
Class began with the instructor going over ways to search the Internet using search engines. The instructor described different shortcuts that the students will use to find the answers to the
scavenger hunt. Each student was paired with another student. The pairs searched the Internet to find the answers to each question. The instructor offered a prize to the group that finished first
with all the correct answers. One question asked the students to give the boiling point of Iron. The instructor walked around to help groups individually if they got stuck trying to solve a problem.
The students worked very hard to finish the scavenger hunt and win the ending prize.
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©1994-2014 Shodor
Scavenger Hunt
Topic: Scavenger Hunt
Time Duration: 1 hour activity and 20 minute lesson discussion
Grades: K-7th
Class began with the instructor going over ways to search the Internet using search engines. The instructor described different shortcuts that the students will use to find the answers to the
scavenger hunt. Each student was paired with another student. The pairs searched the Internet to find the answers to each question. The instructor offered a prize to the group that finished first
with all the correct answers. One question asked the students to give the boiling point of Iron. The instructor walked around to help groups individually if they got stuck trying to solve a problem.
The students worked very hard to finish the scavenger hunt and win the ending prize.
Topic: Scavenger Hunt Time Duration: 1 hour activity and 20 minute lesson discussion Grades: K-7th
Class began with the instructor going over ways to search the Internet using search engines. The instructor described different shortcuts that the students will use to find the answers to the
scavenger hunt. Each student was paired with another student. The pairs searched the Internet to find the answers to each question. The instructor offered a prize to the group that finished first
with all the correct answers. One question asked the students to give the boiling point of Iron. The instructor walked around to help groups individually if they got stuck trying to solve a problem.
The students worked very hard to finish the scavenger hunt and win the ending prize. | {"url":"http://www.shodor.org/cyberpathways/archive2007/spring/WDHill/overviews/","timestamp":"2014-04-17T03:50:54Z","content_type":null,"content_length":"7948","record_id":"<urn:uuid:a817e982-bb8d-4183-b71e-757926293e17>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00141-ip-10-147-4-33.ec2.internal.warc.gz"} |
small presheaf
Category Theory
Universal constructions
Topos Theory
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
A small presheaf on a category $C$ is a presheaf which is determined by a small amount of data. If $C$ is itself small, then every presheaf on $C$ is small, but this is no longer true when $C$ is
large. In many cases, when $C$ is large, it is the small presheaves which seem to be more important and useful.
Let $C$ be a category which is locally small, but possibly large. A presheaf $F\colon C^{op}\to Set$ is small if it is the left Kan extension of some functor whose domain is a small category, or
equivalently if it is a small colimit of representable functors.
Of course, if $C$ is itself small, then every presheaf is small.
Categories of small presheaves
We write $P C$ for the category of small presheaves on $C$. Observe that although the category of all presheaves on $C$ cannot be defined without the assumption of a universe, the category $P C$ can
be so defined, using small diagrams in $C$ as proxies for small colimits of representable presheaves. Moreover $P C$ is locally small, and there is a Yoneda embedding $C\hookrightarrow P C$.
Of course, if $C$ is small, then $P C$ is the usual category of all presheaves on $C$.
Since small colimits of small colimits are small colimits, $P C$ is cocomplete. In fact, it is easily seen to be the free cocompletion of $C$, even when $C$ is not small. It is not, in general,
complete, but we can characterize when it is (cf. Day–Lack).
$P C$ is complete if and only if for every small diagram in $C$, the category of cones over that diagram has a small weakly terminal set, i.e. there is a small set of cones such that every cone
factors through one in that set.
If $C$ is either complete or small, then $P C$ is complete.
We also have:
If $C$ and $D$ are complete, then a functor $F\colon C\to D$ preserves small limits if and only if the functor $P F\colon P C \to P D$ (induced by left Kan extension) also preserves small limits.
These results can all be generalized to enriched categories, and also relativized to limits in some class $\Phi$ (which, for some purposes, we might want to assume to be “saturated”). See the paper
by Day and Lack.
The results of Chorny–Dwyer are cited by Rosicky in Accessible categories and homotopy theory, http://www.math.yorku.ca/~tholen/HB07Rosicky.pdf | {"url":"http://ncatlab.org/nlab/show/small+presheaf","timestamp":"2014-04-20T03:20:24Z","content_type":null,"content_length":"34506","record_id":"<urn:uuid:f2a12607-f480-45f2-bccd-af37a2dc1bdd>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00197-ip-10-147-4-33.ec2.internal.warc.gz"} |
Claremont Center for the Mathematical Sciences
May/02/2012 Feedback, Lineages and Cancer John Lowengrub (University of California, Irvine)
Apr/25/2012 From flocking to phase transitions: the mathematics of social dynamics Alethea Barbaro (University of California, Los Angeles)
Apr/18/2012 Image Analysis in Vision Research Exploring Presbyopia Kathryn Richdale (SUNY College of Optometry)
Apr/11/2012 Parabolic Classical and Path-Dependent Partial Differential Equations: Stochastic Henry Schellhorn, CGU
Mar/28/2012 Two-phase Flow in Porous Media: Sharp Fronts and the Saffman-Taylor Instability Michael Shearer (NCSU)
Mar/26/2012 Body weight, diet, and exercise: A dynamic energy balance approach Diana Thomas, PhD, Montclair State University, New Jersey
Mar/21/2012 A Primer of Swarm Equilibria Andrew Bernoff (HMC)
Mar/07/2012 Dynamics of particle-laden, thin-film flow Aliki Mavromoustaki (UCLA)
Feb/22/2012 Coincidence Degree Theory and Semilinear Problems at Resonance Adolfo Rumbos, Pomona College
Feb/15/2012 In the beginning was Green... Pablo Amster (University of Buenos Aires)
Feb/08/2012 Self-Intersections of Two-Dimensional Equilateral Random Walks Nicholas Pippenger, HMC
Feb/01/2012 Math modeling challenges for the nanowire transistor Shigeyasu Uno (Department of Photonics, College of Science and Engineering, Ritsumeikan University
Jan/26/2012 Graphical models for sparse estimation and neuroscience Alyson Fletcher (University of California, Berkeley)
Jan/25/2012 Modeling drug kinetics in slow-release tablets: mathematical approaches and Ami Radunskaya, Pomona College
Jan/18/2012 From swarming and self-assembly to vortex interaction: Applications of nonlocal PDE David Uminsky (UCLA) | {"url":"http://ccms.claremont.edu/Applied-Math-Seminars","timestamp":"2014-04-21T04:45:10Z","content_type":null,"content_length":"44322","record_id":"<urn:uuid:1fa792b8-9db7-4f4a-abd0-43a29cc921ac>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00428-ip-10-147-4-33.ec2.internal.warc.gz"} |
Matrix Mechanics
What is Matrix Mechanics
Because of Heisenberg's assertion regarding the commutation relation between certain variables
it is obvious that the properties of particles such as position and momentum can no longer be represented by functions of time (as in the case presented on the previous page.) Rather they must be
represented by mathematical objects in which the order of operation IS important. Matricies and Operators are two mathematical creatures which have this property. Heisenberg chose to pursue the
Matrix pathway and started to associate matricies with the properties of matter. When two matricies, A and B, are multiplied tgether, the product AB is in general not the same as BA (though the
situation can arise when they are the same). It is said that they do not commute, or that their commutator (as given above) is non-zero. Heisenberg constructed matricies so that they would obey the
above rule. Such is matrix mechanics.
The other choice has also been developed - that is by using Operator Algebra. Indeed, Schrödinger's Equation's is now envisioned as being an operator equation. One way of choosing operators is as
follows. Let the operator for position be x, defined as being the operation of multiplying by x. (This is no real change over the definition of it as a funciton.) Linear momentum, on the other hand,
is to be interpreted as the operation of differentiation with respect to x and multiplication by a constant, specifically
This choice of operator satisfies the Uncertainty Principle as required.
Following these ideas, is the direction taken by Heisenberg, as assisted by Max Born and Pascual Jordan. Looked at on the surface, this approach compared to that of Schrödinger appears to be
completely different. When these two approaches were brought up almost simultaneously, considerable argument arose concerning which was correct. David Hilbert was one of several mathematicians who
soon helped to settle the argument.
Author: Dan Thomas email: <thomas@chembio.uoguelph.ca>
Last Updated: Friday, July 5, 1996 | {"url":"http://www.cobalt.chem.ucalgary.ca/ziegler/educmat/chm386/rudiment/tourquan/matmech.htm","timestamp":"2014-04-19T22:12:01Z","content_type":null,"content_length":"3328","record_id":"<urn:uuid:8633f1e0-ae77-4b62-b650-e2fda760585f>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00035-ip-10-147-4-33.ec2.internal.warc.gz"} |
Danh sách liên kết và danh sách nhảy cóc
Bài giảng cho lớp cấu trúc dữ liệu. Bạn nào rảnh đọc thì góp ý giùm, nhất là phần C++.
Bài trước: Lớp Vector và phân tích khấu hao
Bài sau: cây nhị phân và cây tìm kiếm nhị phân.
1. Singly linked lists
The UBVector class and dynamic arrays in general have several disadvantages:
• Inserting and removing an element somewhere in the middle of the sequence takes linear time. Furthermore, if the items we hold in the sequence are large, the cost of shifting a suffix of the
array down one position is proportionally large.
• A vector wastes $\Omega(n)$ space in the worst case, where the constant inside $\Omega(n)$ is propotional to the item size. We have to allocate space to hold about $n$ items.
• A vector is stored in a contiguous block of memory in the freestore (there’s a subtle difference with the heap but we couln’t care less). Sometimes, it is possible that the freestore has
sufficient free space to store all elements of a vector, but it doesn’t have a sufficiently large contiguous block of memory to store the vector. This is called the fragmentation problem, which
is very common in all kinds of memory/disk allocation tasks in operating system design.
• Inserting and deleting from a vector are highly global operations: we shift an entire suffix of the underlying array. List updates are highly local: we only need to lock two adjacent elements.
This advantage cannot be overlooked in a concurrent environment because we may want multiple threats to modify/access a llong list at the same time. In fact, locality is also a key advantage of
linked lists over more elaborate search structures such as various types of binary search trees.
Linked list is a classic container data structure which overcomes those drawbacks. A singly linked list is a sequence of “nodes” chained together by pointers: one node has a pointer to the next node.
The last node typically points to NULL which is a pseudonym for 0. We can’t store things at this NULL address. Each node stores a “payload” which could be of any type, just as an item in a vector can
be of any type. Linked lists are the core data structure in the LISP programming language. (LISP stands for, well, LISt Processing.”) Herbert Simon won the Turing award in 1975 partly due to the
invention of linked lists. He also predicted in the 60s that a computer will play chess better than the best human within 10 years. His prediction was off by about 20 years, but his linked list lives
much longer than that.
While a vector allows for random access to elements in the vector (using the subscripting operator, indexing elements is efficient), a linked list does not have that capability. If we know the
address of the top element of a vector, we can compute the address of (and thus access) the kth element in a vector easily using pointer arithmetic. This is possible because all elements of a vector
are stored contiguously in memory. Accessing a linked list is intrinsically a sequential operation. In a linked list, elements do not have to be stored contiguously. Addresses of elements are
basically arbitrary locations in the freestore. To get to a particular element, we typically have to traverse the list down the pointer chain until we meet the wanted element.
The following example should clarify the structure of a singly linked list.
// sll.cpp: singly linked list
// want to: search, insert, delete, maintain sortedness, compute some function
#include <iostream>
using namespace std;
struct Node {
Node* next;
string payload;
Node(string pl="", Node* n_ptr=NULL) :
payload(pl), next(n_ptr) {};
void print_list(Node* ptr);
int main() {
Node* head = new Node("deep");
head = new Node("the", head);
head = new Node("in", head);
head = new Node("rolling", head);
return 0;
// print members of the list, starting from 'ptr'
void print_list(Node* ptr) {
while (ptr != NULL) {
cout << ptr->payload << " ";
ptr = ptr->next;
cout << endl;
The above program prints "rolling in the deep", and the data structure as illustrated with the following figure:
In general, with a singly linked list we are still wasting $\Omega(n)$ space because of the extra pointer per element. However, this space wasting is not propotional to the size of the items stored
in the list as in the vector case. Thus, a linked list typically will require less space overhead than a vector. And, linked list does not suffer from the fragmentation problem.
1.1. Search
Searching for an item in a linked list is inherently a linear-time operation. We
just keep going down the list until we find it or encounter NULL. (Assuming the list is NULL-terminated. See a problem below!)
* -----------------------------------------------------------------------------
* search for the first occurrence of name in the list, starting from ptr
* return NULL if not found, pointer to containing node if found
* -----------------------------------------------------------------------------
Node* search(string key, Node* ptr) {
while (ptr != NULL && ptr->payload != key)
ptr = ptr->next;
return ptr;
1.2. Delete
To delete a specific node in a singly linked list which is not the head node, we must have a pointer to the parent node.
* -----------------------------------------------------------------------------
* delete the successor node of the node pointed to by ptr
* -----------------------------------------------------------------------------
void del_successor(Node* ptr) {
if (ptr == NULL || ptr->next == NULL) return;
Node* temp = ptr->next;
ptr->next = temp->next;
delete temp; // always remember this
The logic of the code is simple, but if we want to delete the head node we will have to write a slightly different routine; for this reason, many implemenations introduce a dummy sentinel node which
is never removed. We are not concerned with a clean implementation of the singly linked list data structure in this lecture.
Exercise: write a function which takes a string str and a head pointer, finds and delete the first node with str payload, and returns a pointer to either NULL or the successor of the removed node.
The following free all nodes down the list starting from ptr
* -----------------------------------------------------------------------------
* free the memory of all nodes starting from ptr down
* -----------------------------------------------------------------------------
void free_list(Node* ptr) {
Node* temp;
while (ptr != NULL) {
temp = ptr;
ptr = ptr->next;
delete temp;
Here is a classic question regarding singly linked lists: given a head pointer pointing to the head of a singly linked list (which terminates at NULL), write a function which reverses the list and
return a pointer ot the new head node. You should think about it for a little before looking at the solution below.
* -----------------------------------------------------------------------------
* reverse the list with given head pointer, return pointer to the new head.
* -----------------------------------------------------------------------------
Node* reverse_sll(Node* head) {
Node *prev = NULL, *temp;
while (head != NULL) {
temp = head->next;
head->next = prev;
prev = head;
head = temp;
return prev;
// and here's to test it
int main() {
Node* head = NULL;
ostringstream oss;
// build a 10-node singly linked list for testing
for (int i=0; i<10; i++) {
oss.str(""); // clear buffer
oss << "Node" << i;
head = new Node(oss.str(), head);
head = reverse_sll(head);
Exercise: write a recursive function which reverses a singly-linked list. Your function should take two pointers and return one pointer to Node.
1.3. Insert into a sorted list
A linked list is a pretty good data structure for maintaining a sorted list of items. All we have to do is to go down the list, find the right spot and insert the new node in between. The search
takes linear time (in the worse case), but the insertion only takes constant time. This is by contrast to a sorted vector where searching takes logarithmic time but the actual insertion takes linear
time in the worst case. The linear time in a vector insert might involve moving large items down the line and thus should be less efficient than the linear search time for a linked list.
* -----------------------------------------------------------------------------
* assume the list is already sorted, insert a new node with the given payload
* into the list; return a pointer to the (potentially) new head
* assume node_ptr points to an allocated node
* -----------------------------------------------------------------------------
Node* insert_into_sorted_list(Node* head, Node* node_ptr) {
if (head == NULL || node_ptr->payload < head->payload) {
node_ptr->next = head;
return node_ptr;
Node *prev = head, *temp = head->next;
while (temp != NULL && temp->payload < node_ptr->payload) {
prev = temp;
temp = temp->next;
prev->next = node_ptr;
node_ptr->next = temp;
return head;
We can test the function in several ways. For example, we could do
int main() {
Node* head = NULL;
ostringstream oss;
// build a 10-node singly linked list for testing
for (int i=9; i>=0; --i) {
oss.str(""); // clear buffer
oss << "Node" << i;
head = new Node(oss.str(), head);
// print a sorted list
// now delete "Node4"
Node* temp = search("Node3", head);
// finally insert "Node4" back
head = insert_into_sorted_list(head, new Node("Node4"));
return 0;
Or, we could build a sorted list from scratch.
int main() {
Node* head = NULL, *temp;
ostringstream oss;
srand(static_cast<unsigned int>(time(0)));
// build a 10-node singly linked list for testing
for (int i=0; i<10; i++) {
oss.str(""); // clear buffer
oss << "Node" << (rand() % 10);
temp = new Node(oss.str());
head = insert_into_sorted_list(head, temp);
return 0;
1.4. Compute
There are many other questions we can ask on traversing and computing things with a singly linked list. For example, suppose the Node structure holds numbers instead of strings:
struct Node {
Node* next;
int payload;
Node(int pl=0, Node* ptr=NULL) : payload(pl), next(ptr) {};
Here are a few examples of questions one might encounter. In what follows we assume that the final node in the list points to NULL
1. Given the two heads of two sorted singly linked list, write a function which returns the head pointer to a sorted singly linked list which merges the two given lists.
2. Write a function which takes a head pointer and returns the sum of squares of the payloads.
3. Write a function which takes a head pointer and returns the number of nodes in the list.
4. Swap two sub-blocks of two lists. This is a constant time operation. We cannot do that with a vector/array.
5. Remove consecutive duplicate elements from a singly linked list which is already sorted
6. Remove elements equal to a given value
7. Sort a given list (using insertion sort)
There is one classic question which is quite different from what we have seen:
In general, in a singly linked list the last node does not have to point to NULL. It might point back to one of the predecessor nodes. Given a head pointer, write a function which returns true if
the list terminates at NULL and false if the list cycles back. Make sure your function uses as little space as possible.
It is possible to write such a function which takes constant space and linear time.
2. Doubly linked lists and C++ list template class
2.1. Doubly linked lists, XOR list
A singly linked list has several disadvantages: (1) we can never traverse backward, (2) we cannot delete a node if we do not have a pointer to the parent node, and (3) we can only insert after a node
we have a pointer too. We can fix these disadvantages by having two pointers with a node, one forward and one backward. The price we have to pay is the complexity in code, and the space needed for
all the extra pointers.
There is, in fact, one very interesting way to represent a doubly-linked list with only one pointer: an XOR linked list. Each node holds the bit-wise XOR of the addresses of the two adjacent nodes.
The XOR operation has the nice property that a^(a^b) = (a^b)^a = b. Thus, suppose we have three pointers: prev, cur, and prev^next (this is the pointer stored inside the node pointed to by cur), then
we can compute next by doing prev^(prev^next). This way, we can traverse down the list. To traverse backward, we do the reverse.
This idea is rarely used in practice due to code complexity. But, in a small device where memory is precious such as a sensor node, or in some kernel module, I would not be surprised if it is used.
On a related note, here’s another classic interview question:
how do you swap two integers without using a third variable?
2.2. C++’s list class and iterator
The standard template library provides a singly linked list template class called forward_list, and a doubly linked list template class called list. We navigate a list using an iterator, which is an
object that allows programmers to traverse a container. A container is an object which is meant to contain other objects. STL’s containers include stack, list, set, map, deque, forward_list, and the
An iterator can be thought of as an abstraction for pointers. Iterators allow us to traverse a container and access members of the container. Iterators are directly tight to the internal
representation of the container, and thus they are “private data types” of the container. We can access the item that an iterator “points to” by using the dereferencing operator. It is probably best
to look at a couple of examples.
// iter_list.cpp: navigating a list using iterators
#include <iostream>
#include <list>
using namespace std;
// print all members of the list, assuming << makes sense for T
template <typename T>
void print_list(list<T>& mylist) {
typename list<T>::iterator it = mylist.begin();
while (it != mylist.end()) {
cout << *(it++) << " ";
cout << endl;
* -----------------------------------------------------------------------------
* main body
* -----------------------------------------------------------------------------
int main() {
list<int> mylist;
// build a 10-node linked list for testing
for (int i=0; i<10; i++) {
return 0;
The keyword typename has to be put before defining the variable it because we have to tell the compiler that list<T>::iterator is a type that’s dependent on the template parameter.
There are two methods erase() and remove() for deleting elements of a list. erase can be used to remove an element pointed to by an iterator. After erasing, the iterator becomes invalid (think: the
pointer points to a memory location which was freed). However, the erase function returns an iterator to the next element down the list; hence, if you want to know where we are after erasing, just do
it = mylist.erase(it). The remove method removes from the list all elements equal to a given value.
We don’t have to physically reverse the list. We can use a reverse iterator to iterate from the end:
template <typename T>
void print_list(list<T>& mylist) {
// typename list<T>::iterator it = mylist.begin();
typename list<T>::reverse_iterator it = mylist.rbegin();
while (it != mylist.rend()) {
cout << *(it++) << " ";
cout << endl;
A reversed iterator is an iterator where the meaning of the operators +, -, ++, -- and the likes are reversed. We should start from rbegin() which is the iterator to the end of the list. We cannot
start from end() since mylist.end() has the semantic of a NULL pointer.
2.3. Implementing a UBList class
Implementing a list class is not difficult. The technical hurdle is mostly syntactical because we will have to define the iterator types ourselves. Our textbook has a section on implementing doubly
linked list. And, the TAs will explain that implementation in some details.
3. Implementing stacks and queues with lists
Linked lists are excellent choices for the underlying data structure for stacks and and queues. We have seen stacks before, which is a container where elements are inserted and accessed in a
last-in-first-out (LIFO) order. A queue is a container where elements are accessed and inserted in a first-in-first-out (FIFO) order. C++ provides a double-ended queue (deque) where elements can be
accessed and inserted from both ends.
In any case, it should be obvious why linked lists are a good choice for implementing stacks and queues. However, in practice (double ended) queues are often implemented with a circular buffer.
4. Skip lists
Going beyond stacks and queues, let us consider other operations that we may want to perform with a sorted (doubly) linked list:
Operation Time Complexity
Search O(N)
Insertion O(N)
Removal O(N)
Insertion and removal from the ends O(1)
Insertion and removal a given node O(1)
The run times for insertion and removal are pretty horrible, which is dominated by the search time. In a sorted vector we can do search in $O(\log n)$ time. Can we design a linked list with better
search time? In 1990, Professor William Pugh of Maryland proposed a very cute data structure called skip list that can achieve the desired $O(\log n)$ search time. If we can search in $O(\log n)$
time, then we can do insert and delete in the same asymptotic time because after searching we have a pointer to the node (assumming we don’t have to update too many pointers after insertion or
deletion of a given node).
4.1. Search only
Let’s first assume we do not do any insertion or deletion. The idea of a skip list is very simple. We add additional pointers to the nodes in the list so that we can jump quicker into the middle of
the list. Say we have n elements in the list. At level 0, each element has a next pointer as usual. At level 1, element numbered 2k has a pointer to element numbered 2(k+1). At level 2, element
numbered 4k has a pointer to element numbered 4(k+1), and so forth. This way, we have about $O(\log n)$ levels as the following picture illustrates. (Picture taken from this link.)
One more level
• To search for an element key, we find two consecutive elements at the top level where key is in between and then move down one level and repeat. The search obviously takes $O(\log n)$-time, much
as binary search.
• How much is the storage overhead we have to pay? At the ith level, there are $\frac{n}{2^i}$ extra pointers. Thus, overall we are wasting at most a big-O of
$\sum_{i=0}^{\log n} \frac{n}{2^i} \leq n\left(\sum_{i=0}^\infty 1/2^i\right) = 2n = O(n)$
of space overhead. The space overhead is still O(n) in terms of pointers (as opposed to O(n) in terms of the item sizes). A skip list wastes about twice as many pointers as a normal linked list,
but it allows for searching for a given element in time $O(\log n)$. This is a very nice feature to have at a small price.
4.2. Insert and delete
Inserting into a skip list will destroy the balancing structure of the list as described in the above idealized situation. If we insert or delete too many elements, we will probably have to
re-organize the list so that it becomes “balanced”, and balancing a skip list takes O(n) time. There are two choices:
1. we only rebalance the list once in a while and amortize that cost to the operations in between
2. build into each operation (insert/delete) some structure so that the list still has the nice $O(\log n)$ search time without re-balancing.
The recomended option is option 2. The idea is also extremely simple: when we insert a new element into the list, we first find (hopefully in logarithmic time) the correct place at level 0 to insert
the new element. Then, we flip a coin and if it comes up head the element will “float up” to the next level. If it does float up, we flip another coin, and so forth, until some max_level is reached.
On average, about half of the elements will float up one level, a quater two levels, one eighth three levels, and so on. Thus, similar to our coin-flipping in the quick sort algorithm, the skip list
will work very well on average.
Programming project: implement a skip list of integers with the randomized insertion and deletion strategy discussed above. Insert randomly 50K integers in to the list. Then, perform 50K random
searches. Compare the total run time of insertion and search with our UBVector class.
6 Comments
1. [Off-topic] Nếu implementation linked list bằng C, thì ở trong linux kernel đã có một implementation tuyệt đẹp, rất đáng tham khảo cho những lập trình viên C
□ @Hoangtran, thanks! Bạn có link nào nhanh đến implementation đó không?
☆ Em nghĩ chắc đây là cái hoangtran nói đến: http://isis.poly.edu/kulesh/stuff/src/klist/
○ Cool! Thanks!
2. Dung C++ 11 voi for_each va lambda ta co the viet cac ham print_list ngan hon 1 chut:
void print_list(const std::list& mylist) {
//std::for_each(mylist.begin(), mylist.end(), [] (const T& item) { std::cout << item << " "; });
std::for_each(mylist.rbegin(), mylist.rend(), [] (const T& item) { std::cout << item << " "; });
std::cout << std::endl;
Tat nhien neu muc dich khong phai la code ngan ma la de sinh vien hieu duoc cach hoat dong ben trong cua linked list thi dung iterator hop ly hon =)
(Xin loi cac ban, khong go duoc tieng Viet vi may bi truc trac…)
□ @Duong: cảm ơn. Sẽ có một bài giảng riêng về các thuật toán trong C++’s #include <algorithm>.
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Chủ đề : C++, Cấu trúc dữ liệu, Lập trình and tagged C, iterator, linked list, skip list. Bookmark the permalink. Trackbacks are closed, but you can post a comment. | {"url":"http://www.procul.org/blog/2012/03/12/danh-sach-lien-k%E1%BA%BFt-va-danh-sach-nh%E1%BA%A3y-coc/comment-page-1/","timestamp":"2014-04-19T20:08:39Z","content_type":null,"content_length":"97568","record_id":"<urn:uuid:47702ca2-6b17-4ac0-8a74-6c08bd97c234>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00080-ip-10-147-4-33.ec2.internal.warc.gz"} |
EXAMPLE: Simple_mean
Mean Value Calculation Example
This example shows how output equations can be used to calculate the mean value of a measurement.
The schematic shown is a simple BJT transistor. The Gain of the transistor is measured, and then the mean (or average) value of the gain is calculated between two reference points. The reference
points (start and stop point for mean calculation) are defined by the user and can be tuned by simulating and opening up the tuner.
The start and stop points are displayed on the graph as circles and the mean value over the specified range is displayed on the graph as a horizontal line.
The actual setup and mean calculation can be seen on the "Output Equations" page.
NEC HBT Schematic
This schematic shows an NEC HBT transistor and bias network. The transistor bias can be adjusted by starting the tuner and tuning on "I".
Gain Graph
This graph shows S21 of the NEC HBT, the start and stop points for the mean calculation, and the mean value over the specified range.
Mean Gain
This table shows the actual mean value numerically instead of graphically. | {"url":"https://awrcorp.com/download/faq/english/examples/Simple_mean.aspx","timestamp":"2014-04-17T15:27:04Z","content_type":null,"content_length":"9071","record_id":"<urn:uuid:56bc5025-7c96-4ff4-96de-19f5a7f06734>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00243-ip-10-147-4-33.ec2.internal.warc.gz"} |
st: RE: do file: t-score, dfuller, to sw regress
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st: RE: do file: t-score, dfuller, to sw regress
From Steven Samuels <sjsamuels@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject st: RE: do file: t-score, dfuller, to sw regress
Date Thu, 9 Dec 2010 22:12:40 -0500
Here are just a few references, containing others, culled from a quick Google search for "stepwise selection problems bootstrap". If I recall, Gail Gong studied a strategy very much like yours,
although for logistic regression. Frank Harrell's book "Regression Modeling Strategies" is a good resource for alternative strategies.
B Efron and G Gong (1983) A leisurely look at the boostrap, the jackknife, and cross-validation. Am Stat 37, 36-48
Gail Gong, 1986, Cross--validation, the jackknife, and the boostrap, Excess error in forward logistic regression, JASA 81, 108-113.
Peter C. Austina, Jack V. Tua Automated variable selection methods for logistic regression produced unstable models for predicting acute myocardial infarction mortality Journal of Clinical
Epidemiology 57 (2004) 1138?1146 http://uncwddas.googlecode.com/files/article2.pdf
Derksen S. and Keselman, H. J. ?Backward, forward and stepwise automated subset selection algorithms: Frequency of obtaining authentic and noise variables?, British Journal of Mathematical and
Statistical Psychology, 45, 265-282 (1992).
Frank E. Harrell Jr., Kerry L. Lee And Daniel B. Mark . Tutorial In Biostatistics. Multivariable Prognostic Models: Issues In Developing Models, Evaluating Assumptions And Adequacy, And Measuring And
Reducing Errors. Statistics In Medicine, Vol. 15,361-387 (1996) http://www.unt.edu/rss/class/Jon/MiscDocs/Harrell_1996.pdf
On Dec 9, 2010, at 3:13 PM, steven quattry wrote:
Thank you Nick for your comments, and apologies to all for being
unclear. I fully understand if this leads many to ignore my original
post. However if I may re-attempt to explain, essentially I have a
do-file created with the help of Statlist contributors that performs
bi-variate regressions, sorts the independent variables by t-score
and removes those below a certain threshold. It then runs a Dfuller
test and further removes variables that do not pass the critical
level, and finally there is code that essentially removes any
variables that have blanks. I would like to be able to learn of a way
to then take this output and sort the resulting variables by t-score,
then keep only the 72 variables with the highest t-score, and run a sw
regress with those variables. My current code is below. Again, I
sincerely apologize for being unclear and would appreciate any
feedback but understand if I do not receive any.
Also Nick, I assume you do not have the time to go into the
spuriousness of the above process, but if you were able to direct me
to a certain chapter in a well known stats text, or even an online
resource I would be quite thankful, however I fully understand it is
not your role.
Thank you for your consideration,
I am using Stata/SE 11.1 for Windows
* 2.1 T-test and Dickey-Fuller Filter
drop if n<61
tsset n
tempname memhold
tempname memhold2
postfile `memhold' str20 var double t using t_score, replace
postfile `memhold2' str20 var2 double df_pvalue using df_pvalue, replace
foreach var of varlist swap1m-allocglobal uslib1m-infdify
dswap1m-dallocglobal6 {
qui reg dhealth `var'
matrix e =e(b)
matrix v = e(V)
local t = abs(e[1,1]/sqrt(v[1,1]))
if `t' < 1.7 {
drop `var'
else {
local mylist "`mylist' `var'"
post `memhold' ("`var'") (`t')
postclose `memhold'
foreach l of local mylist {
qui dfuller `l', lag(1)
if r(p) > .01 {
drop `l'
else {
local mylist2 "`mylist2' `l'"
post `memhold2' ("`l'") (r(p))
postclose `memhold2'
keep `mylist2'
use t_score,clear
gsort -t
use df_pvalue, clear
log off
* 2.2 Missing data Filter
drop if n<61
foreach x of varlist `mylist2' {
qui sum `x'
if r(N)<72 {
di in red "`x'"
drop `x'
else {
local myvar "`myvar' `x'"
sum date
keep if date==r(max)
foreach x of varlist `myvar' {
if `x'==. {
drop `x'
else {
local myvar2 "`myvar2' `x'"
d `myvar2'
log off
* 2.3 Stepwise Regressions
drop if n<61
*Simultaneous Model
local x "Here is where I paste in variables after sorting by
t-score and keeping only 72 highest"
sw reg dhealth `x', pe(0.05)
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2010-12/msg00403.html","timestamp":"2014-04-19T22:25:23Z","content_type":null,"content_length":"13596","record_id":"<urn:uuid:f919227b-da22-478c-90f2-8018291c2a2f>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00485-ip-10-147-4-33.ec2.internal.warc.gz"} |
Lesson Plan
Geometry: Congruent Figures
Students will learn to identify congruent figures and segments.
• Students will review the basics of calculating measurements.
• Students will learn to define the term congruent.
• Students will learn the symbol for the term congruent.
• Students will identify congruent segments.
• Students will practice these skills in their independent work.
Suggested Grades:
Seventh Grade - Eighth Grade - Ninth Grade - Tenth Grade - including special education students
Print the classroom lesson plan and worksheet questions (see below).
Lesson Excerpt:
Note: This is a shorter lesson and should be combined with construction practice using a straightedge, compass and a pencil. Students need a lot of practice constructing congruent figures and line
We have been learning about some of the basic building blocks of geometry. Some of these things have included terms and descriptions. In our last lesson, we learned about calculating measurements.
Let's review a little of what we learned in our last lesson.
When points are collinear, meaning that they are on the same line, we can use this information and our algebra skills to figure out the lengths of different line segments. Let's look at an example: | {"url":"http://www.instructorweb.com/les/congruentfigure.asp","timestamp":"2014-04-18T20:44:52Z","content_type":null,"content_length":"18743","record_id":"<urn:uuid:54667237-74cd-4b22-8558-97daf023cefc>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00221-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: July 1998 [00069]
[Date Index] [Thread Index] [Author Index]
Re: Non-commutative algebra
• To: mathgroup at smc.vnet.net
• Subject: [mg13129] Re: [mg13053] Non-commutative algebra
• From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
• Date: Tue, 7 Jul 1998 03:44:32 -0400
• References: <199807042044.QAA02851@smc.vnet.net.>
• Sender: owner-wri-mathgroup at wolfram.com
I replied to David Withoff's message before I had a chance to carefully
read the message from Daniel Lichtbau. Since this message essentially
deals with all the points I raised I am now quite satisfied.
At 4:29 PM -0500 98.7.6, Daniel Lichtblau wrote:
>Andrzej Kozlowski wrote:
>> I have a question which (probably) can only be answered by a wri
>> insider. Since it concerns a "obsolete" version of Mathematica I
>> suppose it will not be seen as terribly important, but still I would be
>> very grateful if someone would satisfy my curiosity.
>> Recently I was showing some computations in non-commutative algebra to
>> my students. This is very easy to do in Mathematica 3.0. Basically all
>> you need to do is to Clear the attribute Orderless in Times and use
>> Mathematica as usual. However, my university (for various reasons)
>> still has Mmma 2.2 installed on all the computers in the class where I
>> teach, so after making my notebook using 3.0 I tried the same
>> computations in 2.2. To my surprise the answers came out wrong! I soon
>> realized that Expand in Mathematica 2.2 has commutativity "built in"
>> quite independently from the Orderless attribute of Times. To see this
>> all you need to do is to evaluate:
>> In[2]:=
>> Unprotect[Times]
>> Out[2]=
>> {Times}
>> In[3]:=
>> ClearAttributes[Times,Orderless]
>> In[4]:=
>> Protect[Times]
>> Out[4]=
>> {Times}
>> In[5]:=
>> Expand[(a+b)^2]
>> In versions 3.0 and 2.2. In 3.0 you will correctly get:
>> b*a + a^2 + a*b + b^2
>> but in 2.2 you get
>> a^2+2a*b+b^2
>> I have looked through the documentation for Expand in both versions, and
>> through various accounts of the changes in Version 3.0 (e.g.
>> Mathematica Journal Vol. 6 Issue 4) but cannot find any mention of
>> commutativity. It seems to me very odd to deliberately "hard-wire"
>> commutativity in Expand. My question is: Why this behavious in version
>> 2? Was it ever recognized as a bug and fixed v. 3 or was the fix just
>> a side-effect of some other changes?
>It was indeed recognized as a bug and fixed. Well, partly fixed. Power
>is still in need of some work, because it uses Times.
>In[9]:= ee = a*b;
>In[10]:= ClearAttributes[Times, Orderless]
>In[11]:= ee * ee
>Out[11]= a b a b
>In[12]:= ee^2
> 2 2
>Out[12]= a b
>This matters e.g. if want to Expand the square of (x^2 + x*y + y^2).
>> (Fortunately by using Distribute I was able to define my own Expand in
>> 2.2 which does not assume commutativity, so this problem no longer has
>> a practical significance for me, it's just a matter of wanting to
>> understand what happened).
>> Andrzej Kozlowski
>> ...
>We recommend that you use, say, NoncommutativeMultiply and attach rules
>for integer powers to it to do the expansion. It is virtually never safe
>to change the Orderless attributes of low-level operations such as
>Plus/Times/Power, and moreover (as you found) it is hard for us to
>obtain the correct functionality when you do make such changes.
>Daniel Lichtblau
>Wolfram Research
Andrzej Kozlowski
• References: | {"url":"http://forums.wolfram.com/mathgroup/archive/1998/Jul/msg00069.html","timestamp":"2014-04-18T08:21:21Z","content_type":null,"content_length":"38138","record_id":"<urn:uuid:be6de591-89d4-4b27-912d-bf0f8f59cdef>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00545-ip-10-147-4-33.ec2.internal.warc.gz"} |
Probability Puzzles
Q: You have an array of items that you want to shuffle. You pick an element and then pick a random number uniformly between \([1,n]\) and swap it with the element following which you move on to the
next element and repeat. Would this result in a fair shuffle?
Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
A: Some people shuffle a deck of playing cards in the above manner. They tend to pick a small (random sized) slab of cards from under the deck, place it on top and do so several times over.
The above shuffle would not result in a fair shuffle. Here is why. Assume you have \(n\) items in the list. When you pick an element and swap it out with another element there are \(n\) positions you
could swap it with, including itself in which case there is no swap. This implies that the first swap would result in \(n\) arrangements. The second swap follows the same algorithm and would result
in the same number of arrangements, that is \(n\). This leads to a total of \(n^{n}\) possible arrangements. However in a fair shuffle, the first element would be chosen from \([1,n]\), the second
from \([1,n-1]\) and so on, leading to \(n!\) arrangements in total. Clearly, any strategy that gives a fair shuffle has to yield a number of arrangements \(N\) such that
$$ N \pmod{n!} = 0 $$
$$ n^{n}\pmod{n!} \ne 0 $$
As \(n^{n}\) does not divide \(n!\) it implies that the algorithm would end up favouring some combinations over the other resulting in a biased shuffle.
If you looking to learn probability & algorithms here are some good books.
Introduction to Algorithms This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists
Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems) This one is a must have if you want to learn machine learning. The
book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some
great examples and test data in the text book too. Introduction to Probability Theory
An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
Introduction to Probability, 2nd Edition
The Mathematics of Poker Good read. Overall Poker/Blackjack type card games are a good way to get introduced to probability theory Quantum Poker Well written and easy to read mathematics. For the
Poker beginner.
Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5) An excellent resource (students/engineers/
entrepreneurs) if you are looking for some code that you can take and implement directly on the job.
Understanding Probability: Chance Rules in Everyday Life A bit pricy when compared to the first one, but I like the look and feel of the text used. It is simple to read and understand which is vital
especially if you are trying to get into the subject | {"url":"http://bayesianthink.blogspot.com/2013/03/a-shuffling-puzzle.html","timestamp":"2014-04-16T10:09:55Z","content_type":null,"content_length":"68120","record_id":"<urn:uuid:b2be016a-ceb7-496e-8f0c-aa42e38aa528>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00174-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Review Of Prospect Theory
Introduction (Via SSRN)
Prospect theory is a descriptive theory of how individuals choose among risky alternatives. The theory challenged the conventional wisdom that economic decision makers are rational expected
utility maximizers. We present a number of empirical demonstrations that are inconsistent with the classical theory, expected utility, but can be explained by prospect theory. We then discuss the
prospect theory model, including the value function and the probability weighting function. We conclude by highlighting several applications of the theory.
Excerpts (Via SSRN)
The objective of prospect theory is to describe how people make decisions when there is uncertainty about the consequences of their choices. Decision theorists distinguish between: decision under
risk, situations in which the likelihood of events are known or objective such as a spin of a roulette wheel); and decision under uncertainty, situations in which the decision maker must assess
the probability of the uncertain events and hence the likelihood of events are subjective (such as the outcome of a sports game) [3]. Although prospect theory applies to both risk and uncertainty
we will focus on risk here for simplicity.
Example (Via SSRN)
In Situation 1, you are first given $1,000. You must now choose between two options. If you choose A, you will receive an additional $500 for sure. If you choose B, there is an equal chance that
you will receive either an additional $1,000 or nothing.
Now consider Situation 2. This time you are first given $2,000. Again, there are two options. Perhaps you would like C, a sure loss of $500? Or, maybe you would prefer D, a 50% chance at losing
$1,000 and a 50% chance at losing $0?
When confronted with Situation 1, most people prefer A, the sure $500, to the gamble B. On the other hand, when posed with a choice between C and D, most choose D, the uncertain loss, over the
sure loss C. While both choices seem reasonable in isolation, Situation 1 and 2 are identical in terms of final consequences, reducing to a choice between $1,500 for sure (A and C), and a lottery
that offers an even chance at $1,000 or $2,000 (B and D).
Leave a Reply Cancel reply | {"url":"http://www.simoleonsense.com/a-review-of-prospect-theory/","timestamp":"2014-04-20T10:47:02Z","content_type":null,"content_length":"25451","record_id":"<urn:uuid:28c44c7c-46e7-4e74-b685-b9f998a0aa32>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00620-ip-10-147-4-33.ec2.internal.warc.gz"} |
March 22nd 2011, 04:38 PM #1
Junior Member
Feb 2011
I have to show that the following sets of stochastic processes are not equal:
$L^2:=\{(X_t)|(X_t)$ is progessive measurable and $\mathbb{E}[\int\limits_0^T X_t^2dt]< \infty \}$
$H^2:=\{(X_t)|(X_t)$ is progessive measurable and $\int\limits_0^T X_t^2dt< \infty$ a.s. $\}$
I know that these sets aren't equal.
But how can I show that $\int\limits_0^T X_t^2dt< \infty$ a.s. doesn't imply $\mathbb{E}[\int\limits_0^T X_t^2dt]< \infty$
Does anyone know a proof?
I also know that the Itô-integral $\int\limits_0^T X_t dW_t$ where $W_t$ is a standard Brownian motion and $X_t \in L^2$ is not just a local martingale, it's also a martingale.
But is $\int\limits_0^T X_t dW_t$ also a martingale if $X_t \in H^2$?
Can anybody help me?
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Tildes and micro sign decompositions
Sent: Thursday, May 27, 1999 10:35 AM
Subject: Tildes and micro sign decompositions
Please forward to the UTC for the June meeting.
I hope this e-mail format is sufficient. If not, please tell
me which form to fill in.
Kind regards
/Kent Karlsson
NOT TILDE (2241) should be tied to the tilde operator in the
decomposition, not the spacing tilde accent:
2241;NOT TILDE;Sm;0;ON;007E 0338;;;;Y;;;;; --->
2241;NOT TILDE;Sm;0;ON;223C 0338;;;;Y;;;;;
The NOT TILDE is like the other negated math characters.
The 007E TILDE, however, is not normally seen as a math
character, even though it is used in several programming
lanugages as a (usually unary) operator. But so is
HYPHEN-MINUS, which is not classified as a math character.
TILDE is often printed/displayed *above* the math operator
vertical centre line, whereas TILDE OPERATOR and NOT
TILDE are set *on* the math operator vertical centre line,
just like =, <, etc.
The spacing tilde accent, TILDE (007E), is not decomposed like
the other spacing versions of the accents, as it should.
007E;TILDE;Sm;0;ON;;;;;N;;;;; --->
007E;TILDE;Sk;0;ON;<compat> 0002 0303;;;;N;;;;;
02DC;SMALL TILDE;Sk;0;ON;<compat> 0020 0303;;;;N;SPACING TILDE;;;; --->
02DC;SMALL TILDE;Sk;0;ON;<small> 007E;;;;N;SPACING TILDE;;;;
This is probably closest to the original intent and use of the TILDE in ASCII.
It is also closer to the *current* view of tilde in many keyboard mappings,
where a "dead key" tilde can be followed either by a letter (producing
õ, ñ, etc.) or by space. The latter produces ~, 007E TILDE. Indeed, that
is the only way of producing TILDE on many keyboards (hex code input
(SMALL TILDE can be made canonically equivalent to TILDE.)
Alternatively (to the above), continue to regard TILDE as a
math operator, but then make TILDE OPERATOR canonically
equivalent to it:
223C;TILDE OPERATOR;Sm;0;ON;;;;;Y;;;;; --->
223C;TILDE OPERATOR;Sm;0;ON;007E;;;;Y;;;;;
This, however, departs from the common view of 007E TILDE.
The micro sign should be *canonically* equivalent to Greek mu:
00B5;MICRO SIGN;Ll;0;L;<compat> 03BC;;;;N;;;;; --->
00B5;MICRO SIGN;Ll;0;L;03BC;;;;N;;;;;
Just because it was thought important enough (for use in units) to include
this character in Latin-1 does not make this mu any different from
the mu encoded in the Greek block. The situation is very similar
to that for the Kelvin sign, the Ångström sign, and the Ohm sign,
which are all canonically equivalent to the corresponding letter.
This has mistakenly lead to some slight appearance differences
between these equivalenced characters, even though they are or
should be canonically equivalent. This should be regarded as font
mistakes, and not grounds for not making them canonically equivalent. | {"url":"http://www.unicode.org/L2/L1999-UTC/u1999-013.htm","timestamp":"2014-04-21T10:02:51Z","content_type":null,"content_length":"6995","record_id":"<urn:uuid:fc6ca84a-e689-4829-ba3b-974fab82a1db>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00188-ip-10-147-4-33.ec2.internal.warc.gz"} |
Contracting a curve of negative self-intersection on a surface
up vote 4 down vote favorite
It is easy to show using birational factorization that the only curves on a surface which can be contracted to get an algebraic, smooth surface are smooth $(-1)$-curves. Furthermore, I know of
examples of smooth curves of some other negative self-intersection which can be contracted in the algebraic category to result in a singular surface (where the order of the singularity is equal to
the negative self-intersection?)
Question 1: Given a curve of negative self-intersection on a complex surface, what is the construction (in the analytic category) of its contraction?
Question 2: Given the curve, what conditions on it that determine whether the contraction is algebraic or not?
Good, clear, elementary references would be fine, as an alternative to an answer!
algebraic-surfaces ag.algebraic-geometry
Added the tag ag.algebraic-geometry – Francesco Polizzi Mar 2 '13 at 13:04
Hi Philip, this article (of mine) gives a necessary and sufficient criterion for algebraicity in a special case: arxiv.org/abs/1301.0126 PS: I myself am interested in your Question 1, and I don't
know of any other reference other than Grauert's original article, which is in German and therefore I can't read :( – auniket Apr 27 '13 at 1:07
add comment
1 Answer
active oldest votes
The best numerical criterion of contrability for curves on surfaces is perhaps the following result, due to Michael Artin.
Proposition. Let $V$ be a surface and $X=\bigcup X_i \subset X$ be a connected curve. Then the following are equivalent:
$\boldsymbol{(i)}$ $X$ is contractible and if $\pi \colon V \to \bar{V}$ is the contraction map, then $\chi(\mathcal{O}_V) = \chi(\mathcal{O}_{\bar{V}})$;
$\boldsymbol{(ii)}$ the intersection matrix $|(X_i \cdot X_j)|$ is negative definite and for any cycle $Z$ supported in $X$ one has $p_a(Z) \leq 0$.
up vote 8 Moreover, under these conditions, if $V$ is a normal projective surface then $\bar{V}$ is also projective.
down vote
In other words, if $V$ is normal, projective and $X$ is "sufficiently rational and negative", then $X$ is contractible and the contraction is algebraic. For further detais, see
M. Artin: Some Numerical Criteria for Contractability of Curves on Algebraic Surfaces,American Journal of Mathematics Vol. 84, No. 3 (Jul., 1962), pp. 485-496,
in particular Theorem 2.3 p. 491.
Thanks for your answer, it partially resolves the case of my second question, by giving criteria for contractibility in the algebraic category in certain cases. I mainly wanted an
explicit construction {\it in the analytic category} of the contraction. (The conditions would of course be weaker if we allow the contraction not to be an algebraic surface). Since the
proposition above seems to be the best result about contractibility, I assume it is hard then to determine whether the resulting surface is algebraic... – Philip Engel Mar 6 '13 at 22:25
add comment
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Re: st: Structural Break Type Test with Fixed Effects IV and GMM
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Re: st: Structural Break Type Test with Fixed Effects IV and GMM
From Dominic Soon <dominic.soon.etc@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: Structural Break Type Test with Fixed Effects IV and GMM
Date Sun, 11 Apr 2010 22:12:26 +0800
Yes, my mistake!
I'm intending to use three approaches, from -xtreg- , -xtivreg- and
-xtabond- (i.e. dynamic panel) Unfortunately, I don't have any other
instrumental variables, let alone instrumental variables which impact
S "differentially" in the early and the late period. In these
circumstances, would the following make sense? SxL refers to S*L.
xtreg Y L S SxL, fe
xtivreg Y L (S SxL = L.S L.SxL) /* Assume that lagged S does not
affect shocks in current Y */
xtabond Y L , lags(1) endogenous(L SxL)
(Other covariates are omitted for ease of discussion)
My problem is that clearly the idea that lagged SxL is probably going
to be a very poor instrument for SxL close to the beginning of the
"late" period. For instance, if "late" refers to 2001, lagged SxL will
be zero.
On Sun, Apr 11, 2010 at 3:07 AM, Austin Nichols <austinnichols@gmail.com> wrote:
> Dominic Soon <dominic.soon.etc@gmail.com>
> -fe- is not an option for -xtabond-. Are you thinking you have a
> dynamic panel model, or S is endogenous (i.e. you mean -xtivreg-)? If
> the latter, what instruments are you using? You probably want L and
> L*S included as regressors, and you then have 2 endog vars: S and L*S.
> Then you want instruments that differentially affect S in the early
> and late periods.
> On Fri, Apr 9, 2010 at 11:34 PM, Dominic Soon
> <dominic.soon.etc@gmail.com> wrote:
>> Dear Statalisters,
>> I am trying to estimate a regression of two variables output (Y) on
>> R&D capital stock (S), as well as some other variables (e.g. labour,
>> capital, so on and so forth). For simplicity, let's say the model is:
>> Y_it = a_i + beta * S_it + error term
>> I am trying to see whether the coefficient beta is different between
>> an (assumed) "early" and "late" period. I'm also attempting to run
>> the regression using both fixed effects and GMM.
>> The question is - are there any issues with this methodology,
>> particularly when running a GMM estimation. Suppose I ran something
>> like:
>> xtabond Y S S_late, fe
>> where S_late is equal to L times S, with L being a dummy variable that
>> is equal to 1 if t is later than my (assumed) breakpoint, can the
>> t-statistics be interpreted sensibly?
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What is the square root symbol in Word format
The Square Root symbol in Word is the same as for any application and even as
hand-written, and means "A divisor of a quantity that when squared gives the
"Where" the symbol is in Word is simple, either under Insert, Symbol, any
font, Mathematical Operators, or shortcut of 221A, Alt-X.
Hope this helps.
"Veena" wrote:
What is the square root symbol in Word format | {"url":"http://www.wordbanter.com/showthread.php?t=131016","timestamp":"2014-04-18T01:24:57Z","content_type":null,"content_length":"46143","record_id":"<urn:uuid:3c6c7bbd-fb98-4a97-9915-265c68b2b1a3>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00293-ip-10-147-4-33.ec2.internal.warc.gz"} |
Assign Decimal Values
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(1) The code in Reformat is:
out::reformat(in) =
let int i = 0;
let int count = (lookup_count("lookup",
let decimal ("{^}") [] RATE = for (i, i<count): lookup_next("lookup").rate;
out.* :: in.*;
out.RATE :: for (i, i<5): (if (i<count) RATE else "0");
I am getting the below error:
While executing transform reformat:
Wrong length in assignment for RATE.
Trying to assign 0 elements to a vector of length 5.
Source is the variable "RATE"
Rejected input record 1.
The values in the lookup for rate is: 2.5, 3.5, 4.5, 5.67, etc...
(2) I wanted to convert Vector to normal column, i.e. In Reformat 1 output dml I have defined as "decimal ("{^}") Rate [5]" which is the input of Reformat 2.
I wanted the output dml of Reformat 2 as "decimal ("{^}") Rate". How can this be done?
Could someone guide me on this?
Rajasad replied Jan 1, 0001
Instead of Lookup_next , please use lookup_count
Arijit Duttagupta replied Jan 1, 0001
Hi Srinivas,
out.RATE :: for (i, i<5): (if (i<count) RATE else "0");
Is this in your 2nd reformat?
If so out.RATE is scalar here. So you cannot assign a vector to a scalar.
Something like this should work -
out.RATE :: for (i, i<5): (if (i<count) RATE else "0");
But what are you trying to achieve with this. If you do this you will always get the value of the last element in the vector RATE[] while assigning in in out.RATE, as it is overwriting the previous
element values through the for loop.
Are you trying to get separate records for each of the elements in RATE from your 1st Reformat?
If so then the 2nd transform component would not be a Reformat but it should be a Normalize.
In the Normalize length function assign - out :: length_of(in.RATE);
In the Normalize transform function assign - out.RATE :: in.RATE[index];
Hope this helps ...
Arijit Duttagupta replied Jan 1, 0001
Correction to my previous post --
Something like this should work -
out.RATE :: for (i, i<5): (if (i<count) RATE else "0");
dwh replied Jan 28, 2013
How the out.RATE is defined in the output dml of Reformat 1?
In the business rule assignment for loop , Use RATE to refer to the value.
When you say "I wanted the output dml of Reformat 2 as "decimal ("{^}") Rate".". What do you mean exactly? You want to assign only one value (say RATE[3]) from the array to the output dml or you want
to normalize the data?
Naveen Nimmala replied Jan 28, 2013
You have declared RATE as a vector with 5 occurrences(RATE[1], RATE[2],RATE[3],RATE[4],RATE[5]
As long as you don't provide values to all the occurrences ((RATE[1], RATE[2],RATE[3],RATE[4],RATE[5]) you will always get this error.
"Wrong length in assignment for RATE.
Trying to assign 0 elements to a vector of length 5."
Solution 1: initialize the vector.
Solution 2: define the length of vector dynamically based on the data.
Solution 3: don't use a variable declaration as vector (if that is not your output format),try writing the data as a string and use vector functions and string functions to get the desired result.
Naveen Nimmala
“Mark this answer as helpful. If it really helped”
Arijit Duttagupta replied Jan 28, 2013
Looks like Toolbox is spoiling the vector -
Hope this time the correction works -
Something like this should work -
out.RATE :: for (i, i<5): (if (i<count) RATE [ i ] else "0");
dwh replied Jan 30, 2013
Oops.. Don't know why it is trimming automatically, But i meant it to be this:
In the business rule assignment for loop , Use RATE [ i ] to refer to the value.
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Provide a 150 word response to the following questions.
Imagine that a line on a Cartesian graph is approximately the... - Homework Help - eNotes.com
Provide a 150 word response to the following questions.
Imagine that a line on a Cartesian graph is approximately the distance y in feet a person walks in x hours.
What does the slope of this line represent? How is this graph useful?
A Cartesian plane whose x-axis represents the number of hours a person walks while the y-axis represents the distance travelled by the person can be used to examine a lot of things related to the
motion of the person. For instance, by simply looking at the x-coordinate, one can immediately tell, by looking at the corresponding y-value of the line at the given x-coordinate, the distance
travelled at time x hours.
Getting the slope, on the other hand, would reveal to us the speed of the person. Slope is, generally, given by the formula `m = (y_2 - y_1)/(x_2 - x_1) = (Delta y)/(Delta x)` . Delta here denotes
change, and we see that the slope is simply the change in the distance travelled for every given change in the number of hours - or the amount of travelled after a span of time - in other words,
speed (Note the units: distance per time can be meters per second or kilometers per hour, of course, depending on the units of the values in your y- and x-axes.)
Join to answer this question
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User j0equ1nn
bio website quaternion.ws.gc.cuny.edu
location New York, United States
age 35
visits member for 2 years, 11 months
seen Apr 8 at 18:58
stats profile views 130
After a brief stint as a double major in math and film-making at Carnegie Mellon, I was an independent artist for most of my 20s (to the dismay of my parents). When I was 27 a crazy turn of events
inspired me finish my BA, which then became my BA/MA, in pure math at Hunter College. I'm grateful to Prof. John Loustau at Hunter for encouraging me to go further. I'm currently a PhD candidate at
the CUNY Graduate Center.
Algebra-related stuff comes most easily to me but I'm more interested in applications of it to other fields, especially topology and prime number theory (preferably both). Other topics I've studied
at post MA-level fall under: differential geometry, set theory, model theory, algebraic topology, algebraic number theory, and dynamical systems. My current focus is the classification of hyperbolic
3-manifolds using quaternion algebras. My adviser is Abhijit Champanerkar.
MathOverflow 305 rep 19
Mathematics 188 rep 8
12 Votes Cast
all time by type
12 up 3 question
0 down 9 answer | {"url":"http://mathoverflow.net/users/14835/j0equ1nn","timestamp":"2014-04-21T04:40:44Z","content_type":null,"content_length":"59993","record_id":"<urn:uuid:0479f06e-e5bb-4ede-b6f7-a20f9cc82a93>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00058-ip-10-147-4-33.ec2.internal.warc.gz"} |
A generic "ziggurat algorithm" implementation. Fairly rough right now.
There is a lot of room for improvement in findBin0 especially. It needs a fair amount of cleanup and elimination of redundant calculation, as well as either a justification for using the simple
findMinFrom or a proper root-finding algorithm.
It would also be nice to add (preferably by pulling in an external package) support for numerical integration and differentiation, so that tables can be derived from only a PDF (if the end user is
willing to take the performance and accuracy hit for the convenience).
data Ziggurat v t Source
A data structure containing all the data that is needed to implement Marsaglia & Tang's "ziggurat" algorithm for sampling certain kinds of random distributions.
The documentation here is probably not sufficient to tell a user exactly how to build one of these from scratch, but it is not really intended to be. There are several helper functions that will
build Ziggurats. The pathologically curious may wish to read the runZiggurat source. That is the ultimate specification of the semantics of all these fields.
The X locations of each bin in the distribution. Bin 0 is the infinite one.
In the case of bin 0, the value given is sort of magical - x[0] is defined to be V/f(R). It's not actually the location of any bin, but a value computed to make the algorithm more concise and
slightly faster by not needing to specially-handle bin 0 quite as often. If you really need to know why it works, see the runZiggurat source or "the literature" - it's a fairly standard setup.
The ratio of each bin's Y value to the next bin's Y value
The Y value (zFunc x) of each bin
An RVar providing a random tuple consisting of:
□ a bin index, uniform over [0,c) :: Int (where c is the number of bins in the tables)
□ a uniformly distributed fractional value, from -1 to 1 if not mirrored, from 0 to 1 otherwise.
This is provided as a single RVar because it can be implemented more efficiently than naively sampling 2 separate values - a single random word (64 bits) can be efficiently converted to a double
(using 52 bits) and a bin number (using up to 12 bits), for example.
The distribution for the final "virtual" bin (the ziggurat algorithm does not handle distributions that wander off to infinity, so another distribution is needed to handle the last "bin" that
stretches to infinity)
A copy of the uniform RVar generator for the base type, so that Distribution Uniform t is not needed when sampling from a Ziggurat (makes it a bit more self-contained).
The (one-sided antitone) PDF, not necessarily normalized
A flag indicating whether the distribution should be mirrored about the origin (the ziggurat algorithm in its native form only samples from one-sided distributions. By mirroring, we can extend it
to symmetric distributions such as the normal distribution)
(Num t, Ord t, Vector v t) => Distribution (Ziggurat v) t
mkZigguratRec :: (RealFloat t, Vector v t, Distribution Uniform t) => Bool -> (t -> t) -> (t -> t) -> (t -> t) -> t -> Int -> (forall m. RVarT m (Int, t)) -> Ziggurat v tSource
Build a lazy recursive ziggurat. Uses a lazily-constructed ziggurat as its tail distribution (with another as its tail, ad nauseam).
• flag indicating whether to mirror the distribution
• the (one-sided antitone) PDF, not necessarily normalized
• the inverse of the PDF
• the integral of the PDF (definite, from 0)
• the estimated volume under the PDF (from 0 to +infinity)
• the chunk size (number of bins in each layer). 64 seems to perform well in practice.
• an RVar providing the zGetIU random tuple
mkZiggurat :: (RealFloat t, Vector v t, Distribution Uniform t) => Bool -> (t -> t) -> (t -> t) -> (t -> t) -> t -> Int -> (forall m. RVarT m (Int, t)) -> (forall m. t -> RVarT m t) -> Ziggurat v t
Build the tables to implement the "ziggurat algorithm" devised by Marsaglia & Tang, attempting to automatically compute the R and V values.
Arguments are the same as for mkZigguratRec, with an additional argument for the tail distribution as a function of the selected R value.
mkZiggurat_ :: (RealFloat t, Vector v t, Distribution Uniform t) => Bool -> (t -> t) -> (t -> t) -> Int -> t -> t -> (forall m. RVarT m (Int, t)) -> (forall m. RVarT m t) -> Ziggurat v tSource
Build the tables to implement the "ziggurat algorithm" devised by Marsaglia & Tang, attempting to automatically compute the R and V values.
• flag indicating whether to mirror the distribution
• the (one-sided antitone) PDF, not necessarily normalized
• the inverse of the PDF
• the number of bins
• R, the x value of the first bin
• V, the volume of each bin
• an RVar providing the zGetIU random tuple
• an RVar sampling from the tail (the region where x > R)
findBin0 :: RealFloat b => Int -> (b -> b) -> (b -> b) -> (b -> b) -> b -> (b, b)Source
I suspect this isn't completely right, but it works well so far. Search the distribution for an appropriate R and V.
• Number of bins
• target function (one-sided antitone PDF, not necessarily normalized)
• function inverse
• function definite integral (from 0 to _)
• estimate of total volume under function (integral from 0 to infinity)
Result: (R,V) | {"url":"http://hackage.haskell.org/package/random-fu-0.2.3.0/docs/Data-Random-Distribution-Ziggurat.html","timestamp":"2014-04-18T19:23:27Z","content_type":null,"content_length":"19297","record_id":"<urn:uuid:54547e7f-d83f-4292-82e1-710da8775645>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00077-ip-10-147-4-33.ec2.internal.warc.gz"} |
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these are identified in the Course Schedule. Prerequisite: Written consent of instructor.
110T. Conference Course: Texas Department of Insurance Internship.
Supervised internship at the Texas Department of Insurance. May be repeated for credit. Admission by application only. Students must apply to the director of the concentration in actuarial studies
the semester before they take the course.
112K. Actuarial Laboratory on Calculus and Linear Algebra.
Problems and supplementary instruction in calculus, matrix algebra, and linear algebra, especially as required for the Society of Actuaries and Casualty Actuarial Society Exam 100. Three laboratory
hours a week for one semester. Prerequisite: Mathematics 408D with a grade of at least C, and credit or registration for Mathematics 341 (or 311) or 340L.
112L. Actuarial Laboratory on Interest Theory.
Problems and supplementary instruction in interest theory, especially as required for the Society of Actuaries Exam 140. Three laboratory hours a week for one semester. Prerequisite: Actuarial
Foundations 309 with a grade of at least C and consent of the director of the concentration in actuarial studies.
112M. Actuarial Laboratory on Probability and Statistics.
Problems and supplementary instruction in probability and statistics, especially as required for the Society of Actuaries and Casualty Actuarial Society Exam 110. Three laboratory hours a week for
one semester. Prerequisite: Mathematics 362K, credit or registration for Mathematics 378K, and consent of the director of the concentration in actuarial studies.
112N. Actuarial Laboratory on Life Contingencies.
Problems and supplementary instruction in actuarial mathematics and contingency theory, especially as required for the Society of Actuaries Exam 150. Three laboratory hours a week for one semester.
Prerequisite: Credit or registration for Mathematics 469L, and consent of the director of the concentration in actuarial studies.
Mathematics: M
Lower-Division Courses
301. College Algebra.
Topics include a brief review of elementary algebra; linear, quadratic, exponential, and logarithmic functions; polynomials; systems of linear equations; applications. Three lecture hours a week for
one semester. Usually offered only in the summer session. May not be counted toward the major requirement for the Bachelor of Arts, Plan I, degree with a major in mathematics or toward the Bachelor
of Science in Mathematics degree. In some colleges of the University, Mathematics 301 may not be counted toward the Area C requirement or toward the total number of hours required for a degree.
Credit for Mathematics 301 may not be earned after a student has received credit for any calculus course with a grade of C or better. Prerequisite: A passing score on the mathematics section of the
Texas Academic Skills Program (TASP) test.
302. Introduction to Mathematics.
Intended primarily for general liberal arts students seeking knowledge of the nature of mathematics as well as training in mathematical thinking and problem solving. Topics include number theory and
probability; additional topics are chosen by the instructor. Mathematics 302 and 303F may not both be counted. A student may not earn credit for Mathematics 302 after having received credit for any
calculus course. May not be counted toward the major requirement for the Bachelor of Arts, Plan I, degree with a major in mathematics or toward the Bachelor of Science in Mathematics degree. May be
used to fulfill the Area C requirement for the Bachelor of Arts, Plan I, degree or the mathematics requirement for the Bachelor of Arts, Plan II, degree. Prerequisite: Three units of high school
mathematics at the level of Algebra I or higher, and a passing score on the mathematics section of the Texas Academic Skills Program (TASP) test.
303D. Applicable Mathematics.
An entry-level course for the nontechnical student, dealing with some of the techniques that allow mathematics to be applied to a variety of problems. Topics include linear and quadratic equations,
systems of linear equations, matrices, probability, statistics, exponential and logarithmic functions, and mathematics of finance. Mathematics 303D and 303F may not both be counted. May not be
counted toward the major requirement for the Bachelor of Arts, Plan I, degree with a major in mathematics or toward the Bachelor of Science in Mathematics degree. A student may not earn credit for
Mathematics 303D after having received credit for Mathematics 305G or any calculus course. Prerequisite: A satisfactory score on the SAT II: Mathematics Level I or Level IC test, or Mathematics 301
with a grade of at least C.
403K. Calculus I for Business and Economics.
Differential and integral calculus of algebraic, logarithmic, and exponential functions with applications; introduction to mathematics of finance. Three lecture hours and two discussion sessions a
week for one semester. May not be counted by students with credit for Mathematics 408C, 308K, or 308L. May not be counted toward the major requirement for the Bachelor of Arts, Plan I, degree with a
major in mathematics or toward the Bachelor of Science in Mathematics degree. Prerequisite: A satisfactory score on the SAT II: Mathematics Level I or Level IC test, Mathematics 301 with a grade of
at least B, or Mathematics 304E or 305G with a grade of at least C.
403L. Calculus II for Business and Economics.
Differential and integral calculus of functions of several variables with applications, infinite series, improper integrals; introductions to probability, differential equations, matrices, systems of
linear equations, and linear programming. Three lecture hours and two discussion sessions a week for one semester. Only one of the following may be counted: Mathematics 403L, 408D, 308M. May not be
counted toward the major requirement for the Bachelor of Arts, Plan I, degree with a major in mathematics or toward the Bachelor of Science in Mathematics degree. Prerequisite: Mathematics 403K,
408C, or 308L with a grade of at least C.
305G. Elementary Functions and Coordinate Geometry.
Study of elementary functions, their graphs and applications, including polynomial, rational, and algebraic functions, exponential, logarithmic, and trigonometric functions. Mathematics 301, 305G,
and equivalent courses may not be counted toward the total number of hours required for the Bachelor of Arts, Plan I, degree with a major in mathematics or the Bachelor of Science in Mathematics
degree. Credit for Mathematics 305G may not be earned after a student has received credit for any calculus course with a grade of C or better. Prerequisite: A satisfactory score on the SAT II:
Mathematics Level I or Level IC test, or Mathematics 301 with a grade of at least C.
408C. Differential and Integral Calculus.
Certain sections are designated as honors sections for well-prepared students of mathematics and mathematically oriented sciences who wish to investigate more thoroughly the foundations of calculus.
Introduction to the theory and applications of differential and integral calculus of functions of one variable; topics include limits, continuity, differentiation, the mean value theorem and its
applications, integration, the fundamental theorem of calculus, and transcendental functions. Three lecture hours and two discussion hours a week for one semester. May not be counted by students with
credit for Mathematics 403K, 308K, or 308L. Prerequisite: Four years of high school mathematics and a satisfactory score on the SAT II: Mathematics Level I or Level IC test, or Mathematics 305G with
a grade of at least C.
408D. Sequences, Series, and Multivariable Calculus.
Certain sections are designated as honors sections for well-prepared students of mathematics and mathematically oriented sciences who wish to investigate more thoroughly the foundations of calculus.
Introduction to the theory and applications of sequences and infinite series, including those involving functions of one variable, and to the theory and applications of differential and integral
calculus of functions of several variables; topics include parametric equations, sequences, infinite series, power series, vectors, vector calculus, functions of several variables, partial
derivatives, gradients, and multiple integrals. Three lecture hours and two discussion hours a week for one semester. Only one of the following may be counted: Mathematics 403L, 408D, 308M.
Prerequisite: Mathematics 408C or the equivalent with a grade of at least C.
308K. Differential Calculus.
Introduction to the theory and applications of differential calculus of functions of one variable; topics include limits, continuity, differentiation, and the mean value theorem and its applications.
This course is available for transfer credit but is not taught in residence. Only one of the following may be counted: Mathematics 403K, 408C, 308K.
308L. Integral Calculus.
Introduction to the theory and applications of integral calculus of functions of one variable; topics include integration, the fundamental theorem of calculus, transcendental functions, parametric
equations, and sequences. Only one of the following may be counted: Mathematics 403K, 408C, 308L. Prerequisite: Mathematics 308K or the equivalent with a grade of at least C.
308M. Multivariable Calculus.
Introduction to the theory and applications of infinite series, including those involving functions of one variable, and to the theory and applications of differential and integral calculus of
functions of several variables; topics include infinite series, power series, vectors, vector calculus, functions of several variables, partial derivatives, gradients, and multiple integrals. Certain
sections are designated as honors sections for well-prepared students of mathematics and mathematically oriented sciences who wish to investigate more thoroughly the foundations of calculus. Only one
of the following may be counted: Mathematics 403L, 408D, 308M. Prerequisite: Mathematics 308L or the equivalent with a grade of at least C.
110, 210, 310, 410. Conference Course.
Supervised study in mathematics, with hours to be arranged. May be repeated for credit when the topics vary. Some sections are offered on the pass/fail basis only; these are identified in the Course
Schedule. Prerequisite: Written consent of instructor. Forms are available in the department office or in the Mathematics, Physics, and Astronomy Advising Center.
210E. Emerging Scholars Seminar.
Restricted to students in the Emerging Scholars Program. Supplemental problem-solving laboratory for precalculus, calculus, or advanced calculus courses for students in the Emerging Scholars Program.
Three two-hour laboratory sessions a week for one semester. May be repeated for credit. Offered on the pass/fail basis only.
315C. Functions and Modeling.
Study in depth of topics in secondary school mathematics that are used in teaching precalculus and in the transition to calculus. Modeling with linear, exponential, and trigonometric functions; curve
fitting; discrete and continuous models. Four laboratory hours a week for one semester. Prerequisite: Enrollment in the UTeach program or consent of instructor.
316. Elementary Statistical Methods.
Graphical presentation, frequency functions, distribution functions, averages, standard deviation, variance, curve-fitting, and related topics. Only one of the following may be counted: Mathematics
316, 360K (Topic 1: Applications of Probability Theory), 362K. May not be counted toward the major requirement for the Bachelor of Arts, Plan I, degree with a major in mathematics or toward the
Bachelor of Science in Mathematics degree. Prerequisite: A satisfactory score on the SAT II: Mathematics Level I or Level IC test, or Mathematics 301 with a grade of at least C.
316K. Foundations of Arithmetic.
An analysis, from an advanced perspective, of the concepts and algorithms of arithmetic, including sets; numbers; numeration systems; definitions, properties, and algorithms of arithmetic operations;
and percents, ratios, and proportions. Problem solving is stressed. May not be counted toward the major requirement for the Bachelor of Arts, Plan I, degree with a major in mathematics or toward the
Bachelor of Science in Mathematics degree. Credit for Mathematics 316K may not be earned after the student has received credit for any calculus course with a grade of C or better, unless the student
is registered in the College of Education. Prerequisite: Mathematics 302, 303D, 305G, or 316 with a grade of at least C.
316L. Foundations of Geometry, Statistics, and Probability.
An analysis, from an advanced perspective, of the basic concepts and methods of geometry, statistics, and probability, including representation and analysis of data; discrete probability, random
events, and conditional probability; measurement; and geometry as approached through similarity and congruence, through coordinates, and through transformations. Problem solving is stressed. May not
be counted toward the major requirement for the Bachelor of Arts, Plan I, degree with a major in mathematics or toward the Bachelor of Science in Mathematics degree. Credit for Mathematics 316L may
not be earned after the student has received credit for any calculus course with a grade of C or better, unless the student is registered in the College of Education. Prerequisite: Mathematics 316K
with a grade of at least C.
318M. Introduction to Scientific Computing.
Introduction to the computer as a basic tool in the sciences, mathematics, and engineering. Fundamentals of computer hardware, operating systems, networks, algorithms, and software. Beginning
programming in an advanced language. Introduction to applications of numerical methods and computation in the mathematical sciences. Prerequisite: Mathematics 408C or the equivalent with a grade of
at least C.
119S, 219S, 319S, 419S, 519S, 619S, 719S, 819S, 919S. Topics in Mathematics.
This course is used to record credit the student earns while enrolled at another institution in a program administered by the University's Study Abroad Office. Credit is recorded as assigned by the
study abroad adviser in the Department of Mathematics. University credit is awarded for work in an exchange program; it may be counted as coursework taken in residence. Transfer credit is awarded for
work in an affiliated studies program. May be repeated for credit when the topics vary.
Upper-Division Courses
325K. Discrete Mathematics.
Provides a transition from the problem-solving approach of Mathematics 408C and 408D to the rigorous approach of advanced courses. Topics include logic, set theory, relations and functions,
combinatorics, and graph theory and graph algorithms. Prerequisite: Mathematics 408D with a grade of at least C, or consent of instructor.
326K. Foundations of Number Systems.
Intended to provide future teachers with an understanding of certain concepts in school mathematics. Includes place value and arithmetic operations (including historical perspectives and analysis of
both standard and nonstandard algorithms); prime factorization and other properties of integers; irrational and transcendental numbers; complex numbers; properties of polynomials; and connections of
these topics with other areas of mathematics. Emphasis on conceptual understanding, developing both formal proofs and informal explanations, looking at concepts from multiple perspectives, and
problem solving involving these topics. Prerequisite: Mathematics 408D or the equivalent with a grade of at least C.
427K. Advanced Calculus for Applications I.
Ordinary and partial differential equations and Fourier series. Five class hours a week for one semester. Prerequisite: Mathematics 408D with a grade of at least C.
427L. Advanced Calculus for Applications II.
Matrices, elements of vector analysis and calculus of functions of several variables, including gradient, divergence, and curl of a vector field, multiple integrals and chain rules, length and area,
line and surface integrals, Green's theorems in the plane and space, and, if time permits, complex analysis. Five class hours a week for one semester. Prerequisite: Mathematics 408D with a grade of
at least C.
328K. Introduction to Number Theory.
Provides a transition from the problem-solving approach of Mathematics 408C and 408D to the rigorous approach of advanced courses. Properties of the integers, divisibility, linear and quadratic
forms, prime numbers, congruences and residues, quadratic reciprocity, number theoretic functions. Mathematics 328K and 360K (Topic 2: Number Theory) may not both be counted. Prerequisite:
Mathematics 341 (or 311) with a grade of at least C.
129S, 229S, 329S, 429S, 529S, 629S, 729S, 829S, 929S. Topics in Mathematics.
This course is used to record credit the student earns while enrolled at another institution in a program administered by the University's Study Abroad Office. Credit is recorded as assigned by the
study abroad adviser in the Department of Mathematics. University credit is awarded for work in an exchange program; it may be counted as coursework taken in residence. Transfer credit is awarded for
work in an affiliated studies program. May be repeated for credit when the topics vary.
329W. Cooperative Mathematics.
This course covers the work period of mathematics students in the Cooperative Education program, which provides supervised work experience by arrangement with the employer and the supervising
instructor. Forty laboratory hours a week for one semester. The student must repeat the course each work period and must take it twice to receive credit toward the degree; at least one of these
registrations must be during a long-session semester. No more than three semester hours may be counted toward the major requirement; no more than six semester hours may be counted toward the degree.
The student's first registration must be on the pass/fail basis. Prerequisite: Application through the College of Natural Sciences Career Services Office; Mathematics 408D; a grade of at least C in
two of the following courses: Mathematics 325K, 427K, 341 (or 311), 362K, 378K; and consent of the undergraduate adviser.
333L. Structure of Modern Geometry.
Axiom systems, transformational geometry, introduction to non-Euclidean geometries, and other topics in geometry; use of these ideas in teaching geometry. Prerequisite: Mathematics 408D with a grade
of at least C, or upper-division standing and consent of instructor.
439J. Probability Models with Actuarial Applications.
Probability models with actuarial applications, including Markov chains, Brownian motion, the Black-Scholes formula, frequency-of-loss and severity-of-loss random variables, compound distributions,
and ruin theory. With Mathematics 339U and 439V, covers the syllabus for exam #3 of the Society of Actuaries and the Casualty Actuarial Society. Four lecture hours a week for one semester.
Prerequisite: Mathematics 362K.
339U. Actuarial Contingent Payments I.
Simulation of random samples; single-status models; present-value random variables for life insurance and annuities. With Mathematics 439J and 439V, covers the syllabus for exam #3 of the Society of
Actuaries and the Casualty Actuarial Society. Prerequisite: Mathematics 362K. Actuarial Foundations 309 is recommended.
439V. Actuarial Contingent Payments II.
Mathematical analysis of insurance premiums, reserves, multiple-status survival models, multiple-decrement survival models; applications to such areas as life insurance and property/casualty
insurance. With Mathematics 439J and 339U, covers the syllabus for exam #3 of the Society of Actuaries and the Casualty Actuarial Society. Four lecture hours a week for one semester. Prerequisite:
Mathematics 339U.
340L. Matrices and Matrix Calculations.
Techniques of matrix calculations and applications of linear algebra. Only one of the following may be counted: Mathematics 311, 340L, 341. Prerequisite: One semester of calculus with a grade of at
least C or consent of instructor.
341. Linear Algebra and Matrix Theory.
Vector spaces, linear transformations, matrices, linear equations, determinants. Some emphasis on rigor and proofs. Only one of the following may be counted: Mathematics 311, 340L, 341. Mathematics
majors are expected to take Mathematics 341 immediately after 408D. Prerequisite: Mathematics 408D with a grade of at least C.
343K. Introduction to Algebraic Structures.
Elementary properties of groups and rings, including symmetric groups, properties of the integers, polynomial rings, elementary field theory. Students who have received a grade of C or better in
Mathematics 373K may not take Mathematics 343K. Prerequisite: Mathematics 341 (or 311) with a grade of at least C and either 325K or 328K with a grade of at least C.
343L. Applied Number Theory.
Basic properties of integers, including properties of prime numbers, congruences, and primitive roots. Introduction to finite fields and their vector spaces with applications to encryption systems
and coding theory. Prerequisite: Mathematics 328K or 343K with a grade of at least C.
344K. Intermediate Symbolic Logic.
Same as Philosophy 344K. A second-semester course in symbolic logic: formal syntax and semantics, basic metatheory (soundness, completeness, compactness, and Lowenheim-Skolem theorems), and further
topics in logic. Prerequisite: Philosophy 313K or consent of instructor.
346. Applied Linear Algebra.
Emphasis on diagonalization of linear operators and applications to dynamical systems and ordinary differential equations. Other subjects include inner products and orthogonality, normal mode
expansions, vibrating strings and the wave equation, Fourier series and Fourier integrals, and Green's functions. Prerequisite: Mathematics 341 (or 311) or 340L with a grade of at least C.
348. Scientific Computation in Numerical Analysis.
Introduction to mathematical properties of numerical methods and their applications in computational science and engineering. Introduction to object-oriented programming in an advanced language.
Study and use of numerical methods for solutions of linear systems of equations; nonlinear least-squares data fitting; numerical integration; and solutions of multidimensional nonlinear equations and
systems of initial value ordinary differential equations. Prerequisite: Mathematics 318M, 427K, and 341 (or 311) or 340L with a grade of at least C.
349P. Actuarial Statistical Estimates.
Statistical estimates for frequency-of-loss and severity-of-loss random variables; credibility theory; statistics of simulation. Covers 40 percent of the syllabus for exam #4 of the Society of
Actuaries and the Casualty Actuarial Society. Prerequisite: Mathematics 439J, and Mathematics 358K or 378K.
349T. Time Series and Survival-Model Estimation.
Introduction to the probabilistic and statistical properties of time series; parameter estimation and hypothesis testing for survival models. Covers 30 percent of the syllabus for exam #4 of the
Society of Actuaries and the Casualty Actuarial Society. Prerequisite: Mathematics 339U, and Mathematics 358K or 378K.
358K. Applied Statistics.
Exploratory data analysis, correlation and regression, data collection, sampling distributions, confidence intervals, and hypothesis testing. Prerequisite: Mathematics 362K with a grade of at least
360M. Mathematics as Problem Solving.
Discussion of heuristics, strategies, and methods of evaluating problem solving, and extensive practice in both group and individual problem solving. Communicating mathematics, reasoning, and
connections among topics in mathematics are emphasized. Prerequisite: Mathematics 408D with a grade of at least C and written consent of instructor.
361. Theory of Functions of a Complex Variable.
Elementary theory and applications of analytic functions, series, contour integration, and conformal mappings. Prerequisite: Mathematics 427K or 427L with a grade of at least C or consent of
361K. Introduction to Real Analysis.
A rigorous treatment of the real number system, of real sequences, and of limits, continuity, derivatives, and integrals of real-valued functions of one real variable. Students who have received a
grade of C or better in Mathematics 365C may not take Mathematics 361K. Prerequisite: Mathematics 341 (or 311) with a grade of at least C and either 325K or 328K with a grade of at least C.
362K. Probability I.
An introductory course in the mathematical theory of probability, fundamental to further work in probability and statistics. Only one of the following may be counted: Mathematics 316, 360K (Topic 1:
Applications of Probability Theory), 362K. Prerequisite: Mathematics 408D with a grade of at least C.
362M. Introduction to Stochastic Processes.
Introduction to Markov chains, birth and death processes, and other topics. Prerequisite: Mathematics 362K with a grade of at least C.
364K. Vector and Tensor Analysis I.
Invariance, vector algebra and calculus, integral theorems, general coordinates, introductory differential geometry and tensor analysis, applications. Prerequisite: Mathematics 427K or 427L with a
grade of at least C.
364L. Vector and Tensor Analysis II.
Continuation of Mathematics 364K, with emphasis on tensor and extensor analysis. Riemannian geometry and invariance. Prerequisite: Mathematics 364K with a grade of at least C.
365C. Real Analysis I.
A rigorous treatment of the real number system, Euclidean spaces, metric spaces, continuity of functions in metric spaces, differentiation and Riemann integration of real-valued functions of one real
variable, and uniform convergence of sequences and series of functions. Students who have received a grade of C or better in Mathematics 365C may not take Mathematics 361K. Prerequisite: Mathematics
341 (or 311) with a grade of at least C and either 325K or 328K with a grade of at least C. Students who receive a grade of C in 325K or 328K are advised to take 361K before attempting 365C.
365D. Real Analysis II.
Recommended for students planning to undertake graduate work in mathematics. A rigorous treatment of selected topics in real analysis, such as Lebesgue integration, or multivariate integration and
differential forms. Prerequisite: Mathematics 365C with a grade of at least C.
367K. Topology I.
An introduction to topology, including sets, functions, cardinal numbers, and the topology of metric spaces. Prerequisite: Mathematics 361K or 365C or consent of instructor.
367L. Topology II.
Various topics in topology, primarily of a geometric nature. Prerequisite: Mathematics 367K with a grade of at least C or consent of instructor.
368K. Numerical Methods for Applications.
Continuation of Mathematics 348. Topics include splines, orthogonal polynomials and smoothing of data, iterative solution of systems of linear equations, approximation of eigenvalues,
two-point-boundary value problems, numerical approximation of partial differential equations, signal processing, optimization, and Monte Carlo methods. Only one of the following may be counted:
Computer Sciences 367, Mathematics 368K, Physics 329. Prerequisite: Mathematics 348 with a grade of at least C.
372. Fourier Series and Boundary Value Problems.
Discussion of differential equations of mathematical physics and representation of solutions by Green's functions and eigenfunction expansions. Prerequisite: Mathematics 427K with a grade of at least
372K. Partial Differential Equations and Applications.
Partial differential equations as basic models of flows, diffusion, dispersion, and vibrations. Topics include first- and second-order partial differential equations and classification (particularly
the wave, diffusion, and potential equations), and their origins in applications and properties of solutions. Includes the study of characteristics, maximum principles, Green's functions, eigenvalue
problems, and Fourier expansion methods. Prerequisite: Mathematics 427K with a grade of at least C.
373K. Algebraic Structures I.
A study of groups, rings, and fields, including structure theory of finite groups, isomorphism theorems, polynomial rings, and principal ideal domains. Students who have received a grade of C or
better in Mathematics 373K may not take Mathematics 343K. Prerequisite: Mathematics 341 (or 311) with a grade of at least C and either 325K or 328K with a grade of at least C, or consent of
instructor. Students who receive a grade of C in Mathematics 325K or 328K are advised to take 343K before attempting 373K.
373L. Algebraic Structures II.
Recommended for students planning to undertake graduate work in mathematics. Topics from vector spaces and modules, including direct sum decompositions, dual spaces, canonical forms, and multilinear
algebra. Prerequisite: Mathematics 373K with a grade of at least C.
374. Fourier and Laplace Transforms.
Operational properties and application of Laplace transforms; some properties of Fourier transforms. Prerequisite: Mathematics 427K with a grade of at least C.
374G. Linear Regression Analysis.
Fitting of linear models to data by the method of least squares, choosing best subsets of predictors, and related materials. Prerequisite: Mathematics 358K or 378K with a grade of at least C,
Mathematics 341 or 340L, and consent of instructor.
374K. Fourier and Laplace Transforms.
Continuation of Mathematics 374. Introduction to other integral transforms, such as Hankel, Laguerre, Mellin, Z. Prerequisite: Mathematics 374 with a grade of at least C.
474M. Introduction to Mathematical Modeling and Industrial Mathematics.
Some of the problems encountered in current industry, and how mathematics can help solve them. Basic material in theory and computation of ordinary and partial differential equations, integral
equations, calculus of variations and control theory. Specific industrial applications. Three lecture hours and two laboratory hours a week for one semester. Prerequisite: Mathematics 318M or the
equivalent, Mathematics 427K, and Mathematics 341 (or 311) or 340L with a grade of at least C in each.
175, 275, 375, 475. Conference Course.
Supervised study in mathematics, with hours to be arranged. May be repeated for credit when the topics vary. Prerequisite: Upper-division standing in mathematics and written consent of instructor.
375C. Conference Course (Computer-Assisted).
Supervised study in mathematics on material requiring use of computing resources, with hours to be arranged. May be repeated for credit when the topics vary. Prerequisite: Upper-division standing in
mathematics and written consent of instructor.
376C. Methods of Applied Mathematics.
Variational methods and related concepts from classical and modern applied mathematics. Model s of conduction and vibration that lead to systems of linear equations and ordinary differential
equations, eigenvalue problems, initial and boundary value problems for partial differential equations. Topics may include a selection from diagonalization of matrices, eigenfunctions and
minimization, asymptotics of eigenvalues, separation of variables, generalized solutions, and approximation methods. May be repeated for credit when the topics vary. Prerequisite: Mathematics 318M or
the equivalent, Mathematics 427K, and Mathematics 341 (or 311) or 340L with a grade of at least C in each.
378K. Introduction to Mathematical Statistics.
Estimation of parameters and testing of hypotheses. Mathematics 362K and 378K form the core sequence for students in statistics. Prerequisite: Mathematics 362K with a grade of at least C.
379H. Honors Tutorial Course.
Directed reading, research, and/or projects, under the supervision of a faculty member, leading to an honors thesis. Conference course. Prerequisite: Admission to the Mathematics Honors Program, a
grade point average of at least 3.50 in Mathematics 365C and 373K, and approval of the honors adviser. | {"url":"http://www.utexas.edu/student/registrar/catalogs/ug00-02/ch09/courses/ch0908m.html","timestamp":"2014-04-20T09:49:33Z","content_type":null,"content_length":"50817","record_id":"<urn:uuid:73047800-93b4-4c91-9f70-ad29f9f8636a>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00532-ip-10-147-4-33.ec2.internal.warc.gz"} |
SQL Server Helper - Date Functions - Get First Day of the Week Function
It is quite common for applications to produce a report that contains transactions or records beginning from the start of the week until the current day. Getting the current day is easy because this
is achieved from the GETDATE() function but getting the start of the week is a little bit tricky. The user-defined function below accepts a date input and returns the first day of the week for that
input date.
CREATE FUNCTION [dbo].[ufn_GetFirstDayOfWeek]
( @pInputDate DATETIME )
SET @pInputDate = CONVERT(VARCHAR(10), @pInputDate, 111)
RETURN DATEADD(DD, 1 - DATEPART(DW, @pInputDate),
Getting the first day of the week is basically straight-forward once you know the trick. Each day of the week is represented by a number from 1 to 7, with 1 being the first day of the week. Given any
day of the week, to get back to the first day, you simply have to subtract from the current day the number of days equal to the day of the week then add 1 day.
For example, if today is the fifth day of the week, to get back to the first day, subtract 5 days from the current day and then add 1. So, if today is '09/01/2005', a Thursday, which is the fifth day
of the week in the U.S. English calendar, subtracting 5 days from this date becomes '08/27/2005' then adding 1 day becomes '08/28/2005', a Sunday, which is the first day of the week for the date '09/
1. SET @pInputDate = CONVERT(VARCHAR(10), @pInputDate, 111) - The first step is to get the date part of the input parameter because since it is defined as a DATETIME data type, it may contain a time
part. Converting the input date into VARCHAR(10) with a format of 101, which is in "YYYY/MM/DD" performs the task of getting the date part of the input date. Since the receiving variable is
defined as a DATETIME data type, there is no need to explicitly CAST this back to DATETIME because SQL Server will implicitly convert it to DATETIME data type.
2. RETURN DATEADD(DD, 1 - DATEPART(DW, @pInputDate), @pInputDate) - This step is responsible in deriving the first day of the week for the given input date. It basically subtracts from the input
date the number of days equal to the day of the week of the input date then adds 1 day to it. It may be confusing to see how the day of the week is subtracted from the input date then another day
is added with the formula above. Another way of writing the above statement is DATEADD(DD, -DATEPART(DW, @pInputDate) + 1, @pInputDate), which is the equivalent of subtracting the day of the week
plus 1 day.
Getting the date part only of any DATETIME or SMALLDATETIME data type is quite a common task that it is recommended to create a separate user-defined function just for this task. The user-defined
function below is the same as the one above with the exception that it assumes that a user-defined function that extracts the date part only of a DATETIME or SMALLDATETIME data type exists with a
name of [dbo].[ufn_GetDateOnly].
CREATE FUNCTION [dbo].[ufn_GetFirstDayOfWeek] ( @pInputDate DATETIME )
SET @pInputDate = [dbo].[ufn_GetDateOnly] ( @pInputDate )
RETURN DATEADD(DD, 1 - DATEPART(DW, @pInputDate), @pInputDate)
Here's an example of how to use this user-defined funcation:
SELECT * FROM [dbo].[Orders]
WHERE [OrderDate] >= [dbo].[ufn_GetFirstDayOfWeek] ( GETDATE() )
This statement returns all orders from the start of the week.. | {"url":"http://www.sql-server-helper.com/functions/get-first-day-of-week.aspx","timestamp":"2014-04-18T00:12:53Z","content_type":null,"content_length":"15185","record_id":"<urn:uuid:b460bfc6-d8a2-4edf-a1f5-048ca224f355>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00370-ip-10-147-4-33.ec2.internal.warc.gz"} |
CMSC 250 BFS
Breadth-first search: the basic algorithm
Given a graph with N vertices and a selected vertex A:
for(i = 1; there are unvisited vertices ; i++)
Visit all unvisited vertices at distance i
(i is the length of the shortest path between A and currently processed vertices)
As we saw, the algorithm for single source shortest path uses breadth-first search
For a graph with N vertices the shortest path between two vertices can be
at most N-1. The search is implemented by storing the neighbors in a queue.
BFS algorithm
1. Store source vertex S in a queue and mark as processed
2. While queue is not empty
Read vertex v from the queue
For all neighbors w:
If w is not processed
Mark as processed
Append in the queue
Record the parent of w to be v
(parent is necessary only if we need the shortest path tree)
We can use a distance table to mark vertices as processed by storing the distance to the source vertex.
Adjacency lists
A: B, C, E
B: A, C, E
C: A, B, D
D: C, E
E: A, E
Let's select A as the starting node.
Distance table:
1. After reading A from the queue and processing its neighbors:
2. Read B. Its neighbors are A, C and E. All are processed.
3. Read C. Its neighbors are A, B and D. A and B are processed. D is not. After processing D we have;
4. Read E. Its neighbors are processed
5. Read D. Its neighbors are processed
The queue is empty and the algorithm ends.
The table defines the shortest path tree, rooted at A.
Complexity of the BFS algorithm
Consider the algorithm’s basic loop:
While queue is not empty:
1. Read vertex v from the queue
2. For all neighbors w:
If w is not processed
Mark w as processed
Append w in the queue
In step 1 we read a node from the queue. The question to be answered is: how many nodes are stored in the queue? The answer is: each node is stored only once – if it is not processed. Thus the
reading step will be performed O(V) times, where V is the number of the nodes in the graph.
In step 2 we examine all neighbors, i.e. we examine all edges of the currently read node. Since the graph is not oriented, we will examine 2*E edges, where E is the number of the edges in the graph.
Hence the complexity of BFS is O(V + 2*E)
If we use adjacency matrix instead of adjacency lists, the complexity will be O(V^2)
Attention: a typical error in determining the complexity is to multiply the number of the nodes by the number of the edges. Why we should not multiply the number of the edges by the number of the
Depth-First Search: Basic Idea
For a selected vertex A let B[1], B[2], .. B[m] be adjacent vertices. For all adjacent vertices, if B[k] is not visited, visit B[k] and all vertices reachable from B[k] before proceeding with B[k+1].
General algorithm:
Procedure dfs(s)
mark all vertices in the graph as not reached
invoke scan(s)
Procedure scan(s)
mark and visit s
for each neighbor w of s
if the neighbor is not reached
invoke scan(w)
Relation between DFS and BFS
BFS and DFS are very closely related to each other. The difference is that BFS uses a queue while DFS uses a stack. Queues can be described as lists where we write at the end and read from the
beginning. Stacks can be described as lists where we write at the end and read from the end. Thus we can describe BFS and DFS in the following way:
list L = empty
tree T = empty
choose a starting vertex x
while(L nonempty)
remove edge (v,w) from beginning of L
if w not visited
add (v,w) to T
list L = empty
tree T = empty
choose a starting vertex x
while(L nonempty)
remove edge (v,w) from end of L
if w not visited
add (v,w) to T
visit(vertex v)
mark v as visited
for each edge (v,w)
add edge (v,w) to end of L | {"url":"http://faculty.simpson.edu/lydia.sinapova/www/cmsc250/LN250_Weiss/L24-BreadthDepth.htm","timestamp":"2014-04-19T07:09:14Z","content_type":null,"content_length":"16030","record_id":"<urn:uuid:46161ade-2b6b-4ee5-96de-3cc5622cda1c>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00505-ip-10-147-4-33.ec2.internal.warc.gz"} |
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An Introduction to Lie Groups and Lie Algebras
Cambridge University Press
Book Description:
With roots in the nineteenth century, Lie theory has since found many and varied applications in mathematics and mathematical physics, to the point where it is now regarded as a classical branch of
mathematics in its own right. This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of
Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Written in an informal style, this is a contemporary introduction to the subject which emphasizes
the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of
New York at Stony Brook, the book includes numerous exercises and worked examples and is ideal for graduate courses on Lie groups and Lie algebras. | {"url":"http://www.math.princeton.edu/books/introduction-lie-groups-and-lie-algebras","timestamp":"2014-04-20T21:00:44Z","content_type":null,"content_length":"12437","record_id":"<urn:uuid:7a2de4f8-657e-4da0-b78e-a04a5af6fb9c>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00303-ip-10-147-4-33.ec2.internal.warc.gz"} |
cellfun – undocumented performance boost
Matlab’s built-in cellfun function has traditionally enabled several named (string) processing functions such as ‘isempty’. The relevant code would look like this:
data = cellfun('isempty',cellArray);
In recent years, newer Matlab releases has added support for function handles, so the previous code snippet can now be written as follows:
data = cellfun(@isempty,cellArray);
The newer function-handle format is “cleaner” and more extensive than the former format, accepting any function, not just the limited list of pre-specified processing function names (‘isreal’,
‘islogical’, ‘length’, ‘ndims’, ‘prodofsize’). Some have even reported that the older format has limitations vis-a-vis compilation etc.
All this is well known and documented. However, it turns out that, counter-intuitively (and undocumented), the older format is actually much faster than the newer format for those pre-specified
processing function names. The reason appears to be that ‘isempty’ (as well as the other predefined string functions) uses specific code-branches optimized for performance:
>> c = mat2cell(1:1e6,1,repmat(1,1,1e6));
>> tic, d=cellfun('isempty',c); toc
Elapsed time is 0.115583 seconds.
>> tic, d=cellfun(@isempty,c); toc
Elapsed time is 7.493989 seconds.
Perhaps a future Matlab release will improve cellfun’s internal code, to check for function-handle equality to the optimized functions, and use the optimized code branch if possible. When I posted
this issue today as a correction to a reader’s misconception, Matlab’s Loren Shure commented as follows:
“We could improve cellfun to check function handles to see if they match specified strings. Even then MATLAB would have to be careful in case the user has overridden the built-in version of whatever
the string points to.”
While this comment seems to imply that the performance boost feature will be maintained and possibly improved in future releases, users should note that this is not guarantied. One could even argue
that future code optimizations would be applied to the new (function-handle) format rather than the old string format. The performance pendulum might also change based on user platform. Therefore,
users for whom performance is critical should always test both versions on their target system: ‘isempty’ vs. @isempty etc. (note that the corresponding function for ‘prodofsize’ is @numel).
Related posts:
12 Responses to cellfun – undocumented performance boost
1. They seem to already have improved it quite a bit in R2009a; here are my results from running your code:
>> c = mat2cell(1:1e6,1,repmat(1,1,1e6));
>> tic, d=cellfun(‘isempty’,c); toc
Elapsed time is 0.032880 seconds.
>> tic, d=cellfun(@isempty,c); toc
Elapsed time is 0.563284 seconds.
□ Ashish – actually your results show a factor of 17 between the slower @isempty and the faster ‘isempty’, consistent with the results I posted above (my reported factor of 65 is almost the
same order of magnitude as 17, and may be due to external platform-dependent factors).
The absolute values of the results of course depend on the platform: my results were for a run-down heavily-loaded laptop… The important thing here is the factor between @isempty and
‘isempty’ – not the absolute values. And a factor of 17 is still high enough to be taken into consideration in a performance-critical application.
□ Yair,
I wasn’t disputing your results. Just wanted to show that the factor has improved significantly in newer version (65 to 17). Of course, 17 times faster is still very significant as you
pointed out.
- Ashish.
2. wow. that’s a pretty significant unnecessary slowdown. at least this would be easy to catch with the profiler.
3. Yair-
As I noted to you on my blog, MATLAB doesn’t convert from FH to string method because the user might have overridden whatever the method, e.g., isempty. MATLAB could, at runtime, see if it’s
overridden, and if not, call the optimized version. But it can’t do that blindly without risk of wrong answers.
4. I know this is old, but I just noticed it gets even worse if you use the other way of calling cellfun (which I’ve been using a lot):
>> c = mat2cell(1:1e6,1,repmat(1,1,1e6));
>> tic, d=cellfun('isempty',c); toc
Elapsed time is 0.034638 seconds.
>> tic, d=cellfun(@isempty,c); toc
Elapsed time is 0.859156 seconds.
>> tic, d=cellfun(@(x) isempty(x),c); toc
Elapsed time is 7.961039 seconds.
□ I looked at your example, and I noticed that in Octave we hadn’t quite optimised this as much as possible. I went ahead and committed a change to fix this:
On my laptop with Intel Core 2 Duo @ 2.20G, I see the following:
octave:1> c = mat2cell(1:1e6,1,repmat(1,1,1e6));
octave:2> tic, d=cellfun('isempty',c); toc
Elapsed time is 0.0171831 seconds.
octave:3> tic, d=cellfun('isempty',c); toc
Elapsed time is 0.0182698 seconds.
octave:4> tic, d=cellfun('isempty',c); toc
Elapsed time is 0.0223808 seconds.
octave:5> tic, d=cellfun(@isempty,c); toc
Elapsed time is 0.0193319 seconds.
octave:6> tic, d=cellfun(@isempty,c); toc
Elapsed time is 0.01612 seconds.
octave:7> tic, d=cellfun(@isempty,c); toc
Elapsed time is 0.019449 seconds.
This should be part of our 3.6 release that should happen very soon!
Sadly, your preferred method of calling cellfun cannot be easily optimised:
octave:8> tic, d=cellfun(@(x) isempty(x),c); toc
Elapsed time is 0.924903 seconds.
octave:9> tic, d=cellfun(@(x) isempty(x),c); toc
Elapsed time is 0.873197 seconds.
octave:10> tic, d=cellfun(@(x) isempty(x),c); toc
Elapsed time is 0.875425 seconds.
Note that Octave still is single-threaded, so this does not benefit from any parallelisation right now. There’s work to build parallelisation into Octave, so perhaps we can see more dramatic
speedups in the future.
5. This is very interesting. Our independent implementation of cellfun in Octave actually behaves very similarly! However, I did optimise it to check function handles for built-in string cases.
We have a thread about it:
□ @Jordi – thanks. If you have any other comparisons to Octave for any of the other articles here, please do post a comment.
6. MATLAB 2011b result:
>> c = mat2cell(1:1e6,1,repmat(1,1,1e6));
>> tic, d=cellfun('isempty',c); toc
Elapsed time is 0.032773 seconds.
>> tic, d=cellfun(@isempty,c); toc
Elapsed time is 2.100385 seconds.
>> 2.100385/0.032773
ans = | {"url":"http://undocumentedmatlab.com/blog/cellfun-undocumented-performance-boost","timestamp":"2014-04-24T22:01:52Z","content_type":null,"content_length":"257551","record_id":"<urn:uuid:97a1af05-7d2e-4fe0-9e4f-716821267fea>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00310-ip-10-147-4-33.ec2.internal.warc.gz"} |
need help please (physics)
July 26th 2007, 02:17 PM #1
Jun 2007
need help please (physics)
In college homecoming competition, 16 students lift sports car while holding the car off the ground, each student exerts an upward force of 400 N. (a) what is the mass of the car in kilograms?
(b) what is its weight in pounds?
a) Each of the 16 students is exerting an upward force on the car and (I presume) the car is stationary and off the ground. So the students are supplying the only upward force, which is
counteracting the weight of the car. Thus
$w = 16F = 16 \cdot (400~N) = 6400~N$
(where w is the weight of the car.)
So the mass of the car is
$m = \frac{w}{g} = \frac{6400~N}{9.8~m/s^2} = 653.061~kg$
b) 1 lb = 4.45 N, so
$\frac{6400~N}{1} \cdot \frac{1~lb}{4.45~N} = 1439.75~lb$
July 26th 2007, 04:21 PM #2 | {"url":"http://mathhelpforum.com/math-topics/17246-need-help-please-physics.html","timestamp":"2014-04-21T12:25:55Z","content_type":null,"content_length":"35711","record_id":"<urn:uuid:3ccdf2c5-f282-4b6b-9855-3d3f7554562e>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00182-ip-10-147-4-33.ec2.internal.warc.gz"} |
From New World Encyclopedia
│ Regular hexagon │
│ │
│ A regular hexagon, {6} │
│ Edges and vertices │ 6 │
│ Schläfli symbols │ {6} │
│ │ t{3} │
│ Coxeter–Dynkin diagrams │ │
│ │ Image:CDW_ring.png Image:CDW_ring.png │
│ Symmetry group │ Dihedral (D[6]) │
│ Area │ $A = \frac{3 \sqrt{3}}{2}t^2$ │
│ (with t=edge length) │ $\simeq 2.598076211 t^2.$ │
│ Internal angle │ 120° │
│ (degrees) │ │
In geometry, a hexagon is a polygon with six edges and six vertices. A regular hexagon has the Schläfli symbol {6}.
Regular hexagon
The internal angles of a regular hexagon (one where all sides and all angles are equal) are all 120° and the hexagon has 720 degrees. It has six lines of symmetry. Like squares and equilateral
triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive
honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of
The area of a regular hexagon of side length $t\,\!$ is given by $A = \frac{3 \sqrt{3}}{2}t^2 \simeq 2.598076211 t^2.$
The perimeter of a regular hexagon of side length $t\,\!$ is, of course, $6t\,\!$, its maximal diameter $2t\,\!$, and its minimal diameter $t\sqrt{3}\,\!$.
There is no platonic solid made of regular hexagons. The archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and
fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron.
Hexagons: natural and artificial
• North polar hexagonal cloud feature on Saturn, discovered by Voyager 1 and confirmed in 2006 by Cassini [1] [2] [3]
• Naturally formed basalt columns from Giant's Causeway in Ireland; large masses must cool slowly to form a polygonal fracture pattern
See also
External links
All links retrieved June 19, 2013.
Triangle • Quadrilateral • Pentagon • Hexagon • Heptagon • Octagon • Enneagon (Nonagon) • Decagon • Hendecagon • Dodecagon • Triskaidecagon • Pentadecagon • Hexadecagon • Heptadecagon • Enneadecagon
• Icosagon • Chiliagon • Myriagon
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CC-by-sa 3.0 License (CC-by-sa), which may be used and disseminated with proper attribution. Credit is due under the terms of this license that can reference both the New World Encyclopedia
contributors and the selfless volunteer contributors of the Wikimedia Foundation. To cite this article click here for a list of acceptable citing formats.The history of earlier contributions by
wikipedians is accessible to researchers here:
Note: Some restrictions may apply to use of individual images which are separately licensed.
Research begins here... | {"url":"http://www.newworldencyclopedia.org/entry/Hexagon","timestamp":"2014-04-21T14:46:27Z","content_type":null,"content_length":"30659","record_id":"<urn:uuid:4dfb0077-65e5-4936-b243-0bb46bafce05>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00068-ip-10-147-4-33.ec2.internal.warc.gz"} |
Normal distribution help
January 11th 2010, 11:15 AM #1
Nov 2009
Normal distribution help
Q: To avoid a complete shutdown of the assembly line if a component fails, major manufacturing companies operate on the principle of preventative maintenance. the lifetime of one component is
normally distributed with a mean of 325h and a standard deviation of 20h. How frequently should the component be replaced so that the probability of its failing during operation is less than
A: I don't exactly understand it.
What exactly do I do? I tried stuff like P(x<0.01) etc but that doesnt work.
Is it something to do with making it =0.01?
Q: To avoid a complete shutdown of the assembly line if a component fails, major manufacturing companies operate on the principle of preventative maintenance. the lifetime of one component is
normally distributed with a mean of 325h and a standard deviation of 20h. How frequently should the component be replaced so that the probability of its failing during operation is less than
A: I don't exactly understand it.
What exactly do I do? I tried stuff like P(x<0.01) etc but that doesnt work.
Is it something to do with making it =0.01?
Did you try to solve for $a$ where
$P\left( Z<\frac{a-325}{20}\right)= 0.01$
z*=0.01 @ 0.5040
0.5040 = a-325 / 20
mm, still don't get it
This is the problem.
$z^{*} =0.01$ at $-2.3263$
Look here Z table - Normal Distribution
Where did you get -2.3263 from?
Yes, I looked there... but it just has 2 graphs and I played around and I do not get -2.3263 anywhere.
Go to Z table - Normal Distribution
Under the second curve, under "Shadded Area" put 0.01 and tick "Below" A figure will pop up next to "Below"
Which then solves to a=278.474?
Which means: the component should be replaced every 278.5 hours so it fails less than 1%?
Thats how I read it.
January 11th 2010, 12:23 PM #2
January 11th 2010, 12:44 PM #3
Nov 2009
January 11th 2010, 12:51 PM #4
January 11th 2010, 12:57 PM #5
January 11th 2010, 01:11 PM #6
Nov 2009
January 11th 2010, 01:17 PM #7
January 11th 2010, 01:32 PM #8
Nov 2009
January 11th 2010, 01:39 PM #9
January 11th 2010, 01:41 PM #10
Nov 2009
January 11th 2010, 01:47 PM #11
January 11th 2010, 01:52 PM #12
Nov 2009
January 11th 2010, 01:54 PM #13
January 11th 2010, 02:00 PM #14
Nov 2009
January 11th 2010, 02:02 PM #15 | {"url":"http://mathhelpforum.com/advanced-statistics/123297-normal-distribution-help.html","timestamp":"2014-04-19T00:17:49Z","content_type":null,"content_length":"80837","record_id":"<urn:uuid:663d8a85-ed32-4de1-80f8-962d9e2a9710>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00524-ip-10-147-4-33.ec2.internal.warc.gz"} |
From A Random World to a Rational Universe
Copyright © University of Cambridge. All rights reserved.
Randomness, Luck, Astragali and Dice
In the time before the mathematical idea of randomness was discovered, people thought that everything that happened was part of the will of supernatural beings, the gods, who looked down upon human
affairs and decided to 'tip the balance' one way or another to influence events. Hence, sacrifices were made and rituals performed to discover the 'will of the gods' or to try to influence human
affairs. This idea still prevails, and many people all over the world use lucky charms, engage in superstitious practices, use horoscopes, and still have some kind of belief that there are such ways
of influencing their lives. The gods may be dead, but 'Lady Luck' still survives.
The astragalus is a small bone, about an inch cube, found in the heel of hoofed mammals. Astragali have six sides but are not symmetrical, so there is no way of knowing which way they will eventually
come to rest. For many ancient civilizations, astragali were used by priests to discover the opinions of their gods. It was customary in divination rites to roll, or cast, five astragali. Typically,
each possible configuration was associated with the name of a god and carried with it the sought-after advice.
Astragali from the heel of a sheep showing the four positions of rest.
The small one in the foreground is made from pottery
Showing the four positions of rest. The small one in the foreground is made from pottery. Astragali found in excavations typically have their sides numbered or engraved. They were also used in board
games in the First Dynasty in Egypt, c 3500 BCE; archaeological evidence consists of boards, counters, and astragali for various games, including one similar to Snakes and Ladders, still popular
The game of Hounds and Jackals dating from 1800 BCE found in an Egyptian tomb
The astragali have been used from classical times for gambling, and similar stones are still in use today for games like 'fivestones' or 'jacks'.
Gradually, over thousands of years, astragali were replace by dice [see note 1 below], and pottery dice have been found in Egyptian tombs. The earliest die known was made from pottery and excavated
in Northern Iraq dating from about 3,000 BCE. It has dots arranged as in (Die A).
Die (B), from about 1400 BCE found in a tomb in Egypt, shows consecutive numbers opposite each other.
Dice with other markings like the names or portraits of gods have been found, probably used for special games or rituals, and others where some numbers are repeated, or 'loaded', for special purposes
or possibly for cheating (Die C).
Once the Greeks had worked out the geometry of the polyhedra, dice of other shapes began to be constructed. However, whether cube or polyhedral, the shapes were not entirely regular and were
therefore biased.
Over time, gamblers would get used to using the same dice, and have an intuitive idea of how they would fall, but given another set of dice, the odds would be different. Later, as the manufacture of
dice became more exact, some ideas of the possible combinations of number began to emerge.
The Earth and The Cosmos
There were many other forms of rituals hoping to overcome the randomness of nature and man's condition. A few of these which became of particular mathematical interest are geomancy, the nine square
grid or magic square, and temple designs, the ancestors of board games.
Geomancy means divination of or by the earth , and is a system of 16 mathematically related arrangements of stones, beans or other available small objects used to make decisions, answer questions, or
foretell the future. The stones are cast upon the ground and the pattern formed is interpreted. The symbols represent a series of binary 'opposites' like good and evil, male or female, sadness and
happiness, etc. Combinations of these opposites can be used to represent odd and even numbers.
The sixteen figures of the Geomancy system of Divination. The headings of the columns are: "The greater fortune" and "The lesser fortune". From a Book of Occult Philosophy published 1655. Notice that
each pair of shapes are associated with the traditional signs for the planets and that each configuration could be interpreted from the throws of two dice.
As in all methods of divination, each of these figures has a number of interpretations depending on its relation to other figures shown, and many other circumstances like the time of day, the
weather, and the kind of person who is asking the question.
The Grid of Nine Squares
The Nine square grid is said to come from an ancient system for the division of land, probably from feudal India. In China the nine-square configuration was supposed to be an ideal arrangement, with
eight farmers' fields surrounding a central well. The grid of nine squares, or a circle divided into nine sections by straight lines often appears as a central form in Tibetan sacred diagrams. In
Scotland, the pattern was used at Beltane (the eve of May) where eight squares were cut out from the turf, and a bonfire lit on the central square.
In this way, from practical beginnings in different cultures, the nine-square grid acquired mystic importance and symbolised divine order, and the representation of control by the gods.
Magic Squares are directly related to the Sacred Grid, supposedly being the numerical mystery which underlies their physical form. The simplest magic square is the square of nine, ascribed to Saturn,
where each row and column adds up to 15; the total of the rows and the columns is 45, and the diagonals 30. The 4x4 square with row and column numbers 34 is assigned to Jupiter, the 5x5 with row or
column numbers 65 to Mars, and so on for the Sun, Venus, Mercury and the 9x9 square with row or column numbers 369, to the Moon [an NRICH article on Magic Squares can be found here at
As with other devices, these magic squares are all said to have correspondences to different numbers, various deities, days of the week, natural objects, different qualities, and so on. In the Hindu
Temple Yantra [see note 2 below] you can see the nine squares, the 'sacred space', or source of energy, in the centre.
This is a Yantra from a Hindu Temple. Yantras (or Mandalas) are used as a focus for mystical contemplation and often for the basis of design for a temple. This one is based on a 5 x 5 square, with
the 'sacred space' of the 3 x 3 square in the centre.
Board Games are clearly linked with divination, astrology and sacred geometry, and the designs of the boards can show their sacred or occult origins. The popular game of 'snakes and ladders' is
controlled by the throw of dice, and the ladders and snakes originally referring to good and bad fortune, now refer to good and bad 'luck' in the progress of the game. In some cases the designs of
the boards are the same as the plans of temples and holy cities with a 'sacred space' in the centre.
This is the board for the ancient Korean game of 'Yut' or 'Nyout' The board can be made of cloth or paper, or can be drawn on the floor. It is played with four 'Yut Sticks' of semi-circular section,
and the way they fall determines the move of a token. The shape can be square or circular and represents the division of the world into twenty outer regions and nine central spaces.
The game of 'Nine Men's Morris' is played with counters on the dots on this board. The design is said to represent the four elements, (earth, air, fire and water) the four winds, or the four cardinal
points of the compass, and the central sacred area was a symbol of rebirth or renewal.
The game was supposed to have originated in Egypt, and was known to the Romans. This is a picture from the 13th century of the game being played in England.
This is a traditional diagram used for the Horoscope of Robert Burton from his tomb in Christchurch in Oxford. This clearly has a link with the diagrams from sacred architecture and board games.
Mathematics and Magic
In ancient times, few people could understand even the simplest arithmetic and geometry, and the confusion of mathematics with magic has a long history.
People who had knowledge of the regular movements of the heavens were able to predict the position of planets, and the particular the times when astronomical events appeared in certain sections of
the sky. In ancient civilisations these were highly skilled technicians, called 'priests', and their activities were partly scientific, and partly religious. In Europe, after the arrival of
Christianity, the religious aspect of these practices was condemned as superstition. Because numbers were used in these processes, anyone who used numbers was regarded with considerable suspicion. In
this way genuine mathematicians were looked upon with suspicion by the ignorant, and the titles of Astrologer, Mathematician and Conjurer were virtually synonymous.
An early Bishop of the Church, St. Augustine of Hippo (354-430 CE) once said:
"The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and
confine man in the bonds of Hell."
Augustine was arguing that belief in astrology denies the freedom of the will.
Roger Bacon (1214 - 1292), often called England's first Scientist, had a reputation as a 'great necromancer' because of his ingenious experiments and John Dee (1527 - 1609) probably one of the
foremost mathematicians in Europe of his time, gained a reputation as a 'Conjuror' while he was at Oxford because he was respnsible for developing a simple mechanical device by which an actor
appeared to fly, and people claimed he was in league with the devil. [see note 3 below]
John Dee
Robert Recorde
During the sixteenth century in England, mathematicians like Robert Recorde (1510-1558) and Thomas Digges (1546-1595) published many works showing the everyday practical usefulness of mathematical
knowledge for ordinary people clearly showing that mathematics was not an occult practice
Following the foundation of the Oxford chairs in mathematics and astronomy in 1619, some parents kept their sons away from the university in fear of them becoming contaminated by the 'Black Art'.
As the predictive power of astronomy and other practical uses of mathematics became apparent, mathematicians were able to dispel the idea that many events were not controlled by the goddess Fortuna,
but could be explained in a rational way.
This is the title page of Robert Recorde's The Castle of Knowledge published in 1556. It is his fourth book for the self-education of craftsmen and artisans and shows how to make the instruments for
astronomy and navigation. The blind goddess Fortuna stands on the unstable sphere holding the wheel of chance, while the Spirit of Knowledge stands on a stable cube holding the navigator's dividers
and the sphere of destiny.
The Beginnings of Probability
Since dice were used in gambling, in religious ceremonies and for divination, it is believed that those who used the dice had a good intuitive idea of the likely frequency of various number
combinations. The first printed document showing the possibilities with three dice was the Latin poem De Vetula , which shows all the combinations for the fall of three dice, and is believed to have
been written in the early 13th century. The idea of using binomial coefficients to calculate the possibilities appears in the poem, but is not taken up until much later [see note 4 below].
Part of the poem De Vetula written in the 13th century, shows the different combinations of three dice. Notice the written numerals on the left of the text.The written numerals for the number
combinations appear to the left of the poem.
(1501- 1576)
Girolamo Cardano
Since the Christian Church was against gaming, and there was much superstition about divination, it is not surprising that a theory of probability did not begin to appear until the 16th century.
Cardano, writing with considerable personal knowledge of gambling, recognised that if the die was honest, each face would have an equal chance of appearing. His manuscript, Liber De Ludo Aleae , was
written about 1526 but only found after his death, and not published until 1663. He gave tables of the results for one, two and three dice, but these are not all correct. However, Cardano is credited
with recognising that the abstraction of the 'honest die' is the key to a theory of probability based on mathematical principles.
Nicolo Tartaglia (1500-1557)Tartaglia (1500 - 1557) and others discuss various versions of the division of the stakes when a gambling game is stopped, called the 'problem of points', and this shows
that Cardano's ideas were likely to be common knowledge among scholars of the later 16th and early 17th century.
Galileo (1564 - 1642) wrote on probability but his work was not published until1718. He stated that with three dice there can only be one way of obtaining a 3 (1,1,1) and an 18 (6,6,6) but there are
three combinations for obtaining a 6 (4,2,1), (3,2,1) and (2,2,2) which can occur in different orders making 10 possibilites and four combinations for a 7 (5,1,1), (4,2,1), (3,3,1) and (3,2,2) which
lead to fifteen possibilities. However, although 9 and 12 could be made up in the same number of ways as 10 and 11, from their experience, gamblers claimed that the occurrence of 10 and 11 were more
likely! Galileo showed that the total number of possible throws with three dice are 216, and he gave a table of the number of possible throws for a total of 10, 9, 8, 7, 6, 5, 4 and 3, showing that
the throws for 11 to 18 were symmetrical with these. In this way he showed that there were 27 possible throws to obtain a 10, and 25 for a 9.
This is a copy of the table Galileo published in his Sopra le Scoperte de I Dadi (Concerning an Investigation on Dice). Here he shows clearly how to count the different combinations of the various
His work showed that by this time there was no doubt about the general method for calculating chances with a die, and it was clear that the mathematical concepts of the equal probability of the throw
of a die, and the procedures to analyse the results were well known.
Pascal's Triangle
By the mid 16th century the theory of probability became established on a rigorous basis with the work of Pascal and Fermat. However, as we have seen, the idea of the application of 'Pascal's
Triangle' had been suggested as early as the 13th century but forgotten for some 200 years. The triangle itself was known and published before, by Stifel (Arithmetica Integra 1543) Tartaglia
(Trattato 1556) Stevin (Arithmetic 1625) Pierre Herigone (Cours Mathematique 1634), and we also know it was known to the Chinese and the Arabs by the mid 13th century, but Pascal was the first to
apply it to probability.
For pedagogical notes:
Use the notes tab at the top of this article or click here .
1. The word 'die' (plural 'dice') come from the Latin verb 'dare' (pron. da-ray) to give its participle dadus means 'given by the gods'.
2. The word Yantra is a Sanscrit word meaning a mystical diagram or picture. They contain geometric items and archetypal shapes and patterns of squares, triangles, circles and other floral patterns.
In contrast, a Mantra is a spoken verse or poem.
3. At this time, 'Mathematics' included the applied mathematics of physics, statics, mechanics, hydraulics, and other practical arts. This is clear from John Dee's famous 'Preface' to the 1570
English Edition of Euclid.
4. It is possible that the scholar who wrote this poem might have been aware of the 'number triangle' of the Arabs.
General Background and History
David, F. N. Games, Gods and Gambling . New York. Dover Books
This is the well-known classic book on the subject still full of interesting and reliable information.
De Moivre, A. (1967) The Doctrine of Chances or A method of Calculating the Probabilities of Events in Play . London. Frank Cass.
A facsimilie of the 1738 second edition of Abraham de Moivre'??s classic where he makes corrections, expands the explanations, and gives more details of the solutions of the problems than in the
first edition of 1718.
Ore, O. (1953) Cardano, The Gambling Scholar .
A good story and biography of Girolamo Cardano by the Danish Mathematician Oystein Ore. Contains a translation of Cardan'??s Liber de Ludo Aleae (Book on Games of Chance).
Hacking, I. (1975) The Emergence of Probability . C.U.P. Cambridge UK
Hacking's book contains many references to original works in the period up to and including the 18th century.
Hald, A. (2003) A History of Probability and Statistics and their Applications Before 1750 . New York. Wiley
Pennick, N. (1988) Games of the Gods . London. Hutchinson
Religious Beliefs, Superstition, Astrology and Divination in may cultures and their connections to many games of dice and on the board
Some Books for the Classroom
Jenkins, G.W. & Slack J.L. (1979) Classroom Experiments with Dice . St Albans. Tarquin Publications
Woods, G. Symmetry Dice . St Albans. Tarquin Publications
Benson, S. (2005) Ways to think about Mathematics:Activities and Investigations for Grade 6 . California. Corwin Press. This is a useful book with many examples of activities. There are sections
about probability and binomial coefficients.
Here is a shop for all kinds of dice : large; small; all colours; with numbers; with spots; blank; arithmetic symbols; money symbols; polyhedral; round (yes round!); loaded; and for cheating! http://
This interesting Board Game
site has many traditional games like Nine Men's Morris, various types of strategy games like Solitaire, Fox and Geese; Mancala, Ludo; and Snakes and Ladders. All of them have well-researched
historical notes. | {"url":"http://nrich.maths.org/6120/index?nomenu=1","timestamp":"2014-04-19T06:54:03Z","content_type":null,"content_length":"29461","record_id":"<urn:uuid:9ea59804-f695-436c-b6af-57e95db823dc>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00492-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Portability GHC only
Stability experimental
Maintainer ekmett@gmail.com
Safe Haskell None
This module provides reverse-mode Automatic Differentiation using post-hoc linear time topological sorting.
For reverse mode AD we use StableName to recover sharing information from the tape to avoid combinatorial explosion, and thus run asymptotically faster than it could without such sharing information,
but the use of side-effects contained herein is benign.
grad :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f aSource
The grad function calculates the gradient of a non-scalar-to-scalar function with reverse-mode AD in a single pass.
>>> grad (\[x,y,z] -> x*y+z) [1,2,3]
grad' :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f a)Source
The grad' function calculates the result and gradient of a non-scalar-to-scalar function with reverse-mode AD in a single pass.
>>> grad' (\[x,y,z] -> 4*x*exp y+cos z) [1,2,3]
gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f bSource
grad g f function calculates the gradient of a non-scalar-to-scalar function f with reverse-mode AD in a single pass. The gradient is combined element-wise with the argument using the function g.
grad = gradWith (_ dx -> dx)
id = gradWith const
gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f b)Source
grad' g f calculates the result and gradient of a non-scalar-to-scalar function f with reverse-mode AD in a single pass the gradient is combined element-wise with the argument using the function g.
grad' == gradWith' (_ dx -> dx)
jacobian :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f a)Source
The jacobian function calculates the jacobian of a non-scalar-to-non-scalar function with reverse AD lazily in m passes for m outputs.
>>> jacobian (\[x,y] -> [y,x,x*y]) [2,1]
>>> jacobian (\[x,y] -> [exp y,cos x,x+y]) [1,2]
jacobian' :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (a, f a)Source
The jacobian' function calculates both the result and the Jacobian of a nonscalar-to-nonscalar function, using m invocations of reverse AD, where m is the output dimensionality. Applying fmap snd to
the result will recover the result of jacobian | An alias for gradF'
ghci> jacobian' ([x,y] -> [y,x,x*y]) [2,1] [(1,[0,1]),(2,[1,0]),(2,[1,2])]
jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f b)Source
'jacobianWith g f' calculates the Jacobian of a non-scalar-to-non-scalar function f with reverse AD lazily in m passes for m outputs.
Instead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g.
jacobian = jacobianWith (_ dx -> dx)
jacobianWith const = (f x -> const x <$> f x)
jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (a, f b)Source
jacobianWith g f' calculates both the result and the Jacobian of a nonscalar-to-nonscalar function f, using m invocations of reverse AD, where m is the output dimensionality. Applying fmap snd to the
result will recover the result of jacobianWith
Instead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g.
jacobian' == jacobianWith' (_ dx -> dx)
hessian :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f (f a)Source
Compute the hessian via the jacobian of the gradient. gradient is computed in reverse mode and then the jacobian is computed in reverse mode.
However, since the grad f :: f a -> f a is square this is not as fast as using the forward-mode jacobian of a reverse mode gradient provided by hessian.
>>> hessian (\[x,y] -> x*y) [1,2]
hessianF :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f (f a))Source
Compute the order 3 Hessian tensor on a non-scalar-to-non-scalar function via the reverse-mode Jacobian of the reverse-mode Jacobian of the function.
Less efficient than hessianF.
>>> hessianF (\[x,y] -> [x*y,x+y,exp x*cos y]) [1,2]
diff :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> aSource
Compute the derivative of a function.
>>> diff sin 0
>>> cos 0
diff' :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> (a, a)Source
The diff' function calculates the value and derivative, as a pair, of a scalar-to-scalar function.
>>> diff' sin 0
diffF :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f aSource
Compute the derivatives of a function that returns a vector with regards to its single input.
>>> diffF (\a -> [sin a, cos a]) 0
diffF' :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f (a, a)Source
Compute the derivatives of a function that returns a vector with regards to its single input as well as the primal answer.
>>> diffF' (\a -> [sin a, cos a]) 0
Unsafe Variadic Gradient
class Num a => Grad i o o' a | i -> a o o', o -> a i o', o' -> a i oSource
Num a => Grad (AD Kahn a) [a] (a, [a]) a
Grad i o o' a => Grad (AD Kahn a -> i) (a -> o) (a -> o') a | {"url":"http://hackage.haskell.org/package/ad-3.4/docs/Numeric-AD-Mode-Kahn.html","timestamp":"2014-04-17T16:16:50Z","content_type":null,"content_length":"28441","record_id":"<urn:uuid:f6e5df7f-816a-4ff0-a859-ea517a696f06>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00434-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - Skew Lines Conceptual Problem
knowLittle Feb26-12 07:39 PM
Skew Lines Conceptual Problem
1 Attachment(s)
1. The problem statement, all variables and given/known data
Suppose L1 and L2 are skew lines. Is it possible for a non-zero vector to be perpendicular to both L1 and L2? Give reasons for your answers.
2. Relevant equations
I know that skew lines are not parallel or intersect. Also, they don't lie on the same plane.
3. The attempt at a solution
I say that it is possible.
Picture a line on the x axis on (x,0,0) and a line on the y-axis exactly above the previous line but with height or z=5.
I can easily draw a vector that is perpendicular to both L1 and L2.
Am I correct?
Re: Skew Lines Conceptual Problem
yes you are correct
suppose you have the equations of two skew lines, you can always deduce the equation of the line passing through and perpendicular to them.
knowLittle Feb27-12 02:15 PM
Re: Skew Lines Conceptual Problem
Re: Skew Lines Conceptual Problem
Given [tex]
(l_1 ):r = \left( {\begin{array}{*{20}c}
{a_1 } \\
{a_2 } \\
{a_3 } \\
\end{array}} \right) + t_1 \left( {\begin{array}{*{20}c}
{b_1 } \\
{b_2 } \\
{b_3 } \\
\end{array}} \right) \\
(l_2 ):r = \left( {\begin{array}{*{20}c}
{a^' _1 } \\
{a^' _2 } \\
{a^' _3 } \\
\end{array}} \right) + t_2 \left( {\begin{array}{*{20}c}
{b^' _1 } \\
{b^' _2 } \\
{b^' _3 } \\
\end{array}} \right) \\
Let A, B be 2 arbitrary points lying on (l1) and (l2) respectively, then write the position vector of them.
Write the equation of [tex]
\overrightarrow {AB}[/tex]
Since AB is perpendicular to (l1) and (l2): [tex]
\overrightarrow {AB} .\left( {\begin{array}{*{20}c}
{b_1 } \\
{b_2 } \\
{b_3 } \\
\end{array}} \right) = 0
and [tex]
\overrightarrow {AB} .\left( {\begin{array}{*{20}c}
{b^' _1 } \\
{b^' _2 } \\
{b^' _3 } \\
\end{array}} \right) = 0
Solving the equations simultaneously, you'll find t1 and t2, which are later be used to compute the coordinates of A and B. The vector [tex]
\overrightarrow {AB}[/tex] is what you're after.
knowLittle Feb28-12 12:52 PM
Re: Skew Lines Conceptual Problem
What reasons can I give to my answer?
I know that it can be done, because it's geometrically possible to picture. But, how do I put it into academic words that can be considered a correct answer?
Also, I know how to find a point A and B in L1 and L2. It's just giving an arbitrary value to their parameters. In addition, I know how to find the segment uniting points AB perpendicular to both
But, how do I solve the equations simultaneously?
Do you mean:
and then find "t" and "s"? How do I later find the coordinates of AB?
Re: Skew Lines Conceptual Problem
Quote by knowLittle (Post 3789177)
It's just giving an arbitrary value to their parameters.
It's not. You are supposed to write down the position vectors of them in terms of t1 and t2. Then you'll have 2 equations of 2 unknowns (t1 and t2), which is solvable. Finding the coordinates is just
a matter of substitution now.
For example, given (l): r=(1 2 3) + t(3 2 1). For every point M lying on (l), its coordinate is of the form (1+3t, 2+2t, 3+t).
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Romeoville Calculus Tutor
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BCD (Binary Coded Decimal) Number System
You should now be familiar with the Binary, Decimal and Hexadecimal Number System. If we view single digit values for hex, the numbers 0 - F, they represent the values 0 - 15 in decimal, and occupy a
nibble. Often, we wish to use a binary equivalent of the decimal system. This system is called Binary Coded Decimal or BCD which also occupies a nibble. In BCD, the binary patterns 1010 through 1111
do not represent valid BCD numbers, and cannot be used.
Conversion from Decimal to BCD is straightforward. You merely assign each digit of the decimal number to a byte and convert 0 through 9 to 00000000 through 00001001, but you cannot perform the
repeated division by 2 as you did to convert decimal to binary.
Let us see how this works. Determine the BCD value for the decimal number 5,319. Since there are four digits in our decimal number, there are four bytes in our BCD number. They are:
Thousands Hundreds Tens Units
Since computer storage requires the minimum of 1 byte, you can see that the upper nibble of each BCD number is wasted storage. BCD is still a weighted position number system so you may perform
mathematics, but we must use special techniques in order to obtain a correct answer. | {"url":"http://melabs.com/resources/articles/bcdnumbers.htm","timestamp":"2014-04-17T00:50:56Z","content_type":null,"content_length":"22527","record_id":"<urn:uuid:ab57dfe9-fa85-4c49-aa08-5db4554d51bf>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00565-ip-10-147-4-33.ec2.internal.warc.gz"} |
Expanding Visualization of published system edges (R)
February 28, 2012
By Intelligent Trading
I happened to be looking over a revised text of a systems author I happen to follow. I will be a bit vague about specifics, as the system itself is based on well know ideas, but I'll leave the reader
to research related systems. The basic message illustrated in this post, is that I often make an effort to look at different viewpoints of system related features that are not always explored in the
Fig 1. BarGraph Illustration showing 0.48% average weekly gain given conditional system parameters, over arbitrary trades giving 0.2% average return per week over same 14 yr. period.
For example, the following system is based upon buying at pullbacks of a certain equity series and holding for a week. In the book, a bargraph is shown illustrating the useful edge of about 0.48%/
trade vs. simply buying and holding for 0.2%/trade. Although, the edge is useful and demonstrated well in the bargraph illustration, it can be useful to look at the system performance from various
different perspectives.
As an example, we might wonder how the system unfolded over time. In order to look at this, we can plot a time series representation of the system's equity curve (assuming 100% compounding, no
slippage, and no fractional sizing). The curve is shown compared to a straw-broom plot of 100 monte carlo simulation paths of the true underlying data, comprised of randomly and uniformly selected
data over the same period.
Figure 2. Plot of edge based equity path vs. simulated Monte Carlo Straw-Broom Plots of randomly selected series based upon true underlying data.
Looking at the 'unwinding' of the actual system's 14yr. time series path, we can make a few observations.
1) The edge, in terms of terminal wealth only, far outperforms several randomly simulated data paths built from the actual instrument.
2) Unfortunately, the edge also has very wild swings and variation (resulting in a very large drawdown).
Had we blindly selected the system itself based upon the bar graph alone, it's very possible, that we could have entered at the worst possible time (just prior to the drawdown). It also illustrates
an issue I personally have with using simple monte carlo analysis (with IID assumptions) as a proxy to underlying data. Namely, that the auto-correlation properties have been filtered out, making the
system based edge appear much better in comparison. I have spent a lot of time thinking about ways to deal with this issue; but it's a discussion for another day. But it's not necessarily a bad
result at all. Rather it gives us some features (persistence and superior edge/trade) that we can use as a spring-board for further optimization; for example, we might think about adding a
conditional filter to mitigate the large drawdown based upon underlying features that may have coincided during that period.
Data was plotted using R ggplot2. Although I think the plotting tool is excellent, I find that the processing time is a bit consuming.
for the author, please follow the link and comment on his blog:
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Leonard Jimmie Savage
Born: 20 November 1917 in Detroit, Michigan, USA
Died: 1 November 1971 in New Haven, Connecticut, USA
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Leonard Jimmie Savage's parents, Louis Ogashevitz and Mae Rugawitz, were Jewish; Jimmie was the first of their four children. The first question that the reader will naturally ask, therefore, is why
Jimmie Savage is named Savage. In fact the reason he was called 'Jimmie Savage' is even more complicated. Louis Ogashevitz, who was in the real estate business, was born in Detroit in 1897 to parents
who had emigrated to the United States from Russia. After Jimmie was born, his mother, who had a high school education and was a trained nurse, was seriously ill and, as a consequence, there was a
delay in giving him a name. A nurse at the hospital where he was born put the name "Jimmie" against his entry in the hospital records but when his mother recovered he was given the name Leonard
Ogashevitz. However, the name 'Jimmie' stuck and he was known as Jimmie as he was growing up. In 1920 his father changed his name from Ogashevitz to Savage but this name change did not apply to his
children. Many years later, when Jimmie was undertaking classified war work, he had his name legally changed from Leonard Ogashevitz to Leonard Jimmie Savage. Although known as Jimmie all his life,
he wrote his papers under the name Leonard Savage.
Allen Wallis writes in [12] about Jimmie's father:-
His father, Louis, was the most significant figure in Savage's life. His respect and love for his father had extraordinary depth and devotion. Except for Louis's unwavering devotion and support
... Jimmie would not have had as productive a life nor as happy a one.
Jimmie had a difficult time growing up. In part this was due to poor eyesight, caused by nystagmus (involuntary eye movement) and extreme myopia, which meant that he had only very limited vision. But
there was another problem which was a direct consequence of his parent's fear that their children might be kidnapped - a real possibility in Detroit at this time. Jimmie was confined to his home
(which was protected by a surrounding wall) and educated by a governess but this caused much stress and arguments between Jimmie and his two sisters Joan (born 1921) and Barbara (born 1922). His
parents tried to improve matters by sending Jimmie to boarding school but he later described the year spent there as one of the worst in his life. He then attended Central High School, Detroit, but
his teachers were unimpressed with him and would not recommend him for university studies. Richard, Jimmie's brother born in 1925 who also became a well-known statistician, said in a 1999 interview [
Central High was a very academic-oriented public high school with mostly Jewish students. Both Jimmie and Joan had rather unpleasant experiences there.
Richard also explained why Jimmie had so much trouble at school [10]:-
Jimmie was truly a polymath from a very young age. He was a brilliant child, but he paid no attention to what was going on in school because he couldn't see what was going on in school. The
teachers thought he was more or less feebleminded.
Jimmie's father Louis did everything he could to get his son an education and persuaded one of his friends to recommend Jimmie to Wayne University in Detroit. He studied engineering for a year at
Wayne and did well enough that he was allowed to enter the University of Michigan, Ann Arbor, to study chemical engineering.
At the University of Michigan things again went badly for Savage who, because of his poor eyesight, caused a fire in the chemistry laboratory. He was expelled but again his father intervened and
Jimmie was allowed to take some mathematics courses with the aim of majoring in physics. His grades began to improve: C in analytic geometry; B in calculus; B in differential equations; A in Raymond
Wilder's foundations of mathematics; and A in Raymond Wilder's point set topology course. Inspired by Wilder he received all A grades from that point on and changed to major in mathematics. He
received his BS from the University of Michigan in 1938. In that year he married Jane Kretschmer; they had two sons, Sam Linton and Frank Albert. Their eldest son, Samuel Linton Savage, received his
Ph.D. in computer science from Yale University in 1973. He has worked in the Management Science Department at Chicago and at Stanford. He is the author of The Flaw of Averages: Why we underestimate
risk in the face of uncertainty and Decision making with insight.
In 1941 Savage received his PhD with a thesis was on metric and differential geometry. Writing in 1950 he described his doctoral thesis The Application of Vectorial Methods to the Study of Distance
Spaces as follows (see [12]):-
My dissertation for the Ph.D. degree at the University of Michigan was on applications of vectorial methods to metric geometry (in the sense of the Menger school), especially with a view to the
merging of metric geometry in that sense with differential geometry. Professor S B Myers at the University of Michigan sponsored my dissertation, but I was particularly close to R L Wilder there.
He spent session 1941-42 at the Institute for Advanced Study at Princeton as a Rackham fellow, and there he continued to work on pure mathematics. While at the Institute he solved an open problem in
the calculus of variations suggested to him in discussions with John von Neumann and Marston Morse. He published this result in On the crossing of extemals at focal points (1943). He was then
appointed as an Instructor in Mathematics at Cornell University where he spent the academic year 1942-43, following which he spent a year at Brown University as a Research Mathematician in the
classical mechanics group as part of his contribution to the war effort. In 1944, still undertaking war work, he joined the Statistical Research Group at Columbia University as a Research Associate -
this move into statistics was suggested by von Neumann who had recognised Savage's talents when he was at Princeton.
Following the end of World War II in 1945, Savage spent a year working with Richard Courant at the Institute of Applied Mathematics at New York University and then he was awarded a Rockefeller
Fellowship to spend time at the Institute of Radiobiology and Biophysics at the University of Chicago. He went to Chicago in the autumn of 1946 and began one of the most productive periods of his
life being appointed as a Research Associate at Chicago in 1947. He published a joint paper with the economist Milton Friedman The Utility Analysis of Choices Involving Risk in 1948 in the Journal of
Political Economy and, in the following year, a joint paper with Paul Halmos Application of the Radon-Nikodym theorem to the theory of sufficient statistics. Remaining at Chicago, Savage was one of
the founders of the Statistics Department there in 1949. An interesting snapshot of Savage at this time can be had from a reference written to support his application for a Guggenheim Foundation
Fellowship in 1950 (see [2]):-
Dr Savage is brilliant and scholarly, has broad and varied interests, and knows a number of fields rather deeply. He is in addition a remarkable personality, vitally stimulated and interested by,
and stimulating and interesting to, others. His basic training is in pure mathematics, but fundamentally he is equally interested in empirical science, whether physical, biological, or social.
Statistics, in which he began to work about 1944, has provided a suitable meeting ground for his formal-abstract and his empirical-inductive interests. At least two of the several contributions
he has already made to statistics are of major importance, and have stimulated a flow of papers by others. In addition, he has published significant papers in economics, biology, and medicine. It
is quite possible, though obviously too early to predict, that he will become one of the great figures of his generation in the field of statistics.
He was awarded the Guggenheim Foundation Fellowship and, in addition, was a Fulbright grantee allowing him to spend the academic year 1951-52 in Paris and in Cambridge, England. In 1954 Savage was
promoted to professor at Chicago and he served as Chairman of the Statistics Department from 1956 to 1959.
Savage wrote on the foundations of statistics which led him into deep philosophical questions both about statistics and knowledge in general. The other main direction of his work was to study
gambling as a source to stimulate problems in probability and decision theory. Savage's book The Foundations of Statistics (1954) is perhaps his greatest achievement. It shows von Neumann's influence
and also that of Ramsey. The book starts with six axioms, which are both motivated and discussed, and from these are deduced the existence of a subjective 'personal' probability and a utility
function. A special case of a utility function had been introduced by von Neumann and Morgenstern in their theory of games. The Foundations of Statistics sets out Savage's ideas on Bayesian
statistics and, in particular, explains his theory of subjective and personal probability. These important ideas did, however, lead to difficult relations with his colleagues. William Kruskal, one of
these Chicago colleagues, wrote [11]:-
In his development of personal probability, Savage moved more and more to a proselytizing position. Personal probability was not only useful and interesting to study; it became for him the only
sensible approach to probability and statistics. Thus, orthodoxy of neoradicalism developed: if one were not in substantial agreement with him, one was inimical, or stupid, or at the least
inattentive to an important scientific development. This attitude, no doubt sharpened by personal difficulties and by the mordant rhetoric of some anti-Bayesians, exacerbated relationships
between Jimmie Savage and many old professional friends. The problem had a special poignancy for those who, like myself, took an eclectic point of view.
Savage left Chicago in 1960 and took up a professorship at the University of Michigan. Chicago had tried hard to keep him and, in a letter he wrote before leaving Chicago, he expressed his
appreciation for the Chicago Department (see [12]):-
For a person who wants to do original, realistic, and critical work in statistics there is no atmosphere anywhere in the world today to compare with this Department.
He remained at Michigan for four years before moving to Yale University where he was named Eugene Higgins Professor of Statistics. An important work by Savage, published in 1965 after he took up the
chair at Yale, is How to gamble if you must : Inequalities for stochastic processes, written jointly with Lester Dubins. They describe their motivation in the Introduction:-
Imagine yourself at a casino with $1,000. For some reason, you desperately need $10,000 by morning; anything less is worth nothing for your purpose. What ought you to do? ... As is well known,
any policy of compounding bets that are subfair to you must decrease your expected wealth. Consequently, no matter how you play, your chance of converting $1,000 into $10,000 will be less than ^1
/[10]. How close to ^1/[10] can you make it and by what strategy?
Other articles written by Savage relate to statistical inference, in particular the Bayesian approach. He introduced Bayesian hypothesis tests and Bayesian estimation. His Bayesian approach, however,
opposed the views of Fisher and Neyman. In his later years he wrote on the philosophy of statistics.
Jimmie Savage was divorced in 1964 and, on 10 July of that year, he married Jean Strickland. Sadly, Savage only had seven years at Yale before he died at the early age of fifty-three. However, these
were good years [12]:-
Yale proved much to his liking, especially the close association with Frank Anscombe, whom he had held in high personal and professional regard from the time they first met and to whom, in fact,
Savage had enthusiastically offered a professorship at Chicago. The agreeable professional circumstances during the seven Yale years, and above all his great happiness with Jean, combined to make
the last period of Savage's life personally the happiest.
He was president of the Institute of Mathematical Statistics (1957-58), invited to give the Fisher lecture On rereading R A Fisher in 1970, and due to give the 1972 Wald lectures at the time of his
death. In 1963 he was awarded an doctorate by the University of Rochester. His early death, however, prevented him receiving many honours which would almost certainly have been given to him. He has
been honoured with the establishment in 1977 of the Savage Award made each year to two outstanding doctoral dissertations in Bayesian econometrics and statistics. The American Statistical Association
and the Institute of Mathematical Statistics decided to sponsor the publication of a memorial volume [2] for Savage which was published in 1981. Also in his honour was Studies in Bayesian
econometrics and statistics published as Contributions to Economic Analysis No 86 (1975).
Finally, let us quote Milton Friedman writing in 1964 (see [2]):-
Jimmie is one of the few really creative people I have met in the course of my intellectual life. He has an original, independent mind capable of throwing new light on whatever problems he looks
at. He also has a wide-ranging curiosity. In whatever fields he turns his mind to, he gets new insights, ideas, and approaches. ... Here is one of those extraordinary people of whom there are
only a handful in any university at any time.
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 (12 books/articles)
Mathematicians born in the same country
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JOC/EFR © November 2010 School of Mathematics and Statistics
Copyright information University of St Andrews, Scotland
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Millbourne, PA Prealgebra Tutor
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How can one generate all possible permutations of r elements out of n where r<n?
Hi PerlMonks,
It might be a silly question. But I donot find a simple perl code to solve this problem. I have an array like @a=qw/A B C D/; and I'm interested to get all possible permutations of say 3 elements out
of 4 in the array and their total numbers i.e. 4p3=4!/(4-3)!=24 permutations along with 4 other uniform permutations (i.e. AAA,BBB, CCC,DDD). Thus total number of permutations will be 24+4=28. I need
the permutations like ABC,ABD,BCD .. DDD and their total numbers (28). I have gone through some posts but nowhere I find the answer to this type of basic question. May I request the Perlmonks to
provide suggestions for this problem? I am sure that I have put a very silly question. Moreover, the code must work for permutations of 2 or 4 elements out of 4 i.e. 4p2 and 4p4 along with their
uniform permutations with little change in the code. | {"url":"http://www.perlmonks.org/index.pl?node_id=1020674","timestamp":"2014-04-17T03:54:35Z","content_type":null,"content_length":"25965","record_id":"<urn:uuid:65692ddc-cc38-44b6-8903-1f83b1aa119e>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00262-ip-10-147-4-33.ec2.internal.warc.gz"} |
How Big Is That Picture?
How Big Is That Picture?
Students gain a perspective of field of vision from a great height.
Grade Level: 4-5
Time Duration: Two or three class periods
Concepts Explored: Scale and structure
Suggested grouping: class demo with student helpers
• toilet paper or paper towel roll
• small flash light with a strong beam (a magnetite or halogen variety
• that will fit in the roll
• piece of opaque paper to cover bottom of the roll; punch a hole in the
• middle with a paper punch; attach the opaque cover to the bottom of
• the roll with a rubber band
• rope
• tape measure
• dark marker
• tape
• darkened room ideally a gym with a basketball hoop
1. Secure the flashlight in the paper roll with tape. Cover the open end with the opaque paper with the hole to mask the bottom of the roll.
2. Have students lay out the rope, measure off one foot sections and clearly mark them. If you are using a basketball hoop you will need about 25 feet of rope.
3. Tape the flashlight-roll assembly very securely to the rope.
4. During this time the other students can prepare a worksheet with three columns: height, diameter circumference.
5. Review the concept of diameter with the students.
6. Throw the selvage end of the rope through the hoop.
7. Turn on the flashlight and have the students observe the size of the circle of light when it is just above the floor.
8. Have a student pull the rope so the tube moves up one foot (to the first mark). Have another student measure and record the diameter of the circle and the height of the flashlight.
9. Continue to measure foot by foot to 15 feet. A pattern will emerge.
10. Teach the children to find the circumference of the circles. It is okay to approximate and multiply by 3 (instead of the full value of pi). Find the circumference of each and chart it.
11. Using proportion, find how big the circle would be at 100 feet 1,000 feet 10,000 feet and so forth.
12. Aircraft used for remote sensing fly at altitudes between 9,000 and 12,000 feet for low-altitude aircraft, 65,000 feet for high-altitude aircraft and satellites as high as 400 or more miles.
13. How many feet are in 400 miles? What would the diameter of the circle cast by the flashlight be at that height?
Have your students convert all the measures in this exercise to metric units.
Return to Top Down Project Menu | {"url":"http://geo.arc.nasa.gov/sge/jskiles/top-down/how_big_picture/how_big_is_that_picture.html","timestamp":"2014-04-18T13:39:14Z","content_type":null,"content_length":"3207","record_id":"<urn:uuid:b6bb79b3-ebad-403b-8c71-0cecebc16810>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00130-ip-10-147-4-33.ec2.internal.warc.gz"} |
Design for quantum computer proposed | EE Times
News & Analysis
Design for quantum computer proposed
Design for quantum computer proposed
SAN FRANCISCO Work at IBM Corp. on the theory and practice of quantum computing suggests that the industry may be closer to practical CPUs that could process information in the form of quantum
bits, or "qubits," rather than conventional binary bits.
The new thinking was discussed today (Dec. 11) in a plenary lecture at the IEEE International Electron Devices Meeting here. David DiVincenzo of IBM's T. J. Watson Research Center (Yorktown Heights,
N.Y.) surveyed the prospects for quantum computing, concluding that practical, solid-state devices may soon emerge to support the theoretical projections of vast computing power arising from this
"The principles and promise of quantum computers," DiVincenzo said, lie in the "requirements for the physical implementation of quantum computers in atomic physics, quantum optics, nuclear and
electron magnetic-resonance spectroscopes, superconducting electronics and quantum-dot physics."
Proving a point
DiVincenzo already has a track record in quantum theory. For instance, in 1995 he mathematically proved that two-qubit operations were sufficient to execute any quantum algorithm. Thus, engineers
need not design more than two-qubit physical devices to reap all the parallel-processing benefits of any future quantum algorithm.
In DiVincenzo's view, all quantum devices will require a new formal basis to express the kind of algorithmic parallelisms that could be realized with quantum computers. Unlike conventional encodings
of information, quantum devices allow the superposition of multiple discrete states simultaneously on the same qubit. Thus, multiplying two qubits together is equivalent to simultaneously multiplying
every possible string of values that a conventional computer register could hold.
That kind of operation demands a new mathematical formalism in order to craft effective quantum algorithms, said DiVincenzo. "Making bits that obey the quantum-mechanical principles [and] efficient
algorithms for some otherwise intractable problems, like prime factoring, becomes possible," he said.
DiVincenzo listed a sevenfold set of requirements for physical implementation of quantum computing. They are scalable well-defined qubits, resettable states, long superposition times, a universal set
of quantum gates, easy qubit measurements, easy qubit-to-digital conversions and easy qubit telecommunications.
Today, laboratory implementations of quantum devices concentrate on perhaps one or two of the seven requirements, he said, but to create commercial devices all seven must be met.
DiVincenzo summarized current attempts at building a quantum "transistor," most of which only have a superficial resemblance to today's silicon transistors. Using a voltage-controlled gate to switch
the qubit may be the only recognizable commonfeature.
The most striking new feature, according to DiVincenzo, is the fact that a single atomic element will embody the memory element in a microscopic domain, probably in the spin direction (either up or
down) of a single atom or electron. The quantum gate will most likely be switched with a voltage-controlled pulse at the gate, which superimposes a new state into the currently executing quantum
Aside from these generalizations, there is little similarity among the various current proposals for implementing quantum computers. Methodologies, so far, are based on basic physics rather than chip
technologies, DiVicenzo said, but advances in quantum dots embrace standard solid-state physics for integrated circuits. In particular, DiVincenzo described an ion-trap computer that holds qubits in
pairs of energy levels of ions held in a linear electromagnetic trap.
Other atomic-physics-based proposals use the position of atoms in a trap or lattice or the vibrational quanta of trapped electrons, ions or atoms as their qubits. The presence or absence of a
photon in an optical cavity has also been proposed as the basic qubit mechanism, and superconducting devices are being proposed that store qubits as charge or "flux."
On the solid-state chip side, impurities can introduce well-characterized discrete energy-level spectra directly onto silicon chips. Separately, researchers are at work on various quantum-dot
approaches, storing the qubit in the spin state, the orbital state or the charge state of quantum dots, according to DiVincenzo.
"Every implementation detail of a qubit from its initialization to its interaction with neighboring bits, the errors that it might be subjected to and its readout have to be thought out and
investigated from scratch," he said.
DiVincenzo has narrowed the overall development effort, however, by applying his seven criteria to the known theories today. The result is to single out, by elimination, a model for the solid-state
implementations of the future.
The right spin
In DiVincenzo's model, qubits are represented by the spins of individual electrons trapped in an array of quantum dots. Many proposals for building quantum-dot arrays are on record, but DiVincenzo's
model will reset the system by cooling the device to a predetermined temperature. Basic two-bit gates would be built by changing the height of an electrostatic barrier between quantum dots.
According to Divincenzo, the superposition time of such arrays has already been determined in preliminary laboratory experiments measuring spins in semiconductors, to be long enough to allow quantum
algorithms to execute. And other laboratory measurements, he said, also suggest that spin can be conveniently converted into directly measurable electron position. | {"url":"http://www.eetimes.com/document.asp?doc_id=1142758","timestamp":"2014-04-17T07:00:38Z","content_type":null,"content_length":"126273","record_id":"<urn:uuid:09759d6f-26fb-4756-a14a-21eb44266622>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00511-ip-10-147-4-33.ec2.internal.warc.gz"} |
Integration of the product of pdf & cdf of normal distribution
up vote 2 down vote favorite
Denote the pdf of normal distribution as $\phi(x)$ and cdf as $\Phi(x)$. Does anyone know how to calculate $\int \phi(x) \Phi(\frac{x -b}{a}) dx$? Notice that when $a = 1$ and $b = 0$ the answer is
$1/2\Phi(x)^2$. Thank you!
probability-distributions integration
Is $a>0$? In this case, write the integral, do a substitution on the integral which defines $\Phi((x-b)/a)$ in order to get a bound $x$ in this integral. Then integrate by parts, and complete the
squares. – Davide Giraudo Jul 6 '12 at 14:07
Thank you very much for the answer. Yes, a>0, but I am not clear about what you mean.. Can you explain that in details? – user9836 Jul 8 '12 at 7:10
In fact I made a miscomputation. I used below a different approach. – Davide Giraudo Jul 9 '12 at 9:12
add comment
closed as off-topic by Did, Yemon Choi, David White, David Roberts, Andres Caicedo Jul 12 '13 at 5:52
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Yemon Choi, David White, David Roberts, Andres Caicedo
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protected by Community♦ Jun 27 '13 at 5:00
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3 Answers
active oldest votes
We have $\phi(x)=\frac 1{\sqrt{2\pi}}\exp\left(-\frac{†^2}2\right)$ and $\Phi(x)=\int_{-\infty}^x\phi(t)dt$. We try to compute $$ I(a,b):=\int\phi(x)\Phi\left(\frac{x-b}a\right)dx.$$ Using the
dominated convergence theorem, we are allowed to take the derivative with respect to $b$ inside the integral. We have $$\partial_bI(a,b)=\int\phi(x)\left(-\frac 1a\right)\phi\left(\frac{x-b}a\
right)dx$$ and 2\pi\phi(x)\phi\left(\frac{x-b}a\right)&=\exp\left(-\frac 12\left(x^2+\frac{x^2}{a^2}-2\frac{bx}{a^2}+\frac{b^2}{a^2}\right)\right)\\\ &=\exp\left(-\frac 12\frac{a^2+1}{a^2}\
up left(x^2-2\frac b{a^2+1}x+\frac{b^2}{a^2+1}\right)\right)\\\ &=\exp\left(-\frac 12\frac{a^2+1}{a^2}\left(x-\frac b{a^2+1}\right)^2-\frac 12\frac{a^2+1}{a^2}\left(\frac{b^2}{a^2+1}-\frac{b^2}
vote 2 {(a^2+1)^2}\right)\right)\\\ &=\exp\left(-\frac 12\frac{a^2+1}{a^2}\left(x-\frac b{a^2+1}\right)^2\right)\exp\left(-\frac{b^2}{2a^2}\frac{a^2+1-1}{a^2+1}\right)\\\ &=\exp\left(-\frac 12\frac{a
down ^2+1}{a^2}\left(x-\frac b{a^2+1}\right)^2\right)\exp\left(-\frac{b^2}{2(a^2+1)}\right). Integrating with respect to $x$, we get that $$\partial_b I(a,b)=-\frac 1{\sqrt{a^2+1}}\phi\left(\frac b
vote {\sqrt{a^2+1}}\right).$$ Since $\lim_{b\to +\infty}I(a,b)=0$, we have I(a,b)&=\int_b^{+\infty}\frac 1{\sqrt{a^2+1}}\phi\left(\frac s{\sqrt{a^2+1}}\right)ds\\\ &=\int_{b\sqrt{a^2+1}}^{+\infty}\
phi(t)dt. This can be expressed with the traditional erf function.
Thank you very much for the answer but it seems that the "Integrating with respect to x" part is not correct. Can you double-check or add some more explanation on that part? Thank you! –
user9836 Jul 10 '12 at 1:48
The result of the integration step should contains some erf function of x instead of $\phi$ in the result, and the exp(-b^2/2/(a^2+1)) part should not be omitted. – user9836 Jul 10 '12 at
I integrate on the whole real line with respect to $x$, and I do the substitution $\frac{a^2+1}{a^2}\left(x-\frac b{a^2+1}\right)^2=t^2$. I didn't omit the exponential term, since I wrote it
using $\phi$. – Davide Giraudo Jul 10 '12 at 9:35
Thank you but I still think that you miss something. I think the result is not correct as $\int_{b\sqrt{a^2+1}}^{+\infty}\phi(t)dt$ is a constant that does not depends on $x$ or $t$. Would
you like to double check your result? Thank you very much! – user9836 Jul 12 '12 at 6:10
The only fixed parameters are $a$ and $b$, we integrate with respect to $x$. Hence it's normal that $x$ doesn't appears in the final result. – Davide Giraudo Jul 12 '12 at 9:53
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This might be a setting where relying on the probabilistic meaning of the functions $\phi$ and $\Phi$ saves ink and tedious computations.
Let $X$ and $Y$ denote standard normal random variables. Then $\int\limits_{-\infty}^\infty u(x)\phi(x)\mathrm dx=E(u(X))$ for every suitable function $u$ and $\Phi(x)=P(Y\leqslant x)$ for
every real number $x$. Using this for the function $u:x\mapsto\Phi((x-b)/a)$ and assuming furthermore that $X$ and $Y$ are independent, one sees that the integral to be computed is $$ (\ast)=
up vote E(\Phi((X-a)/b))=P(Y\leqslant(X-b)/a). $$ Thus, $$ (\ast)=P(Z\geqslant b), $$ where $Z=X-aY$ (this step uses the fact that $a\gt0$). Now, the random variable $Z$ is normal as a linear
2 down combination of independent gaussian random variables, with mean $0$ and variance $1+a^2$, hence $Z=\sqrt{a^2+1}\cdot T$, where $T$ is standard normal. Thus, $$ (\ast)=P(T\geqslant b/\sqrt{a^
vote 2+1})=1-\Phi\left(b/\sqrt{a^2+1}\right). $$ Likewise, if $a\lt0$, then $(\ast)=\Phi\left(b/\sqrt{a^2+1}\right).$
In particular, if $b=0$ then, for every $a\ne0$, $(\ast)=\frac12$.
1 As already explained. – Did Jul 1 '13 at 7:36
add comment
1. The last equation should be integral from $b/\sqrt{a^2+1}$
2. In I(a, b), a is supposed to be positive. When $a<0$, the answer will be $\int_{-\infty}^{b/\sqrt{a^2+1}} \phi(t) dt$
up vote 0 down vote
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About This Tool
The online Relative Standard Deviation Calculator is used to calculate the relative standard deviation (RSD) of a set of numbers.
Relative Standard Deviation
In probability theory and statistics, the relative standard deviation (RSD or %RSD) is the absolute value of the coefficient of variation. It is often expressed as a percentage. It is useful for
comparing the uncertainty between different measurements of varying absolute magnitude.
The following is the relative standard deviation calculation formula:
s = sample standard deviation
RSD = relative standard deviation
x[1], ..., x[N] = the sample data set
x̄ = mean value of the sample data set
N = size of the sample data set | {"url":"http://www.miniwebtool.com/relative-standard-deviation-calculator/","timestamp":"2014-04-18T23:21:59Z","content_type":null,"content_length":"5780","record_id":"<urn:uuid:c27f6e3e-1673-472f-af2d-5dd64e96e926>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00091-ip-10-147-4-33.ec2.internal.warc.gz"} |
convergent/divergent series
March 7th 2008, 02:48 PM #1
convergent/divergent series
How to check whether $\sum_{n=1}^{\infty}{n\sin}\frac{1}{n^3}$ is convergent/divergent? I could show that $\sum_{n=1}^{\infty}{n\sin}\frac{1}{n^2}$ is divergent, but I don't know how to start
with the first one... Won't someone drop a hint, please?
Input appreciated.
Since $\sin(x)\approx{x}$ when $x\rightarrow{0}$
we have $n\cdot{\sin\left(\frac{1}{n^3}\right)}\approx{\fra c{1}{n^2}}$ when $n\rightarrow{+\infty}$ ( $\lim_{n\rightarrow{+\infty}}\frac{n\cdot{\sin\left (\frac{1}{n^3}\right)}}{\frac{1}{n^2}}=
1$ )
But: $\sum_{n=1}^{\infty}{\frac{1}{n^2}}$ does converge, so, by the Limit Comparison Test, $\sum_{n=1}^{\infty}{n\cdot{\sin\left(\frac{1}{n^3} \right)}}$ converges
Again: $\lim_{n\rightarrow{+\infty}}\frac{n\cdot{\sin\left (\frac{1}{n^2}\right)}}{\frac{1}{n}}=1$
and $\sum_{n=1}^{\infty}{\frac{1}{n}}$ diverges, thus: $\sum_{n=1}^{\infty}{n\cdot{\sin\left(\frac{1}{n^2} \right)}}$ diverges
Great! $\alpha$.
$<br /> \sum_{n=1}^{\infty}{n\sin}\frac{1}{n^\alpha}<br />$
Last edited by disclaimer; March 8th 2008 at 01:37 AM.
March 7th 2008, 03:37 PM #2
March 8th 2008, 12:07 AM #3 | {"url":"http://mathhelpforum.com/calculus/30279-convergent-divergent-series.html","timestamp":"2014-04-18T16:52:58Z","content_type":null,"content_length":"38834","record_id":"<urn:uuid:7b3becaa-5d5c-42a5-a1f3-00d9a8bdd5bc>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00501-ip-10-147-4-33.ec2.internal.warc.gz"} |
Binomial Distributions, Geometric Distributions, and Sampling Distributions Multiple Choice Practice Problems for AP Statistics
Duane C. Hinders
— McGraw-Hill Professional
Updated on Feb 4, 2011
Review the following concepts if necessary:
1. A binomial event has n = 60 trials. The probability of success on each trial is 0.4. Let X be the count of successes of the event during the 60 trials. What are μ[x] and σ[x]?
a. 24, 3.79
b. 24, 14.4
c. 4.90, 3.79
d. 4.90, 14.4
e. 2.4, 3.79
2. Consider repeated trials of a binomial random variable. Suppose the probability of the first success occurring on the second trial is 0.25. What is the probability of success on the first trial?
a. 1/4
b. 1
c. 1/2
d. 1/8
e. 3/16
3. To use a normal approximation to the binomial, which of the following does not have to be true?
a. np ≥ 5, n(1 –p)≥ 5 (or: np ≥ 10, n (1–p) ≥10).
b. The individual trials must be independent.
c. The sample size in the problem must be too large to permit doing the problem on a calculator.
d. For the binomial, the population size must be at least 10 times as large as the sample size.
e. All of the above are true.
4. You form a distribution of the means of all samples of size 9 drawn from an infinite population that is skewed to the left (like the scores on an easy Stats quiz!). The population from which the
samples are drawn has a mean of 50 and a standard deviation of 12. Which one of the following statements is true of this distribution?
5. A 12-sided die has faces numbered from 1–12. Assuming the die is fair (that is, each face is equally likely to appear each time), which of the following would give the exact probability of
getting at least 10 3s out of 50 rolls?
6. In a large population, 55% of the people get a physical examination at least once every two years. An SRS of 100 people are interviewed and the sample proportion is computed. The mean and
standard deviation of the sampling distribution of the sample proportion are
a. 55, 4.97
b. 0.55, 0.002
c. 55, 2
d. 0.55, 0.0497
e. The standard deviation cannot be determined from the given information.
7. Which of the following best describes the sampling distribution of a sample mean?
a. It is the distribution of all possible sample means of a given size.
b. It is the particular distribution in which μ[] = μ and σ[] = σ.
c. It is a graphical representation of the means of all possible samples.
d. It is the distribution of all possible sample means from a given population.
e. It is the probability distribution for each possible sample size.
8. Which of the following is not a common characteristic of binomial and geometric experiments?
a. There are exactly two possible outcomes: success or failure.
b. There is a random variable X that counts the number of successes.
c. Each trial is independent (knowledge about what has happened on previous trials gives you no information about the current trial).
d. The probability of success stays the same from trial to trial.
e. P(success) + P(failure) = 1.
9. A school survey of students concerning which band to hire for the next school dance shows 70% of students in favor of hiring The Greasy Slugs. What is the approximate probability that, in a
random sample of 200 students, at least 150 will favor hiring The Greasy Slugs?
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From 5 Steps to a 5 AP Statistics. Copyright © 2010 by The McGraw-Hill Companies. All Rights Reserved.
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Three Claims Function Carnival Makes About Online Math Education
January 27th, 2014 by Dan Meyer
Today Desmos is releasing Function Carnival, an online math happytime we spent several months developing in collaboration with Christopher Danielson. Christopher and I drafted an announcement over at
Desmos which summarizes some research on function misconceptions and details our efforts at addressing them. I hope you'll read it but I don't want to recap it here.
Instead, I'd like to be explicit about three claims we're making about online math education with Function Carnival.
1. We can ask students to do lots more than fill in blanks and select from multiple choices.
Currently, students select from a very limited buffet line of experiences when they try to learn math online. They watch videos. They answer questions about what they watched in the videos. If the
answer is a real number, they're asked to fill in a blank. If the answer is less structured than a real number, we often turn to multiple choice items. If the answer is something even less
structured, something like an argument or a conjecture … well … students don't really do those kinds of things when they learn math online, do they?
With Function Carnival, we ask students to graph something they see, to draw a graph by clicking with their mouse or tapping with their finger.
We also ask students to make arguments about incorrect graphs.
I'd like to know another online math curriculum that assigns students the tasks of drawing graphs and arguing about them. I'm sure it exists. I'm sure it isn't common.
2. We can give students more useful feedback than "right/wrong" with structured hints.
Currently, students submit an answer and they're told if it's right or wrong. If it's wrong, they're given an algorithmically generated hint (the computer recognizes you probably got your answer by
multiplying by a fraction instead of by its reciprocal and suggests you check that) or they're shown one step at a time of a worked example ("Here's the first step for solving a proportion. Do you
want another?").
This is fine to a certain extent. The answers to many mathematical questions are either right or wrong and worked examples can be helpful. But a lot of math questions have many correct answers with
many ways to find those answers and many better ways to help students with wrong answers than by showing them steps from a worked example.
For example, with Function Carnival, when students draw an incorrect graph, we don't tell them they're right or wrong, though that'd be pretty simple. Instead, we echo their graph back at them. We
bring in a second cannon man that floats along with their graph and they watch the difference between their cannon man and the target cannon man. Echoing. (Or "recursive feedback" to use Okita and
Schwartz's term.)
When I taught with Function Carnival in two San Jose classrooms, the result was students who would iterate and refine their graphs and often experience useful realizations along the way that made
future graphs easier to draw.
3. We can give teachers better feedback than columns filled with percentages and colors.
Our goal here isn't to distill student learning into percentages and colors but to empower teachers with good data that help them remediate student misconceptions during class and orchestrate
productive mathematical discussions at the end of class. So we take in all these student graphs and instead of calculating a best-fit score and allowing teachers to sort it, we built filters for
common misconceptions. We can quickly show a teacher which students evoke those misconceptions about function graphs and then suggest conversation starters.
A bonus claim to play us out:
4. This stuff is really hard to do well.
Maybe capturing 50% the quality of our best brick-and-mortar classrooms at 25% the cost and offering it to 10,000% more people will win the day. Before we reach that point, though, let's put together
some existence proofs of online math activities that capture more quality, if also at greater cost. Let's run hard and bury a shoulder in the mushy boundary of what we call online math education,
then back up a few feet and explore the territory we just revealed. Function Carnival is our contribution today.
23 Responses to “Three Claims Function Carnival Makes About Online Math Education”
1. on 27 Jan 2014 at 12:29 pm1
Cool stuff.
1. Nice combination of graphs and open answer features. Some of it I know is in DME but not in the way it is combined here. There seems to be more overlap with the -not online- features that
modelling software like Coach have. Will Desmos be looking at other examples than a man and cannon, because there are some examples with trajectories and quadratics of course.
2. Like the take on feedback. “though that’d be pretty simple.”, no it would be hard, I think, because WHEN is a graph correct or incorrect. Maybe even harder than mirroring. Which isn’t to say
that both aren’t useful. Having played with it, I do wonder whether having multiple men when there are more function values for a given x really reveals enough. It will reveal that a student has
done something wrong because, well, I have one man and the feedback shows several men, but of course the important question is why.
3. This is very powerful. How are the misconceptions assigned?
4. Yes, totally agree. Good design costs a lot of time and money.
2. on 27 Jan 2014 at 12:50 pm2
You’re my hero. My one thought is that with exercises like this physics and math are getting harder and harder to keep separate (which might be a good thing).
3. on 27 Jan 2014 at 12:52 pm3
Nathan Amrine
Brilliant stuff.
4. on 27 Jan 2014 at 1:50 pm4 Joe
Two related design decisions that I really like:
* Height in the graph corresponds directly (and is right next to) height in the animation.
* Time in the graph corresponds to the moving time bar on the bottom.
An excellent way to scaffold out the task of scaling.
5. on 27 Jan 2014 at 3:38 pm5
Mark James
Really awesome – and by that I mean the module itself as well as the well-articulated and justified claims. After doing a walkthrough the Carnival myself today, I immediately emailed the link to
my entire HS math dept (14 teachers). Then, after dismissal I went and physically dragged 3 colleagues into my room so I could show them how amazing it is (as I had done with Penny Circle, et al
– I’m still chipping away at the “I’m too busy to surf the MTBoS” mindset in my building). Looking forward to doing Function Carnival this Friday (if the Chromebooks arrive before then as
promised) with my classes. I’ll let you know how it goes. Thanks Dan, Christopher, and Desmos!
6. on 27 Jan 2014 at 5:11 pm6
l hodge
Very nice. Very clean and clear.
Interesting to note that multiple jumpers appear if you draw a graph that indicates the jumper needs to be in more than one place at a particular time. Relevant for functions, intersections of
two functions, parallel lines, etc.
Would be great if it were possible to tweak some of the features or questions. In particular, I would like to be able to adjust the equation controlling the projectile(s).
7. on 27 Jan 2014 at 5:59 pm7
Chris Painter
I am vastly behind in the technological realm in comparison; however, I wonder if there is a way to start pairing animations such as these along with short films as there are on the graphing
stories webpage. I imagine an animate figure being imposed on the video footage itself. I am sure there is a way. Just imagine how much this would free up time during the construction of such
problems (assuming of course that the animation overlay isn’t a nightmare to pull off).
8. I can’t wait to try this myself. All the math education I ever got has come through following this blog for the last few years! May it go on forever!
9. Have you seen the PARCC sample questions? It’s got some ingredients of what you’re doing in there. (I’m guessing Smarter Balanced is doing something similar, but I haven’t been keeping up to
It’s also got some super-frustrating parts of the interface, but that’s a different ballpark of worry.
10. This is a great tool. I love the way that it shows the reality of what a student’s graph shows. The fact that both the green “reality of the story” and the blue “interpretation of the story” are
both visible on the playback is a wonderful gift that helps the use to tweak their interpretation.
I found myself getting the general shape of the graph, but my blue item reached the max height sooner than the real green item. I had to continue to tweak my graph to get it as “correct” as I
wanted to.
My frustration was with my own inadequacy in being able to physically manipulate the graph I drew. I assume that this would improve with practice. Perhaps if I had a touch screen, it would be
easier. I was doing it using my MacBook Pro touch pad.
This site is a great contribution to moving math education into the 21st century. Thanks and kudos to you and Christopher for creating it and also for sharing it freely.
11. Very nice !
I have one suggestion:
When using the line segment option, it would be easier for the user to click on the newly positioned dot and the line from the previous dot to be drawn automatically.
Keep it up!
12. on 28 Jan 2014 at 10:41 am12
Bryan Bazilauskas (@nobackswing)
this is big time awesome. stuff like this is a huge part of the future of math education.
13. @Jason, just checked them out on your recommendation. Not a lot that dazzles me. They’re duplicating existing hard copy tools online (the protractor, the ruler), they have students clicking
coordinates on a grid and dragging items around (which I can’t get too worked up over), and then there’s a pile of fairly ill-considered interactions. Like making a kid click 48 separate boxes in
a grid to indicate an answer. Or the dropdown menus to create the parts of an equation.
14. @Dan: The equation editor is especially painful. I challenge anyone to type in the geometry proof they are asking for without tearing their hair out and/or having the browser crash and lose all
their data.
The interesting thing is, philosophically, it makes all the same points as your #1, yet somehow it comes off worse.
15. Jason:
The interesting thing is, philosophically, it makes all the same points as your #1, yet somehow it comes off worse.
I see how it aims in the same direction. It’s an instance where the tools get in the way of thought, where the same task (constructing a geometric argument) would be easier on paper than
digitally. It doesn’t enhance thought. It’s like running an already-pretty-tough race with leg weights.
16. Jason, I tried them out too. Ouch! I would not agree that the tasks aim in the same direction as Dan’s and Christopher’s tasks. I answered all of the items and was told that 4 of the 6 did not
have answers. This was because I evidently didn’t know the nuances, the correct format, of the system. It was not intuitive at all. It also did not include the feedback loop that the blue and
green icons supplied in Function
Carnival. Granted that the PARCC tasks were not necessarily functions. Considering only functions with an independent and dependent variable allowed allowed Function Carnival to be more focused.
Thanks, though, for sending the link. I am hoping that there is a major overhaul before this becomes the real assessment!
17. on 29 Jan 2014 at 8:32 am17
Holly Brie Thomas
Gorgeous! It’s wonderful to see someone actually designing lessons (and software) that adds value by using technology, instead of essentially duplicating a paper-and-pen experience digitally.
I checked out the PARCC sample items yesterday and had the exact same reaction as Dan’s responses to Jason — they are basically old-school tasks with digital data entry, and with a clunky
interface at that — Who clicks 48 separate boxes adjacent boxes when they want to select them in a spreadsheet? Why must I structure my work as a paragraph proof because the entry box is
text-based & linear? Why is there no opportunity to SKETCH anything anywhere? Why didn’t the interface CONNECT the plotted points in the “graph 3 vertices of a rectangle and find the fourth”,
which would at least add _some_ value beyond pen-and-paper graphing. And don’t get me started on the quality of the tasks as actual assessment items capable of providing meaningful, nuanced
feedback about what a student can actually do…
Dan, if you have ANY ability to get what you are doing in front of the PARCC/SBAC folks, please do! Tasks like Function Carnival beautifully illustrate a paradigm shift from using technology to
duplicate “the way we’ve always done it”, to a using technology as a tool to give students a deeper experience that takes them further into the math. The teacher feedback with miniature versions
of the graphs students created is wonderful — much more powerful than any kind of alpha-numeric data would be.
This reminds me of a methods course I took focused on teaching functions using graphing calculators. I’m just old enough that graphing functions in high school was all pencil & paper, and there
was still resistence to letting students use standard calculators in lower-level courses like Algebra. Consequently, it took a while to graph a quadratic function because you had to calculate 5-7
points and then plot them, so exposure to parabolic functions was mostly limited to those that could be expressed as x^2 + bx + c or -x^2 + bx + c, and which were located relatively close to the
origin. However, with graphing calculators, the focus could shift from calculating and plotting individual functions to examining permutations within a class of functions (i.e. viewing a
quadratic function through the lens of standard form, vertex form or factored form, as well as looking at how changes to variables impact the graph), and easily comparing classes of functions.
The calculators could have been used to look at old-school functions faster (HOW the material was presented), but the real power actually came from changing WHAT material was presented — the
automaticity of graphing the calculators afforded allowed for deeper exploration of functions (and hopefully deeper understanding, as well).
Since so much of what is happening in education right now is being driven by high-stakes assessment, it would be amazing if the folks designing the high-stakes assessments were thinking along the
lines of Function Carnival, rather than drag-and-drop, point-and-click, and type-the-number-in-the-box, as they currently (sadly) appear to have the ability to reach the widest audience.
18. on 29 Jan 2014 at 12:44 pm18
That WAS fun! I was mesmerized for a good ten minutes, and embarrassed by my first attempt at the cars activity. I should really know better ;). After fixing my mistake, it was very satisfying to
see the “feedback” problem that I now knew how to fix! If go this excited over “being wrong and then right” and only imagine how my students would feel!
19. My 8th grade math intervention class just did this, and these thoughts need to get written while they’re fresh:
1.) The user interface is clean and easy-to-use, even on iPads (though it automatically selected blocks of text when students tapped-and-held too long).
2.) I gave easy corrections while the class was working, viewing individual graphs (projected on the wall) and offering feedback in real time. Many who were stuck would just watch the screen as I
critiqued others, then apply my advice to their own.
3.) The Carnival supported students working at their own pace. Three finished the whole 6 steps (nearly perfectly) before 1/4 of the class finished cannon man.
4.) That class–and probably all my classes–need remediation on “the graph of a line is a series of points all close together”. I had the same conversation 15 times one-on-one.
5.) More students were engaged with this activity than during my lesson the hour previous.
Well done.
20. Spot on! This is actually VERY similar to how motion is controlled in 3D animation software (except in 3 separate axes: x,y and z – which kind of hurts your brain in a good way!)
The feedback tool is exceptionally well thought out – I like how you can move the play-head and place a dot at key locations etc. all the time seeing both the video and your own version.
As Chris Painter mentioned: using this tool in conjunction with other videos would add another level of challenge. Could pupils upload videos of their own and then use the tool to graph them?
Maybe that’s a big ask? : )
21. Kudos! I like how the image of Cannon Man appears when you hover over the graph. Even before committing to any answer, I get instant feedback. Also love the diversity of drawing tools – line,
dots, freehand.
How many different “filters for common misconceptions” did you put in for each puzzle? And what kind of AI or sorting algorithms are you using to detect misconceptions?
22. on 24 Feb 2014 at 8:54 pm22
Great activity! My students were pretty into how you can see the man or car described by your graph laid over the original. They were a little bummed because they couldn’t figure out how to save
orreturn to their work (they started after they took a quiz, and had about 30 min in class to work). Is this possible or do they need to complete the exercise in one sitting?
23. Hi Zachary, great critique. We’re aware of the issue and will be resolving it in future updates. Thanks for taking the time. | {"url":"http://blog.mrmeyer.com/2014/three-claims-function-carnival-makes-about-online-math-education/","timestamp":"2014-04-17T04:13:04Z","content_type":null,"content_length":"59076","record_id":"<urn:uuid:4345e199-ebb4-4fb1-b209-216fc798489b>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00471-ip-10-147-4-33.ec2.internal.warc.gz"} |
On the likelihood of having all three of your kids share a birthday
In his Guardian column yesterday, Dr Ben "Bad Science" Goldacre schooled us (and the
Daily Express
) on some basic statistical numeracy:
Often one data point isn't enough to spot a pattern, or even to say that an event is interesting and exceptional, because numbers are all about context and constraints. At one end there are the
simple examples. "Mum beats odds of 50 million-to-one to have 3 babies on same date" is the headline for the Daily Express on Thursday. If that phenomenon was really so unlikely, then since there
are less than a million births a year in the UK, this would genuinely be a very rare event.
Their number is calculated as 365 x 365 x 365 = 48,627,125. But in reality, of course, it's out by an order of magnitude: one in 50 million are the odds of someone having 3 siblings sharing one
particular prespecified birth date that the editors of the Daily Express sealed in an envelope and gave to a lawyer 50 years ago. In reality there is no constraint on which day the first baby
gets born on, so after that, the odds of two more babies sharing that birthday are 365×365=133,225. And they might even be a bit lower, if you two feel friskier in winter and have more babies in
the autumn, for example.
Then there is the context. Living on your street, hanging out with the people from work, it's easy to miss the sheer scale of humanity on the planet. In England and Wales there were 725,440
births last year. From the ONS Statistical Bulletin "Who is having babies" 14% were third births, and another 9% were fourth or subsequent births. So there are 102,000 third children born a year,
167,000 third or more-th children, and if we include the rest of the Kingdom there are even more, so on average, three shared birthdays will happen once or twice a year in the UK (although to be
written about in the Express it would need to be a birth within a marriage, making 55,000 chances a year, or once every two years)
Guns don't kill people, puppies do
59 Responses to “On the likelihood of having all three of your kids share a birthday”
1. Lady Strathconn says:
My son was born on my birthday last year. 9 days early. I have friends who are 5 years and 5 hours apart and a student whose brother is 2 year and 1 hour younger.
The birthday problem says chances are 99% that if 57 or more people are together 2 of them will share a birthday. When I started my job there were 45 people working there. 6 of us shared
birthdays with one other person (6 people = 3 birthdays). Also, we had a disproportionate number of birthdays in Sept., Mar., and May. Even through changes in staffing those are our 3 biggest
2. Shelby Davis says:
My second sister’s (early September) due date was the same as my first sister’s birthday, but she ended up four days late. And yep, I never thought about it before reading this post, but my
parents’ anniv is indeed at the very end of December. (and come to think of it, my brother’s bday is the start of October. Early-to-bed winters, indeed. And I, as the sole Spring birth, could be
explained by being my parents’ first, and therefore originating when they were newlyweds and more frisk–gah, statistics giving me more inferences on parents’ sex lives than I ever wanted!! Make
it stop!
3. Anonymous says:
couldn’t the odds be tipped in their favor if they consciously became pregnant on the same night as the previous one, making the likelihood of the next baby being born on the same day drastically
4. Anonymous says:
My birthday is August 18, my sisters is August 20th, and my brothers is August 22. What are the coincidences of that? haha
5. Anonymous says:
I share a birthday with my two-year-younger sister, and that’s considered surprising to most people, even though the odds really aren’t all that bad.
6. takeshi says:
The odds of a person being born on any given day is approximately 1 in 365.25. It doesn’t matter how many siblings you have who share the same birthday.
When we’re talking about one sexually active couple, it’s never going to be completely up to chance. It’s not the same as asking, “what are the odds of 3 siblings being picked by a blindfolded
stranger from a group of 1095.75 people?”
The couple’s sexual activity (or lack thereof) is obviously a factor. The probability of three siblings being born on the same date is not the same as that of three randomly selected people. If a
couple only has sex once a year, in February, the odds of three children being born in February are infinitesimal. However, the odds of their three children being born on the same day in November
increase significantly.
7. Aurini says:
Something that should be pointed out to more people is that winning the lottery (at 1 in 14 million odds) is an event so improbable that it effectively *never* happens.
Compare, say, to your probability assessment of aliens blowing up the sun (I take a rather cynical view of the Drake Equation, so personally I’d probably rate this one lower than winning the
lottery)? What about the probability of the LHC destroying the earth? I’m sure that’s higher than 0.0000000714.
When I point this out to people they always insist that “Well, derp, my coworker’s cousin once won the lottery…” Selection Bias! Of course you’ll ‘know’ somebody who already won, but out of the
two- or three-hundred people you’re actually aquainted with already, none of them are ever going to win.
I trust my life to worse odds every time I drive on the freeway.
8. rebdav says:
or cheat and have triplets
9. Anonymous says:
My mom had both of my brothers on the same day, but not me. In both cases she started labouring the day prior, which is my grandfather’s birthday. I was born on a cousin’s birthday, which is also
our great-grandfather’s birthday. Needless to say, there’s a lot of weird birthday sharing in our family.
10. Anonymous says:
My parents have children born in every month. Two were born in the same year- one in January the next in December. Two share a birthday. Two were born a year and a day apart. My 7th sister was
born on my dad’s birthday- a nice gift- and he was born on his aunt’s. That aunt considered my Dad a gift to her and my sister a gift from him.
We have several birthdays on holidays and now with spouses and grandkids the doubled and tripled up birthdays are growing every year.
11. Michael_GR says:
Since there is quite a gap between their kids’ birth years, and assuming they did not abstain from sex in the intervening years, it’s safe to say they were using contraceptives. Although they
claim to not intentionally set out to have all their kids’ births at the same date, they must have planned for about the same season, at least. This brings the chances down even further.
12. Anonymous says:
I have a friend who had her two children, on her birthday, and my daughter, niece, friend, and cousin’s adopted daughter all share the same b-day.
13. morcheeba says:
I think “seasonal friskiness” could be narrowed down even further… my sister and I shared the same due date, which was 9 months after a certain major holiday. Biology conspired to put 2 weeks
between us, which is a tighter distribution than the norm of 13 weeks.
14. Miss Cellania says:
My former boss had four children, each several years apart, and three had the same birthday. We all joked that the date must be nine months after their wedding anniversary.
15. Anonymous says:
I am not a mathematician, but it seems that (a) you cannot apply 365x365x365 to this ‘coincidence’, since the parents conceived on a certain date and that would place the normal birth
approximately 270 days away. It’s feasible that birth may happen the day after conception(not resultant in a healthy baby, obviously, nor a date that the parents might celebrate year after year)
but could it also happen 365 days from conception? Unless there is absolutely no control over the results, pure math doesn’t work here; and (b), labor can always be induced.
16. Anonymous says:
Ah, Bull. You can elect to have a child born just about any date you want. Doctors induce labor all the time, and if it’s going to be a C-section there is even less of an issue. It’s not too hard
to arrange for full term to be about the date you want. That being said, you’d have to be a pretty sick bastard to want to do that.
17. Wirelizard says:
“although to be written about in the Express it would need to be a birth within a marriage”
Excellent subtle snark, and an interesting stats lesson. Hey, I was both amused and educated! Excellent!
18. Sally says:
Well, if you or ANYONE could figure THIS amazing coincidence of same birthday’s I would be thankful, and pass this statistical chance fact to my little ones. We have 5 children. One was
biological now age 16. Born August 2nd. Our next child is adopted through Foster care (reminding that we have no control over a baby being placed to us by the Agency and cleared for adoption) She
is now 3 and her birthday is August 25th. Our next child also adopted through Foster care is ALSO 3 yrs old, and his birthday is THREE days later on August 28th! Now it gets interesting….we have
our twin babies who are now 9 months old. A Boy and Girl…ALSO foster care and pending adoptions to us soon! THEIR BIRTHDAY’s are the same day as our Daughter AUGUST 25th!!!! HOW CAN THAT BE EVEN
CALCULATED ON THE CHANCES????? If anyone could figure it out and let me know -I would be very thankful!
19. Courtney says:
My Nana and her brother (fraternal twins) and her older (fraternal) twin sisters were born on the same day, 5 years apart.
20. Anonymous says:
And what’s the likelihood of this being posted on my birthday?
21. Nylund says:
Entirely agree. Before I got too far in the post, I was already thinking, “but the first date doesn’t matter!” which already makes it 365×365 AS AN UPPER BOUND (or should that be lower bound?).
All sorts of behavioral and seasonal effects should be taken into account as well that could significantly change the probability.
Plus, there is the story of my great aunt. Her first two children were both born on Christmas day. For her third child, she did everything she could regarding every scientific, pseudo-scientific,
and folkish thing she could think of…have sex, run up and down stairs, etc. to have that third baby on Christmas as well! (and it worked).
And, as you point out, when events happen in very large numbers, rare events will happen. Maybe winning the lottery is a 1 in a million chance, but get 10′s of millions of people together and its
going to happen to some of them (and it does!).
22. theturtle says:
Dear humans:
just please stop making more humans, we have far more than we need already. When we need more, we’ll let you know.
23. Anonymous says:
My sisters and I wasn’t born on the same day, but we all were born on the 23rd of the month we were born, my sister had her son on her birthday so another 23rd, my maternal Grandmother was also
23rd and I have a second cousing born on the 23rd of a month too…
24. markaa says:
I have 2 daughters born on the same day 7 years apart. The 2nd, our 7th born on the 7th of the 7th on our other daughter’s 7th birthday. Also 2 sons born within a few hours of being exactly 14
year apart.
25. Marchhare says:
So, then what is the probability of two people being born on the same day? It’s not 1/365*365. This model suggests that the probability of having three people born on the same day isn’t 1/3(365).
□ japroach says:
huh, two would be 1/365.
□ sapere_aude says:
The probability that two randomly selected people will share the same birthday is approximately 1/365.25 (the .25 is due to Leap Year, and the “approximately” is due to the possibility that
birthdays may not be randomly distributed across the population). The probability that three randomly selected people will share the same birthday is approximately (1/365.25)*(1/365.25).
But the probability that, in a group of size n, at least two people in that group will share the same birthday is 1-((365!)/((365^n)*(356-n))). Yeah, that’s a bit hard to follow; but what it
basically means is that in a group of at least 23 people, there is a 50% chance that at least two of them will share the same birthday; and in a group of at least 57 people, there is more
than a 99% chance that at least two of them will share the same birthday.
26. Anonymous says:
The other commentators are correct, biology has made women tend to get pregnant the same time of year and be due the same time of year (for themselves, not necessarily in correlation to other
women). In a woman’s set of progeny most of her offspring will be born around the same time of year. Some women are better timed, like caribou.
27. Anonymous says:
Also, consider that the female reproductive cycle is in fact, cyclical – therefore one would naturally expect a higher degree of periodicity within births. Of course, most armchair statisticians
ignore these things.
□ Anonymous says:
Well, over a sufficiently large population (of women), I guess “fertility days” shall be uniformly distributed (law of large numbers, or something like that)
For only one woman, I’m not sure that the fertility is predictible one year after. I heard menstruations may often be irregular.
28. Anonymous says:
Good remark from Nylund; actually 365×365 is not an upper bound, but an uninformative prior. The true probability distribution should take account of strong biases (such as, here, the mother
aiming at a specific date after the birth of her first child). Which makes the true probability sharply peaked around the “target” date. And subsequently makes the odds much less impressive :)
29. Anonymous says:
1 in 8000 births result in triplets. Higher if you factor in fertility treatments. So this should happen at least 30 times a year.
31. Samantha says:
Obviously this post is about having birthdays on the same days, but I want to put my late grandmother up for honorable mention. With two different fathers and a 20 year span amongst them, my
grandma managed to have my mom and her two sisters on consecutive days. The oldest was born Feb 1, the middle child was born 7 years later on Jan 31, and 13 years after that (by my grandfather),
my mom was born on Jan 30. I think this definitely speaks to the fact that people have tendencies to do things at the same time!
32. lewis stoole says:
what are the odds of one of the kids being the anti-christ?
now, what if the anti-christ is female?
33. Michael_GR says:
Nylund @6 – Did your great aunt’s kids get both a Christmas present and a birthday present on their (and Jesus’) birthday? Or did she go through all that just so she could skimp on buying
34. kristofer says:
we came really close with two of our kids… Their birthdays are 1-22-01 and 1-21-02
35. Anonymous says:
hi, i gave birth 2 my little boy nov 12 2008 he was 2 weeks overdue! My little girl was born nov 12 2010 10 weeks early! Me and my husband moved in 2gether nov 12 2005 and got engaged nov 12 2006
my dear grandfather died nov 12 1990 WIERD!
36. Anonymous says:
A friend of mine was complaining about a novel that was so filled with unlikely coincidences that his credulity was shot. He said that you can have ONE amazingly unlikely coincidence in a book
because THEY HAPPEN and after all, that’s what the book is about.
37. Anonymous says:
My son was born naturally on my birthday, which was also his due date. I cannot find any statistics on a child being born on its due date which is also the birthdate of the mother. Has anyone
ever heard of this or am I the first??
38. Anonymous says:
Also, the odds depend on how many kids she has. If she has six kids (assuming no twins,) the odds are about one out of 6661 that three will have the same birthday.
39. Anonymous says:
My birthdy is 8 20, my sisters is 8 18, and my brothers is 8 22
40. Anonymous says:
Lets talk statistics.
Your basic assumption (as you even point out yourself) is wrong. You assume that having a baby born on any given day is equally likely. That’s like saying that winning the lotterey carries a 50%
chance, since you can either win or lose.
If you consider the odds of a woman trying to have three children in a row (Her trying means that she’s doing everything possible each day, then we can actually make the rough assumption that the
propability is the same each day), then the propability comes out much higher simply due to the fact that the pregnancy period takes away a large amount of the year from consideration.
A woman i typically pregnant for 38 weeks, or 266 days.
If we define n as the number of days one has to try to get pregnant before success, then the probability is 0 if n < 99 (which is 365-266 )
but when n is above or equal to 99, the probability is just 1/n^2 which has the upper bound of just 1 in 9801. The lower bound for n is ofcause pushed lower for every day the womans waits before
trying to get pregnent, so if she waits four and a half weeks the upper bound could be as much as 1 in 1000.
Ofcause then you have to figure out what the propability is that a woman fits this pattern of having three children “in a row”.
Anyways, the point is that the statistics are much more influenced by the individual person and their patterns, which lead to the propability of having a child on any given day, which for most
people will be anything but 1/365 for each day. Don’t spend your money on that 50% chance of wining the lottery, as we know tough there are only two options, they carry wieldly different
41. Marshall says:
My roommate, her mother and her grandmother all share the same birthday. I’ve always thought that was somehow amazing and as a perpetual birthday forgetter, convenient.
42. Anonymous says:
Well. I had both of my daughters on MY birthday. It was never planned and I actually unfortunately had a few miscarriages in the mix. 28 hour labour with my first which lead to an emergency
cesarean and my second decided to come 10 days early NATURALLY. Still blows me away and I have to say after just sharing OUR birthdays a month or so ago… it’s goin g to be interesting in our
household in a few years.
43. Clemoh says:
In our family, my two cousins, who are ten years apart(And the eldest and youngest siblings), share the same birthday. This is also the same birthday as their father. Another has the same
birthday as her mother, and still another shares her birthday with me: same month, day and year!
44. Bryan Price says:
I’ve got four (step, not that it matters really) kids with their birthdays one week apart.
Two sets of twins, 5 years apart.
The funny(?) thing about it is that the conception date for both sets is their father’s birthday! The ONLY two times that those two had unprotected sex. Go figure.
45. Anonymous says:
My husband and his first wife had their first child, a boy, who was born on my husband’s birthday. Two and a half years later, their baby girl was born on her mother’s birthday. (No planning, no
C-section, etc.) They divorced 6 years later, he was granted custody of both kids, we met and married, raised his children, and had one child together………no, not on anybody else’s birthday!
46. aixwiz says:
Here’s a good one:
My brother, my father and my father’s father were all:
1. Born on October 14th
2. Born on a Monday
3. And were the first son in their family
Don’t believe me; check out “Ripley’s Believe It Or Not” 19th edition, page 109.
47. ericreiss says:
I have 3 cousins, all siblings, no twins who all have a birthday today. They are their only three kids, and oddly enough, their anniversary is 9 1/2 months ago. All natural births also, no
cheating c-sections.
48. trevoranderson says:
@30 True enough that there are other factors that could be taken into account to get more accurate chances, but that doesn’t make the figure totally invalid.
Suppose we have a bag with two red balls and two white balls in it, and my friend and I each draw one without looking and keep it closed in our hand. Without knowing what my friend pulled, I say
my chance of drawing red are 2/4=50%. But if my friend shows me he pulled red, then I now know my chances of having a red in my hand are only 1/3=33%. I open my hand and there’s a red and now I
know it was actually 100% all along.
In general, you can think of “randomness” in everyday life as an expression of our level of knowledge in a given situation. Is there a certain minimum level of knowledge where it’s fair to make a
probability statement? I don’t think so. How would you even judge that? All you can say is “based on this and this, I estimate X probability”. That’s what the original poster did, nothing wrong
with that.
49. Anonymous says:
Your statistics seem to assume that a woman can get pregnant any day of the month, which is incorrect. If she’s on a regular cycle, she’s only ovulating a short time during every 28 day
increment. That window of fertility, not your intentions at romantic babymaking, is what really decides if the pregnancy starts or not.
50. Anonymous says:
My parents had three of us on the same day but different years. My older sister and younger brother and I were all born November 10. I asked my dad once how they managed to have three of us on
the same day and he quipped “I don’t know, but every February your mother would send me away for a month”. We may have fought against it when we were younger, not wanting to share anything, but
as we’ve learned, it is a terrific story to share whenever our birthday comes around.
51. Anonymous says:
My uncle is born on December 8. My aunt (his sister), Jan 8. My boyfriend, Feb 8. My grandfather’s companion, March 8. My cousin, April 8. My sister, May 8. My childhood best friend, June 8.
Another close friend: September 8. I’m looking for ways to complete the set :)
52. Anonymous says:
I have 3 kids in all My youngest and oldes are both girls and share the same B-Day may 28th and were both born on a Thursday, were 11 years apart and if my youngest would have been born 11
minutes later it would have been exactly 11 years apart. My son who is the middle child shares my cousins B-day September 10th and my 2 girls also share their Great Great Grandmother’s B-Day.
My youngest was May 28th 2009- hers sister’s B-day
Borna at 2:24 Pm(I was born 2-24-1976)
and she weighed 6 lbs 7 oz- the same as her great uncle.
53. klossner says:
“Please don’t let them have my birthday” pleaded my eight-year-old as I rushed my wife to the hospital. But the twins were born that day, three months early.
So that each might have a special day all to herself, we celebrated on the anniversary of the day each twin came home from the hospital. That spread the days out to one per month.
54. Anonymous says:
This is the crummiest statistical analysis I’ve ever read. In order for it to be truly probabilistic, you have to eliminate confounding personal variables such as menstrual cycle and frequency of
The original calculation would be sound if an opportunity for conception occurred every day of every year, but it simply doesn’t. First of all, because a woman cannot conceive a child on everyday
of every year. Secondly, because couples aren’t having sex everyday of every year.
The date of conception often seems random, but it’s not and often has more to do with personal choices and circumstances.
55. Anonymous says:
The majority of women in any one populated area ovulate around the same time. This is much more noticeable if you break it down to social groups. A woman and her girl friends who spend a lot of
time together likely all ovulate within a week of each other. Research suggests this is due to a primal instinct to compete for mates and produce viable offspring. Someone on birth control who’s
“spotting” or “bleeding through” is probably experiencing this adjustment where their body is trying to match the reproductive cycle of another female they spend a lot of time with.
Funny, but unfortunately very true. It’s also a factor and probability reducer in the equation.
56. Scary_UK says:
Me and my sister have the same birthdate 2 years apart.
As kids we always thought this was an odd co-incidence but then it occurred to us that our birthday was at the end of June and our parents anniversary was at the end of October! | {"url":"http://boingboing.net/2010/02/13/on-the-likelihood-of.html","timestamp":"2014-04-20T16:12:18Z","content_type":null,"content_length":"121426","record_id":"<urn:uuid:b2153f82-a288-4a34-bf8a-df87a7c34e83>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00068-ip-10-147-4-33.ec2.internal.warc.gz"} |
Top Ten: Ways to Tell if You're Really an Engineer
10. You have no life - and you can PROVE it mathematically.
9. You enjoy pain.
8. You know vector calculus but you can't remember how to do long division.
7. You chuckle whenever anyone says "centrifugal" force.
6. You've actually used every single function on your graphing calculator.
5. It is a beautiful fall day, and you are working on a computer.
4. You frequently whistle the theme song to "MacGyver."
3. You know how to integrate a chicken and can take the derivative of water.
2. You think in "math."
1. You've calculated that the World Series actually diverges.
Adapted from | {"url":"http://www.columbia.edu/cu/moment/features/engineer.html","timestamp":"2014-04-18T10:44:40Z","content_type":null,"content_length":"7461","record_id":"<urn:uuid:39ec5219-182a-481b-b824-504f3f1d075b>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00487-ip-10-147-4-33.ec2.internal.warc.gz"} |
American Mathematical Monthly - February 2002
FEBRUARY 2002
Words and Pictures: New Light on Plimpton 322
By Eleanor Robson
In the half-century since its publication, Plimpton 322 has become one of the most famous mathematical objects in the world. It is a small clay tablet from ancient Mesopotamia (modern-day Iraq) made
nearly 4,000 years ago, bearing a mathematical table of Pythagorean triples and related numbers. Several attempts have been made to explain its purpose, from astronomy to trigonometry and number
theory. But are any of those interpretations correct? And how can we tell? Can we even begin to know who wrote it, and why? To answer these questions we need to examine not just the numbers on the
tablet but the words and the layout as well.
Finite Quantum Chaos
Audrey Terras
We compare the statistics of the spectrum for various linear operators --- Schroedinger operators from quantum mechanics, Laplace operators of manifolds arising as quotients of arithmetic groups
acting on the upper half plane, and finally adjacency operators of finite upper half plane graphs. These graphs may be viewed as a finite model for arithmetical quantum chaos. C.L. Chai and W.-C. W.
Li have shown that the histograms of the spectra of the finite upper half plane graphs approach the Wigner semi-circle distribution as the number of vertices of the graph approaches infinity. That
is, the spectral distribution approximates that of a random real symmetric matrix. The histograms for differences of adjacent eigenvalues for finite upper half plane graphs appear to approach the
Poisson density, as do the level spacings for arithmetical quantum chaos. The eigenfunctions for finite upper half planes have contours that roughly show the same short of chaos as those for
arithmetical quantum chaos, at least for Maass wave forms for the modular group of 2x2 integer matrices of determinant one.
Finding topology in a factory: configuration spaces
A. Abrams and R. Ghrist
abrams@math.uga.edu, ghrist@math.gatech.edu
As a Source of interesting examples of topological spaces, the modern automated warehouse is surprisingly productive. We discuss a class of configuration spaces that arises naturally in the solution
of industrial problems associated with safe multiple-agent robot control. The configuration spaces are easy to define, yet quite difficult to visualize. One particularly pleasant example is the
configuration space of two labeled robots on a complete graph of five vertices: this space is topologically equivalent to a genus six surface.
Constructions using a compass and twice-notched straightedge
Arthur Baragar
It is not possible to trisect an arbitrary angle or double the cube using only a straightedge and compass. However, if we are clumsy enough to scratch our straightedge in two places, then those
notches can be used to trisect an arbitrary angle or find the cube root of an arbitrary length. These results were known to the ancient Greeks. In this paper, we will place these results in a modern
setting, demonstrate that the twice-notched straightedge can be used to construct the regular 7-gon, 9-gon, and 13-gon, and show that it can even be used to construct points whose coordinates are the
roots of quintic polynomials that are not solvable by radicals. We will also demonstrate some of its limitations; for example, one cannot construct the regular 23-gon or 29-gon using only a compass
and a twice-notched straightedge.
Hexagonal Economic Regions Solve the Location Problem
Frank Morgan and Roger Bolton, Williams College
The Location Problem in economics asks where to place centers of production in order to minimize the average distance from a consumer to the nearest center. We show that in a certain mathematical
sense the ideal solution places them at the centers of regular hexagonal tiles.
Finite Groups of Matrices Whose Entries Are Integers
James Kuzmanovich and Audrey Pavlichenkov
kuz@wfu.edu, A.Pavlichenkov@hotmail.com
Finite groups of matrices appear early as examples in a first course in abstract algebra, and most of these example are given with integral entries. While these groups provide a setting in which to
illustrate new concepts and to pose problems, they also have surprising and beautiful properties. For example, Minkowski proved the unexpected result that GL(n,Z), the group of n x n matrices having
inverses whose entries are also integers, has only finitely many isomorphism classes of finite subgroups. As a consequence, there are only finitely many possible orders for elements of GL(n,Z);
fortunately, the possible orders can be determined using linear algebra. In general, the number of possible orders increases as n increases, but even here we have the surprising result that no new
possible orders are obtained when going from GL(2k,Z) to GL(2k+1,Z). This paper is an exposition of these and other related results and questions.
Problems and Solutions | {"url":"http://www.maa.org/publications/periodicals/american-mathematical-monthly/american-mathematical-monthly-february-2002","timestamp":"2014-04-20T02:11:12Z","content_type":null,"content_length":"103188","record_id":"<urn:uuid:5789167b-86fb-4326-b4bc-2683e5c5d4f7>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00067-ip-10-147-4-33.ec2.internal.warc.gz"} |
Newest 'linear-optimization computational-complexity' Questions
There is a problem that I can not solve. Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...
asked Oct 15 '13 at 16:34 | {"url":"http://mathoverflow.net/questions/tagged/linear-optimization+computational-complexity","timestamp":"2014-04-21T09:59:23Z","content_type":null,"content_length":"37267","record_id":"<urn:uuid:ec63ab85-c137-42b7-8267-1871cadd9232>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00315-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is integral (ln 2 to ln 3) 1/(e^x - 1)dx ? - Homework Help - eNotes.com
What is integral (ln 2 to ln 3) 1/(e^x - 1)dx ?
Given definite integral is `int_ln2^ln3(1/(e^x - 1))dx`
`=int_ln2^ln3(e^x-(e^x-1))/(e^x - 1))dx`
`=int_ln2^ln3(e^x)/(e^x-1)dx -int_ln2^ln3(e^x-1)/(e^x-1)dx`
`=int_ln2^ln3(d(e^x-1))/(e^x-1)dx -int_ln2^ln3 1dx`
Join to answer this question
Join a community of thousands of dedicated teachers and students.
Join eNotes | {"url":"http://www.enotes.com/homework-help/what-integral-ln-2-ln-3-1-e-x-1-dx-442534","timestamp":"2014-04-24T10:42:00Z","content_type":null,"content_length":"24663","record_id":"<urn:uuid:b24f8868-048b-4c16-87e8-4626d3d26abb>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00392-ip-10-147-4-33.ec2.internal.warc.gz"} |
nesc0326 AIROS-2A, Space-Independent Reactor Kinetics and Space-Dependent Heat Transfer, Mass Transfer
nesc0858 APACHE, 2-D Chemical Reactive Fluid Flow Dynamic for CW Chemical Lasers
nesc0152 ARGUS, Transient Temperature Distribution Cylindrical Geometry, Space-Dependent or Time-Dependent Heat Generator
nea-1368 ARIANNA-2, Sub-Compartment Thermo-Hydraulic Transients in LOCA
nesc0580 ASTEM, Evaluation of Gibbs, Helmholtz and Saturation Line Function for Thermodynamics Calculation
nesc9564 AYER, 2-D Thermal Conduction by Finite Element Method
nesc0767 BEACON/MOD3, 1-D and 2-D 2 Phase Flow and Heat Transfer in Containment, LWR LOCA
psr-0377 BLOCKAGE2.5R, Plug of Emergency Core Cooling Suction Strainers by Debris BWR
nea-0683 BLOK, Turbulent Flow in Pipes and Channels with Rectangular Obstruction
nea-0159 BWCAL, Void Distribution and Flow Velocity in BWR
nea-1313 BWRDYN, Thermal Hydraulic Analysis of a BWR Plant
nea-1044 BWRPLANT/ZERO, Dynamic Model for BWR Nuclear Plant
nesc0892 CCC, Heat Flow and Mass Flow in Liquid Saturated Porous Media
nea-0553 CEDRAZAL, Steady-State Heat Transfer in HTR with Multifuel Region
ests0663 CFDLIB, Computational Fluid Dynamics Library
nea-0631 CLAPTRAP, Pressure Transients in LWR Containment During LOCA
nea-0864 CLUHET, Steady-State Thermohydraulics of Rod Bundles with 1 Phase Flow
nea-0255 CLUS, Heat Transfer and Fuel Power in Liquid Cooled 7 Rod Fuel Elements Cluster
nesc0432 COBRA, Transient Thermohydraulics Fuel Elements Clusters, Subchannel Analysis Method
nesc9978 COBRA-3C/RERTR, Thermohydraulic Low Pressure Subchannel Transients Analysis
ests0135 COBRA-SFS CYCLE3, Thermal Hydraulic Analysis of Spent Fuel Casks
nesc1091 COBRA-SFS, Thermal Hydraulics of Spent Fuel Storage System
nesc0702 COMPARE, Transient Subcompartment Thermodynamics Analysis with 2 Phase Vent Flow
nesc0776 COMPARE-MOD1 COMPARE-MOD1A, 2 Phase Flow Thermodynamics, Pressure in LWR Containment
ests0023 COMPBRN3, Modelling of Nuclear Power Plant Compartment Fires
ests0680 CONCHAS-SPRAY, Reactive Flows with Fuel Sprays
nea-0946 CONDN-63B, Thermohydraulics of Nuclear Power Plant Condenser
nea-1305 COOLOD, Steady-State Thermal Hydraulics of Research Reactors
nea-0567 CORAN, PWR and BWR Containment Response to LOCA
nea-0398 COSTAX-BOIL, Transient Dynamic Analysis of BWR and PWR in Axial Geometry
nesc0737 DIFFUSER, 2-D and 3-D Diffuser Performance, Boundary Layer and Turbulent Flow
nea-0215 DRUCKSCHALE-44, Pressure and Temperature Transients in Blowdown Accident
nea-0839 DRUFAN-01/MOD2, Transient Thermohydraulics of PWR Primary System LOCA
nesc0784 DSNP, Program and Data Library System for Dynamic Simulation of Nuclear Power Plant
nesc0440 DYNAM, Once Through Boiling Flow with Steam Superheat, Laplace Transformation
ests0219 ECO2N, a TOUGH2 fluid property module for mixtures of water-NaCl-CO2
nea-0261 EQUSTA, Thermodynamics Analysis and Mechanical Analysis for Fast Reactor Accident
nesc9952 EVENT, Explosive Transients in Flow Networks
nesc1046 FED, Geometry Input Generator for Program TRUMP
ests0198 FEM-3, Heavy Gas Dispersion Incompressible Flow
nesc1092 FIRAC, Nuclear Power Plant Fire Accident Model
nesc0395 FLAC FLAC-SI, Steady-State Flow and Pressure Distribution, 1-D Incompressible Flow Equation
nesc9597 FLODIS, Thermal Response of FSV HTGR Core
nesc0246 FLOW-MODEL, Multichannel 2-D 2 Phase Flow for Open Matrix Flow BWR
nesc9592 FLOWPLOT2, 2-D, 3-D Fluid Dynamic Plots
nea-0396 FRANCESCA, 2 Phase Flow Dynamic in Boiling Test Channel and Heat Elements Conduction
nea-0397 FRANCESCA-BWR, 2 Phase Flow Dynamic for BWR Cooling Channel
nesc0694 FRAP-S3 FRAP-S1, Steady-State LWR Oxide Fuel Elements Behaviour
nesc0694 FRAPCON2, Steady-State LWR Oxide Fuel Elements Behaviour, Fission Products Gas Release, Error Analysis
nea-0073 GHT, 3-D Steady-State and Transient Heat Conduction
nesc9911 GRAY CNVUFAC, Black-Body Radiation View Factors with Self-Shadowing
ests0576 GRIDMAKER, 2-D, 3-D Finite Element Method Grid Generation for Ground Water and Pollutant Transport
nesc0618 GTR2 GAPCON-THERMAL2, Steady-State Fuel Rod Thermal Behaviour and Fission Products Gas Release
nea-0876 H2O, Calculation of Thermodynamics Properties of Steam and H2O
nea-0682 H2OTP, Temperature Dependent and Pressure Dependent Thermodynamics Properties, Transport Properties of H2O
nea-0547 HASSAN, Time-Dependent Temperature Distribution and Stress and Strain in HTR Fuel Pins
nea-1292 HEATHYD, Steady-State Thermal Hydraulic Analysis of Low-Enriched U Fuel Reactor
psr-0199 HEATING-7, Multidimensional Finite-Difference Heat Conduction Analysis
nesc0434 HEATMESH, Geometry Data Generator for Heat Transfer Calculation in Axisymmetric System
nea-1095 HEATP, Steady-State and Transient Heat Transfer in PWR
nea-0303 HEATRAN, 2-D Heat Diffusion for X-Y or R-Z Geometry with Heat Transfer Across Gaps
nea-0536 HERA-1A, Steady-State Thermohydraulics of Na Cooled Fuel Rod Bundles
ests0648 HTRATE, Power Plant Heat Rate Improvement from Condenser Retubing
nea-0518 HUBBLE-BUBBLE, Transient Subcooled or Superheated H2O Bubble Flow
iaea1377 HYDMN, Thermal Hydraulics of Miniature Neutron Source Reactor
nesc9553 HYDRA-2, 3-D Heat Transport for Spent Fuel Storage System
ests0405 HYFRAC3D, 3-D Hydraulic Rock Fracture Propagation by Finite Element Method
ests0406 HYFRACP3D, 3-D Hydraulic Fracture Propagation by Finite Element Method
nea-0100 HYTHEST, Dependence of Fuel Fabrication Tolerances on Hydraulics of BWR, PWR
nea-0216 HYTRAN, Open Channel Thermal and Hydraulic Transients in LOCA
nesc9683 ICARUS-LLNL, 1-D Heat Transfer in Planar, Cylindrical, Spherical Geometry Using Finite Element Method
nesc9473 IMPSOR, 3-D Boundary Problems Solution for Thermal Conductivity Calculation
nea-0554 INSUL, Calculation of Thermal Insulation of Various Materials Immersed in He
ests0219 ITOUGH2, Inverse Modeling for TOUGH2 Multiphase Flow Simulators
nea-1838 JASMINE V.3, Steam explosion simulation
nea-0624 JOSHUA, Neutronics, Hydraulics, Burnup, Refuelling of LWR
nea-0154 JPHYDRO, Voids and Flow Velocity in Steady-State BWR System
nesc0877 K-FIX(3D), Transient 2 Phase Flow Hydrodynamic, X-Y-Z and Cylindrical Geometry, Eulerian Method
nesc0727 K-FIX, Transient 2 Phase Flow Hydrodynamic in 2-D Planar or Cylindrical Geometry, Eulerian Method
nesc0876 K-TIF, Thermohydraulic Dynamic of PWR in Steady-State and Transient Flow Conditions
nea-1002 KINE, 1-D PWR Dynamic with Partial Core Boiling
iaea1339 KINETIC, Time-Dependent Heat and Mass Transfer
ests0463 LDEF-SS, Solve Equation Two Phase Fluid Flow in Spray Dryers
nea-0623 LOCA-MARK-2, Fuel Temperature and Clad Temperature in HWR Steam Generator LOCA
nea-0185 LOOP-3, Hydraulic Stability in Heated Parallel Channels
nea-0250 LUPO, Temperature and Void Rate and Pressure Drop and Flow Rate in Pressure Loop
ests0386 MAGNUM-2D, Heat Transport and Groundwater Flow in Fractured Porous Media
nesc0256 MANTA, Heat Transfer Fuel Elements Cluster to Single-Phase Steady-State Fluid Flow
nea-0448 MATTEO, BWR Subchannel Steady-State and Transient Thermohydraulics
nea-0362 MEDEA, Steady-State Pressure and Temperature Distribution in He H2O Steam Generator
nea-1534 MESYST, Simulation of 3-D Tracer Dispersion in Atmosphere
ests0143 MINET, Transient Fluid Flow and Heat Transfer Power Plant Network Analysis
nea-1005 MOBIDIC, Fast Reactor Hexagonal Infinite Lattice 2 Component Fuel Pin Diffusion Coefficient
nesc0551 MOXY-MOD32, Thermal Analysis Swelling and Rupture of BWR Fuel Elements During LOCA
nesc0551 MOXY/MOD-1, Thermal Analysis Swelling and Rupture of BWR Fuel Elements During LOCA
nea-0816 MUENSTER, 2-D R-Z Geometry Thermohydraulics Calculation for Pebble-Bed Reactor
nesc9489 NACHOS2, Incompressible Viscous Fluid Dynamic
nesc9644 NASA-VOF2D, 2-D Transient Free Surface Incompressible Fluid Dynamic
nesc9568 NASA-VOF3D, 3-D Transient, Free Surface, Incompressible Fluid Dynamic
ests0262 NORIA, 2-D Non-Isothermal 2-Phase Flow Through Porous Media
nesc0703 ORCENT-2, Full Load Steam Turbine Cycle Thermodynamics for LWR Power Plant
nesc0525 ORTHAT, Transient Heat Conduction in 2-D X-Y, R-Z and R-Theta Geometry
nesc0525 ORTHIS, Steady-State Heat Conduction in 2-D X-Y, R-Z and R-Theta Geometry
nesc1102 ORTURB, HTGR Steam Turbine Dynamic for FSV Reactor
nea-0802 OWEN-1, LOCA Transient and Steady-State 2 Phase Flow in Heated Channel
nesc0865 PELE-IC, 2-D Eulerian Incompressible Hydrodynamic and Bubble Dynamic after LWR LOCA
nea-1612 PIN99W, Modelling of VVER and PWR Fuel Rod Thermomechanical Behaviour
ests0650 PIPE-ESTSC, Friction Factor for 3-D Turbulent Flow in Rough Tubes
nesc0586 PLENUM, Bulk Flow Distribution in Cylindrical Reactor Coolant Inlet Plenum, Potential Flow
nea-0251 PREST, Pressure Temperature Transients, I Inhalation in Containment Building from LOCA
nesc1023 PROGRAM-H, Analysis of Transonic Airfoils with Turbulent Boundary Layer Correlation
ests0790 PROGRAM-K, Transonic Airfoil, Turbine, Compressor Blade Design
nesc0552 PWR-PPM, Boration-Dilution Tables Generator for PWR Operation
nea-0632 RAPVOID, H2O Flow and Steam Flow in Pipe System with Phase Equilibrium
nesc0369 RELAP-4, Transient 2 Phase Flow Thermohydraulics, LWR LOCA and Reflood
nesc0917 RELAP-5, Transient 2 Phase Flow Thermohydraulics, LWR LOCA Accidents
nea-0437 RELAP-UK, Thermohydraulic Transients and Steady-State of LWR
nea-0821 RELAP/REFLA, Core Reflooding During PWR LOCA
ests0185 RIPPLE, Incompressible Fluid Dynamics with Free Surfaces
nesc0900 SALE-2D, 2-D Fluid Flow, Navier Stokes Equation Using Lagrangian or Eulerian Method
nesc1069 SALE-3D, 3-D Fluid Flow, Navier Stokes Equation Using Lagrangian or Eulerian Method
ccc-0785 SCALE 6.1.2, Modular system for criticality, shielding, source term, fuel depletion/decay, inventories, reactor physics
nea-0431 SCEPTIC, Pressure Drop, Flow Rate, Heat Transfer, Temperature in Reactor Heat Exchanger
nesc0802 SCHAFF, Single-Phase Flow, Heat Transfer in Porous Media, Geothermal Energy System
nea-0537 SCOTCH, 1-D 2 Group HTGR Core Kinetics with Temperature Transients and Fluid Dynamic
nea-0865 SCRIMP, Steady-State Thermohydraulics of HTGR Subchannel
nesc0893 SHAFT-79, 2 Phase Flow in Porous Media for Geothermic Energy System
nesc9593 SIMPLE, 2-D Hydrodynamic, Heat Flow Benchmark
ests0767 SIMSOL, Multiphase Fluid and Heat Flow in Porous Media
nesc0521 SOCOOL-2, Molten Materials Na Coolant Interaction, Temperature and Pressure Transient
nesc0832 SOLA-DF, Time-Dependent 2-D 2 Phase Flow, Eulerian Method with Various Boundary Conditions
nesc0723 SOLA-ICE, Compressible Fluid Flow Transients, 2-D Planar, Cylindrical Geometry, Eulerian Method
nesc0859 SOLA-LOOP, Transient 2 Phase Flow in Networks of 1-D Components
nesc0651 SOLA-SURF, 2-D Plane, Axisymmetric, Incompressible Flow Navier Stokes Equation for Transient
nesc0948 SOLA-VOF, 2-D Transient Hydrodynamic Using Fractional Volume of Fluid Method
iaea0895 SPAGAF, PWR Fuel, Cladding Behaviour with Fission Products Gas Release
nea-0055 STDY-3, Steady-State Parallel Channel Thermal Analysis of PWR
nea-0703 STEADY-ACE, 3-D Neutronics and Multichannel Thermohydraulics Analysis of BWR
nesc0487 STEAM-67, Thermodynamics Properties of H2O and Steam from ASME Tables (1967)
iaea0900 STOFFEL-1, Steady-State In-Pile Behaviour of Cylindrical H2O Cooled Oxide Fuel Rod
nea-0253 STYLE, Steam Cycle Heat Balance for Turbine Blade Design in Marine Operation
nesc0853 SURGTANK, Steam Pressure, Saturation Temperature or Reactor Surge Tank
nesc0973 SWIFT, 3-D Fluid Flow, Heat Transfer, Decay Chain Transport in Geological Media
nesc9766 T-HEMP3D, 3-D Time-Dependent Elastic Plastic Flow
ests0219 T2VOC, H2O, Air, VOC Flow Simulation in Porous Multidimensional Media
nesc0408 TAC-2D, Steady-State and Transient Heat Transfer in X-Y, R-Z or R-Theta Geometry
nesc0414 TAC-3D, 3-D Steady-State and Transient Heat Transfer in X-Y-Z and R-Theta-Z Geometry
nesc9838 TAC0-3D, 3-D Linear or Nonlinear, Steady-State or Transient Heat Transfer
nea-0532 TAFE, 2-D Steady-State Heat Conduction for Structure with Gas Gaps
nea-0531 TAFEST, 2-D Transient Heat Conduction
nesc9566 TAP-LOOP, Steady-State and Transient Thermal Analysis of Closed Test Loops
nea-0556 TAPIR, Thermal Analysis of HTGR with Graphite Sleeve Fuel Elements
nea-0570 TEMP, Steady-State and Transient Heat Conduction in Planar or Cylindrical Geometry
iaea0836 TEMPELS, Heat Conduction for Arbitrary Geometry by Finite Element Method (FEM)
nesc9808 TEMPEST-BNW, Transient 3-D Thermohydraulics for FBR
iaea1338 TEMPUL, Temperature Distribution in Fuel Element after Pulse
nea-1098 THARC-S, Rod Bundle Thermohydraulic Transients of LMFBR for Single Phase Conditions
nesc9940 THERMIT, 3-D Thermo-Hydraulics of BWR and PWR
nea-0411 THESEE-3, Orgel Reactor Performance and Statistic Hot Channel Factors
nea-0377 THREAT, 3-D Steady-State or Transient Heat Diffusion in Multi-Region Prism
ests0219 TMVOCV1.0, Multicomponent, multiphase, nonisothermal flows of water, soil gas, volatile organic chemicals (VOCs)
nesc9801 TOPAZ, 2-D Plane or Axisymmetric Heat Conduction Analysis
nesc9599 TOPAZ-3D, 3-D Steady-State or Transient Heat Transfer by Finite Element Method
nesc9669 TOPAZ-SNLL, Transient 1-D Pipe Flow Analysis
nesc9801 TOPAZ2D, 2-D Finite Element Method Heat-Transfer and Electrostatic and Magnetostatic (E&M) Potential Field Program
nesc1093 TORAC, Flows, Pressure, Materials Transport within Structure During Tornado
ests0219 TOUGH2, Unsaturated Ground Water and Heat Transfer
ests0219 TOUGHREACTV1.2, Chemically reactive non-isothermal flows of multiphase fluids in porous and fractured media
iaea1209 TRANSV2, LOCA and Steady-State Thermohydraulic Analysis of MTR
nea-0807 TRAWA, LWR Dynamic by Coupled Neutron Diffusion and Thermohydraulics Calculation
iaea1337 TRISTAN-IJS, Steady-State Axial Temperature and Flow Velocity in Triga Channel
psr-0522 TRUMP, Steady-State and Transient 1-D, 2-D and 3-D Potential Flow, Temperature Distribution
nea-0233 TURBINA, Reheat Steam Turbine Generator Design with Preheater and Condenser
nea-0581 TURBPLANT, 1-D Steady-State Model of Power Reactor Steam Turbine Components
nesc0809 TVENT, 1-D Incompressible Flow for Pressure Transients in Ventilation System
ests0404 UHS, Ultimate Heat Sink Cooling Pond Analysis
ests0333 USINT, High Temperature Heat and Mass Transfer on Concrete Surfaces in LMFBR
nea-0587 UTSG, Steady-State and Transients of Vertical U-Tube Steam Generator
nesc0755 VARR2 VARRLXSG, 2-D Transient Fluid Flow and Heat Transfer in X-Y and Cylindrical Geometry
iaea1324 VITEK, Non Stationary Navier-Stokes Solver for Compressible, Turbulent Flow
nea-0655 VSOP, Neutron Spectra, 2-D Flux Synthesis, Fuel Management, Thermohydraulics Calculation
nea-0506 WAKE, Navier Stokes Equation with 2-D Turbulence, Stream Function, Vorticity
ests0160 WELBORE, Transient Wellbore Fluid Flow Model | {"url":"http://www.oecd-nea.org/tools/abstract/list/category/h","timestamp":"2014-04-20T03:15:33Z","content_type":null,"content_length":"52125","record_id":"<urn:uuid:07fa54bc-ceb3-41da-9709-0d42d0bd4a6a>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00468-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Projectile Fired At A 45 Degree Angle
this excerise is a bit tricky, cause as you say, it's "the other way around". but we know, considering the equations for the two components, that;
6m= - gt^2+V0*sinθ*t (the relevant velocity is here the vertical hence the sine)
100m= V0*cosθ*t (the horizontal velocity calls for the use of a cosine)
everybody concur? :)
Since we have 2 equations and 2 unknowns, finding the answer for V0, the initial velocity, is just some algebraic puzzlework. | {"url":"http://www.physicsforums.com/showthread.php?t=536524","timestamp":"2014-04-18T10:51:19Z","content_type":null,"content_length":"68967","record_id":"<urn:uuid:e85dd11c-52e7-4ddc-b464-267e7f7f881e>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00027-ip-10-147-4-33.ec2.internal.warc.gz"} |
Results 1 - 10 of 294
- Artificial Intelligence , 2000
"... Agent-oriented techniques represent an exciting new means of analysing, designing and building complex software systems. They have the potential to significantly improve current practice in
software engineering and to extend the range of applications that can feasibly be tackled. Yet, to date, there ..."
Cited by 480 (23 self)
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Agent-oriented techniques represent an exciting new means of analysing, designing and building complex software systems. They have the potential to significantly improve current practice in software
engineering and to extend the range of applications that can feasibly be tackled. Yet, to date, there have been few serious attempts to cast agent systems as a software engineering paradigm. This
paper seeks to rectify this omission. Specifically, it will be argued that: (i) the conceptual apparatus of agent-oriented systems is well-suited to building software solutions for complex systems
and (ii) agent-oriented approaches represent a genuine advance over the current state of the art for engineering complex systems. Following on from this view, the major issues raised by adopting an
agent-oriented approach to software engineering are highlighted and discussed. 1.
- Journal of the ACM , 1997
"... Temporal logic comes in two varieties: linear-time temporal logic assumes implicit universal quantification over all paths that are generated by system moves; branching-time temporal logic
allows explicit existential and universal quantification over all paths. We introduce a third, more general var ..."
Cited by 448 (47 self)
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Temporal logic comes in two varieties: linear-time temporal logic assumes implicit universal quantification over all paths that are generated by system moves; branching-time temporal logic allows
explicit existential and universal quantification over all paths. We introduce a third, more general variety of temporal logic: alternating-time temporal logic offers selective quantification over
those paths that are possible outcomes of games, such as the game in which the system and the environment alternate moves. While linear-time and branching-time logics are natural specification
languages for closed systems, alternating-time logics are natural specification languages for open systems. For example, by preceding the temporal operator "eventually" with a selective path
quantifier, we can specify that in the game between the system and the environment, the system has a strategy to reach a certain state. Also the problems of receptiveness, realizability, and
controllability can be formulated as model-checking problems for alternating-time formulas.
- Information and Computation , 1994
"... We investigate extensions of temporal logic by connectives defined by finite automata on infinite words. We consider three different logics, corresponding to three different types of acceptance
conditions (finite, looping and repeating) for the automata. It turns out, however, that these logics all ..."
Cited by 250 (55 self)
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We investigate extensions of temporal logic by connectives defined by finite automata on infinite words. We consider three different logics, corresponding to three different types of acceptance
conditions (finite, looping and repeating) for the automata. It turns out, however, that these logics all have the same expressive power and that their decision problems are all PSPACE-complete. We
also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem. 1 Introduction For many
years, logics of programs have been tools for reasoning about the input/output behavior of programs. When dealing with concurrent or nonterminating processes (like operating systems) there is,
however, a need to reason about infinite computations. Thus, instead of considering the first and last states of finite computations, we need to consider the infinite sequences of states that the
program goes through...
- Logics for Concurrency: Structure versus Automata, volume 1043 of Lecture Notes in Computer Science , 1996
"... Abstract. The automata-theoretic approach to linear temporal logic uses the theory of automata as a unifying paradigm for program specification, verification, and synthesis. Both programs and
specifications are in essence descriptions of computations. These computations can be viewed as words over s ..."
Cited by 217 (23 self)
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Abstract. The automata-theoretic approach to linear temporal logic uses the theory of automata as a unifying paradigm for program specification, verification, and synthesis. Both programs and
specifications are in essence descriptions of computations. These computations can be viewed as words over some alphabet. Thus,programs and specificationscan be viewed as descriptions of
languagesover some alphabet. The automata-theoretic perspective considers the relationships between programs and their specifications as relationships between languages.By translating programs and
specifications to automata, questions about programs and their specifications can be reduced to questions about automata. More specifically, questions such as satisfiability of specifications and
correctness of programs with respect to their specifications can be reduced to questions such as nonemptiness and containment of automata. Unlike classical automata theory, which focused on automata
on finite words, the applications to program specification, verification, and synthesis, use automata on infinite words, since the computations in which we are interested are typically infinite. This
paper provides an introduction to the theory of automata on infinite words and demonstrates its applications to program specification, verification, and synthesis. 1
, 1999
"... Software and knowledge... In this article, we argue that intelligent agents and agent-based systems offer novel opportunities for developing effective tools and techniques. Following a
discussion on the classic subject of what makes software complex, we introduce intelligent agents as software struc ..."
Cited by 193 (17 self)
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Software and knowledge... In this article, we argue that intelligent agents and agent-based systems offer novel opportunities for developing effective tools and techniques. Following a discussion on
the classic subject of what makes software complex, we introduce intelligent agents as software structures capable of making "rational decisions". Such rational decision-makers are well-suited to the
construction of certain types of software, which mainstream software engineering has had little success with. We then go on to examine a number of prototype techniques proposed for engineering agent
systems, including formal specification and verification methods for agent systems, and techniques for implementing agent specifications
- in E.W. Mayr and C. Puech (Eds), Proc. STACS'95, LNCS 900 , 1995
"... Abstract. This paper presents algorithms for the automatic synthesis of real-time controllers by nding a winning strategy for certain games de ned by the timed-automata of Alur and Dill. In such
games, the outcome depends on the players ' actions as well as on their timing. We believe that these res ..."
Cited by 190 (20 self)
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Abstract. This paper presents algorithms for the automatic synthesis of real-time controllers by nding a winning strategy for certain games de ned by the timed-automata of Alur and Dill. In such
games, the outcome depends on the players ' actions as well as on their timing. We believe that these results will pave theway for the application of program synthesis techniques to the construction
of real-time embedded systems from their speci cations. 1
- In 25th International Colloqium on Automata, Languages and Programming, ICALP ’98 , 1998
"... Abstract. The p-calculus can be viewed as essentially the "ultimate" program logic, as it expressively subsumes all propositional program logics, including dynamic logics, process logics, and
temporal logics. It is known that the satisfiability problem for the p-calculus is EXPTIMEcomplete. This upp ..."
Cited by 129 (12 self)
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Abstract. The p-calculus can be viewed as essentially the "ultimate" program logic, as it expressively subsumes all propositional program logics, including dynamic logics, process logics, and
temporal logics. It is known that the satisfiability problem for the p-calculus is EXPTIMEcomplete. This upper bound, however, is known for a version of the logic that has only forward modalities,
which express weakest preconditions, but not backward modalities, which express strongest postconditions. Our main result in this paper is an exponential time upper bound for the satisfiability
problem of the p-calculus with both forward and backward modalities. To get this result we develop a theory of two-way alternating automata on infinite trees. 1
"... In this work we tackle the following problem: given a timed automaton, restrict its transition relation in a systematic way so that all the remaining behaviors satisfy certain properties. This
is an extension of the problem of controller synthesis for discrete event dynamical systems, where in addi ..."
Cited by 123 (14 self)
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In this work we tackle the following problem: given a timed automaton, restrict its transition relation in a systematic way so that all the remaining behaviors satisfy certain properties. This is an
extension of the problem of controller synthesis for discrete event dynamical systems, where in addition to choosing among actions, the controller have the option of doing nothing and let the time
pass. The problem is formulated using the notion of a real-time game, and a winning strategy is constructed as a fixed-point of an operator on the space of states and clock configurations.
- In: Proceedings of the First International Workshop on Agent-Oriented Software Engineering , 2000
"... Abstract. Software engineers continually strive to develop tools and techniques to manage the complexity that is inherent in software systems. In this article, we argue that intelligent agents
and multi-agent systems are just such tools. We begin by reviewing what is meant by the term “agent”, and c ..."
Cited by 106 (0 self)
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Abstract. Software engineers continually strive to develop tools and techniques to manage the complexity that is inherent in software systems. In this article, we argue that intelligent agents and
multi-agent systems are just such tools. We begin by reviewing what is meant by the term “agent”, and contrast agents with objects. We then go on to examine a number of prototype techniques proposed
for engineering agent systems, including methodologies for agent-oriented analysis and design, formal specification and verification methods for agent systems, and techniques for implementing agent
specifications. 1
- Third Int. Conf. on Algebraic Methodology and Software Technology, AMAST'93, Twente , 1993
"... This paper is a survey of our specification and verification techniques, in a very general, language independent, framework. Section 1 introduces a simple model of synchronous input/output
machines, which will be used throughout the paper. In section 2, we show how such a machine can be designed to ..."
Cited by 101 (10 self)
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This paper is a survey of our specification and verification techniques, in a very general, language independent, framework. Section 1 introduces a simple model of synchronous input/output machines,
which will be used throughout the paper. In section 2, we show how such a machine can be designed to check the satisfaction of a safety property, and we discuss the use of such an observer in program
verification. In section 3, we use an observer to restrict the behavior of a machine. This is the basic way for representing assumptions about the environment. Applications to modular and inductive
verification are considered. In modular verification, one has to find, by intuition, a property of a subprogram that is strong enough to allow the verification of the whole program without fully
considering the subprogram. In section 4, we consider the automatic synthesis of such a property, and in section 5, we investigate the possibility of deducing the subprogram from such a synthesized | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=86318","timestamp":"2014-04-18T02:03:52Z","content_type":null,"content_length":"39292","record_id":"<urn:uuid:401ee674-494d-40cb-947c-814a536a9274>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00309-ip-10-147-4-33.ec2.internal.warc.gz"} |
Inconsistency between mathematics and our universe.
So basically you're saying that 10*-1=-10, at which point i argue the same thing as before. Why wouldnt the 10 make the -1 go into the positive direction instead. why is it the negative that take
It's simpy by definition. We have define 10*-1 as -10, and we notice that everything works fine.
We could also have define 10*-1=10 and then we would obtain another number system which we can calculate with.
So the question "Why is the result -10 instead of 10" is easily answered by noticing that we chose it to be this way.
The question that you should be asking is "why did we choose it this way and not the other way".
There are many answers to this. One answer is that otherwise the integers wouldn't be a ring, and being a ring is a very desirable property.
That is, the following are true for natural numbers:
a*1=a=1*a for nonzero a
Now we adjoin for each element a, an element -a such that a+(-a)=0
We want all the above properties to still hold because they are familiar for natural numbers.
From this we get:
Adding -10 to both sides gives us
So IF the above identities are all true, then it immediately FORCES us to accept -10=10*(-1), we have no other choice.
This is a pure mathematical way of looking at things, but there are other ways as well!!
For example, one may motivate the existence of negative numbers with a bank account.
In this case, if you have -10$ then this means that you owe the bank 10$. So you are in debt.
What happens if your debt is twice as big?? Then we should look at 2*(-10$). It makes sense that this value should be -20$ rather than 20$. Because if 2*(-10$)=20$, then there is no easy way to
describe your debt being doubled!! Things like 2*(-10$)=-20$ happen a lot in real life, things like 2*(-10$)=20$ are much less occuring and thus not interesting. | {"url":"http://www.physicsforums.com/showthread.php?p=3884662","timestamp":"2014-04-18T10:50:15Z","content_type":null,"content_length":"91788","record_id":"<urn:uuid:b77ed6ba-d544-4f43-92d5-e674d9cb77f5>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00577-ip-10-147-4-33.ec2.internal.warc.gz"} |
XL Fortran for AIX 8.1
Language Reference
An array is an ordered sequence of scalar data. All the elements of an array have the same type and type parameters.
A whole array is denoted by the name of the array:
! In this declaration, the array is given a type and dimension
REAL, DIMENSION(3) :: A
! In these expressions, each element is evaluated in each expression
PRINT *, A, A+5, COS(A)
A whole array is either a named constant or a variable.
Each dimension in an array has an upper and lower bound, which determine the range of values that can be used as subscripts for that dimension. The bound of a dimension can be positive, negative, or
+-------------------------------IBM Extension--------------------------------+
In XL Fortran, the bound of a dimension can be positive, negative or zero within the range -(2**31) to 2**31-1. The range for bounds in 64-bit mode is -(2**63) to 2**63-1.
+----------------------------End of IBM Extension----------------------------+
If any lower bound is greater than the corresponding upper bound, the array is a zero-sized array, which has no elements but still has the properties of an array. The lower and upper bounds of such a
dimension are one and zero, respectively.
When the bounds are specified in array declarators:
• The lower bound is a specification expression. If it is omitted, the default value is 1.
• The upper bound is a specification expression or asterisk (*), and has no default value.
Related Information:
The extent of a dimension is the number of elements in that dimension, computed as the value of the upper bound minus the value of the lower bound, plus one.
INTEGER, DIMENSION :: X(5) ! Extent = 5
REAL :: Y(2:4,3:6) ! Extent in 1st dimension = 3
! Extent in 2nd dimension = 4
The minimum extent is zero, in a dimension where the lower bound is greater than the upper bound.
+-------------------------------IBM Extension--------------------------------+
The theoretical maximum number of elments in an array is 2**31-1 elements in 32-bit mode, or 2**63-1 elements in XL Fortran 64-bit mode. Hardware addressing considerations make it impractical to
declare any combination of data objects whose total size (in bytes) exceeds this value.
+----------------------------End of IBM Extension----------------------------+
Different array declarators that are associated by common, equivalence, or argument association can have different ranks and extents.
The rank of an array is the number of dimensions it has:
INTEGER, DIMENSION (10) :: A ! Rank = 1
REAL, DIMENSION (-5:5,100) :: B ! Rank = 2
According to Fortran 95, an array can have from one to seven dimensions.
+-------------------------------IBM Extension--------------------------------+
An array can have from one to twenty dimensions in XL Fortran.
+----------------------------End of IBM Extension----------------------------+
A scalar is considered to have rank zero.
The shape of an array is derived from its rank and extents. It can be represented as a rank-one array where each element is the extent of the corresponding dimension:
INTEGER, DIMENSION (10,10) :: A ! Shape = (/ 10, 10 /)
REAL, DIMENSION (-5:4,1:10,10:19) :: B ! Shape = (/ 10, 10, 10 /)
The size of an array is the number of elements in it, equal to the product of the extents of all dimensions:
INTEGER A(5) ! Size = 5
REAL B(-1:0,1:3,4) ! Size = 2 * 3 * 4 = 24
Related Information
• These examples show only simple arrays where all bounds are constants. For instructions on calculating the values of these properties for more complicated kinds of arrays, see the following
• Related intrinsic functions are SHAPE(SOURCE), and SIZE(ARRAY, DIM). The rank of an array A is SIZE(SHAPE(A)).
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Errors involving a node with insufficient inputs.
The Sifflet visual functional programming language and aid to understanding recursive functions (http://mypage.iu.edu/~gdweber/software/sifflet/home.html)
#46Errors involving a node with insufficient inputs.
Given this incorrect definition of the power function:
Power x n = if (zero? n)
(* x (Power (sub1 n)))
-- note that in the recursive step, the call to Power has only one argument -- and giving inputs x = 3, n = 6, trying to expand the Power node (the recursive call) causes the program to crash.
O: 2012 Nov 12 R: A Starks P: high
• Saving the function definition and evaluating the function call are okay (Sifflet displays stack overflow errors on the diagram, which is right because the second argument to Power never
changes). On trying to expand the recursive Power node, sifflet crashes with this error message:
sifflet-devel: buildFooterText: mismatched lists
• 2013 July 4.
Sifflet infers the type of this function as (Num -> (Num -> Num)), which seems wrong because of the missing argument to the recursive Power node.
Indeed, ghci reports an "occurs check" error for the following definition:
power x n =
if n == 0
then 0
else x * power (n - 1)
Occurs check: cannot construct the infinite type: a0 = a0 -> a0
In the return type of a call of `power'
Probable cause: `power' is applied to too few arguments
In the second argument of `(*)', namely `power (n - 1)'
In the expression: x * power (n - 1)
□ added tag p:medium
□ removed tag p:high
Partly fixed. Sifflet no longer crashes in this case, but displays the arguments without corresponding inputs as, e.g., "x = ?".
This is no longer high priority, but there are a couple of concerns:
1. The type checker should reject function definitions of this kind.
2. Even if it does not, the evaluator should produce a different kind of error than stack overflow, since the value of partially applied Power is not a number, but a function of type (Num ->
Num), and therefore unsuitable as input to the '*' function.
□ summary changed to "Errors involving a node with insufficient inputs."
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Negative value of latency
Negative value of latency
Hello all,
I've done some measurement on some system using Linux about latency. Measurement is based on periodic activation of processes.
Latency = (Current Activation TIme - Previous Activation TIme) - Period Of Activation
I would like to know if this formula is valid. If not, please correct me. Another issue is the results I've obtained contains negative latency value. I've looked around the web but unfortunately
can't find the explanation of the existence of the negative value. I really would like to know what causes the negative value. It would be very appreciated if anyone can explain about this.
Correct me if I'm wrong, but isn't latency the delay between 2 things, like, for example, the milliseconds in between connections to the server when running a network game? So, shouldn't the
latency be calculated by subtracting the first time from the current time? say:
Current time - previous time = latency
where "previous time" is when the last occurrence of the event you want to test was.
The lower the resulting value, the faster, or less delay, there was between the events. Ideally, it should be as close to zero as possible, but it can never be zero, or below of course, unless
you were testing your system in a black hole or something... :p
Im sorry for the mistake. I've found out that the negative values are the results of wrong formula. As I'm running periodic task, the calculation should be like this.
latency = current_time - (index_of_activation x period_of_activation - first_activation_time)
This will never yield negative value. It is a periodic activation. The delay should not be measured based on previous activation time, instead, it should be measured by the time scale, that is
the how-many-time-the-process-have-been-activated times the period between activation.
For example, 250ms activation period and 200 activation points.
At activation point 1, the delay is 30ms. At activation point 2, the delay is 0ms.
Using this formula,
Latency(Index) = (Current Activation TIme - Previous Activation TIme) - Period Of Activation
Latency(1) = (280-0)-250 = 30ms
Latency(2) = (500-280)-250 = -30ms
latency(index) = current_time - (index_of_activation x period_of_activation - first_activation_time)
latency(1) = 280 - (1x250-0) = 30ms
latency(2) = 500 - (2x250-0) = 0ms
so, the first one is invalid. The second one is the valid latency. Hence, negative value of latency will never exist.
For Themer, sorry for the mistake. The equation on the first post should be like this.
Latency = (Current Activation TIme - Previous Activation TIme) - Period Of Activation
I'll edit it. Thanks anyway. | {"url":"http://www.linuxforums.org/forum/kernel/negative-value-latency-print-146116.html","timestamp":"2014-04-19T15:00:13Z","content_type":null,"content_length":"6369","record_id":"<urn:uuid:c7e63b30-f710-4273-a8eb-77069e339bbd>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00140-ip-10-147-4-33.ec2.internal.warc.gz"} |
Bigravity theory
The bigravity theory: a bimetric cosmological model
The bigravity theory is a bimetric cosmological model with two CPT-symmetric universes interacting through gravitation, including joint variations of the speed of light and all physical constants
through time. The model is based on two coupled Einstein field equations, and the geometrical support is a Riemannian manifold with two conjugated metrics having their own geodesics.
The developments of this model led to various papers published in academic journals through peer review and specialized international meetings since 1977. It is described with classical yet powerful
tools ranging from Newtonian cosmology, though General relativity, to group theory. This web site, bigravitytheory.com, is a professional place to find, download and study related preprints and
published papers about the theory (see footnotes) and some popular initiation courses to standard and twin cosmologies as well as group theory (coming soon).
The bigravity theory represents an alternative to the standard model. Both share the same basic physical laws (the general relativity for example) but the bimetric model describes several
obervational data otherwise, and can propose explanations to problems the standard model cannot answer. For example the bimetric model explains the apparent absence of primordial antimatter and
describes what could be the physical expression of the "time reversal" of a particle. The acceleration of the cosmic expansion, classically charged to a negative pressure "cosmological constant" or
another kind of exotic "dark energy" whose nature remains unknown, is due to the repulsive effect of the twin universe on ours. There is also no need to resort to any "dark matter": the invisible
repulsive "twin matter" in the twin universe gives gravitational lensing effects; 2D bigravity computer simulations with joint gravitational instabilities showed the self-formation of a lacunar
large-scale structure, with twin matter conglomerates located inside giant hollow cells. They also showed the self-formation of barred spiral structures, stable on great numbers of turns, which could
explain the abnormal flatness of the galaxy rotation curve by a twin confinment effect, preventing galaxies from bursting under the effect of their high centrifugal force, with no need to add any
ingredient as "cold dark matter" nor ad hoc local adjustements of parameters. The same twin matter halos would also explain the "Pioneer anomaly" i.e. robotic spacecrafts decelerating while they
approach such an antigravitational barrier surrounding the outer solar system. The model also proposes a working principle for the mysterious quasars. Finally, due to the joint variations of physical
constants in the Radiation-Dominated Era of the universe and with high local energy densities, the model offers an alternative explanation for redshift and apparent farthest "dwarf galaxies". It also
suggests a "hyperspace transfer" transient mechanism for neutron stars reaching their critical limit (alternative to the "frozen" black hole theory) where a hypertoric bridge would briefly link the
geodesics of the two metrics, through which the matter in excess would escape before a quick closure of the spacetime surgery.
Author's career in astrophysics
Perhaps some readers know that, before publishing in the theoretical field of astrophysics and cosmology from 1987, I had a 15 years long activity in experimental research about magnetohydrodynamics
(MHD) from which I published various papers (if you want more information about this subject you can browse the web site MHDprospects.com devoted to my MHD career). Initially I was indeed a
specialist in fluid dynamics, graduated from Supaero (French National Higher School of Aeronautics and Space). I worked seven years at the Marseille Institute of Fluid Mechanics, France, between 1965
and 1972.
In 1972 I defended my EngD thesis,^1 whose subject linked both engineering and astrophysical aspects of ionized gases in two parts: at first, a presentation of the basis for the kinetic theory of
non-equilibrium plasmas, and thereafter its application to galactic dynamics (there I resumed the work from the astrophysicist Subrahmanyan Chandrasekhar, with a more compact matrix rewriting). I
integrated the French National Center for Scientific Research (CNRS) as an astophysicist at the Marseille Observatory, where I worked until retirement in 2003. At that time galactic dynamics was
precisely my first applied work following my thesis, where I made the Friedmann equations emerge from an elliptic solution of the Vlasov equation coupled with Poisson's equation. At the very begining
in this field, I published a rewriting of the Newtonian cosmology using my kinetic theory of non-equilibrium plasmas, resuming the work made by Arthur Milne and William McCrea (1934). This solution
besides allowed to recover the rotating universe of Otto Heckmann and Engelbert Schücking.^2
In 1977, I published two papers of what would then become my main work: the construction of a cosmological model with not only one universe, but two universes born from the same initial Big bang
singularity: the twin universe theory.^3,^4 This model intended at the beginning to propose an answer to the complete absence of primordial antimatter from the Big bang and the apparent baryon
assymetry of the universe. In this idea, matter and antimatter would have mainly populated two different, parallel universes. The twin universe with its antiparallel arrow of time would be the
T-symmetric of our universe. It would be also enantiomorph (P-symmetry) and mainly populated by antimatter (C-symmetry) because its CP-violation would be opposite. So the two metrics would present
each other as a complete CPT symmetry, and considering them as a whole, the universe would globally keep an unviolated parity. In 1984 I discovered Andrei Sakharov's scientific work for the first
time, collected outside USSR in a book.^5 I had the surprise to learn he also published the same idea before.^6,7,8,9 Alas Sakharov died before we could meet and discuss about it.
In 1988, I was the first to introduce a variable speed of light (VSL) cosmology,^10,11,12,13 with joint variations of all physical constants, as an alternative to the cosmic inflation theory. Unlike
other preceding authors who tried to vary the gravitational constant G without changing the speed of light c, and unlike other following authors who introduced a variable speed of light without
touching anything else, the model I propose preserves the Lorentz invariance and the energy. In the Radiation-Dominated Era of the universe, the time varies like the conformal time log t, and the
metric is conformally flat. The joint variations of all physical constants are combined to space and time scale factors changes, so that all equations and measurements of these constants remain
unchanged through the evolution of the universe. The only true "absolute constant" is the ratio G/c^2, as stipulated by Einstein's constant in the Einstein field equation.
From 1995 I published for the first time these parameters within a mature twin universe model, which approached its modern form.^15
Since that time the bigravity theory is developed from a mathematical physics underlying ground. The model is now fully geometrized, for example with the description of the twin matter naturally
arising from the group theory using a method invented in 1974 by the mathematician Jean-Marie Souriau.^16 I produced a series of preprints between 1996 and 1998 on these grounds.^17,18,19,20,21
Using this work I presented the most complete synthesis of the bimetric cosmological model at an international meeting on astrophysics and cosmology in 2001.^31
For years, the great majority of cosmologists did not understand the model cause of quite difficult topological and chronological concepts, or did not take the time to evaluate it. Except for Andrei
Sakharov who easily handled T-symmetric topologies and proposed the first bimetric CPT-symmetric cosmological model (but without bigravity), we can cite Michael Green and John Henry Schwarz, Robert
Foot and Ray Volkas, Nima-Arkani Ahmed, Savas Dimopoulos and Georgi Dvali, as well as the Nobel laureate Abdus Salam, who pushed towards descriptions of multiple universes interacting through
gravitation (but wihout any T-symmetry). As to other VSL theories, they change the speed of light through time but do not jointly modify other physical constants through a gauge process, so they
break Lorentz invariance and do not preserve energy. One can say although its great proposals, the bigravity theory has not successfully interested the cosmological community yet. But in 2007, I
incorporated a very selective club made of the best mathematicians and geometers, working in the field of mathematical physics through functional analysis. An international meeting took place in
August 2007. There, I could finally talk to an assembly made of specialists who not only were aware of the basis involved, but were also for the first time firing questions after questions, being
strongly interested in the model thanks to its mathematical physics basis and its links with observational data.^32,33,34,35,36,37
In 2008 I was invited at Imperial College London to present my work about variable speed of light cosmology, at Physical Interpretations of Relativity Theory conference.^38
Thanks to this good professional contact with the world of geometers, future developments are coming. Among other things we plan to finally make bigravity 3D computer simulations with powerful modern
CPUs. We also hope to interest some clever scientists in English-speaking parts of the world, who are working on exciting new ideas such as LQG (Loop Quantum Gravity). We do not think the superstring
theory could ever bring any concrete improvement in our knowledge of the universe. Not only asphyxiating other alternative theories, it has above all divorced long ago with true Popper's
falsifiability, betraying one of the most fundamental principle of scientific research.
You can read and study the bigravity theory with documents to download as PDF files below. The most complete description of this bimetric model to date is presented in the summary from 2001,^31 and
in the paper series from 2007.^32,33,34,35,36,37
^1 J.P. Petit (1972). "Applications of the kinetic theory of gases to plasma physics and galactic dynamics". Doctor of Engineering thesis, Aix-Marseille University, France.
^2 J.P. Petit (16–20 Septembrer, 1974). "Proceedings" in International meeting on spiral glaxies dynamics. Institut des Hautes Études Scientifiques (IHES), Bures-sur-Yvette, France.
^3 J.P. Petit (May 23, 1977). "Enantiomorphic universes with opposite time arrows". Comptes rendus de l'Académie des Sciences 263: 1315–1318. Paris: French Academy of Sciences.
^4 J.P. Petit (June 6, 1977). "Universes interacting with their opposite time-arrow fold". Comptes rendus de l'Académie des Sciences 284: 1413–1416. Paris: French Academy of Sciences.
^5 A.D. Sakharov (1982). "Collected Scientific Works" (tr. D. Ter Haar, D. V. Chudnovsky et al.). Marcel Dekker, NY. ISBN 0824717147.
^5 A.D. Sakharov (1984). "Œuvres scientifiques" (in French, tr. L. Michel, L.A. Rioual). Anthropos (Economica), Paris. ISBN 2715710909.
^6 A.D. Sakharov (1967). "CP violation and baryonic asymmetry of the Universe". ZhETF Pis'ma 5 (Tr. JETP Lett. 5, 24–27) (5): 32–35.
^7 A.D. Sakharov (1970). "A multisheet Cosmological model". preprint. Moscow, Russia: Institute of Applied Mathematics.
^8 A.D. Sakharov (1972). "Topological structure of elementary particles and CPT asymmetry". Problems in theoretical physics, dedicated to the memory of I.E. Tamm. Nauka, Moscow, Russia.
^9 A.D. Sakharov (1980). "Cosmological model of the Universe with a time vector inversion". ZhETF (Tr. JETP 52, 349-351) (79): 689–693.
^10 J.P. Petit (1988). An interpretation of cosmological model with variable light velocity. Modern Physics Letters A, 3 (16): 1527.
^11 J.P. Petit (1988). Cosmological model with variable light velocity: the interpretation of red shifts. Modern Physics Letters A, 3 (18): 1733.
^12 J.P. Petit; M. Viton (1989). Gauge cosmological model with variable light velocity: Comparizon with QSO observational data. Modern Physics Letters A, 4 (23): 2201–2210.
^13 P. Midy; J.P. Petit (June 1989). Scale invariant cosmology. The International Journal of Modern Physics D, 8: 271–280.
^14 J.P. Petit (July 1994). The missing mass problem. Il Nuovo Cimento B, 109: 697–710.
^15 J.P. Petit (1995). Twin Universes Cosmology. Astrophysics and Space Science (226): 273–307.
^16 J.M. Souriau (1997). "Structure of dynamical systems". Birkhäuser. ISBN 0817636951.
^16 J.M. Souriau (1970). Structure des systèmes dynamiques (in French, free download). Dunod. ISSN 0750-2435.
^17 J.P.Petit; P. Midy (1998). Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 1: Charges as additional scalar components of the momentum of a group
acting on a 10D-space. Geometrical definition of antimatter. Preprint.
^18 J.P. Petit; P. Midy (1998). Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 2: Geometrical description of Dirac's antimatter. Preprint.
^19 J.P. Petit; P. Midy (1998). Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 3: Geometrical description of Dirac's antimatter. A first
geometrical interpretation of antimatter after Feynmann and so-called CPT-theorem. Preprint.
^20 J.P. Petit; P. Midy (1998). Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 4: The twin group. Geometrical description of Dirac's antimatter.
Geometrical interpretation of antimatter after Feynmann and so-called CPT-theorem. Preprint.
^21 F. Henry-Couannier; G. d'Agostini, J.P. Petit (2005). I- Matter, antimatter and geometry. II- The twin universe model: a solution to the problem of negative energy particles. III- The twin
universe model plus electric charges and matter-antimatter symmetry. Preprint. arXiv:0712.0067
^22 J.P. Petit; P. Midy (1998). "Repulsive dark matter". Preprint.
^23 J.P. Petit; P. Midy (1998). "Matter ghost matter astrophysics. 1: The geometrical framework. The matter era and Newtonian approximation". Preprint.
^24 J.P. Petit; P. Midy (1998). "Matter ghost matter astrophysics. 2: Conjugated steady state metrics. Exact solutions". Preprint.
^25 J.P. Petit; P. Midy (1998). "Matter ghost-matter astrophysics. 3: The radiative era : The problem of the "origin" of the universe. The problem of the homogeneity of the early universe". Preprint.
^26 J.P. Petit; P. Midy (1998). "Matter ghost matter astrophysics. 4: Joint gravitational instabilities". Preprint.
^27 J.P. Petit; P. Midy, F. Landsheat (1998). "Matter ghost matter astrophysics. 5: Results of numerical 2d simulations. VLS. About a possible schema for galaxies' formation". Preprint.
^28 J.P.Petit; F.Landsheat (1998). "Matter ghost matter astrophysics. 6: Spiral structure". Preprint.
^29 J.P. Petit; P. Midy (1998). "Matter-ghost matter astrophysics. 7: Confinment of spheroidal galaxies by surounding ghost matter". Preprint.
^30 J.P. Petit; P. Midy (1998). "Questionable black hole". Preprint.
^31 J.P. Petit; P. Midy, F. Landsheat (June 2001). Twin matter against dark matter in International Meeting on Atrophysics and Cosmology. "Where is the matter?", Marseille, France.
^32 J.P. Petit; G. d'Agostini (August 2007). Bigravity as an interpretation of the cosmic acceleration. Colloque International sur les Techniques Variationnelles CITV, tr. International Meeting on
Variational Techniques. arXiv:0712.0067
^33 J.P. Petit; G. d'Agostini (August 2007). Bigravity: a bimetric model of the Universe. Exact nonlinear solutions. Positive and negative gravitational lensings. Colloque International sur les
Techniques Variationnelles CITV, tr. International Meeting on Variational Techniques. arXiv:0801.1477
^34 J.P. Petit; G. d'Agostini (August 2007). Bigravity: a bimetric model of the Universe with variable constants, inluding VSL (variable speed of light). Colloque International sur les Techniques
Variationnelles CITV, tr. International Meeting on Variational Techniques. arXiv:0803.1362
^35 J.P. Petit; G. d'Agostini (August 2007). "Bigravity: Bimetric model of the universe. Very large structure". Colloque International sur les Techniques Variationnelles CITV, tr. International
Meeting on Variational Techniques.
^36 J.P. Petit; G. d'Agostini (August 2007). "Bigravity: Bimetric model of the universe. Joint gravitational instabilities". Colloque International sur les Techniques Variationnelles CITV, tr.
International Meeting on Variational Techniques.
^37 J.P. Petit; G. d'Agostini (August 2007). "Bigravity: spiral structure". Colloque International sur les Techniques Variationnelles CITV, tr. International Meeting on Variational Techniques.
^38 J.P. Petit; G. d'Agostini (12-15 September 2008). Bigravity Variable Constants Model (A bimetric model of the Universe. Interpretation of the cosmic acceleration. In early time a symmetry
breaking goes with a variable speed of light era, explaining the homogeneity of the early Universe. The c(R) law is derived from a generalized gauge process evolution). 11^th international conference
on Physical Interpretations of Relativity Theory (PIRT XI), Imperial College, London. | {"url":"http://www.bigravitytheory.com/","timestamp":"2014-04-16T21:52:23Z","content_type":null,"content_length":"33656","record_id":"<urn:uuid:79d70cae-24d0-49a4-8427-218cd93782a3>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00480-ip-10-147-4-33.ec2.internal.warc.gz"} |
Remote Pairs, sudokuwiki.org
I coined Remote Pairs (back in 2005) to distinguish this new strategy/test/check, whatever, from the simpler more obvious pairs. Since first writing about it the strategy has been expanded in several
directions and is more common than first thought.
Pairs occur where two or more cells have the same two possible numbers. A locked pair is two such cells which lock each other in. For example, in the diagram on the right in the top row: 6 and 9
occur twice (labelled A and B) as a pair on the same row. This means the number 6 and the number 9 MUST both occur in these two cells. We can therefore eliminate other 6s and 9s from the same row.
Same rule applies to boxes and columns. This forms the basis of Test 3 in my solver.
On the diagram on the right I have marked all locked pairs with a red line. These are AB, AC, BD, CD and DE. You can load and view the whole of this board from the main Sudoku page. Look for Remote
Pair Test in the drop-down list of examples.
The point of this test/strategy is that we want to eliminate the 9 from cell Z and leave the 8. Can we do this logically, or must we guess? What about cell X which also looks like a candidate for
removing the 6 and 9?
Remote Pair 1 | {"url":"http://www.sudokuwiki.org/Print_Remote_Pairs","timestamp":"2014-04-19T01:48:25Z","content_type":null,"content_length":"9476","record_id":"<urn:uuid:75ca7ef9-9946-45ec-97a5-8e681b94af45>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00308-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Formula Of Inertia
its not quantifiable
I have to disagree there; the inertia for a specific body is definitely quantifiable. A particle's inertia is simply it's mass, whereas one can express the inertia of an extended body as an inertia
matrix. Either way, one can definitely quantify the inertia of a specified mass distribution. | {"url":"http://www.physicsforums.com/showthread.php?t=226732","timestamp":"2014-04-21T09:49:41Z","content_type":null,"content_length":"38654","record_id":"<urn:uuid:25c07571-ccef-4d93-b403-d28ea484ba65>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00372-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Convergence of the Poisson solver
2.1 PASS: Convergence of the Poisson solver
Stéphane Popinet
sh poisson.sh poisson.gfs
Required files
poisson.gfs (view) (download)
poisson.sh res-7.ref error.ref order.ref
Running time
52 seconds
This is one of the test cases presented in Popinet [31]. We solve the Poisson equation in a square domain with Dirichlet boundary conditions on all sides and the right-hand-side:
∇·U^⋆⋆(x,y) = −π^2(k^2+l^2)sin(π kx)sin(π ly)
with k = l = 3. The exact solution of the Poisson equation with this source term is
φ(x,y)=sin(π kx)sin(π ly).
Figure 1 illustrates the evolution of the maximum residual as a function of CPU time. Figure 2 illustrates the average residual reduction factor (per V-cycle). The evolution of the norms of the error
of the final solution as a function of resolution is illustrated on Figure 3. The corresponding order of convergence is given on Figure 4.
The curves labeled "Hypre" were obtained using the hypre module solver rather than Gerris’ built-in multilevel solver.
2.1.1 PASS: Convergence with a refined circle
Stéphane Popinet
sh ../poisson.sh circle.gfs
Required files
circle.gfs (view) (download)
res-7.ref error.ref order.ref
Running time
3 minutes 33 seconds
Same as the previous test but in order to test the accuracy of the gradient operator at coarse/fine boundaries, two levels of refinement are added in a circle centered on the origin and of radius
The solver still shows second-order accuracy in all norms (Figure 8).
2.1.2 PASS: Dirichlet boundary condition
Stéphane Popinet
sh ../poisson.sh dirichlet.gfs
Required files
dirichlet.gfs (view) (download)
res-7.ref error.ref order.ref
Running time
36 seconds
Similar to the previous test but with embedded solid boundaries and a non-trivial Dirichlet boundary condition. This test case was proposed by [21]. The boundary of the domain is defined by
The Poisson problem to solve in this domain is
which has the exact solution
which is used as boundary condition. | {"url":"http://gerris.dalembert.upmc.fr/gerris/tests/tests/poisson.html","timestamp":"2014-04-19T05:07:20Z","content_type":null,"content_length":"11020","record_id":"<urn:uuid:35ec9821-bc62-4df2-a57e-9beb1b4cd60e>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00526-ip-10-147-4-33.ec2.internal.warc.gz"} |
San Geronimo Algebra 1 Tutor
I hold a B.A. in molecular and cell biology from University of California at Berkeley and a B.S. in health Science from University of the Sciences in Philadelphia. I love teaching and enjoy
working with students of all ages. I served as a teacher's assistant in my math, English and Chinese classes since I excelled in those classes during middle and high school.
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49 Subjects: including algebra 1, calculus, physics, geometry
...My goal as a tutor is to help students acquire the skills that will help them excel at math. I am primarily interested in working during the summer but I am willing to work during the school
year.I have 5 years of experience teaching ESL in Asia. From August of 2001 to October 2004 I worked for TCD Tutorial Services in Bangkok Thailand.
10 Subjects: including algebra 1, geometry, statistics, accounting
...I currently teach Executive Functioning (organizational and study skills) through a tutoring agency that I work with. Besides my experience directly tutoring it, I have received approximately
10 hours of direct training in this area from a tutor/mentor. I have years of experience tutoring, including tutoring philosophy.
29 Subjects: including algebra 1, English, reading, writing
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Peak oil statistics
February 4th 2010, 11:20 AM #1
Feb 2010
Peak oil statistics
I was hoping someone might be able to help me with some statistical problems I'm having!
I am writing about Hubbert's Peak Theory and I was looking to do some stastical work on oil production figures. Basically, Hubbert's theory is that oil production will peak in some given year and
then decline after that. The curve will roughly be represented by the probability density function of a logistic distribution.
If I have a dataset of yearly oil production figures, how would I go about using excel or similar package to produce a best fit curve and then test this curve for it's credentials - skewness,
kurtosis, etc so that I can determine whether it really is a logistic distribution or otherwise?
I can calculate total oil production for past and future and put a figure on it - is this the same as the area under the graph showing yearly production figures? If it is, then I presume I have
the value of the integral between t(start of oil production) and t(end of oil production)? Will this be useful for construction my own graph?
Is there any way for me to draw my graph alongside a 'model' gaussian curve and a model logistic distribution PDF?
Thanks EVER so much!
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Lecture notes
Set one
: Basics of continuum mechanics - Continuum approximation, Tensor algebra and calculus, Frames, Material derivatives, Reynolds Transport Theorem, Conservation of Mass, Conservation of Momentum,
Conservation of Energy
Set two
: Basics of continuum mechanics - Cauchy stress, Shear and Rotation, Constitutive equations, Navier Stokes equations, Boundary conditions.
Homework one
: Due September 25th. | {"url":"http://www.math.utah.edu/~choheneg/fluids.html","timestamp":"2014-04-18T23:16:08Z","content_type":null,"content_length":"8254","record_id":"<urn:uuid:aa811529-3a8f-4f26-85ce-d17caeba4b36>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00277-ip-10-147-4-33.ec2.internal.warc.gz"} |
Pre-Calculus Help
In this section you'll find study materials for pre-calculus help. Use the links below to find the area of pre-calculus you're looking for help with. Each study guide comes complete with an
explanation, example problems, and practice problems with solutions to help you learn pre-calculus.
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Math Forum - Ask Dr. Math Archives: High School Puzzles
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Selected answers to frequently posed puzzles:
1000 lockers.
Letter+number puzzles.
Getting across the river.
How many handshakes?
Last one at the table.
Monkeys dividing coconuts.
Remainder/divisibility puzzles.
Squares in a checkerboard.
Weighing a counterfeit coin.
What color is my hat?
I'm trying to disprove a proof that says 1 is equal to two. I've been struggling with finding the error in my logic for quite some time now, but can't seem to quite get where I messed up.
When the 10 distances between 5 pairs of points on a line are listed from smallest to largest, the list reads: 2,4,5,7,8,k,13,15,17,19. What is the value of k?
Mrs. Shoe gave her prize winnings to her children in order... and all the prize money was divided equally amongst her children. How many children were there?
Consider the 180 digit number formed by putting the numbers from 10 to 99 together consecutively (1011121314...99). If you divide this number by 99, what is the remainder?
The leftmost digit of an integer of length digits is 3. In this integer, any two consecutive digits must be divisible by 17 or 23. The 2000th digit may be either a or b. What is the value of a+b?
Arrange the digits 0 to 9 such that the number formed by the first digit is divisible by 1, the number formed by the first two digits is divisible by 2, that formed by the first three digits
divisible by 3, and so forth; thus the number formed by the first 9 digits will be divisible by 9 and that formed by all 10 digits divisible by 10.
If I start with a penny and double it daily for 30 days, how many pennies do I have at the end?
How many pennies will we have at the end of 365 days if we begin on January 1 and double our money each day?
Can you help me make a schedule to staff an ice cream parlor?
How many 3 digit numbers have the digital sum of nine?
In the equation EVE/DID = .TALKTALKTALK... each letter corresponds to a different number, and EVE/DID is in lowest terms. What does each letter stand for?
How can I construct magic squares that work for even numbers?
Ten women are fishing in a long, narrow boat. One seat in the center of the boat is empty. The five women in the front of the boat want to change seats with the five women in the back of the
The number's nine digits contain all the digits from 1 to 9. When read from left to right the first two digits form a number divisible by two, the first three digits form a number divisible by
We have 3 pastures with grass of identical height, density and growth rate... How many oxen can be fed for 18 weeks?
Why is the area of our rectangle, formed from a square, 65 when the square's area was 64?
We can cut an 8x8 square with an area of 64 into four pieces and reassemble to get a 5x13 rectangle with an area of 65. Where does the extra 1x1 square come from?
What is the smallest positive cube that ends with the digits 2007?
A general strategy for solving problems such as finding the smallest whole number that when divided by 5, 7, 9, and 11 gives remainders of 1, 2, 3, and 4 respectively.
Given that 1J1 = 2, 3J5 = 34, 6J9 = 117, and 10J14 = 296, conjecture a value for 3J8.
Given six digits in a sequence: two 4's, two 5's, and two 6's, with one digit between the two 4's, two digits between the two 5's; and three digits between the two 6's, write this sequence of
In a 6-digit number, the sum of the digits is 43, and only two of the following three statements about the number are true: (1) It's a perfect square. (2) It's a perfect cube. (3) It's less than
There are 35 different sets of 7 prime numbers that sum to 100. Of those sets, which has the largest product, and which has the largest number? I'm using trial and error and it's very
frustrating. Is there a better way?
Are there any formal or systematic methods for solving problems that ask you to find the next number in a series?
A four digit number N leaves remainder 10 when divided by 21, remainder 11 when divided by 23 and remainder 12 when divided by 25. What is the sum of the digits of N?
Find pairs of positive integers whose greatest common factor is 1 and whose sum is 2000.
How can I determine the 3-digit integers a, b, and c when the ratio a:b:c is 1:3:5 and the digits of a, b, and c are 1,2,3,...,9, each appearing exactly once?
Find the pattern: 3*4=5, 8*4=0, 3*7=2, 1*2=9, and use it to complete the combinations: 5*5=?, 4*4=?, 5*7=?
Two players take turns choosing any number from 1-10, keeping a running sum of all the numbers. The first player to make this sum exactly 100 is the winner. Is there a surefire way to win this
Is there a formula for finding the month and the day on which Easter falls in a given year?
How many triangles can you draw on a square grid of dots of size x*x?
Is there an equation for how many squares there are in a rectangle divided up into 1cm blocks?
I have read the Dr. Math FAQ on finding the day of the week from the date. How do you get the part of the equation that deals with the month?
Please tell us what day of the week the Declaration of Independence was signed on, and the formula to determine it.
How can you get 73 by using 4 fours and any mathematical equation?
Four dogs are at four corners of a field. Each dog chases the dog to its right; all four run at the same speed and no acceleration is assumed. Where will they meet, and how long and how far will
they have run when they meet?
Each door conceals one item: a treasure, a rope, a key, and a lantern. You must find all four items in a particular order to keep the treasure.
Two people are trapped in a small freezer that is slowly getting colder and colder... Is it possible for them to escape? If so, how? What’s the maximum number of minutes required to escape given
ANY initial combination of button states?
What is the least number of jumps needed for all the frogs to trade sides?
Problem 1: A cylindrical hole six inches long is drilled straight through the center of a solid sphere. What is the volume remaining in the sphere? Problem 2: The classical stay-switch problem.
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Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
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Topic review (newest first)
2012-09-29 03:27:03
You are switching problems in the middle of solving one.
mukesh wrote:
sir its a good news i have solved the questions related to 5 prizes,sir its answer will be 10*9*10*9*10=81000
Sir m i right,
We are working on this problem. So please go back here and we will work on it there.
2012-09-29 02:59:25
sir its a good news i have solved the questions related to 5 prizes,sir its answer will be 10*9*10*9*10=81000
Sir m i right,
2012-09-29 02:24:36
sir plse describe it ,i didnt understand
2012-09-23 20:43:56
Hi mukesh;
The vowels are considered to be all alike. There are also 2 T's .Now it is a standard "MISSISSIPPI" problem.
2012-09-23 17:26:30
find the number of arrangements by the letters of the word 'INTERMEDIATE' so that order of vowels do not change.
Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
Hi;You are switching problems in the middle of solving one.
mukesh wrote:sir its a good news i have solved the questions related to 5 prizes,sir its answer will be 10*9*10*9*10=81000Sir m i right,
sir its a good news i have solved the questions related to 5 prizes,sir its answer will be 10*9*10*9*10=81000Sir m i right,
We are working on this problem. So please go back here and we will work on it there.http://www.mathisfunforum.com/viewtopic.php?id=18169
find the number of arrangements by the letters of the word 'INTERMEDIATE' so that order of vowels do not change. | {"url":"http://www.mathisfunforum.com/post.php?tid=18148&qid=233460","timestamp":"2014-04-20T06:13:20Z","content_type":null,"content_length":"17384","record_id":"<urn:uuid:6f5796c1-5007-41ac-8df6-596ad3f2a52a>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00232-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum Discussions - re Easter
Date: Feb 22, 1995 10:05 AM
Author: Calvin E. Piston
Subject: re Easter
While I certainly agree that giving a formula is not a topic for this
list, I do think that a problem such as that of determining the date
for Easter could be useful.
In the upper elementary or middle school grades, it provides a "real-
world" application of modular arithmetic.
For secondary students, it could serve as a basis fro a nice
historical AND mathematical research project: Why does the formula
work? (The answer to this is not beyond a good algebra -level
I apologize for not including this note with my first post.
* Calvin Piston *
* Department of Mathematics *
* John Brown University *
* Siloam Springs, AR 72761 *
* 501 - 524 - 7272 *
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Radar Cross-Section Formulation of a Shell-Shaped Projectile Using Modified PO Analysis
Modelling and Simulation in Engineering
Volume 2012 (2012), Article ID 328321, 9 pages
Research Article
Radar Cross-Section Formulation of a Shell-Shaped Projectile Using Modified PO Analysis
Department of Electrical and Electronic Engineering, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh
Received 30 April 2012; Revised 11 August 2012; Accepted 12 August 2012
Academic Editor: Azah Mohamed
Copyright © 2012 Mohammad Asif Zaman and Md. Abdul Matin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
A physical optics based method is presented for calculation of monostatic Radar Cross-Section (RCS) of a shell-shaped projectile. The projectile is modeled using differential geometry. The paper
presents a detailed analysis procedure for RCS formulation using physical optics (PO) method. The shortcomings of the PO method in predicting accurate surface current density near the shadow
boundaries are highlighted. A Fourier transform-based filtering method is proposed to remove the discontinuities in the approximated surface current density. The modified current density is used to
formulate the scattered field and RCS. Numerical results are presented comparing the proposed method with conventional PO method. The results are also compared with published results of similar
objects and found to be in good agreement.
1. Introduction
Prediction and measurement of Radar Cross-Section (RCS) have been a significant area of research for scientists and engineers for many years. The widespread uses of radar technology since the Second
World War have demanded accurate and efficient prediction of fields scattered by radar targets. The knowledge of echo characteristics of radar targets is of great importance for the design of
high-performance radars, as well as low visibility stealth targets [1, 2].
In radar technology, an antenna radiates electromagnetic (EM) energy. When an object is illuminated by the radar EM field, it reflects back some EM energy, which is received by an antenna. In
monostatic radar system, the transmitting and reception of the EM energy are done by the same antenna or by multiple antennas located very close to each other [1, 3]. In bistatic radar system,
separate antennas are used for transmitting and receiving, and the antennas are usually far away from each other [1]. The radar field reflective nature of an object is specified in terms of RCS. The
RCS is the area that a target would have to occupy to produce the amount of reflected power that is detected back at the radar [2, 4]. The RCS of an object depends on the viewing angles, the size,
geometry, and composition of the object, frequency and polarization of the radar signal, and so forth [3, 4]. Stealth technology concentrates on reducing the RCS of airplanes and missiles to make
them invisible to radars. Conversely, radar engineers are developing more sensitive radars that can detect low RCS targets. In both cases, accurate numerical simulation methods are essential for
design purposes.
Numerical simulation of RCS of an object requires calculation of the scattered field from the object for a given incident field. Several numerical methods exist for scattered EM field calculations.
Pure numerical methods such as Method of Moments (MoM), Finite-Difference Time-Domain (FDTD) method, Fast Multipole Method (FMM), and Transmission-Line Matrix (TLM) have been successfully used in
predicting RCS of radar targets [4, 5]. Conformal FDTD-based methods have also appeared in literature [6]. Recently, some variants of MoM have been developed for monostatic RCS formulation [7, 8] and
other scattering related problems [9]. These methods do not depend on the geometry and can be used for any arbitrary shaped objects. However, these methods are computationally demanding, and the high
simulation time for electrically large objects is not always acceptable for design and optimization problems. High-frequency asymptotic methods, such as Geometrical Optics (GO), Physical Optics (PO),
Uniform Geometrical Theory of Diffraction (UTD), and Physical Theory of Diffraction (PTD), have also been used for RCS formulation [4]. These methods are based on local interaction of EM fields.
Therefore, they are computationally less demanding than the pure numerical methods, and they require far less simulation time [10]. However, these methods are geometry dependent. For complex shaped
objects, the scattered field formulation can be tedious, especially when using UTD method [11]. GO and PO methods do not suffer to the same extent as UTD for complex geometry cases. The GO method is
the fastest among the high frequency techniques, but it is relatively less accurate [11]. The PO method gives much more accurate results compared to GO. It is a well-accepted method for formulating
scattered field from electrically large objects [12, 13].
Because of its relatively high accuracy, the PO method has been widely used for RCS formulation [14, 15]. The PO method is also computationally more efficient than pure numeric methods, making it
relatively faster which makes it an essential tool for aerospace designers [16]. Several modified versions of PO have been developed to further increase the speed and accuracy of the PO method [17,
18]. In this paper, a new modified PO method is used to formulate the monostatic RCS of a shell-shaped projectile.
This paper presents a detailed description and procedure of RCS calculation method of an object using PO method and modified PO method. The procedure includes the geometrical modeling of the
shell-shaped radar object, approximation of the induced surface current on the object, and formulation of the PO radiation integral in parametric space. Although the paper concentrates on RCS
formulation of a specific shell-shaped radar target, the analysis procedure is general and can be applied to objects of any geometry. In spite of the presence of many research and review articles [4
], a complete description of PO method-based procedure for RCS calculation is rare in literature. In addition to the complete calculation procedure, this paper also presents a new modified PO method.
The PO method approximates the induced surface current on the radar target and uses this current to calculate the scattered field [11, 19]. This approximation leads to discontinuous surface current
across the target surface near the shadow boundaries [19]. In the proposed method, a better approximate of the induced surface currents is used to remove the unnatural discontinuities and improve the
accuracy in calculation of the scattered field. The proposed method incorporates filtering inspired Fourier transform-based methods to remove the current discontinuity.
The paper is arranged as follows: in Section 2, the geometrical modeling of the shell-shaped projectile is described. Differential geometry-based definition of the surfaces and normals is derived
here. Section 3 contains approximation of the induced surface current on object and formulation of the scattered field and RCS. Numerical results are presented in Section 4. Finally, concluding
remarks are given in Section 5.
2. Geometrical Modeling
The first step in simulating the RCS of an object is to accurately model the surface of the object. This paper concentrates on the RCS of a shell-shaped object. A shell-shaped object is selected as
most projectiles represent this basic shape. A schematic diagram of the object and the three-dimensional coordinate system is shown in Figure 1. The object can be modeled by three different canonical
surfaces: a half sphere (surface ), a cylinder (surface ), and a flat disc (surface ). Most objects can be similarly modeled by a few common canonical shapes. For this reason, scattering from common
shapes such as flat plates, cones, and cylinders has received attention in literature since the 1960s [20, 21].
The origin of the coordinate system is taken at the center of the half sphere, and axis is taken as the axis of the cylinder. The three surfaces can be expressed by the following equations: Here, =
radius of the half-sphere = radius of the cylinder, and = length of the cylinder. For scattering problems and RCS formulation, it is often more convenient to express the surfaces in differential
geometry format rather than coordinate geometry format [11]. To express the surfaces in differential geometry format, the following parameters are used: Here, and are parameters. Note that the and
for are not the same as the and for or . For each surface the parameters are different and unrelated. Same parameter names are used for simplicity only. Using these parameters, the differential
geometric expressions of the surfaces are [22] The limits of the parameters are given in (2). Using these equations, it is possible to construct a three-dimensional model of the shell-shaped
projectile using computer coding. The computer generated model is shown in Figure 2. For computation, cm and cm are used.
Once the surfaces of the object are defined, it is necessary to define normal vectors on each point of the surface. These normal vectors are necessary for GO-, UTD-, or PO-based scattering
formulations [19]. The normal vectors,, can be calculated from the differential geometric expression of the surfaces using the following equation [11, 22]: The sign of the normal vectors is selected
so that they always point away from the surface. Using (4), the normal vectors on the three surfaces of the object are calculated to be Using (5), the normal vectors are plotted over the wire frame
of the object using computer coding. The results are shown in Figure 3. From visual inspection it is verified that the normals are perpendicular to the surface and points away from it. This verifies
the geometrical modeling performed in this section.
3. RCS Formulation Using Modified PO Method
To formulate the RCS of the object, the incident field must be defined first. As the radar and the target are usually very far away from each other, the incident field can be modeled as a plane wave,
implying that the direction of the wave, the direction of the electric field, and the direction of the magnetic field are perpendicular to each other. With respect to a coordinate system defined at
the source point of the wave, if the wave travels at direction, and the electric field is assumed to be polarized along direction, then the magnetic field will be polarized along direction. However,
in this case, the coordinate system is defined with respect to the object. So, coordinate transformation must be performed to find the expression of the incident field with respect to the defined
coordinate system [23, 24]. For a wave that is incident on the angle (, ), an direction polarized electric field converts to a polarized wave [24]. So, the incident electric field can be expressed as
[11, 24] Here, the amplitude of the incident electric field is assumed to be 1. The direction of the incident ray is given by The incident ray direction, along with electric and magnetic field
polarizations, is shown in Figure 1.
For monostatic RCS calculation, the reflected ray is PO method uses approximate expression of the surface current density induced on the surface of the object due to the incident field to find the
scattered field. The surface current density depends on the incident magnetic field. The incident magnetic field is given by [11, 19] Here, = intrinsic impedance of air. Equation (9) is valid only
for plane wave incidence. As the object can be considered to be very far away from the source for most radar applications, this assumption is justified. The PO approximates the surface current
density, , as [18] Equation (10) holds for a perfectly conduction object. For calculating RCS of a metallic object, this approximation is justified. Using (7), (9) along with (5) in (10), it is
possible to calculate . However, it is necessary to identify which part of the surface is illuminated by the incident field and which part is not for calculation of (10). This can be accomplished by
noticing the angle between and . When the vectors are perpendicular to each other, the incident field is tangent on the surface [22, 23]. These surface points indicate the shadow boundary, beyond
which the surface points will not be illuminated. So, surface points on which the angle between the normal vector and incident ray vector is greater than 90° are in the shadowed region. This
statement can be mathematically expressed as Thus using (10) and (11), the PO approximate of the surface current density can be formulated on , , and surfaces. It should be mentioned that (11) is
true only for objects that only have convex surfaces. If one region of the object creates shadow for another region, then (11) cannot be used to identify the illuminated region and shadowed region.
For the relatively simple geometry presented in this paper, (11) is sufficient for calculating the shadow boundaries.
From , the magnetic vector potential, can be calculated using the radiation integral [12, 19]: Here, is the distance from the object to the receiver. As monostatic radar cross-section is considered,
the source and the receiver are located at the same position. The position of the source point is expressed in polar coordinates as with meter. can easily be calculated from the coordinates of the
source and the object. The surface integrals can be performed over parametric space. The differential surface element can be expressed in terms of the parameters as [22, 23] Evaluating (13) for the
three surfaces and using it in (12): The scattered field, , can be formulated from as [12] Here, = wavelength of the incident field. The RCS, of the object is calculated from the scattered field
using (6) and (15) [1, 4]: The method described so far is the conventional PO method. Although sufficiently accurate, one of the weak points of this method is the approximation of the surface current
density, . From (10), it can be seen that abruptly changes to zero in the boundary between illuminated and shadow region. Practical surface current densities do not have this discontinuous nature.
This discontinuity arises because PO method does not take into account the creeping waves which exist in the boundary between shadowed region and illuminated region. These waves gradually decrease
with distance from the shadow boundaries, and the resulting surface current density decreases gradually.
So, to increase the accuracy of the PO method, the discontinuity in must be removed. In this paper, a filtering-based approach is proposed to remove the discontinuity in current distribution. The
spatial variation in the current distribution over the surface of the object can be compared with temporal variation of an analog signal in time domain. In Fourier expansion of time signals, the time
signal is imagined as superimposition of many sinusoidal signals, termed Fourier spectral components. A sharply varying time signal has high Fourier spectral components. A discontinuous signal has
large high Fourier spectral components. Using this analogy, the spatial variation of the surface current in scattering problems can be imagined as superimposition of many surface current components
with sinusoidal variation. These components can be isolated using Fourier transformation. It can be imagined that the rapidly varying high components of the surface current are responsible for the
sharp discontinuous spatial distribution. The high spectral components add in opposite phase with the other terms in the shadow region to create destructive interference. In absence of these high
spectral components, the other components will not completely cancel each other, and therefore there will be an oscillating distribution of surface current in the shadow region. These oscillating
distributions can be compared to the creeping waves. Thus filtering out high spectral components should create a smooth distribution of surface current densities which may accurately model the actual
current distribution with a higher degree of accuracy.
To perform filtering operations to make continuous, Fourier transform is performed in the parametric space. This produces Fourier domain representation of the PO surface current, . This can be
obtained using the following relation [24]: Here, Fourier transform is performed with respect to , and is another parameter corresponding to the frequency term in conventional Fourier transform of
time series data. The complete information of is implicit within . The discontinuity in with respect to will result in high values for large . This is analogous to high frequency terms in
discontinuous time series data [25]. As these high components contribute to the discontinuity, removing them should result in a smoother surface current density. How many high value components need
to be removed to produce an accurate surface current density cannot be analytically calculated due to the complex nature of the mathematics. Observing the discontinuity in the PO current in many
canonical problems found in literature [11] and testing the filter for different parameters, it is found through trial and error that removing 70% of high value components results in a relatively
accurate surface current distribution. A modified current density, , is constructed in space by removing 70% of high value components:
Now, the modified smoother current density, , in space can be obtained by using inverse Fourier transform [24]: Equation (19) is applicable for canonical geometries and surface of revolution. Now,
the discontinuity in with respect to should no longer be present in . However, may still be discontinuous with respect to . A similar analysis can be performed to remove this discontinuity by taking
Fourier transform of with respect to , eliminating high components and then taking inverse Fourier transform. Then a smoothed surface current density, , can be obtained which is continuous across the
surface of the object.
It is noted that that as some components of the current densities are filtered out, the energy of the conventional PO current may not be equal to the energy of the modified PO current. This may not
be consistent with conservation of energy. To rectify this problem, the filtered current densities must be multiplied by a constant scalar so that the resulting current densities have the same energy
as the PO current. The value of the constant can easily be calculated by calculating the energy of the conventional PO current and the unscaled modified PO current. This scaling ensures that no
changes have occurred in total energy of the surface currents.
Due to harmonic oscillating nature of Fourier transform, has oscillating characteristics near the shadow boundary which accurately represents diffraction patterns. Using instead of in (12)–(16) will
result in a more accurate estimate of RCS.
4. Numerical Results
For numerical analysis, the frequency of the incident field is taken to be 10GHz. The radius of the sphere and cylinder, cm, and the length of the cylinder, cm, are assumed. To perform numerical
analysis, the surface of the object must be divided into discrete points. Discrete values of the parameters and are taken to create discrete points on the surface. Large number of points increase
accuracy but also take considerable simulation time. Here, 22 values per wavelength are considered, and 204 values of per are considered. These values fall in the range of typical selected values for
numerical analysis using PO type method [19]. All numerical results are obtained using computer coding.
Figures 4 and 5 show the surface current density for two different incident field orientations. These current densities are obtained using conventional PO methods using (10). The discontinuity in the
current densities can easily be spotted in both figures. All the current densities are plotted after normalization process. The normalization is performed by dividing the current densities by the
maximum value of the current density. Due to this division, the normalized current density is unitless. The resulting normalized current densities are expressed in decibels by taking logarithm.
Using the proposed modified PO method, the surface current densities are smoothed. Fourier transform is used with respect to both parameters. The resulting smoothed current densities are shown in
Figures 6 and 7. The difference in current densities can easily be observed. Comparing Figure 4 with Figure 6 and Figure 5 with Figure 7, it can be seen that the current densities decrease gradually
when the proposed method is used whereas the conventional PO method creates sharp discontinuities. The oscillating pattern of the currents near the shadow boundary is created when the proposed method
is used. These accurately describe realistic diffracted fields [11].
To verify accuracy of the surface current densities approximated in the proposed filtering method, a simplified canonical problem of scattering from an infinite cylinder is considered. Considering an
infinitely long cylinder with axis parallel to the axis is illuminated by a plane wave. The orientation of the magnetic field of the incident wave is considered along the axis, and the orientation of
the electric field is assumed to be along the axis. The wave propagates along the axis. The PO current given by (10) is constant throughout the illuminated region of the cylinder. There will be no
variation of the surface current along the axis. The plane wave will illuminate only half the surface of the cylinder. If an angle, , is defined along the plane as the angle from the axis, then for a
plane wave propagating along the axis, the illuminated region spans from to . The PO current abruptly falls to zero outside this angle. The proposed filtering method adjusts this discontinuity and
makes the current smooth. The actual angular distribution of the current density can be formulated using a Method of Moments (MoM), which is a well-known benchmark method [19]. The current
distribution obtained from the PO method, the proposed modified PO method, and MoM are shown in Figure 8. It can be clearly observed that the proposed method produces a current distribution which
resembles the results obtained from MoM much closely compared to the PO method. This higher degree of accuracy in estimating the surface current ensures that the proposed method will generate more
accurate RCS results compared to the conventional PO method.
The RCS of the shell-shaped object is formulated using both conventional PO and the modified PO method. The RCS at plane as a function of zenith angle, , is shown in Figure 9.
RCS is expressed in dB with respect to 1m^2 area. This unit is represented as dBm^2 or dBsm [1, 4]. As the object is circularly symmetric around the axis, the scattered field and RCS are
independent of the azimuth angle, [11, 26]. It is seen that both methods give similar results in most angular regions. The variation in result comes for angular regions where the spherical surface is
illuminated. This is expected as the spherical surfaces are affected by the diffracted rays and creeping waves more than the other shaped surfaces [11]. Due to the better approximate of the surface
current, the results obtained from the proposed method are more accurate. It is noted that at only the half-sphere surface is illuminated. The RCS at this angle is around −15dBsm. The projected
area of the half sphere , which is equal to the RCS. This is expected for a spherical shaped object. This consistency verifies the numerical analysis.
The RCS of the projectile at 5.5GHz as a function is shown in Figure 10. The results from the proposed method and MoM are shown in the same plot for comparison. It can be observed that the results
obtained from the proposed method matches with the results obtained from MoM closely for most values of .
For the angles where RCS is high, the results of both methods are almost identical. The deviation arises for angles where RCS value is low. The higher degree of accuracy obtained from MoM comes at a
cost of complex computation and extensive simulation time. The errors produced by the proposed method are relatively small and acceptable for many applications. In most cases the accuracy from PO
method is sufficient, and the proposed method is expected to be more accurate than the PO method.
The RCS of the shell-shaped projectile for frequencies 3GHz, 6GHz, 10GHz, and 14GHz as a function is shown in Figure 11. The obtained RCS pattern is similar to the RCS patterns of objects of
similar shape described in [4, 16].
The computer simulation was performed on an Intel Core i5-2430M 2.4GHz CPU with 2.94 GB usable RAM. The simulation time of conventional PO and modified PO is compared in Table 1.
From Table 1, it can be seen that for lower frequency, the simulation time of conventional PO and the proposed modified PO is comparable. For higher frequency, the modified PO method takes 80% to 90%
longer time to simulation compared to the conventional PO method. This excess simulation time can be expressed by the fact that for high-frequency simulation, the number of discrete surface points
selected is larger. As there are 22 values selected per wavelength and the wavelength is smaller for high-frequency simulation, the overall number of point increases. The filtering of the surface
currents requires this additional simulation time due to large number of points. However, the proposed method is much faster compared to Method of Moments. For each simulation, the Method of Moments
requires over 500 seconds of simulation time.
5. Conclusion
RCS formulation procedure using PO method is described in detail in this paper. The paper covers geometrical modeling of shell-shaped projectile using differential geometry, formulation of surface
current density, evaluation of the scattered field integral in parametric space, and monostatic RCS formulation. A modified PO method is presented which approximates surface current density more
accurately. The additional computational steps require Fourier transform and inverse Fourier transform only. These can be easily incorporated in computer code using well-known Fast Fourier Transform
(FFT) algorithm. Thus the accuracy is increased without significant increase in computational complexity. The obtained results using the modified PO method are consistent with similar results found
in literature.
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26. M. Andreasen, “Scattering from bodies of revolution,” IEEE Transactions on Antennas and Propagation, vol. 13, no. 2, pp. 303–310, 1965. | {"url":"http://www.hindawi.com/journals/mse/2012/328321/","timestamp":"2014-04-21T16:59:55Z","content_type":null,"content_length":"249581","record_id":"<urn:uuid:bed97798-bb4d-4889-915c-eb6648ef370a>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00513-ip-10-147-4-33.ec2.internal.warc.gz"} |
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When can't spaces of correspondences distinguish type $II_{1}$ factors?
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If $M$ is a type $II_{1}$ factor with trace $\tau$, let $Corr(M)$ denote the space of unitary equivalence classes of $M-M$ correspondences (binormal $M-M$ bimodules) equipped with Popa's analogue of
Fell's topology. All details of the above can be found in here.
What are some examples of pairs of nonisomorphic type $II_{1}$ factors $M,N$ such that their associated spaces of correspondences $Corr(M), > Corr(N)$ are homeomorphic?
oa.operator-algebras von-neumann-algebras
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Need Some Calc. Help
November 25th 2009, 02:24 PM #1
Oct 2009
Need Some Calc. Help
Find the value of k so that f(x)=xe^x/k has a critical value at x=10.
I am pretty sure I have to find the derivative here, using the product rule? But I'm not exactly how to find the derivative of e^x/k and after that I'm not sure what to do. Set the derivative
equal to 10 perhaps? Someone help me clear this up, please.
$k$ is a constant, so we can pull it out, getting $\frac{dy}{dx}=\frac{1}{k}(\frac{d}{dx}(xe^x))$. Use the product rule to differentiate $xe^x$, set $\frac{dy}{dx}=0$, and then plug $10$ in for
$x$. Now, solve for $k$.
Find the value of k so that f(x)=xe^x/k has a critical value at x=10.
I am pretty sure I have to find the derivative here, using the product rule? But I'm not exactly how to find the derivative of e^x/k and after that I'm not sure what to do. Set the derivative
equal to 10 perhaps? Someone help me clear this up, please.
assuming you mean ...
$f(x) = xe^{\frac{x}{k}}$
$f'(x) = x \cdot \frac{1}{k}e^{\frac{x}{k}} + e^{\frac{x}{k}}$
$f'(x) = e^{\frac{x}{k}}\left(\frac{x}{k} + 1\right)$
critical values occur where x = 0 ...
$\frac{x}{k} = -1$
at $x = 10$ , $k = -10$
Thank you very much. There are a couple of other problems I'm having trouble with as well.
"The quantity of a drug in the bloodstream t hours after a tablet is swallowed is given by: q(t) = 351(e^-t - e^-2t) in milligrams
When is the quantity of the drug in the bloodstream maximized? That is, find the value of t that maximizes q(t). Round your answer to two decimal places."
This is a surge function problem, but it's not in the normal format of ate^-bt that I'm used to working with. I can do (1/b) to find t, but I don't know how to find b. My first thought would be
to combine e^-t - e^-2t to find b but I'm not sure if I can combine those just because they have the same base.
Also, for part of a problem I need the derivative of e^-kt, would that just be -ke^-kt? Or is k positive?
Thank you very much. There are a couple of other problems I'm having trouble with as well.
"The quantity of a drug in the bloodstream t hours after a tablet is swallowed is given by: q(t) = 351(e^-t - e^-2t) in milligrams
When is the quantity of the drug in the bloodstream maximized? That is, find the value of t that maximizes q(t). Round your answer to two decimal places."
This is a surge function problem, but it's not in the normal format of ate^-bt that I'm used to working with. I can do (1/b) to find t, but I don't know how to find b. My first thought would be
to combine e^-t - e^-2t to find b but I'm not sure if I can combine those just because they have the same base.
Also, for part of a problem I need the derivative of e^-kt, would that just be -ke^-kt? Or is k positive?
you're making this much harder on yourself than it really is ...
$q(t) = 351(e^{-t} - e^{-2t})$
$q'(t) = 351(-e^{-t} + 2e^{-2t})$
$q'(t) = 0$ when $2e^{-2t} - e^{-t} = 0$
$e^{-t}(2e^{-t} - 1) = 0$
solve the single correct factor than can equal 0 for t
next time start a new problem with a new post.
I solved for 2e^-t - 1 = 0 and got t = .69, but when I put that back into the equation I get error so I think I made a mistake somewhere.
November 25th 2009, 02:32 PM #2
Oct 2009
November 25th 2009, 02:33 PM #3
November 25th 2009, 03:18 PM #4
Oct 2009
November 25th 2009, 03:27 PM #5
November 25th 2009, 04:23 PM #6
Oct 2009
November 25th 2009, 07:03 PM #7 | {"url":"http://mathhelpforum.com/calculus/116736-need-some-calc-help.html","timestamp":"2014-04-17T16:06:38Z","content_type":null,"content_length":"54772","record_id":"<urn:uuid:9d9c3a8a-cb80-405d-9c6b-0b24d907dd1f>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00563-ip-10-147-4-33.ec2.internal.warc.gz"} |
9. A Clothesline Has A Linear Mass Density Of 0.28 ... | Chegg.com
9. A clothesline has a linear mass density of 0.28 kg/m, and is stretched with a tension T = 30 N. One end is given a sinusoidal motion with a frequency of 5.0 Hz and amplitude of 0.010 m. At the
time t = 0 the end has zero displacement and is moving in the positive direction.
(a) Find the wave speed, the amplitude, the frequency, period, wavelength, and the wave
number, of the wave motion.
(b) Write a wave function describing the wave.
(c) Find the position of the point at x = 0.25 m at time t = 0.10 sec. | {"url":"http://www.chegg.com/homework-help/questions-and-answers/9-clothesline-linear-mass-density-028-kg-m-stretched-tension-t-30-n-one-end-given-sinusoid-q1263621","timestamp":"2014-04-16T05:55:16Z","content_type":null,"content_length":"21412","record_id":"<urn:uuid:3173b7fe-ed60-47cd-a249-4d7bead0b5f7>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00422-ip-10-147-4-33.ec2.internal.warc.gz"} |
Association submodel
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Contained submodel(s)
Submodel others is a relation submodel for a relation between balls and itself.
Condition for existence of submodel
Units: boolean
effect = hypots<bounce_distance and rightway
bounce_distance is the variable bounce distance in this submodel.
hypots is the variable hypots in this submodel.
rightway is the variable rightway in this submodel.
Units: array(1,2)
comps = forces*[distances]/hypots
forces is the variable forces in this submodel.
hypots is the variable hypots in this submodel.
[distances] is the variable distances in this submodel.
bounce distance
Units: 1
bounce distance = sizes_b1+sizes_b2
sizes_b1 is the variable sizes in balls
sizes_b2 is the variable sizes in balls
Units: array(1,2)
distances = [posns_b1]-[posns_b2]
[posns_b1] is the variable posns in balls
[posns_b2] is the variable posns in balls
Units: 1
hypots = hypot(element([distances],1),element([distances],2))
[distances] is the variable distances in this submodel.
Units: 1
forces = pow(bounce_distance/hypots,4)
bounce_distance is the variable bounce distance in this submodel.
hypots is the variable hypots in this submodel.
Units: boolean
rightway = index(1)<index(2)
Submodel balls is a fixed membership submodel with 32 members.
Units: array(1,2)
Actions = sum({[comps_b1]})-sum({[comps_b2]})
{[comps_b1]} is the variable comps in others
{[comps_b2]} is the variable comps in others
Units: boolean
green? = any(index(1)==[2,3,4,5,7,8,9,12,13,14,17,18,22,23,27,32])
Units: array(1,2)
posns = [p]
[p] is the compartment p in balls/dims
Units: 1
x = element([p],1)
[p] is the compartment p in balls/dims
Units: 1
y = element([p],2)
[p] is the compartment p in balls/dims
Units: 1
sizes = 15
Units: 1
wees = sizes/10
sizes is the variable sizes in this submodel.
Units: 1
weex = x/10+45
x is the variable x in this submodel.
Units: 1
weey = y/10+45
y is the variable y in this submodel.
Contained submodel(s)
Submodel green is a conditional submodel.
Condition for existence of submodel
Units: boolean
cond1 = green_
green_ is the variable green? in balls
Units: 1
size = sizes
sizes is the variable sizes in balls
Units: 1
x = x
x is the variable x in balls
Units: 1
y = y
y is the variable y in balls
Submodel dims is a fixed membership submodel with 2 members.
Units: 1
Initial value: if index(1)==1 then fmod(25*(index(2)-1),112.5) else 16.66*int((index(2)-1)/4.5)
Inflows: move
Units: 1
Initial value: 0
Inflows: flow1
Units: 1
move = v
v is the compartment v in this submodel.
Units: 1
flow1 = force/mass
force is the variable force in this submodel.
mass is the variable mass in this submodel.
Units: 1
gravity = 10*mass*if p<0 then 0-p elseif p>100 then 100-p else 0
mass is the variable mass in this submodel.
p is the compartment p in this submodel.
Units: 1
force = a+element([Actions],index(1))+rand_var(-10,10)
Units: 1
mass = sizes*sizes/100
sizes is the variable sizes in balls
Submodel orange is a conditional submodel.
Condition for existence of submodel
cond1 = missing
Units: 1
size = sizes
sizes is the variable sizes in balls
Units: 1
x = x
x is the variable x in balls
Units: 1
y = y
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Maximum Entropy Distributions
Posted by David Corfield
While we eagerly await Tom’s second post on Entropy, Diversity and Cardinality, here’s a small puzzle for you to ponder. If all of the members of that family of entropies he told us about are so
interesting, why is it that so many of our best loved distributions are maximum entropy distributions (under various constraints) for Shannon entropy?
For example, the normal distribution, $N(\mu, \sigma^2)$, is the maximum Shannon entropy distribution for distributions over the reals with mean $\mu$ and variance $\sigma^2$. And there are others,
including exponential and uniform (here) and Poisson and Binomial (here). So why no famous distributions maximising Tsallis or Renyi entropy? (People do look at maximising these, e.g., here.)
Follow up question: has this property of the normal distribution anything to do with the central limit theorem? This is relevant.
Posted at November 2, 2008 5:43 PM UTC
Re: Maximum Entropy Distributions
As someone who until recently had no clue about entropy, let alone maximizing it, I’d like to share this. It’s a pleasant account of how, if you wanted to model the behaviour of an expert translator
of French (e.g. if you wanted to train a machine to do it?), you’d naturally run into the concept of maximum entropy.
As it says, the maximum entropy method is this:
given a collection of facts, choose a model which is consistent with all the facts, but otherwise as uniform as possible.
Posted by: Tom Leinster on November 3, 2008 12:19 AM | Permalink | Reply to this
Re: Maximum Entropy Distributions
The translation example is a nice one.
It does however also raise some interesting questions.
The translation of a term or word is highly dependent on context, of course. This is something that will be supposedly picked up from a corpus of examples but it is clear that there is not even a
many-many matching between words and phrases between two languages. Somehow there is a local-to-global matching involved. Taking this into account some workers in translation and text analysis seem
to use methods adapted from compression theory (PPM and related methods). They match at various levels character-word-phrase-textblock etc. matching patterns of longer and longer length, including
flags for parsing information etc. I do not know the theory at all well, but have discussed it on many occasions with a colleague, Bill Teahan.
My point in mentioning this is that the actual measures that these text analysis/categorisation practioners use are based on (crossed) entropy at the various textual levels. This always seemed to me
to be not that distant from the idea of hierarchical system and thus of categorical ideas. I know this is vague and it is not at all clear to me how to approach such a measure in a sensible way.
Perhaps Tom has a clearer sense of what might be done.
PS. for PPM see
and for the entropy link the two papers,
# W.J. Teahan, Probability estimation for PPM.
# T. Schürmann and P. Grassberger, Entropy estimation of symbol sequences, Chaos, Vol. 6, pp. 414-427, September 1996.
Posted by: Tim Porter on November 3, 2008 7:37 AM | Permalink | Reply to this
Re: Maximum Entropy Distributions
based on (crossed) entropy at the various textual levels
Crossed modules I know, crossed complexes yes, but crossed entropy……???
Posted by: Tom Leinster on November 3, 2008 8:02 AM | Permalink | Reply to this
Re: Maximum Entropy Distributions
Pinched from
In information theory, the cross entropy between two probability distributions measures the average number of bits needed to identify an event from a set of possibilities, if a coding scheme is
used based on a given probability distribution q, rather than the “true” distribution p.
I found the term in one of the PPM based papers, so am not claiming I understand it!!!!!
I do know that in text categorisation they use a predictive algorithm, which given the text predicts the next textblock (i.e. character word etc depending at what level the analysis is working at
that time). My guess is that asthe various levels are distributed differently, the various predictive values from the levels are somehow compared. Again I do not know this theory merely having talked
with a researcher in an effort to try to apply a categorical viewpoint.
Posted by: Tim Porter on November 3, 2008 9:18 AM | Permalink | Reply to this
Re: Maximum Entropy Distributions
So in terms of our discussion, the cross entropy between $P$ and $Q$ is your expected surprise if you believe the world is distributed according to $Q$, but really it is $P$.
Posted by: David Corfield on November 3, 2008 9:42 AM | Permalink | Reply to this
Re: Maximum Entropy Distributions
I think that is probably a good interpretation.
Posted by: Tim Porter on November 3, 2008 2:06 PM | Permalink | Reply to this
Re: Maximum Entropy Distributions
I am probably talking something that you already know, so feel free to ignore my comments if you think they are not relevant. :)
Cross entropy is a very important concept in information theory and some people think even that a maximum entropy principle should be replaced by a maximum cross-entropy. You can see a detailed
argumentation about that in this book draft: http://arxiv.org/abs/0808.0012
It is also known as the Kullback-Leibler divergence, which is a measure of distance between two probability distributions (although it is not symmetric) with the very interesting property that for
two parametric distributions with infinitely close parameters p and p+dp, it becomes the Fisher information metrics.
I hope this may help somehow…
Posted by: DisorderedBrain on November 5, 2008 3:11 PM | Permalink | Reply to this
Re: Maximum Entropy Distributions
Tim wrote:
I do know that in text categorisation they use a predictive algorithm…
Let’s apply categorification to categorization, and confuse the world!
Posted by: John Baez on November 3, 2008 10:08 PM | Permalink | Reply to this
Re: Maximum Entropy Distributions
From notes I see I once knew that there’s a nice geometric interpretation of the distributions that maximise Tsallis entropy here. If you place a uniform distribution on the $n$-sphere of radius $\
sqrt{n}$, and project onto $m$ of the co-ordinates, Poincaré had observed that in the limit as $n$ tends to infinity, this $m$-vector is distributed according to an $m$-variate Gaussian distribution
with unit covariance matrix. Now these Gaussian distributions are maximisers of the Shannon entropy with this covariance matrix. Some of the Tsallis MaxEnt distributions come from projecting $m$
dimensions from a uniform distribution over an $n$-sphere. You can pick up distributions corresponding to different covariance matrices by looking at uniform distributions over corresponding
Posted by: David Corfield on November 3, 2008 4:58 PM | Permalink | Reply to this
Re: Maximum Entropy Distributions
David wrote, challengingly,
If all of the members of that family of entropies he told us about are so interesting […]
If you’re making the sceptical point I think you are, then I probably agree. My belief is that, in some sense, Shannon entropy $H_1$ has prime position among the family $H_\alpha$ ($\alpha \geq 0$)
of entropy measures that I discussed. The others are still interesting, but not, I think, as important.
By way of analogy, there are many interesting invariants of topological spaces that can be defined via homology: the rank of the $3$rd homology group, for instance. But in some sense it’s the Euler
characteristic that has primacy: it’s the invariant that behaves most like cardinality.
Certainly Shannon entropy has properties not shared by the $\alpha$-entropies $H_\alpha$ for $\alpha eq 1$. Indeed, Rényi made exactly this point when he introduced $H_\alpha$. In particular, he
observed that while $H_\alpha$ shares with $H_1$ the property that the entropy of a product is the sum of the entropies, it does not share the property that the entropy of a convex combination $\
lambda \mathbf{p} + (1 - \lambda)\mathbf{q}$ can be expressed in terms of $\lambda$, $H(\mathbf{p})$ and $H(\mathbf{q})$.
In Part 2 (coming soon!), I’ll define $\alpha$-entropy, $\alpha$-diversity and $\alpha$-cardinality of finite probability spaces in which the underlying set is equipped with a metric. Curiously, in
this extended setting it’s the case $\alpha = 2$ that seems to be best-understood. But I’m sure $\alpha = 1$ must play a special role.
Continuing David’s sentence:
[…] why is it that so many of our best loved distributions are maximum entropy distributions (under various constraints) for Shannon entropy?
A priori, that doesn’t exclude the possibility that they’re also maximum entropy distributions for other entropies $H_\alpha$.
As I understand it, you’re mostly referring to distributions on the real line, and the task is to find the distribution having maximum entropy subject to certain constraints (e.g. ‘mean must be $0$
and standard deviation must be $1$’). But let’s go back to a much simpler case: distributions on a finite set, subject to no constraints. As we saw in Part 1, the distribution with maximum entropy is
the uniform distribution — and that’s true for $\alpha$-entropy, no matter what $\alpha$ is. So in this case, the distribution that maximizes the $\alpha$-entropy is the same for all $\alpha$.
I’ll talk more about maximizing entropy in Part 2. There seem to be some unsolved problems.
Posted by: Tom Leinster on November 3, 2008 11:32 PM | Permalink | Reply to this
Re: Maximum Entropy Distributions
The Pareto distribution is equivalent to a q-exponential, that can be obtained by extremising Tsallis entropy.
Posted by: DisorderedBrain on November 5, 2008 2:23 PM | Permalink | Reply to this
Re: Maximum Entropy Distributions
Interesting. This is explained for generalized Pareto distributions here.
Posted by: David Corfield on November 5, 2008 2:53 PM | Permalink | Reply to this
Re: Maximum Entropy Distributions
Extremizing a Rényi entropy can be used to yield a power-law distribution, if one breaks out the Lagrange multipliers and maximizes a functional with something like a fixed-average-energy condition,
$L_q(p) = \frac{1}{1-q}\log\sum_i p_i^q -\alpha\sum_i p_i - \beta\sum_i p_i E_i.$
See arXiv:cond-mat/0402404, for example. Under a different condition, when a covariance matrix is specified, maximizing the Rényi entropy gives a Student’s-$t$ distribution (arXiv:math.PR/0507400).
Posted by: Blake Stacey on November 8, 2008 12:28 AM | Permalink | Reply to this
Re: Maximum Entropy Distributions
Regarding David’s last question, this is probably the best answer.
Regarding the observation typically ascribed to Poincare which David mentioned here, the history seems a little murky. I haven’t tried checking original sources myself, but this paper, which proves a
stronger version of the statement, claims that it can’t be found in Poincare’s writings.
Posted by: Mark Meckes on November 5, 2008 8:13 PM | Permalink | Reply to this
Re: Maximum Entropy Distributions
Here’s a guess.
Shannon entropy can be derived as the limit of a distribution’s multiplicity as the elements increase to infinity.
Out of all the distributions you are choosing one that can exist in the most number of ways given the constraints. Its like the mean of all possible distributions.
To understand what the other distributions are maximizing exactly one should look at their discrete forms before the limit is taken.
Posted by: Jonathan Fischoff on November 17, 2008 8:43 PM | Permalink | Reply to this
Re: Maximum Entropy Distributions
Whoops. I meant to say “mode” where I said “mean” above. My bad.
Posted by: Jonathan Fischoff on November 18, 2008 9:27 PM | Permalink | Reply to this
Re: Maximum Entropy Distributions
A paper on Maximum Entropy on Compact Groups.
On a compact group the Haar probability measure plays the role as uniform distribution. The entropy and rate distortion theory for this uniform distribution is studied. New results and simplified
proofs on convergence of convolutions on compact groups are presented and they can be formulated as entropy increases to its maximum. Information theoretic techniques and Markov chains play a
crucial role. The rate of convergence is shown to be exponential. The results are also formulated via rate distortion functions.
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Topic: bug in Integrate[]
Replies: 0
bug in Integrate[]
Posted: Dec 12, 1999 8:01 PM
BesselJ[2 i + 1, A Sin[t]] Sin[t], {t, 0, 2 Pi},
Assumptions -> i \[Element] Integers]
It gives 0 if A>0. Rather strange condition huh? Esp. if one
thinks about symmetry of Bessel function.
Anyway, if I put a number (e.g. 3) instead of i, v4.0 gives
other result:
(A^3*Pi*HypergeometricPFQ[{5/2}, {3, 4}, A^2/4])/64
Any clue? | {"url":"http://mathforum.org/kb/thread.jspa?threadID=233636&messageID=783313","timestamp":"2014-04-17T01:06:52Z","content_type":null,"content_length":"13809","record_id":"<urn:uuid:1c9e830a-6e00-4829-8a51-01628617262b>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00245-ip-10-147-4-33.ec2.internal.warc.gz"} |
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This Article
Bibliographic References
Add to:
Some Sequential Algorithms for a Generalized Distance Transformation Based on Minkowski Operations
November 1992 (vol. 14 no. 11)
pp. 1114-1121
ASCII Text x
X. Wang, G. Bertrand, "Some Sequential Algorithms for a Generalized Distance Transformation Based on Minkowski Operations," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14,
no. 11, pp. 1114-1121, November, 1992.
BibTex x
@article{ 10.1109/34.166628,
author = {X. Wang and G. Bertrand},
title = {Some Sequential Algorithms for a Generalized Distance Transformation Based on Minkowski Operations},
journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence},
volume = {14},
number = {11},
issn = {0162-8828},
year = {1992},
pages = {1114-1121},
doi = {http://doi.ieeecomputersociety.org/10.1109/34.166628},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - JOUR
JO - IEEE Transactions on Pattern Analysis and Machine Intelligence
TI - Some Sequential Algorithms for a Generalized Distance Transformation Based on Minkowski Operations
IS - 11
SN - 0162-8828
EPD - 1114-1121
A1 - X. Wang,
A1 - G. Bertrand,
PY - 1992
KW - mathematical morphology; image processing; sequential algorithms; generalized distance transformation; Minkowski operations; binary images; medial axis transformation; point-to-point distance;
image processing; mathematical morphology; transforms
VL - 14
JA - IEEE Transactions on Pattern Analysis and Machine Intelligence
ER -
A generalized distance transformation (GDT) of binary images and the related medial axis transformation (MAT) are discussed. These transformations are defined in a discrete space of arbitrary
dimension and arbitrary grids. The GDT is based on successive morphological operations using alternatively N arbitrary structuring elements: N is called the period of the GDT. The GDT differs from
the classical distance transformations based on a point-to-point distance. However, the well-known chessboard, city-block, and hexagonal distance transformations are special cases of the one-period
GDT, whereas the octagonal distance transformation is a special case of the two-period GDT. In this paper, both one- and two-period GDTs are discussed. Different sequential algorithms are proposed
for computing such GDTs. These algorithms need a maximum of two scannings of the image. The computation of the MAT is also discussed.
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[34] D. Schonfeld, and J. Goutsias, "Morphological representation of discrete and binary images,"IEEE Trans. Acoust. Speech Signal Processing, 1991, vol. 39, pp. 411-414.
[35] D. Schonfeld and J. Goutsias, "On the morphological representation of binary images in a noisy environment,"J. Visual Commun. Image Representation, vol. 2, no. 1, pp. 17-31, 1991.
[36] Z. Zhou and A. N. Venetsanopoulos, "Morphological skeleton representation and shape recognition," inProc. Int. Conf. Acoustics, Speech Signal Processing, vol. II, 1988, pp. 948-951.
[37] Z. Zhou and A. N. Venetsanopoulos, "Pseudo-Euclidean morphological skeleton transform for machine vision," inProc. Int. Conf. Acoustics Speech Signal Processing, 1989, pp. 1695-1698.
[38] D. Schonfeld and J. Goutsias, "Optimal morphological pattern restoration from noisy binary images,"IEEE Trans. Patt. Anal. Machine Intell, vol. 13, no. 1, pp. 14-29, Jan. 1991.
[39] C. Lantuejoul, "Skeletonization in quantitative metallography,"Issues of Digital Image Processing(R. M. Haralick and J. C. Simon, Eds.). Groningea, The Netherlands: Sijthoff and Noordhoff, 1980.
[40] U. Montanari, "A method for obtaining skeletons using a quasi-Euclidean distance,"J. Assoc. Comput. Machinery, vol. 15, pp. 600-624, Oct. 1968.
[41] X. Wang and G. Bertrand, "Computation of the morphological distances," Internal Rep., ESIEE, 1992.
Index Terms:
mathematical morphology; image processing; sequential algorithms; generalized distance transformation; Minkowski operations; binary images; medial axis transformation; point-to-point distance; image
processing; mathematical morphology; transforms
X. Wang, G. Bertrand, "Some Sequential Algorithms for a Generalized Distance Transformation Based on Minkowski Operations," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14,
no. 11, pp. 1114-1121, Nov. 1992, doi:10.1109/34.166628
Usage of this product signifies your acceptance of the
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Illustration 2.5
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Illustration 2.5: Motion on a Hill or Ramp
Animation 1 | Animation 2 | Animation 3
Please wait for the animation to completely load.
A putted golf ball travels up a hill and then down again (position is given in meters and time is given in seconds). Restart. When an object (like a golf ball) travels up or down an inclined ramp or
hill, its motion is often characterized by constant, nonzero acceleration. If the incline of the hill is constant, then the motion of the object can also be considered straight-line motion (or
one-dimensional motion). It is convenient to analyze the motion of the golf ball by defining the +x axis to be parallel to the hill and directed either upward or downward along the hill as shown in
Animation 1.
Here are some characteristics of the motion that you should convince yourself are true:
• In Animation 1 the +x direction is defined to be down the hill. Therefore, when the ball moves down the hill, it is moving in the +x direction and thus v[x] is positive. When the ball moves up
the hill, it is moving in the -x direction and thus v[x] is negative.
• As the golf ball is traveling up/down the hill, is it slowing down or speeding up? Well, the answer to this question depends on what you mean by slowing down and speeding up. As the ball rolls up
the hill its velocity is negative (because of how the x axis is defined) and decreasing in magnitude (a smaller negative number). At the top of the hill, its velocity is zero, and as it travels
down the hill, the ball is speeding up. Therefore, its speed decreases, reaches zero, and then increases. How can this be if v[x] is always increasing? Speed is the magnitude of velocity (and is
always a positive number). As the ball travels up the hill, v[x] increases from -5 m/s to zero; yet its speed decreases from 5 m/s to zero. Note that the phrases "speeding up" and "slowing down"
refer to how the speed changes, not necessarily to how the velocity changes.
• Is the acceleration of the golf ball increasing, decreasing, or constant? To answer this, look at the slope of the graph at every instant of time. The slope of the velocity vs. time graph
(velocity in the x direction) is equal to the acceleration (in the x direction). Does it change or is it the same? Notice that it is constant at all times and is in the positive x direction (as
defined by the coordinates).
• Besides using the graph to calculate acceleration, you can also use the velocity data from the data table. Since average acceleration is the change in velocity divided by the time interval,
choose any time interval, measure v[xi] and v[xf], and calculate the average acceleration a[x avg]. Since the acceleration is constant, the average and instantaneous accelerations are identical.
• The direction of acceleration can also be found by subtracting velocity vectors pictorially. Animation 2 shows the black velocity vectors at t = 0.2 s and t = 1.0 s. To subtract vectors, drag v
[i] away from its original position (you can drag the little circle on the arrow's tail) and then drag the red vector -v[i] into place and use the tip-to-tail method. The direction of the
acceleration is in the same direction as the change-in-velocity vector. Now, try Animation 3, which shows the velocity vectors at t = 1.2 s and t = 2.0 s. Compare the change-in-velocity vector
for each of the two time intervals. You will find that they are the same. Since the acceleration is constant, the change in velocity is constant for any given time interval.
• The area under a v[x] vs. time graph is always the displacement, Δx. You can use the graph to find Δx from t = 0 to t = 3 s. Use the data table to check your answer by determining the
displacement from x - x[0]. What is the displacement from t = 0 to t = 6 s? If you answer anything other than 0 m, you should revisit the definition of displacement.
See Illustration 3.2 for more details on what happens to the acceleration when the angle of the hill is varied.
Illustration authored by Aaron Titus.
Script authored by Aaron Titus and Mario Belloni.
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Transporting jobs through a processing center with two parallel machines
Kellerer, Hans, Soper, Alan J. and Strusevich, Vitaly A. (2010) Transporting jobs through a processing center with two parallel machines. In: Wu, Weili and Daescu, Ovidiu, (eds.) Combinatorial
Optimization and Applications: 4th International Conference, COCOA 2010, Kailua-Kona, HI, USA, December 18-20, 2010, Proceedings, Part I. Lecture Notes in Computer Science (6508). Springer Berlin
Heidelberg, Berlin, Germany, pp. 408-422. ISBN 978364274575 ISSN 0302-9743 (doi:10.1007/978-3-642-17458-2_33)
Full text not available from this repository.
In this paper, we consider a processing system that consists of two identical parallel machines such that the jobs are delivered to the system by a single transporter and moved between the machines
by the same transporter. The objective is to minimize the length of a schedule, i.e., the time by which the completed jobs are collected together on board the transporter. The jobs can be processed
with preemption, provided that the portions of jobs are properly transported to the corresponding machines. We establish properties of feasible schedule, define lower bounds on the optimal length and
describe an algorithm that behaves like a fully polynomial-time approximation scheme (FPTAS).
Item Type: Book Section
Additional [1] Paper published in series: Lecture Notes in Computer Science. Volume 6508, 2010, DOI: 10.1007/978-3-642-17458-2. Book titled Combinatorial Optimization and Applications: 4th
Information: International Conference, COCOA 2010, Kailua-Kona, HI, USA, December 18-20, 2010, Proceedings, Part I. Editors: Weili Wu, Ovidiu Daescu. [2] Also allocated ISBN 978364217458
(Online) and series ISSN 0302-9743. [2] ISSN: 0302-9743 (Print), 1611-3349 (Online)
Uncontrolled scheduling with transportation, parallel machines, FPTAS,
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
School / Department School of Computing & Mathematical Sciences
/ Research Groups: School of Computing & Mathematical Sciences > Department of Computer Science
School of Computing & Mathematical Sciences > Department of Mathematical Sciences
Related URLs: • Publisher
• Organisation
Last Modified: 08 Aug 2013 15:03
URI: http://gala.gre.ac.uk/id/eprint/4316
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