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Mongolia - RHS 2008
Reference ID MNG-NSO-EN-RHS-2008-v1.0
Year 2008
Country Mongolia
Producer(s) National Statistical Office of Mongolia
Sponsor(s) United Nations Population Fund - UNFPA - Financial assistance for conduct of the Survey
Government of Mongolia - - Financial assistance for conduct of the Survey
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Created on Jul 31, 2013
Last modified Jul 08, 2014
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Data Appraisal
Estimates of Sampling Error
Sampling errors are presented in Tables B.02 - B.16 of the report RHS 2008, for variables considered to be of major interest. Results are presented for the whole country, for urban and rural areas
separately, for each of four education groups, for each of five regions, and for each of three age groups. For each variable, the type of statistic (percentage, mean or rate) and the base population
are given in Table B.01. For each variable, Tables B.02 - B.16 present the value of the statistic (R), its standard error (SE), the number of cases (N) where relevant, the design effect (DEFT) where
applicable, the relative standard error (SE/R), and the 95 percent confidence limits (R-2SE, R+2SE).
The confidence limits have the following interpretation. For the percentage of currently married women using the contraceptive intrauterine device (IUD), the overall value for the full sample is
22.3%, and its standard error is 0.6%. Therefore, to obtain the 95 percent confidence limits, one adds and subtracts twice the standard error to the sample estimate, which means that there is a high
probability (95 percent) the true percentage currently using the IUD is between 21.1% and 23.5%.
The relative standard errors for most estimates for the country as a whole are small, except for estimates of very small percentages. The magnitude of the error increases as estimates for
sub-populations such as geographical areas are considered. For the variable IUD, for instance, the relative standard error (as a percentage of the estimated parameter) for the whole country and for
urban and rural areas is 2.7 percent, 4.1 percent, and 4.0 percent, respectively. For the five regions, the relative standard error of the variable IUD varies between 5.4 percent and 10.8 percent.
Special mention should be made of the sampling errors for rates. The denominators are exposure-years, and the numerators are either births or deaths in the population under consideration during the
indicated period of time.
It should be noted that the survey indicates, with a 95 percent level of confidence, that the TFR for the 3-year period prior to the survey lay between 3.046 and 3.31 children per woman, and that the
infant mortality rate for the 3-year period prior to the survey lay between 15.7 and 28.3 per thousand births. The differences between the survey results and registration statistics are not due to
sampling variability.
Table B.01 List of Selected Variables for Sampling Error, Mongolia, 2008
Table B.02 Sampling Error - National Sample, Mongolia, 2008
Table B.03 Sampling Error - Urban Areas, Mongolia, 2008
Table B.04 Sampling Error - Rural Areas, Mongolia, 2008
Table B.05 Sampling Error - Primary or Less, Mongolia, 2008
Table B.06 Sampling Error - Incomplete Secondary, Mongolia, 2008
Table B.07 Sampling Error - Complete Secondary, Mongolia, 2008
Table B.08 Sampling Error - More than Secondary, Mongolia, 2008
Table B.09 Sampling Error - Central Region, Mongolia, 2008
Table B.10 Sampling Error - East Region, Mongolia, 2008
Table B.11 Sampling Error - West Region, Mongolia, 2008
Table B.12 Sampling Error - South Region, Mongolia, 2008
Table B.13 Sampling Error - Ulaanbaatar, Mongolia, 2008
Table B.14 Sampling Error - Age 15-24, Mongolia, 2008
Table B.15 Sampling Error - Age 25-34, Mongolia, 2008
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Power Angle Curve of Synchronous Machine - Electrical Concepts
Power Angle Curve of Synchronous Machine
Power Angle Curve of Synchronous Machine is the graphical representation of electrical output with respect to the power angle. As we know, power angle is also known as load angle, therefore it can be
said that this curve is graphical representation of electrical output of generator with respect to load angle. In this article, we will discuss power angle curve and its importance.
First of all, we should know the mathematical relation between the electrical output of synchronous machine in terms of load angle to get the graph of power versus load angle. The electrical output
of synchronous generator is given as below.
P[e] = (E[f]V[t]/X[s])Sinδ
Where E[f], V[t], X[s] and δ are no load excitation voltage, generator terminal voltage, generator synchronous reactance and load angle respectively. You are requested to read “Power Flow Equation
through an Inductive Load” for getting the detail of derivation part of the above expression of electrical output.
Let us now draw a graph between P[e] and load angle δ assuming rest of the parameters to be constant.
Above graphical representation of power w.r.t δ is called Power Angle Curve. It can be easily seen from the above graph that it is sinusoidal. Thus power angle curve is sinusoidal.
Importance of Power Angle Curve
Power Angle Curve tells us about the electrical power output of synchronous machine when power angle δ is varied. It can be seen from this curve that as we increase δ from 0 to 90°, the output
increases sinusoidally. But a further increase in power angle δ beyond 90°, the generator electrical output decreases. What does this mean?
This simply means that, the generator electrical output is less than the mechanical input. Therefore, the poles of the machine will start to slip and eventually it will lose synchronism. Thus the
machine i.e. generator becomes unstable. Steady state stability limit is the maximum power flows possible through a specific point without lose of synchronism, when the power is increased gradually.
Therefore, steady state stability limit of synchronous machine corresponds to power for load angle δ = 90°. To be accurate, it will be (E[f]V[t]/X[s]).
Not only steady state stability limit rather transient stability limit is also affected by the load angle at which machine is operating. Transient state stability limit is basically the maximum
amount of power flow possible without loss of synchronism when a sudden disturbance occurs. The transient stability limit is determined by Equal Area Criteria which uses power angle curve. Thus power
angle curve is very important for study of stability limit of synchronous machine.
1 thought on “Power Angle Curve of Synchronous Machine”
1. Pa = Pm – Pe in this equation if Pe is increased then del angle will decrease or increases on power angle curve
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RSICC CODE PACKAGE MIS-018
1. NAME AND TITLE
MMRW: Canadian and Early British Energy Reports on Nuclear Reactor Theory (1940-1946).
2. CONTRIBUTORS
M.M.R. Williams, South Croydon Surrey, CR2 0DX, UK through the OECD Nuclear Energy Agency Data Bank, Issy-les-Moulineaux, France.
3. CODING LANGUAGE AND COMPUTER
PDF format; many computers (M00018MNYCP00).
4. NATURE OF PROBLEM SOLVED
MMRW is the collection of reports on the beginning and development of nuclear reactor theory in Canada and in the United Kingdom. In this version, the letter by Alvin Weinberg with comments and
historical background (8 September 2000) has been added.
5. METHOD OF SOLUTION
7. TYPICAL RUNNING TIME
Adobe Acrobat Reader can be used to access the files in Portable Data Format (.pdf).
10. REFERENCES
10a. Included Documentation
- G. Placzek: The Functions of En(x)=Integral(e-xuu-ndu){1,infinity} (MT-1).
- G. Placzek and G.M. Volkoff: Notes on Diffusion of Neutrons without Change in Energy. (MT-4).
- P.R. Wallace and J. LeCaine: Elementary Approximation in the Theory of Neutron Diffusion (Nucleonics 4(2),30-35, 4(3), 48-67, 1949) (MT-12).
- G.M. Volkoff: Analysis of Experimental Data on the Slowing-Down and Thermal Diffusion Lengths in Carbon (MT-14).
- P.R. Wallace: Neutron Distribution in Adjoining Piles of Slightly Different Properties (MT-15).
- G. Placzek: The Neutron Density Near a Plane Surface, I. (MT-16).
- R.E. Marshak: On the Slowing Down Length of Neutrons in Water. (MT-17).
- R.E. Marshak: On The Moments of the Distribution Function of Neutrons Slowed Down in Heavy Elements. (MT-18).
- R.E. Marshak: On The Slowing Down Length of Neutrons in Mixtures. (MT-19).
- G.M. Volkoff: Critical Conditions for a Multiplying Spherical Core with an Infinite Reflector of the Same Scattering Properties. (MT-21).
- J.C. Mark: Milne's Problem for Anisotropic Scattering. (MT-26).
- J. LeCaine: Critical Radius of a Strongly Multiplying Sphere Surrounded by a Non-Multiplying Infinite Medium. (MT-29).
- J. LeCaine and G.M. Volkoff: Application of 'Synthetic' Kernels to the Study of Critical Conditions in a Multiplying Sphere with an Infinite Reflector. (MT-30).
- P.R. Wallace: Boundary Conditions at Thin Absorbing Shells and Plates. (MT-34).
- P.R. Wallace: Neutron Distribution in a Multiplying Exponential Pile with a Plane Source. (MT-35).
- G.M. Volkoff: Critical Radius of a Multiplying Spherical Core Surrounded by a non-Multplying Reflector of Finite Thickness (MT-38).
- E.D. Courant: Production and Consumption of Fissile Materials in One Arrangement of a Producing Unit. (MT-40).
- R.E. Marshak and W.P. Seidel: Milne's Problem for a Sphere. (MT-49).
- R.E. Marshak: A Rigorous formula for the slowing down Length in an Element of Arbitrary Mass. (MT-53).
- R.E. Marshak: On the Slowing Down Length of Neutrons in Water: Supplement to MT-17. (MT-54).
- E.D. Courant: Measurements in Finite System of the Total Number of Neutrons in an Infinite System (MT-59).
- P.R. Wallace: On the Thermal Utilization of Plates in the Presence of Linear Anisotropic Scattering (MT-63).
- E.D. Courant: Diffusion Equations and Critical Conditions in a Medium with Varying Mean Free Path (MT-67).
- F.T. Adler: Theory of the Diffuse Reflection of Neutrons by a Plane Surface of a Semi-infinite Capturing Medium (MT-72).
- B. Davison: Influence of a Small Black Sphere Upon the Neutron Density In an Infinite Non- Capturing Medium. (MT-88).
- J. Stewart and G.M. Volkoff: Estimates of Critical Volume of Polymer for Pilot Plant Under Various Conditions (MT-91).
- J.C. Mark: The Spherical Harmonic Method, I. (MT-92).
- B. Davison: Influence of a Large Black Sphere Upon the Neutron Density in an Infinite Non-Capturing Medium. (MT-93).
- P.R. Wallace: Effect of Scattering on the Capturing Properties of Plates and Spheres. (MT-107).
- B. Davison: Angular Distribution due to an Isotropic Point Source and Spherically Symmetrical Eigensolutions of the Transport Equation. (MT-112).
- B. Davison and G.Placzek: Milne's Problem with Capture and Production. (MT-118).
- B. Davison: Large Spherical Hole In a Slightly Capturing Medium. (MT-124).
- J. LeCaine: A Table of Integrals Involving the Functions En(x) (MT-131).
- B. Davison: Influence of a Large Black Cylinder Upon the Neutron Density in an Infinite Non-Capturing Medium. (MT-135).
- B. Davison: Neutron Density at the Centre of a Small Spherical Cavity. (MT-136).
- E.A. Guggenheim: Compressible flow of perfect gas with heat input distributed symmetrically about middle of channel. (MT-137).
- M.H.L. Pryce: The Critical Size of a Multiplying system Surrounded by a Reflector of Different Scattering Properties. (MT-150).
- M.H.L. Pryce: The Evaluation of the Life Time of a Radioactive Substance from a Small Number of Observations. (MT-151).
- E.D. Courant: Requirements for the Maintenance of Critical Conditions with Changing Material Constants. (MT-164).
- E.A.Guggenheim, M.H.L.Pryce and G.M. Volkoff: Thermal Utilization in Some Systems having Axial Symmetry. (MT-166).
- P.R. Wallace and B. Carlson: Determination of 'f' for spheres. (MT-169).
- E.D. Courant: On Seeding Depleted Piles. (MT-173).
- J.M.G. Fell and G.M.Volkoff: Capture of Neutrons by Cylindrical Absorbing Rods Placed Around a Pile. (MT-196).
- J.M.G. Fell: A Note on the Temperature Distribution Inside Uranium Rods with Variable Heat Source Density and Variable Conductivity. (MT-197).
- E.A.Guggenheim and M.H.L.Pryce: Fast Fission in Tubes. (MT-199).
- F.T. Adler: Reduction of the Two-Group Model to an Equivalent One-Group Model. (MT-201).
- B.Davison, W.P. Seidel and S.Kushneriuk: Influence of a Small Black Cylinder Upon the Neutron Density in an Infinite Non-Capturing Medium. (MT-207).
- M.Goldstein and E.A.Guggenheim: Effect of Xenon Poisoning on Uranium-Graphite Piles. (MT-208).
- B.Davison and S.Kushneriuk: Linear Extrapolation Length for a Black Sphere and a Black Cylinder (MT-214).
- E.A. Guggenheim: Thermal Utilization in a System Con-taining Two Moderators. (MT-220).
- V.H. Rumsey and G.M.Volkoff: Diffusion Theory Expressions for the Thermal Utilization Factor f in Cells with Slab, Cylindrical and Spherical Geometry (MT-221).
- F.T. Adler: Efficiency of Control Rods as a function of their Position in a Cylindrical Pile in the One-Group Picture. (MT-222).
- B. Davison: Influence of an Air Gap Surrounding a Small Black Sphere upon the Linear Extrapolation Length of the Neutron Density in the Surrounding Medium. (MT-232).
- H.H. Clayton: Critical Laplacian and Neutron Densities in the NRX Pile. (MT-240).
- H.H. Clayton: Fate of Neutrons in NRX Pile and Production Rates of 49 and 23. (MT-241).
- M.Goldstein, M.Wales, A.S.Lodge: Fast Fission In Tubes: A Numerical Supplement to MT-199. (MT-242).
- A.S. Lodge: The Temperature Distribution in an Infinite Medium due to a Spatially Uniform Plane Source of Heat (MT-243).
- B. Davison: Linear Extrapolation Length for the Neutron Density at the Surface of a Large Hollow Cylindrical Shaft (MT-245).
- J.M.G. Fell: Effect of the Presence of Fast Neutrons on the Utilization of Neutrons by Absorbing Rods Around a Pile. (MT-249).
- R. Peierls: On the Possibility of a Slow-Neutron Chain Reaction. (1940) (MS-1).
- R. Peierls: Speed of Reaction in a Chain Reactor with Slow Neutrons. (1940) (MS-2).
- R. Peierls: The Effect of Expansion of the Sphere. (1940) (MS-8).
- M.H.L. Pryce: Critical Conditions in Neutron Multiplication. (1943) (MS-P2a).
- K. Fuchs: Effect of a Scattering Container on the Critical Radius and Time Constant. (1943) (MS-85).
- A.H. Wilson: A Comparison of the Approximate Methods of Calculating the Critical Size of a Sphere. (1944) (MS-89).
- B. Davison and K. Fuchs: The Critical Radius and the Time Constant of a Sphere Embedded in a Spherical Scattering Container. (Nov.1943) (MS-97).
- B. Davison: Critical Radius for a Hemisphere with a One-Sided Infinite Container. (Nov. 1944) (MS-100).
- A.H. Wilson: The Effect of Small Departures from the Spherical Shape upon the Critical Size and Time Constant of a Sphere. (March 1944) (MS-102).
- A.H. Wilson: The General Properties of the Transport Equation and its Use in Determining the Multiplication in Bodies Having Spherical Symmetry. (Jan. 1944) (MS-105).
- A.H. Wilson: The Effect of Inelastic Scattering on the Multiplication in a Sphere. (Jan. 1944) (MS-106).
- H. Fairbrother: The Critical Size and Time Constant of a Spheroid. (June 1944) (MS-111).
- P.D. Preston and B. Davison: Critical Radius of a Hemisphere Completely Surrounded by a Container. (July 1944) (MS-112).
- A.H. Wilson: A New Method of Determining the Time Constant of a Sphere in a Container. (Aug. 1944) (MS-115).
- P.A.M. Dirac: Approximate Rate of Neutron Multiplication for a Solid of Arbitrary Shape and Uniform Density, Parts I and II (1944) (MS-D5).
- H. Castle, H. Ibser, G. Sacher and A.M. Weinberg: The effect of fast fission on k (May 4th, 1943) (CP-644).
- Bengt Carlson: The Serber-Wilson Method Formulae and Computation Methods. (LA-756).
- R.E. Marshak: Some neutron diffusion problems. (September 11, 1946) (MDDC-295_LADC-243).
- R.F. Christy: Theoretical Discussion of a Small Homogeneous Enriched Reactor. (June 18, 1946) (MDDC-72).
- C.Mark: Terms in an Expansion of the Wiener-Hopf Solution of Milne's Equation for Small Values of the Argument (October 25, 1943) (TPI-5).
- George Volkoff: TL-1 to TL-6 Volkoff Lectures on Reactor Physics, 1944.
- A.H. Wilson: The Effect of Anisotropic Scattering on the Multiplication in a Sphere. (July 1944) (MS-113).
- G. Placzek: Energy Conversion Factors and Nuclear Constants (MT-51).
- T.H.R. Skyrme: A Reduction in Neutron Density Caused by an Absorbing Disc. (1944) (MS-91).
- Letter by Alvin Weinberg with comments and historical background (8 September 2000).
10b. Background References
MT-5 G. Placzek and W.P. Seidel, Milne's Problem in Transport Theory. (Physical Review, 72 (1947) 550-555).
MT-6 G. Placzek, The Angular Distribution of neutrons Emerging from a Plane Surface. (Physical Review, 72(1947)556-558).
MT-7 G. Placzek, Boundary Conditions in Elementary Diffusion Theory. (Not issued).
MT-13 G. Placzek and G.M. Volkoff, A Theorem on Neutron Multiplication (Canadian Journal of Research, A25 (1947) 276-292).
MT-20 G. Placzek, Diffusion of Neutrons without Velocity Change. (Not Issued).
MT-50 J.C. Mark, The Neutron Density Near a Plane surface II. (Phys Rev 72 (1947) 558-564).
MT-52 R.E. Marshak, Black Utilization. (Not Issued).
MT-55 R.E. Marshak, The Application of Variational Theory to the Determination of Asymptotic Neutron Densities. (Phys Rev 71 (1947) 688-693).
MT-56 J.C. Mark, Constants and Expansions of Wiener-Hopf Method. (Not Issued).
MT-65 G. Placzek, Report on Research in Theoretical Physics Feb. to July 15/44. (Not Issued).
MT-66 F.T. Adler and J.C. Mark, Milne's Problem with Capture. (Not Issued).
MT-89 B. Davison, A remark on the Variational Method. (Physical Rev. 71(1947)694-697).
MT-97 J.C. Mark, The Spherical Harmonic Method, II. (Re-issued as CRT-338). Also AECL-490 (1958).
MT-106 P.R. Wallace, Moments of the Angular Distribution in the Asymptotic Region (Canadian Journal of Research A26(1948)99-114).
MT-119 J. LeCaine, Milne's problem with capture II. (Canadian Journal of Research A28(1950)242).
MT-202 (Anon), Efficiency of Control Rods Absorbing Thermal Neutrons Only. (Not Issued).
MT-223 E.D. Courant and P.R. Wallace, Fluctuations of the number of Neutrons in a Pile. (Physical Rev. 72(1947)1037-1048).
11. CONTENTS OF CODE PACKAGE
The MMRW package is transmitted on CD and includes 92 PDF files.
12. DATE OF ABSTRACT
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Number Talk Examples for Upper ElementaryNumber Talk Examples for Upper Elementary - Hello Learning
Number Talk Examples for Upper Elementary
Are you looking for some number talk examples to use with your upper elementary students?
Number talks are so beneficial to students. They help strengthen number sense, flexibility with numbers, efficiency with procedures and effectively communicating your thinking with others.
Here are 10 number talk examples that you can use with your upper elementary math class, and a freebie just for you at the end!
Dot Talk – Dot Images
The first number talk example you can use is dot talks or dot images.
With dot images, students need to figure out how many dots are shown on a card or slide.
This type of number talk helps to build spatial skills and subitizing skills.
The dots shown on an image can be arranged in familiar patterns or scattered randomly around the page.
Show the image quickly to students (5-10 seconds) and then remove the image. Ask students how many dots there were. You quickly can show the image a second time to have students verify their answer.
After students share how many dots there are, let them know the correct answer.
Although there is only one correct answer for the number of dots on the image, there are multiple ways to get to the answer. Encourage students to share a variety of ways that the dots could be
visualized and grouped to efficiently count the total.
Number Lines
Number lines are another great number talk example that can be used at multiple skill levels. Here are a few ideas:
• Show an open number line with benchmarks at both ends. Ask students where a specific number would best fit on the number line and why.
• Display a number line with marked and labeled intervals, but leave a few numbers out. Students will need to figure out what numbers go in the empty places.
• A number line with all of the intervals marked and labeled, but one (or more) of the numbers is covered up. Students need to use the given information to figure out what the missing numbers are.
All of these number talk examples using a number line can be used with whole numbers, fractions, and decimals. You could even try using two (or three) types of numbers within one number line talk.
Which One Doesn’t Belong?
During this number talk example, students are shown a variety of numbers, images, equations, or statements. They have to figure out which one doesn’t belong with the others.
This type of number talk helps students analyze information and draw conclusions.
The great thing is that there can be multiple correct answers, as long as students can explain and justify their thinking about why it is different from the others.
Which one doesn’t belong number talks help students see that math problems can be viewed in different ways and that sometimes it is possible to have different answers.
Would You Rather?
In this type of number talk, you give students two options to choose from. They have to decide which one they would rather have.
This is a great number talk structure for working on unit price but can be used in many other ways.
Some would you rather number talk examples might include:
• Would you rather have one pound of nickels or one pound of dimes? Why?
• Would you rather have 3 bags of candy with 36 pieces each or 5 bags of candy with 23 pieces each? Why?
• Would you rather buy a box of 6 donuts for $4.50 or one dozen donuts for $8.64? Why?
• Would you rather have 3 $1.00 bills or 18 dimes? Why?
• Would you rather have a pool that is 18 feet long and 9 feet wide or a pool that is 15 feet long and 11 feet wide? Why?
Number Strings
A number string is a series of related problems that guide students to learn and use a specific strategy and highlight number relationships.
This type of number talk helps students find efficient strategies to solve problems.
In a number string, problems are sequenced in a way that leads students to see patterns and connections between the string of problems. All of the problems in a number string progress with a focus on
a specific strategy.
Target Number
When doing a target number talk, students are given a “target number”. They need to create an equation that is equivalent to the target number.
This number talk example can be done in two ways:
• Open ended: students are given a target number, and they can use any combination of numbers and operations they choose to create an equation equal to the target number. Their equation can be as
long, or as short, as they want.
• Specified numbers: students are given a target number and 5 number that they can use in their equation. They can use 2 numbers, 3 numbers, 4 numbers or 5 numbers to create their equation. You can
choose any number for the target number, but I only use the digits 1-9 for the five numbers they have to work with. This type of target number talk is more challenging, but it also gives an entry
point for all students while providing an extra challenge for the kids who are ready for it or want it.
Balance Scales
This number talk example highlights the fact that the equals sign means equivalent and balanced, not just “the answer”.
In this number talk, students are given a scale with a quantity or equation given on one side of the scale.
They have to balance the scale by creating a number, equation, or collection of values for the other side of the scale that is equivalent.
This type of number talk can be used at many levels. You can use whole numbers, decimals, or fractions. (or a combination of them). One side of the balance scale might have one number, or it might
have a combination of numbers or an equation.
The side the students need to balance/solve for might be empty or open-ended, or it might have part of the equation already given.
Number Bonds
Number bonds help students see the relationship between addition and subtraction. Number bonds also help students break numbers apart in different ways and see the relationships between numbers.
Number bonds are frequently used in the primary grades, and with whole numbers, but they also have great applications in the upper grades!
Try using number bonds with fractions or decimals. You can even work with number bonds that have three (or more) parts, or that break numbers down into several layers.
When students work with number bonds they strengthen their ability to break apart numbers, which helps with mental math and computation skills.
Ten Frame
The ten frame number talk example might seem like it only applies to primary classrooms. But did you know ten frames can also help support fractions, decimals, multiplication, and more?
Instead of putting one dot/image in each box of a ten frame, try putting a group of items, like a group of 4 dots. Now students need to look at the ten frame as ___ groups of 4.
Use a fraction in each box of a ten frame. An example could use the number 1/4 or a visual showing 1/4 in each box. Now students need to think of the ten frame as ___ groups of 1/4, or 1/4 __ times.
Fraction Blocks
This type of number talk helps students visualize fractions as a part of a whole, and compare fractional parts of a whole.
I like to use squares or rectangles that are colored using 2 or more colors for these number talks.
Use each color to split the shape into different fractional amounts. Students will need to determine what fraction of the whole each color represents.
You can ask many types of questions all relating to one image. Look at the image below and some of the possible questions that could go with it.
• What fraction of the square is covered by each color?
• What fraction of the square is blue?
• What fraction of the square is green?
• Is the fraction of the square that is yellow greater than or less than the fraction of the square that is blue?
• How many times greater is the blue part of the square than the green part?
• How many times less is the green part of the square than the yellow part of the square?
• Which color covers 6/16 of the square?
• The red and yellow parts of the square combined equal ____ of the square.
The shapes you use can be divided in many ways to increase or decrease the difficulty.
The websites fractiontalks.com and mathforlove.com have a lot of wonderful examples that can be used for this type of number talk.
Now that you have learned about ten different number talk examples, which one will you be trying first?
FREE Number Talk Resource
If you liked the images in this post, then you will LOVE this free set of 20 number talks!
This FREE set of 20 number talks includes all of the number talk examples shown in this post, plus 10 additional number talks. That is enough number talks for 4 weeks of math class!
CLICK HERE or on the image below to grab your free set of number strings to use with your class!
Looking for Some Ready to Go Number Talks?
If you want to get started with number talks right away, and would like some ready-to-go number talk resources, check out my Teachers Pay Teachers store!
CLICK HERE or on the image below to see all of the number talks resources I have created.
Read More About Number Talks
How to do a Number Talk Step by Step
Ways to Encourage Math Talk and Discussion
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OpenStax College Physics for AP® Courses, Chapter 30, Problem 40 (Problems & Exercises)
(a) What is the magnitude of the angular momentum for an $l=1$ electron? (b) Calculate the magnitude of the electron’s spin angular momentum. (c) What is the ratio of these angular momenta?
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Final Answer
a. $1.49\times 10^{-34}\textrm{ J}\cdot\textrm{s}$
b. $9.13\times 10^{-35}\textrm{ J}\cdot\textrm{s}$
c. $1.63$
Solution video
OpenStax College Physics for AP® Courses, Chapter 30, Problem 40 (Problems & Exercises)
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Video Transcript
This is College Physics Answers with Shaun Dychko. The angular momentum for an electron with an angular momentum quantum number of 1 is given by this formula so that's a square root of l times l plus
1 times Planck's constant over 2π. So that's square root of 1 times 1 plus 1 times 6.626 times 10 to the minus 34 joule seconds over 2π and that is 1.49 times 10 to the minus 34 joule seconds of
angular momentum. The electron's spin angular momentum is given by this formula and this spin quantum number for an electron is always one-half and so we have one-half times a half plus 1 all square
rooted times Planck's constant over 2π and that's 9.13 times 10 to the minus 35 joule seconds and then we find the ratio of angular momentum to the spin angular momentum and this works out to 1.63. | {"url":"https://collegephysicsanswers.com/openstax-solutions/what-magnitude-angular-momentum-l1-electron-b-calculate-magnitude-electrons-0","timestamp":"2024-11-08T17:31:58Z","content_type":"text/html","content_length":"196667","record_id":"<urn:uuid:12c16c1d-3779-4b1d-8ff6-801bed302056>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00833.warc.gz"} |
A (visual) Tutorial on Exponential Families
What is this post about?
Exponential family distributions are important for many classical machine learning applications. Their properties form the basis of Expectation Propagation and they are often used in hierarchical
probabilistic models. Because of this, they are well-documented and many tutorials exist. Most resources on exponential families are rather math-heavy and some people learn best with this approach.
However, I personally think and learn primarily in a visual fashion and many others do too. Thus I want to provide a tutorial on exponential families and some of their properties with a focus on
visual interpretations whenever possible.
The Basics
A pdf that can be written in the form
\[\begin{aligned}\label{EF} p(x) &= h(x)\cdot \exp\left(w^\top \phi(x) - \log Z(w) \right) \end{aligned}\]
\[\begin{aligned} Z(w)&:= \int_{\mathbf{X}} h(x)\exp\left(w^\top \phi(x)\right) \,dx \end{aligned}\]
is called an exponential family. \(\phi(x): \mathbb{X} \rightarrow \mathbb{R}^d\) are the sufficient statistics, \(w \in \mathcal{D} \subseteq \mathbb{R}^d\) the natural parameters with domain \(\
mathcal{D}\), \(\log Z(w): \mathbb{R}^d \rightarrow \mathbb{R}\) is the (log) partition function (normalization constant), and \(h(x): \mathbb{X} \rightarrow \mathbb{R}_{+}\) the base measure.
Many distributions can be written as exponential families (see e.g. Wikipedia). Some of the most prominent ones are the Normal, Beta, and Gamma distribution which are displayed for three sets of
parameters respectively.
To show how an exponential family distribution is transformed from standard form to exponential family form, consider the example of the Beta distribution.
\[\begin{aligned} \mathcal{B}(x, \alpha, \beta) &= \frac{x^{(\alpha - 1)} \cdot (1-x)^{(\beta-1)}}{B(\alpha, \beta)} \\ &= \exp\left[(\alpha-1) \log(x) + (\beta-1)\log(1-x) - \log(B(\alpha,\beta)))\
right]\\ &= \frac{1}{x(1-x)}\exp\left[\alpha\log(x) + \beta\log(1-x) - \log(B(\alpha,\beta)))\right] \end{aligned}\]
with exponential family values \(h(x) = \frac{1}{x(1-x)}, \phi(x)=(\log(x), \log(1-x)), w = (\alpha, \beta)\) and \(Z(\alpha, \beta) = \log(B(\alpha,\beta))\) where \(B(\alpha, \beta) = \frac{\Gamma
(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}\) and \(\Gamma(x)\) is the Gamma function.
The sufficient statistics are the statistics of the data that tell you everything interesting about them, i.e. if you know the sufficient statistics another person with the same data is unable to
tell more about the probability distribution than you are. For the normal distribution, for example, the sufficient statistics are \(Y_1 = \sum_i X_i\) and \(Y_2 = \sum_i X_i^2\) for samples \(X_i, i
=1,...,N\). With \(Y_1\) and \(Y_2\) we can compute
\[\begin{aligned} M &= 1/n Y_1 \\ S^2 &= \frac{Y_2 - (Y_1^2/n)}{n-1} \\ &= \frac{1}{n-1} \left[\sum_i X_i^2 - n\overline{X}^2\right] \end{aligned}\]
to rediscover the mean and standard deviation which are also sufficient estimators of the normal distribution.
The natural parameters are the set of parameters \(w\) for which \(p(x)\) is defined. They are related to (but usually not the same as) the parameters of the probability distribution. The parameters
of the normal distribution, for example, are its mean \(\mu\) and standard deviation \(\sigma\). Its natural parameters, however, are \(w_1 = \frac{\mu}{\sigma^2}\) and \(w_2 = -\frac{1}{2\sigma^2}
The base measure is the fuzziest part of exponential families in my experience. This is because we can arbitrarily shift values between the natural parameters and the base measure. The following
three equations, for example, are all possible legitimate definitions of the Beta distribution in exponential family form but they have different base measures and natural parameters.
\[\begin{aligned} \mathcal{B}(x, \alpha, \beta) &= 1 \cdot \exp\left[(\alpha-1)\log(x) + (\beta-1)\log(1-x) - \log B(\alpha, \beta) \right] \\ &= \frac{1}{x(x-1)} \exp\left[(\alpha)\log(x) + (\beta)\
log(1-x) - \log B(\alpha, \beta) \right] \\ &= \frac{1}{x^2(x-1)^2} \exp\left[(\alpha+1)\log(x) + (\beta+1)\log(1-x) - \log B(\alpha, \beta) \right] \end{aligned}\]
However, the most reasonable definition is one where we push all “left-over” parts from the natural parameters into the base measure, e.g. the \(-1\) terms in the Beta distribution (see second
equation above). The reason why it is called “base-measure” is that if you set your natural parameters \(w\) to 0, only your base is left. Similarly, you could say that changing your natural
parameters \(w\) can be seen as updating your base measure \(h(x)\).
The log partition function: is the most underrated part of exponential families. In the beginning, I perceived it as “just a normalizing constant” to ensure that the distribution integrates to one.
However, the fact that it is available in closed form combined with the structure of exponential families yields very powerful properties. I will discuss them further down in more detail but, in
short, the log partition function allows for efficient computation of the moments through derivation instead of integration. This is powerful because derivation is usually much easier than
To understand the normalizing effect of the log partition function consider the following figure where the Gamma distribution has been plotted with and without the normalizing constant.
If you want to find the natural parameters \(w\), sufficient statistics \(\phi(x)\), base-measure \(h(x)\) or log partition function \(\log Z(w)\) and don’t want to calculate them by hand, you can
find a large table of them on Wikipedia.
Important Properties
Exponential Families have a lot of interesting properties. I chose a small selection of the ones I find most important. Since I come from a Machine Learning background, my selection is biased towards
Multiplication and Division
Due to the properties of the exponential function, the product and division of exponential family PDFs is proportional to another instance of the same exponential family. Assume we have an
exponential family with two different parameterizations \(w_1\) and \(w_2\). Then we can write
\[\begin{aligned} p_1(x) \cdot p_2(x) &= h(x)\cdot \exp\left(w_1^\top \phi(x) - \log Z(w_1) \right) \cdot h(x)\cdot \exp\left(w_2^\top \phi(x) - \log Z(w_2) \right) \\ &= \frac{1}{Z(w_1)} \exp((w_1 +
c) \phi(x)) \cdot \frac{1}{Z(w_2)} \exp((w_2 + c) \phi(x)) \\ &= \frac{1}{Z'} \exp((w' + c') \phi(x)) \\ &\propto \frac{1}{Z} h(x) \cdot \exp((w') \phi(x)) \\ \end{aligned}\]
which is again in the form of the same exponential family. I use \(h(x)\) and \(c\) interchangeable, since multiplying with the base measure is similar to adding a constant to the natural parameters
(see section on base measure above). A similar result can be derived for division just that \(w' = w_1 - w_2\) instead of \(w' = w_1 + w_2\). Of course, the \(w'\) have to be valid parameters, e.g.
the variance of a Gaussian cannot be negative. To see this property visually for the Beta and Normal distribution consider the following figure.
The true distributions have been computed by multiplying/dividing distribution 1 and 2 and re-normalizing the results. The update rule for the Beta distribution is simply \(\alpha' = \alpha_1 + \
alpha_2 -1, \beta' = \beta_1 + \beta_2 - 1\) and for the Gaussian distribution
\(\mu' = \frac{\mu_1 \sigma_2^2 - \mu_2 \sigma_1^2}{\sigma_1^2 + \sigma_2^2}\) and \(\sigma' = \sqrt{\frac{1}{\frac{1}{\sigma_1^2} - \frac{1}{\sigma_2^2}}}\).
While this looks like an insignificant property it is actually very important. It implies that we can update the parameters of distributions by simple addition instead of computing costly
approximations. This property is a key component of the Machine Learning technique Expectation Propagation.
Bayesian Inference is done via Bayes Theorem
\[p(\theta \vert X) = \frac{p(X \vert \theta) p(\theta)}{p(X)} = \frac{p(X \vert \theta) p(\theta)}{\int p(X \vert \theta) p(\theta)}\]
Where \(p(\theta \vert X)\) is called posterior, \(p(X \vert \theta)\) likelihood, \(p(\theta)\) prior, and \(\int p(X \vert \theta) p(\theta)\) evidence. In general, the product of two probability
distributions does not yield a new probability distribution in closed-form. Therefore, we have to apply approximations or costly sampling strategies to get the posterior probability density. However,
when the prior distribution is conjugate to the likelihood, the posterior is of the same form as the prior and its parameters can be updated in closed-form. This comes as a blessing since it removes
all of the complexity of Bayesian inference. Most distributions are not conjugate with each other but for commonly used likelihoods, there exists an exponential family conjugate prior. A Bernoulli
likelihood, for example, has a Beta conjugate prior.
I think conjugacy can be easiest understood in visual terms. Thus, in the following, you can see how different data types update their respective conjugate priors. The figures on the left contain the
conjugate distribution and the figure on the right is used to generate the data.
If you find the speed of the GIF too high or low or want to pause it at some point you can always just pull the github repo and step through the HTML files at your own pace.
The Beta distribution describes beliefs about a probability, thus it is plausibly the conjugate prior to Bernoulli and Binomial likelihoods. In both cases we start with a flat Beta prior, i.e. \(\
alpha=\beta=1\). On the right-hand side, the distributions of the likelihood are shown in blue and the samples that simulate the data are drawn in black. Per step, we draw one sample and update our
current Beta distribution accordingly.
For the Bernoulli likelihood this means we compute \(\alpha' = \alpha + \sum_i x_i\) and \(\beta' = \beta + n - \sum_i x_i\), i.e. we add 1 to \(\alpha\) for every success and 1 to \(\beta\) for
every failure.
For the Binomial likelihood we compute \(\alpha' = \alpha + \sum_i x_i\) and \(\beta' = \beta + \sum_i N_i - \sum_i x_i\).
The posterior updates have a very intuitive interpretation. If we have more data, the Beta mean gets closer to the true probability and its uncertainty decreases. The more coinflips we see the better
we know its bias and the more certain we become.
Mean of a Normal (with known variance)
The conjugate prior for the mean of a Gaussian (not the entire Gaussian distribution) is a Gaussian. In other words, the more data points you have from a Gaussian distribution the more certain you
are about its mean. The update equations for this setup are
\[\begin{aligned} v' &= \frac{1}{\frac{1}{\sigma_0^2} + \frac{n}{\sigma^2}} \\ m' &= v' \cdot \left(\frac{\mu_0}{\sigma_0^2} + \frac{\sum_i x_i}{\sigma^2} \right) \end{aligned}\]
Our goal is to estimate the mean of the Gaussian distribution on the right, i.e. the red line. In the beginning, we have no clue about its position and start with a standard normal prior (blue line
on the left). The more samples (black) we see from the true distribution, the more our posterior estimate for the mean (red line on the left) gets closer to the true mean and its variance decreases.
The Gamma is a distribution over positive quantities. It is a conjugate prior to the rate parameter of Poisson likelihoods, which are discrete distributions over rare events, e.g. earthquakes. So if
you have count data of earthquakes and you assume them to be generated by a Poisson distribution you can get a posterior distribution over its rate parameter with
\(\alpha' = \alpha + \sum_i x_i\) and \(\beta' = \beta + n\)
We find that with more data the posterior distribution over the \(\lambda\) rate parameter gets closer to the true value of 0.5 and the uncertainty decreases.
Similar to how the Beta distribution describes believes about one probability, the Dirichlet describes beliefs about a probability vector. It is a conjugate prior to categorical and multinomial
likelihoods. Let’s say, for example, we have three categories (or types) of texts: politics, finance, and sports and we read a newspaper containing three texts every day (they can contain multiple
texts of the same type). Then we might be interested in which kind of focus the publisher has, i.e. what the generating distribution for the texts is. Every individual newspaper can be modeled by a
Binomial distribution with \(n=3\). The generating can be modelled by a Dirichlet and updated with \(\alpha' = \alpha + \sum_i x_i\).
We can see that the mean of the posterior Dirichlet function gets ever closer to the true probabilities generating the Binomial likelihoods and that the uncertainty of our estimate decreases. In our
metaphor, this means that after seeing many individual newspapers we think that the outlet focuses mostly on politics (60%), a bit on finance (25%), and even less on sports (15%).
Inverse Wishart
The inverse Wishart is a distribution over symmetric positive semi-definite matrices. It is a conjugate prior to empirical covariance matrices, e.g. the outer product of vectors drawn from a
zero-mean Gaussian. I always found it hard to get a good grasp of the inverse Wishart distribution and thus first want to introduce it in slightly more detail than the other distributions.
First of all, the inverse Wishart (and the Wishart) distribution is defined on the symmetric positive semi-definite (psd) cone. The psd cone is a subspace of \(\mathbb{R}^d\) on which all psd
matrices lie. A real symmetric matrix \(\begin{pmatrix} a & b \\ b & c \end{pmatrix}\) is psd iff \(a, c \geq 0\) and \(ac - b^2 \geq 0\). Following this stackoverflow post, we will represent this by
\((a, b, c) \in \mathbb{R}^3\). If we set \(a=1\) then \(c \geq b^2\) and if we set \(c=1\) then \(a \geq b^2\). So if we plot \((a,b,c) = (1, b, b^2)\) and \((a,b,c) = (b^2, b, 1)\) for \(-1 \leq b
\leq 1\) respectively and join these points with \((0,0,0)\) we start to see the cone.
If you want to view the cone from different angles you can do so in the respective jupyter notebook.
It is very hard to plot the density of a probability distribution on this cone. Thus we have to fall back on other strategies to plot the inverse Wishart distribution. I will use two different tools
to plot the Wishart distribution.
The first requires a bit of math. For off-diagonal elements of a symmetric psd matrix \(A\) it holds \(a_{ij}^2 \leq a_{ii} a_{jj}\) (see e.g. here). Thus for our 2-dimensional case we know that \(b
= \rho \cdot \sqrt{ac}\) for \(\rho \in (-1, 1)\). So one way to visualize the Wishart is to plot marginals for different values of \(\rho\) with increasing \(a\) and \(c\). The green, purple and
blue line in the above plot show the locations of the marginals for \(\rho\) equal to \(0.5, 0\) and \(-0.99\) respectively.
Computing the pdf of the inverse Wishart for \(\rho = -0.9,-0.8,...,0.8,0.9\) from most blue to most red results in the following figure.
The scale-matrix \(\Psi\) for this figure is \(\begin{pmatrix} 5 & 1 \\ 1 & 2 \end{pmatrix}\). Since the off-diagonal elements are positive, the probability mass is larger for positive values of \(\
rho\) (red curves) than for negative values (blue curves).
The second tool is simpler. It shows histograms of \(a, b\) and \(c\) for samples drawn from the (inverse-) Wishart.
To visualize the likelihood we draw samples from a 2-dimensional Gaussian (top right) and compute their outer products. The update rules for the inverse Wishart are \(\Psi' = \Psi + \sum_i x_i x_i^\
top\) and \(\nu' = \nu + n\).
Combining all this we can inspect how the Wishart distribution behaves when updated with Gaussian outer product likelihoods.
In our first visualization (top left) we find that a) similar to the one-dimensional cases, the probability distribution “wanders” to the true solution. Remember that the figure displays marginals
and the curves thus do not integrate to one. b) The off-diagonal entries of \(\Psi\) are both 0 in the prior and thus \(-\rho\) and \(+\rho\) yield the same marginal distribution. With every update,
the off-diagonal values of \(\Psi\) get more positive and thus \(-\rho\) and \(+\rho\) yield different marginal distributions. Thus, the larger the off-diagonal entries of \(\Psi\) become, the more
probability mass is with the red curves while the blue curves vanish.
In the second visualization (bottom) we find that the histograms of \(a, b\) and \(c\) for 1000 samples of the Wishart get narrower over time. This shows how the uncertainty decreases and we get more
and more certain about the underlying psd matrix.
Overall, I find it hard to think about or explain the inverse Wishart distribution because high-dimensional objects in general – and especially on the psd cone – are hard to visualize.
Moment calculation
The moments of exponential families can be computed by differentiating the log-partition function. In this section, I want to show why this is possible and derive the mean and variance for the Gamma
In general, the moment-generating function (MGF) of a random variable \(X\) is
\[M_X(t) := \mathbb{E}\left[e^{tX} \right], \qquad t \in \mathbb{R}\]
wherever this expectation exists. \(M_X(0)\) always exist and is 1, since probability density functions have to integrate to 1. The cumulant-generating function CGF is defined as the logarithm of the
\[K(t) = \log \mathbb{E} \left[e^{tX} \right], \qquad t \in \mathbb{R}\]
For exponential families, this means that the MGF of the sufficient statistics \(\phi(x)\) is
\[M_{\phi(x)}(u) = \mathbb{E}[e^{u^\top \phi(x)} | w] = \int_x h(x) \exp((w+u)^\top \phi(x) - Z(w)) dx = \exp(Z(w + u) - Z(w))\]
which means that the CGF is
\[K_{\phi(x)}(u) = Z(w + u) - Z(w)\]
which is obviously not \(Z(w)\). However, if we take derivatives of \(K(u)\) w.r.t. \(u\) and then set \(u\) equal to zero (to obtain the coefficients of the Taylor-series expansion), we get the same
answer as is obtained by taking derivatives of \(Z(w)\) w.r.t. \(w\).
In general, the cumulants are not similar to the central moments of a probability distribution. However, the mean, variance, and third central moment are identical and for the rest of this tutorial,
we only care about the first and second central moments.
To illustrate this further, we derive the mean and variance of the Gamma distribution (the same derivation can also be found on Wikipedia). The pdf of the Gamma distribution is
\[p(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}\]
and its natural parameters are \(w_1 = \alpha-1\) and \(w_2 = -\beta\). Their reverse substitutions are \(\alpha = w_1 + 1\) and \(\beta = -w_2\). The sufficient statistics are \((\log x, x)\) and
the log-partition function as a function of the natural parameters is
\[Z(w_1, w_2) = \log \Gamma(w_1 + 1) - (w_1 + 1) \log(-w_2)\]
To compute the mean we need to differentiate the log-partition function w.r.t. \(w_2\) since the second sufficient statistic is \(x\). Thus
\[\begin{aligned} \mathbb{E}[x] &= \frac{\partial Z(w_1, w_2)}{\partial w_2} \\ &= -(w_1 + 1)\frac{1}{-w_2}(-1) = \frac{w_1 + 1}{-w_2} \\ &= \frac{\alpha}{\beta} \end{aligned}\]
To compute the variance we differentiate twice and get
\[\begin{aligned} \mathrm{Var}(x) &= \frac{\partial^2 Z(w_1, w_2)}{\partial w_2} \\ &= \frac{\partial}{\partial w_2} \frac{w_1 + 1}{-w_2} \\ &= \frac{\alpha}{\beta^2} \end{aligned}\]
We can visually confirm, that these approximations are correct by comparing sampling-based and closed-form solutions.
Similar to above we can also compute the moments for the other sufficient statistics, e.g. \(\mathbb{E}[\log x]\) and \(\mathrm{Var}(\log x)\).
All of the moments above can also be computed using integration but it is significantly more complicated than differentiation.
Moment Matching
Let’s say we have two distributions \(p(x)\) and \(q(x)\) where \(p\) is any fixed distribution and \(q\) is a member of an exponential family. Then we can show that the KL-divergence \(KL(p\vert\
vert q)\) between the two distributions is minimized by matching their moments, e.g. setting the mean and variance of \(p(x)\) to that of \(q(x)\). We write
\[\begin{aligned} KL(p||q) &= \int p(x) \log\frac{p(x)}{q(x)} dx \\ &= \int p(x) \log p(x) dx - \int p(x) \log q(x) dx \\ &= \mathbb{E}_{p(x)}[\log p(x)] - \int p(x) \log\exp(w \phi(x) + \log h(x) -
\log Z(w)) dx \\ &= \mathbb{E}_{p(x)}[\log p(x)] - \mathbb{E}_{p(x)}[\log h(x)] - \mathbb{E}_{p(x)}[w \phi(x)] + \mathbb{E}_{p(x)}[\log Z(w)] \end{aligned}\]
which as a function of \(w\) is
\[KL_w(p||q) = - \mathbb{E}_{p(x)}[w \phi(x)] + \log Z(w)\]
where the expectation around \(\log Z(w)\) vanishes as it is independent of \(x\). To get the optimal values for \(w\) we compute the first derivative w.r.t. \(w\) and set it to zero.
\[\begin{aligned} \nabla_w KL_w(p||q) &= - \mathbb{E}_{p(x)}[\phi(x)] + \nabla_w \log Z(w) \overset{!}{=} 0 \\ \Leftrightarrow \nabla_w \log Z(w) &= \mathbb{E}_{p(x)}[\phi(x)] \\ \Leftrightarrow \
mathbb{E}_{q(x)}[\phi(x)] &= \mathbb{E}_{p(x)}[\phi(x)] \end{aligned}\]
where we use the knowledge that the derivatives of the log-partition function yield the moments of the distribution which we derived in the previous section. The visual intuition behind this fact can
be found in the figure below. Our static distribution \(p(x)\) is a normal with \(\mu=4\) and \(\sigma=1\) and we want to fit a Gamma distribution \(q(x)\) such that \(KL(p\vert\vert q)\) is
minimized. To match the moments we compute
\[\begin{aligned} \mu &= \frac{\alpha}{\beta} \\ \sigma^2 &= \frac{\alpha}{\beta^2} \\ \Rightarrow \alpha &= \frac{\mu^2}{\sigma^2} = 16\\ \Rightarrow \beta &= \frac{\mu}{\sigma^2} = 4 \end{aligned}
We can then compare the empirical KL divergences for different pairs of parameters around the theoretically optimal solution to convince ourselves that we have the best fit.
This property of matching the moments (or expectations) gives name to Expectation Propagation since one of its central steps is to fit an exponential family by moment matching.
When I first learned of exponential families I thought “It’s just a different way to write probability densities, why does it matter?”. But after working with and reading about their properties for
about a year now, I learned that they are much more powerful. Exponential families have conjugate priors and thus we can compute posteriors in closed form. This is a huge benefit over sampling-based
approaches such as MCMC. Furthermore, exponential families are “closed” under multiplication and division which - once again - implies fast computations. Their normalizing constants (the
log-partition function) are available in closed-form. Therefore we can compute their moments through differentiation instead of integration saving us a lot of pain. On a related note, we can show
that the KL-divergence between two exponential families is minimized by matching their moments. Overall, a seemingly simple way to write different probability density functions has pretty large
implications and yield the foundations of different algorithms for Bayesian generalized linear models (GLMs), Expectation Propagation, and many more.
My own research is closely linked to exponential families. Maybe you are interested in
1. A cool trick in Bayesian neural networks [blog] [arxiv]
2. A new technique to make GLMs really fast [soon]
3. My current research project [will take a while]
One last note
If you want to get informed about new posts you can subscribe to my mailing list or follow me on Twitter.
If you have any feedback regarding anything (i.e. layout or content) please tell me in a constructive manner via your preferred means of communication. | {"url":"https://www.mariushobbhahn.com/2021-06-10-ExpFam_tutorial/","timestamp":"2024-11-11T23:17:33Z","content_type":"text/html","content_length":"38150","record_id":"<urn:uuid:84432083-64b7-459c-b5f9-bea04dcde3e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00840.warc.gz"} |
PLC Ladder Logic Practice Problems with Solutions PDF – Master the Fundamentals of Automation
Imagine this: you’re in the middle of a critical automation project, and the PLC program you’ve written isn’t behaving as expected. A sinking feeling washes over you as you realize you’ve missed a
critical element in the ladder logic. This stressful situation could have been avoided with some solid practice and a good resource like a PDF containing PLC ladder logic practice problems with
Image: www.youtube.com
Understanding PLC ladder logic is essential for anyone working in industrial automation, robotics, or any field involving programmable logic controllers. It’s the language that allows you to control
machines and processes, from simple operations to complex sequences. But, mastering ladder logic requires practice and a clear grasp of the concepts. In this article, we’ll delve into the world of
PLC ladder logic practice problems, exploring how they can help you become a confident and skilled automation professional.
Why Practice PLC Ladder Logic Problems?
Practice problems are your secret weapon for mastering PLC ladder logic. By working through these problems, you’ll:
• Solidify your understanding of basic and advanced ladder logic concepts.
• Develop your problem-solving abilities and critical thinking skills.
• Gain confidence in your ability to troubleshoot and debug PLC programs.
• Improve your understanding of how different PLC components and instructions interact.
• Prepare for interviews and real-world applications.
Practice problems offer a structured way to apply your theoretical knowledge to real-world scenarios. They expose you to a variety of automation challenges, encouraging you to think creatively and
Understanding the Basics of PLC Ladder Logic
What is PLC Ladder Logic?
PLC ladder logic is a programming language used to control industrial systems and machinery using Programmable Logic Controllers (PLCs). It gets its name from its resemblance to the ladder diagrams
used in electrical engineering. The logic is represented visually, with rungs forming the “rungs” of the ladder, and each rung representing a specific instruction or condition.
Image: www.pinterest.es
Key Elements of Ladder Logic
There are several key elements that make up ladder logic:
• Contacts: These are the “input” elements of a ladder logic. They represent conditions or states, such as the position of a switch, the pressure of a sensor, or the level of a tank.
• Coils: These are the “output” elements. They represent the actions that the PLC will take, such as activating a motor, turning on a light, or controlling a valve.
• Rungs: Each rung of the “ladder” corresponds to a logic statement. Each rung contains one or more contacts and a coil.
• Instructions: Instructions represent various functions that the PLC can perform, such as mathematical operations, timers, counters, and data manipulation.
How Does Ladder Logic Work?
The PLC evaluates each rung of the ladder logic, starting from the top. For a rung to be considered “true,” all the contacts in the rung must be closed. When a rung is true, the coil associated with
that rung is energized, causing the corresponding output to be activated.
Imagine a simple ladder logic rung with a normally open push button contact connected to a motor coil. When the push button is pressed, the contact closes, completing the electrical circuit. The PLC
detects this closed contact and energizes the motor coil, causing the motor to start running. This is just a simple illustration of how ladder logic works, and more complex applications can involve
multiple rungs, complex logic operations, and various instructions to control a wide range of industrial processes.
PLC Ladder Logic Practice Problems: A Deeper Dive
Now, let’s dive deeper into some types of PLC ladder logic practice problems that can help you hone your skills:
1. Basic Control Circuits
These problems focus on understanding the fundamentals of ladder logic, such as controlling motors, lights, and simple on/off operations. These are fundamental exercises that help you understand the
basic elements of ladder logic and their interactions.
Example Problem: Design a ladder logic program to control a pump based on a level sensor. When the liquid level in a tank falls below a certain threshold, the pump should be turned on to fill the
tank. When the level reaches a set point, the pump should be switched off.
2. Timing and Counting
These problems involve using PLC timers and counters to implement delay functions, sequence operations, and count events. These are commonly used for tasks such as controlling conveyor systems,
packaging machines, and other processes requiring timed events.
Example Problem: Create a ladder logic program to control a conveyor belt. The belt should be switched on for 10 seconds, then paused for 5 seconds, and repeat this cycle continuously. You will need
to implement a timer function within your ladder logic program.
3. Data Manipulation
These problems deal with operations like data comparisons, arithmetic calculations, and data movement. These problems involve the use of PLC instructions like add, subtract, multiply, divide,
compare, and data transfer. These concepts are essential for handling data and making decisions within your automation system.
Example Problem: Implement a ladder logic program to measure and display the temperature from a sensor and trigger an alarm if the temperature exceeds a predetermined set point.
4. Communication and Networking
These problems involve communication between PLCs and other devices, such as sensors, actuators, and remote systems. This often involves the use of communication protocols, such as Modbus, Ethernet/
IP, or Profibus.
Example Problem: Design a ladder logic program to receive data from a remote sensor through a Modbus network and display the data on a local HMI.
5. Advanced Control Systems
These problems focus on complex control mechanisms, such as process control, PID (Proportional, Integral, Derivative) control, and sequential logic. These problems involve applying more complex
mathematical calculations and logic operations to accurately control continuous processes or manage complex sequences.
Example Problem: Develop a ladder logic program to implement a PID controller to maintain a constant temperature within a specific range. You would need to use equations and feedback mechanisms
within your program.
Tips for Mastering PLC Ladder Logic
Here are some tips to enhance your PLC ladder logic learning experience:
• Start with the basics: Understand the fundamentals of ladder logic, including the function of each element and instruction. Begin with simple practice problems and gradually progress to more
complex scenarios.
• Visualize the Logic: Before writing ladder logic, visualize how the program will work step-by-step. Sketch out the flow of logic, outlining the necessary conditions and actions.
• Use Simulation Software: PLC simulation software can be invaluable for testing your programs without using real hardware. This allows you to quickly see the effects of your logic changes and
identify problems early on.
• Collaborate and Learn from Experts: Don’t hesitate to seek guidance from experienced professionals, participate in online forums, or join local user groups. Sharing knowledge and learning from
others can accelerate your learning process.
• Practice Makes Perfect: The more practice problems you complete, the more comfortable you’ll become with ladder logic. Start with simple scenarios and gradually work your way up to more complex
These tips are valuable tools for effectively learning PLC ladder logic. Mastering this skill can be a rewarding journey, opening doors to numerous opportunities in the exciting field of automation.
Q: What is the best resource for finding PLC ladder logic practice problems with solutions?
A: There are many online resources available, including websites, forums, and online courses. You can also find textbooks and practice manuals that provide detailed explanations and solutions to
various problems.
Q: What PLC programming software should I use for practice?
A: There are many options available, including free and paid software. Some popular choices include LogixPro, RSLogix 500/5000 Emulate, and Siemens TIA Portal.
Q: Do I need to understand electrical engineering basics to learn PLC ladder logic?
A: While having a background in electrical engineering can be helpful, it’s not strictly necessary. Understanding the fundamental concepts of circuits and basic electrical components will certainly
be beneficial, but dedicated study materials and online courses cater to those with varying levels of technical background.
Plc Ladder Logic Practice Problems With Solutions Pdf
Mastering PLC ladder logic is a crucial step for anyone involved in industrial automation. Practice problems provide a valuable tool for solidifying your knowledge, honing your problem-solving
skills, and gaining confidence in your abilities. By exploring these problems, learning from experts, and actively practicing, you’ll be well on your way to becoming a skilled and sought-after
automation professional.
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How do you write 4 equivalent fractions?
How do you write 4 equivalent fractions?
Divide the numerator and the denominator by the same number.
1. Multiply the Numerator and Denominator by the Same Number. To find the equivalent fractions for any given fraction, multiply the numerator and the denominator by the same number.
2. Divide the Numerator and Denominator by the Same Number.
What is another way of writing 1 4?
We pronounce 1/2 as one-half, 1/4 as one-quarter, and 3/4 as three-quarters.
Is a quarter the same as 1 4?
The fraction one-fourth, written in symbols as 1/4, means “one piece, where it takes four pieces to make a whole.” The fraction one-quarter, written in symbols as 1/4, means “one piece, where it
takes 4 pieces to make a whole.”
How do you write an equivalent fraction?
To create an equivalent fraction, a fraction’s numerator and denominator can both be multiplied or divided by a single number. You can write an equivalent fraction with a given denominator by
finding how that denominator relates to the other fraction. Write down a fraction and the denominator of the proposed equivalent fraction.
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The general number of equivalents formula is. E = MW/charge number . Where MW is the molecular weight of the compound and charge number is the number of proton- or hydroxide-equivalents the compound
contains. Examples with different acids and bases help illustrate how this works in practice.
What is 4 . 10 as a fraction?
To write 4/10 as a fraction you have to get to the number where there is a numerator and a denominator written in whole numbers. 4/10 has already a numerator and denominator so it is a fraction
already. And finally we have: 4/10 as a fraction equals 4/10.
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Meaning and definition for kids of equivalent fractions : Equivalent fractions- Fractions that name the same number or amount. The meaning and definition indicated above are indicative not be used
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How do you write 4 equivalent fractions? Divide the numerator and the denominator by the same number. Multiply the Numerator and Denominator by the Same Number. To find the equivalent fractions
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[Solution] Parameterized Branch Instruction Stream Generation Codeforces Solution[Solution] Parameterized Branch Instruction Stream Generation Codeforces Solution
[Solution] Parameterized Branch Instruction Stream Generation Codeforces Solution
A. Parameterized Branch Instruction Stream Generation
time limit per test
15 seconds
memory limit per test
1024 megabytes
Benchmarks are popular for estimating and comparing the performance of computer systems which has always been a challenging task faced by computer architects and researchers. However, as emerging
applications evolve, benchmark characteristics are known to drift with time making an optimal design using benchmarks of today not optimal for applications of tomorrow. This problem has been aptly
described as, "Designing tomorrow's microprocessors using today's benchmarks built from yesterday's programs.". One of the ways to address these challenges is to supplement application benchmark
suites with synthetic benchmarks.
Some researchers show that with a suitable choice of a few Microarchitecture-Independent-Characteristics (MIC) related to the instruction mix, instruction-level parallelism, control flow behavior,
and memory access patterns, it is possible to generate a synthetic benchmark whose performance directly relates to that of a real-world application. The parameterized nature of this framework enables
the construction of synthetic benchmarks that allow researchers to explore a wider range of the application behavior space, even when no benchmarks yet exist.
Creating a program control flow generation algorithm is one of the most difficult challenges for synthetic benchmarks. Hence, this problem focuses on generating a program control flow for synthetic
benchmarks based on MICs related to the behavior of conditional branch instructions.
To describe the control flow of a program, two components are used:
The control flow graph is a directed graph that consists of internal nodes and 2 special nodes (the start node and end node). Internal nodes are indexed from to , while the index of end node is . The
start node has no input edges and a single output edge. The end node has no output edges and can have multiple input edges. Each of the internal nodes has exactly 2 output edges and can have multiple
input edges. To distinguish between these 2 output edges, they are referred to as left and right edges. Self-loops are possible in the control flow graph. An example of the control flow graph with 3
internal nodes is shown in figure 1.
Figure 1. Example of the control flow graph with 3 internal nodes
The route in the control flow graph begins at the start node and exits on the end node passing all of the internal nodes in the control flow graph at least once. All of the internal nodes can be
passed multiple times on the route. If the route goes through left or right edge from an internal node, we say that the route follows to the left or right on that internal node. The length of the
route is denoted as and does not include the edge from the start node. For each of the internal nodes through which the route goes a vector of the directions (left or right) that the route took on
that node can be constructed. For example, if the vector is [0, 0, 1, 1, 0] for one of the internal nodes with and denoting left and right, respectively, the route passes that node 5 times and goes
left, left, right, right, and left on that node.
There are 3 microarchitecture-independent-characteristics that are used to describe program behavior and generate a program control flow with a given number of internal nodes and a length of the
route :
• Mean and standard deviation of the right output direction ratio of all internal nodes: On the route, a node is passed by multiple times and the output direction of each passing can be either left
or right. Each node has a different taking right output direction ratio on the route equaling the number of times the route went right direction on that node divided by the total number of times
the node was passed on the route. The mean values and population standard deviation values of the right output direction ratio of all internal nodes can therefore be calculated.
• Mean and standard deviation of the node output direction transition ratio of all internal nodes: The transition number of a node is defined as the number of times the output direction changes
between two consecutive passes. If a node always takes the same output direction every time it passes a node on the route, the transition number of the node is . If a node changes output
direction every time it is passed, the transition number is , with equaling the total number of times the node was passed on the route. The transition ratio equals the transition number divided
by . Mean values and population standard deviation values of the output direction transition ratio of all internal nodes can therefore be calculated.
• Mean and standard deviation of the node correlation coefficient of all internal nodes: The output direction of a particular node is highly correlated with the output directions of previously
passed nodes. The node correlation coefficient reflects the predictability of a node output direction based on the output directions of previously passed nodes. The correlation coefficient of a
node is calculated by: first, using an array C[P] to keep the correlative value of a node with nodes previously passed on the route in a sliding correlation window; second, every time a node is
encountered if its output direction is the same with the node in the correlation window, C[i] is increased by , otherwise, C[i] is decreased by ; third, C[i] is normalized to a range [0, 1] by
dividing the absolute value of C[i] by the total number of times the node was passed on the route. At the beginning of the route, if a node has less previous nodes than correlation window size ,
then C[i] is updated only to the index which is not greater than the exact number of previous nodes. Finally, max(abs(C[i])) over is calculated as the node correlation coefficient. The mean
values and population standard deviation values of the node correlation coefficient of all internal nodes can therefore be calculated.
The first input line contains a number of the internal nodes in the control flow directed graph .
The second input line contains the length of the route in the control flow graph.
The third input line contains the mean and population standard deviation of the right output direction of all internal nodes as 2 floating point numbers rounded to 2 decimal places.
The fourth input line contains the mean and population standard deviation of the node output direction transition ratio of all internal nodes as 2 floating point numbers rounded to 2 decimal places.
The fifth input line contains 3 values: first, the mean and population standard deviation of the node correlation coefficient of all internal nodes as 2 floating point numbers rounded to 2 decimal
places, followed by the correlation window size .
The first output line should have the internal node index to show which internal node the start node is connected to. The next lines (one for each internal node) should have the following format:
• First, an integer with a value from to that shows which node (other internal node or the end node) the current internal node is connected to on the left edge.
• Second, an integer with a value from to that shows which node (other internal node or the end node) the current internal node is connected to on the right edge.
• Finally, a sequence of s and s representing a vector of output directions that the route takes on this internal node on each passing with and denoting left and right, respectively.
• Solution will be considered incorrect if the output program control flow does not describe the control flow graph with the provided number of internal nodes and a route with a length .
• Solutions that time out or exceed the memory limit will be considered invalid.
• If the solution is correct, its score is calculated: first, by calculating the microarchitecture-independent-characteristics for the output program control flow generated by the solution,
represented as floating point numbers rounded to 2 decimal places before score formula calculations; and second, how closely the output MIC is compared to the input (original) MIC, using the
following formula:If , then . If , then when , and otherwise.
• For multiple test cases, the final score is calculated as the sum of scores of individual tests.
• If two solutions have the same score, the solution submitted first wins.
• Third-party libraries and multi-thread parallel implementation are not allowed.
No comments: | {"url":"https://www.guptamechanical.com/2022/11/solution-parameterized-branch.html","timestamp":"2024-11-03T00:09:32Z","content_type":"application/xhtml+xml","content_length":"431289","record_id":"<urn:uuid:acf0944d-66d3-4c39-b07c-c913668ceefe>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00190.warc.gz"} |
AMS short couse 2013: Universality for random non-Hermitian matrices.
Roles of random noise in data analysis. (Dec 2012).
Universality for Wigner matrices (July 2012, Medallion lecture, Bernoulli World Congress, Istanbul 2012).
Law of the determinant (the log determinant of a random matrix satisfies CLT)
Random Discrete Matrices (This is a survey I gave at IPAM in 2009, some updated version will appear soon. It contains a fair amount of topics and open questions) | {"url":"https://campuspress.yale.edu/vanvu/presentations/","timestamp":"2024-11-09T12:47:49Z","content_type":"text/html","content_length":"33481","record_id":"<urn:uuid:07107fb2-eb93-4b43-955e-554de78980fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00483.warc.gz"} |
DAQP is a dual active-set solver that solves quadratic programs (QPs) in the form
\[\begin{aligned} &\underset{x}{\text{minimize}}&& \frac{1}{2} x^T H x + f^T x\\ &\text{subject to} && l\:\: \leq \:x \:\:\leq u, \\ & && b_l \leq A x \leq b_u, \\ \end{aligned}\]
where \(H\succ 0\). The case when \(H\succeq 0\) is also handled through proximal-point iterations, which in particular means that DAQP also solves linear programs (LPs). Moreover, DAQP can also
solve mixed-integer QPs by handling binary constraints of the form \(Ax \in \lbrace b_l, b_u \rbrace\) (for which \(x_i \in \lbrace 0,1 \rbrace\) is a special case).
DAQP has been developed for QPs that arise in real-time Model Predictive Control (MPC) applications, with focus on fully condensed MPC formulations. As such, DAQP efficiently solves small/medium
scale, dense, QPs and LPs. If your aim is to solve large-scale sparse problems, consider instead solvers, such as OSQP, that exploit sparsity.
For technical details and numerical results, see the papers
A Dual Active-Set Solver for Embedded Quadratic Programming Using Recursive LDL\(^T\) Updates
D. Arnström, A. Bemporad, D. Axehill
IEEE Transactions on Automatic Control, vol. 67, no. 8, pp. 4362-4369, 2022
A Linear Programming Method Based on Proximal-Point Iterations With Applications to Multi-Parametric Programming
D. Arnström, A. Bemporad, D. Axehill
IEEE Control Systems Letters, vol. 6, pp. 2066-2071, 2022
BnB-DAQP: A Mixed-Integer QP Solver for Embedded Applications
D. Arnström, D. Axehill
Proceedings of the 2023 IFAC World Congress, 2023
Available here.
The solver is written in C and is library free. DAQP can be interfaced to C, Julia, MATLAB, and Python. | {"url":"https://darnstrom.github.io/daqp/","timestamp":"2024-11-12T00:00:33Z","content_type":"text/html","content_length":"9599","record_id":"<urn:uuid:be7dd3de-1c50-4b0b-86f3-6c7fc4c7a4f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00641.warc.gz"} |
10 Most Common SSAT Upper-Level Math Questions
Preparing for the SSAT Upper-Level Math test? Want a preview of the most common mathematics questions on the SSAT Upper-Level Math test? If so, then you are in the right place.
The mathematics section of SSAT Upper Level can be a challenging area for many test-takers, but with enough patience, it can be easy and even enjoyable!
Preparing for the SSAT Upper-Level Math test can be a nerve-wracking experience. Learning more about what you’re going to see when you take the SSAT Upper Level can help to reduce those pre-test
jitters. Here’s your chance to review the 10 most common SSAT Upper-Level Math questions to help you know what to expect and what to practice most. Try these 10 most common SSAT Upper-Level Math
questions to hone your mathematical skills and to see if your math skills are up to date on what’s being asked on the exam or if you still need more practice.
Make sure to follow some of the related links at the bottom of this post to get a better idea of what kind of mathematics questions you need to practice.
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10 Sample SSAT Upper-Level Math Practice Questions
1- If \(x∎y=\sqrt{x^2+y}\), what is the value of \(6∎28\)?
A. \(\sqrt{168}\)
B. 10
C. 8
D. 6
E. 4
2- If \(2≤x<6\), what is the minimum value of the following expression?
A. 9
B. 7
C. 4
D. 2
E. 1
3- What is the answer of \( 0.9÷0.015\)?
A. \(\frac{1}{60}\)
B. \(\frac{1}{6}\)
C. 6
D. 600
E. 60
4- \(\frac{1 \frac{4}{3} + \frac{1}{4}}{2 \frac{1}{2} – \frac{17}{8}}\) is approximately equal to.
A. 3.33
B. 3.6
C. 5.67
D. 6.88
E. 6.97
5- There are four equal tanks of water. If \(\frac{2}{5}\) of a tank contains 300 liters of water, what is the capacity of the four tanks of water together?
A. 1,500 liters
B. 2,000 liters
C. 2,500 liters
D. 3,000 liters
E. 3,500 liters
6- A cruise line ship left Port A and traveled 30 miles due west and then 40 miles due north. At this point, what is the shortest distance from the cruise to port A?
A. 50 miles
B. 55 miles
C. 60 miles
D. 70 miles
E. 110 miles
7- the average weight of 20 girls in a class is 55 kg and the average weight of 42 boys in the same class is 82 kg. What is the average weight of all the 62 students in that class?
A. 70 kg
B. 72.20 kg
C. 73.29 kg
D. 74.44 kg
E.75.20 kg
8- Two-kilogram apple and two-kilogram orange cost $28.4. If a one-kilogram apple costs $5.2, how much does a one-kilogram orange cost?
A. $9
B. $6.5
C. $6
D. $5.5
E. $5
9-David’s current age is 44 years, and Ava’s current age is 4 years old. In how many years David’s age will be 5 times Ava’s age?
A. 4
B. 6
C. 8
D. 10
E. 14
10- Michelle and Alec can finish a job together in 50 minutes. If Michelle can do the job by herself in 2.5 hours, how many minutes does it take Alec to finish the job?
A. 60
B. 75
C. 80
D. 100
E. 150
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1- C
2- B
\(2≤x<6→\) Multiply all sides of the inequality by 3. Then:
Add 1 to all sides. Then: \(→6+1≤3x+1<18+1→ 7≤3x+1<19\)
A minimum value of \(3x+1\) is 7.
3- E
4- D
\(\frac{1 \frac{4}{3}+\frac{1}{4}}{2 \frac{1}{2}-\frac{17}{8}}=\frac{\frac{7}{3}+\frac{1}{4}}{\frac{5}{2}-\frac{17}{8}}=\frac{\frac{28+3}{12}}{\frac{20-17}{8}}=\frac{\frac{31}{12}}{\frac{3}{8}}=\frac
5- D
Let \(x\) be the capacity of one tank. Then, \(\frac{2}{5 } x=300→x=\frac{300×5}{2}=750\) Liters
The amount of water in four tanks is equal to 4×750=3,000 Liters
6- A
Use the information provided in the question to draw the shape.
Use Pythagorean Theorem: \(a^2+b^2=c^2\)
\(30^2+40^2=c^2⇒900+1,600=c^2⇒2500=c^2 ⇒c=50\) miles
7- C
Average \(=\frac{sum \ of \ terms }{number \ of \ terms}\),
The sum of the weight of all girls is: 20×55=1,100 kg, The sum of the weight of all boys is: 42×82=3,444 kg, The sum of the weight of all students is: 1,100+3,444=4,544 kg
The average weight of the 62 students: \(\frac{4,544}{62}=73.29\) kg
8- A
Let \(x\) be the cost of one-kilogram orange, then:\( 2x+(2×5.2)=28.4→\)
9- B
Let’s review the choices provided.
A.4. In 4 years, David will be 48 and Ava will be 8.48 is not 5 times 8.
B.6. In 6 years, David will be 50 and Ava will be 10. 50 is 5 times 10!
C.8. In 8 years, David will be 52 and Ava will be 12.52 is not 5 times 12.
D.10. In 10 years, David will be 54 and Ava will be 14.54 is not 5 times 14.
E.14. In 14 years, David will be 58 and Ava will be 18.58 is not 5 times 18.
10- B
Let b be the amount of time Alec can do the job, (change 2.5 hours to 150 minutes) then:
\(\frac{1}{a}+\frac{1}{b}=\frac{1}{50}→\frac{1}{150}+\frac{1}{b}=\frac{1}{50}→\frac{1}{b}=\frac{1}{50}-\frac{1}{150}=\frac{2}{150}=\frac{1}{75}\) , Then: b=75 minutes
Looking for the best resource to help you succeed on the SSAT Upper-Level Math test?
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What people say about "10 Most Common SSAT Upper-Level Math Questions - Effortless Math: We Help Students Learn to LOVE Mathematics"?
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There are two kinds of rates for private tutoring: standard and discount rates.
Discount rates are for a group of 2 or 3 students.
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PiNUTS offers the following discount rates. For students who register after November 1, 2023:
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Dimitrios G. Giovanis
May 24, 2024
Abstract:This work introduces a comprehensive approach utilizing data-driven methods to elucidate the deposition process regimes in Chemical Vapor Deposition (CVD) reactors and the interplay of
physical mechanism that dominate in each one of them. Through this work, we address three key objectives. Firstly, our methodology relies on process outcomes, derived by a detailed CFD model, to
identify clusters of "outcomes" corresponding to distinct process regimes, wherein the relative influence of input variables undergoes notable shifts. This phenomenon is experimentally validated
through Arrhenius plot analysis, affirming the efficacy of our approach. Secondly, we demonstrate the development of an efficient surrogate model, based on Polynomial Chaos Expansion (PCE), that
maintains accuracy, facilitating streamlined computational analyses. Finally, as a result of PCE, sensitivity analysis is made possible by means of Sobol' indices, that quantify the impact of process
inputs across identified regimes. The insights gained from our analysis contribute to the formulation of hypotheses regarding phenomena occurring beyond the transition regime. Notably, the
significance of temperature even in the diffusion-limited regime, as evidenced by the Arrhenius plot, suggests activation of gas phase reactions at elevated temperatures. Importantly, our proposed
methods yield insights that align with experimental observations and theoretical principles, aiding decision-making in process design and optimization. By circumventing the need for costly and
time-consuming experiments, our approach offers a pragmatic pathway towards enhanced process efficiency. Moreover, this study underscores the potential of data-driven computational methods for
innovating reactor design paradigms. | {"url":"https://www.catalyzex.com/author/Dimitrios%20G.%20Giovanis","timestamp":"2024-11-08T10:41:18Z","content_type":"text/html","content_length":"154878","record_id":"<urn:uuid:9d414b56-f108-4aac-8c91-86bcc22b0176>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00617.warc.gz"} |
first data rsa
Written by
Published: 20 Jan 2021
First Data said it would also work with the merchant to tokenize existing transaction data in the merchant's data warehouse to remove it from the environment. In real-life situations the primes
selected would be much larger; in our example it would be trivial to factor n, 3233 (obtained from the freely available public key) back to the primes p and q. e, also from the public key, is then
inverted to get d, thus acquiring the private key. Learn how to create an effective cloud center of excellence for your company with these steps and best practices. Word . In real scenarios, RSA is
not the perfect match for this kind of password encryption. The process of tokenizing would be added to the transaction time. Privacy Policy The RSA algorithm was first described in the paper: [R.
Rivest, A. Shamir, L. Adleman, "A Method for Obtaining Digital Signatures and Public-key Cryptosystems". [5] In April 1977, they spent Passover at the house of a student and drank a good deal of
Manischewitz wine before returning to their homes at around midnight. Sign-up now. We offer Revenue Sharing Alliance (RSA), Referral Bank and Community Financial Institution … RSA encryption is
mostly used when there are 2 different endpoints are involved such as VPN client and server, SSH, etc. [4], Ron Rivest, Adi Shamir, and Leonard Adleman at the Massachusetts Institute of Technology,
made several attempts over the course of a year to create a one-way function that was hard to invert. Word. When encrypting with low encryption exponents (e.g., If the same clear text message is sent
to, RSA has the property that the product of two ciphertexts is equal to the encryption of the product of the respective plaintexts. While this integration has its benefits, enterprises still need...
After abruptly losing web-hosting services, Parler sues AWS, alleging breach of contract and antitrust behavior. The RSA algorithm involves four steps: key generation, key distribution, encryption,
and decryption. Strong random number generation is important throughout every phase of public key cryptography. Providing security against partial decryption may require the addition of a secure
padding scheme.[26]. This can be done reasonably quickly, even for very large numbers, using modular exponentiation. RSA (cryptography, company) (The initials of the authors) 1. To avoid these
problems, practical RSA implementations typically embed some form of structured, randomized padding into the value m before encrypting it. The prime numbers are kept secret. In October 2017, a team
of researchers from Masaryk University announced the ROCA vulnerability, which affects RSA keys generated by an algorithm embodied in a library from Infineon known as RSALib. The First Data/RSA
service, called First Data Secure Transaction Management, integrates both tokenization and encryption. As a result of this work, cryptographers now recommend the use of provably secure padding
schemes such as Optimal Asymmetric Encryption Padding, and RSA Laboratories has released new versions of PKCS #1 that are not vulnerable to these attacks. In addition, for some operations it is
convenient that the order of the two exponentiations can be changed and that this relation also implies: RSA involves a public key and a private key. The RSA problem is defined as the task of taking
eth roots modulo a composite n: recovering a value m such that c ≡ me (mod n), where (n, e) is an RSA public key and c is an RSA ciphertext. See integer factorization for a discussion of this
problem. [1], In a public-key cryptosystem, the encryption key is public and distinct from the decryption key, which is kept secret (private). For a time, they thought what they wanted to achieve was
impossible due to contradictory requirements. "Tokenization does not replace encryption, but in many scenarios it can help reduce the number of places that card data (or any other type of sensitive
data) is stored – which is invariably a good thing," Pescatore wrote. First Data, a global provider of electronic commerce and payment processing services, and RSA, the security division of EMC, have
teamed up to provide a new service called His discovery, however, was not revealed until 1997 due to its top-secret classification. It is important that the private exponent d be large enough. First
Data performed performance testing to ensure the tokenization would not push transaction times over their set boundary. There are no new hardware deployments or data servers that must be installed in
the merchant location. Heninger says in her blog that the bad keys occurred almost entirely in embedded applications, including "firewalls, routers, VPN devices, remote server administration devices,
printers, projectors, and VOIP phones" from more than 30 manufacturers. Wir entwickeln innovative Lösungen, die die Zahlungsabwicklung … Excel. [8] However, given the relatively expensive computers
needed to implement it at the time, it was considered to be mostly a curiosity and, as far as is publicly known, was never deployed. You are about to leave this website and go to the Investors
section of First Dataâ s US English website. RSA is here to help you manage your digital risk with a range of capabilities and expertise including integrated risk management, threat detection and
response, identity and access management, and fraud prevention. Article Content. It is digital signature algorithm. The result of this computation, after applying Euler's Theorem, is rcd (mod n) and
so the effect of r can be removed by multiplying by its inverse. First Data helps give a competitive edge to financial institutions, including community banks and credit unions. The public key is (n
= 3233, e = 17). It is used for digital signature and its verification. RSA … Just less than five gigabytes of disk storage was required and about 2.5 gigabytes of RAM for the sieving process. Bank
of America Corp. said it will end a payments joint venture with First Data Corp. next June when a contract between the two companies expires. There is no known attack against small public exponents
such as e = 3, provided that the proper padding is used. Both of these calculations can be computed efficiently using the square-and-multiply algorithm for modular exponentiation. RSA organisiert die
RSA Conference, eine jährliche Veranstaltung zum Thema IT-Sicherheit. Currently the most promising approach to solving the RSA problem is to factor the modulus n. With the ability to recover prime
factors, an attacker can compute the secret exponent d from a public key (n, e), then decrypt c using the standard procedure. It is used for secure data transmission. She can use her own private key
to do so. Still, other methods will compete with the First Data-RSA service. Rivest and Shamir, as computer scientists, proposed many potential functions, while Adleman, as a mathematician, was
responsible for finding their weaknesses. where the second-last congruence follows from Euler's theorem. The pairwise similarity measures between response patterns are stored and visualized in a
representational distance matrix (RDM). In 1998, Daniel Bleichenbacher described the first practical adaptive chosen ciphertext attack, against RSA-encrypted messages using the PKCS #1 v1 padding
scheme (a padding scheme randomizes and adds structure to an RSA-encrypted message, so it is possible to determine whether a decrypted message is valid). Telecoms giant announces that data charges
for government-backed academy will be removed. First Data plans to begin rolling out the service in North America. Cookie Preferences We want to show that med ≡ m (mod n), where n = pq is a product
of two different prime numbers and e and d are positive integers satisfying ed ≡ 1 (mod φ(n)). Given m, she can recover the original message M by reversing the padding scheme. The remainder or
residue, C, is... computed when the exponentiated number is divided by the product of two predetermined prime numbers (associated with the intended receiver). The NIST Special Publication on Computer
Security (SP 800-78 Rev 1 of August 2007) does not allow public exponents e smaller than 65537, but does not state a reason for this restriction. The goal of first-level representational similarity
analysis (RSA) is to calculate the similarity of actvity patterns evoked by a set of conditions. Many approaches including `` knapsack-based '' and `` permutation polynomials '' difficulty of the!
Second-Last congruence follows from Euler 's theorem storage was required and about 2.5 of... Openssl to generate and examine a real keypair on implementations transport ) die RSA conference, eine
Veranstaltung., even for very large numbers and the RSA cryptosystem is attributed to Whitfield Diffie and Martin Hellman, published! Users have access gigabytes of RAM for the sieving process
offered by their processor before deciding which way to.! Secure if n is sufficiently large, outside of quantum computing the couch with a certain explained... Of PSS no longer seems to be simple
with fast performance and scalability transaction times over set! Client and server, SSH, etc here are artificially small, but one can also be applied the... Processors also implement simultaneous
multithreading ( SMT ) difficult as the RSA.! Such, OAEP should be used in any new application, and decryption are as essential for the sieving.! To strip credit card numbers on site, '' Heiser said
the Unexpected to leave website! [ 6 ] Rivest, Adi Shamir and Leonard Adleman by choice of a message, security! Way to thwart these attacks is to calculate the similarity of actvity patterns evoked
by a set conditions! Two modular exponentiations both use a spy process to discover ( statistically ) the private exponent d by.. Processing more than $ 1.4 trillion in transactions in 2008 are a
number of attacks against plain RSA as below... Should not be `` too close '', lest the Fermat factorization for a of. A math textbook and started thinking about their one-way function for n be
successful 3233, e =,. Him an encrypted ciphertext c, the `` factoring problem is an example RSA. 2021, CIOs will not only focus on providing greater access to but!, Bleichenbacher showed that this
version is vulnerable to a first predetermined power ( associated the... That it uses encryption and tokenization at virtually every point of the authors ) 1 a construction that to... Encryption in
your mailbox and after email is sent message structure information to Alice bits long merchants! Processor while replacing credit card Data and remove it from their environment while allowin access.
The numbers p and q should not be `` too close '' lest... Key e, corresponding to they thought what they wanted to achieve was impossible due to contradictory.! Other payment processors are likely to
follow, Krikken said address doesn ’ t appear to be simple with performance! In the United States would not push transaction times over their set boundary in 2008 = )..., along with an auxiliary
value rest of the paper ready by daybreak written. By reversing the padding scheme. [ 25 ] how these seven stack! And/Or VAR applications, the service uses RSA 's SafeProxy tokenization technology
into securing payment transactions including E-Guides,,... December 1977 company with these steps and best practices number generation is important that proper... Practical RSA implementations
typically embed some form of structured, randomized padding into the value m before encrypting it company... Rivest, Adi Shamir and Leonard Adleman access service Edge blends network and functions!
When Bob receives the signed message, RSA can also be used to shared. Were reported in 2011 is an example of RSA relies on the practical difficulty of large. Private key ( d ) is to ensure the
tokenization would not have been carefully designed securely... Offered by their processor before deciding which way to thwart these attacks is to calculate similarity. Predetermined power (
associated with the public key e, corresponding to user creates and publishes public... File to be affected seems to be encumbered by patents signing is potentially more secure. 7. Rsa-250 ) in real
scenarios, RSA security estimated that 1024-bit keys were to... 1 have been factored were reported in 2011 m by reversing the padding scheme. 25. May have been legal either 1.5 ) used a construction
that appears to make RSA semantically secure. 7! And trusted platform modules ( TPMs ) were shown to be simple with fast performance scalability! To ciphertext at the encoding terminal by encoding
the message as a m. Lay on the couch with a certain technique explained below ) a representational distance matrix RDM... Healthcare but more equitable access company, see, Importance of strong
random number is! As described below in same order as their paper. [ 35 ] [ 36 ] the if. Encrypting messages signature scheme. [ 26 ] visualized in a non-statistical way and started thinking about
their one-way.. That contain phishing scams, viruses, or malware Dynamics acquired RSA Data security in 1996.
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Mastering Number Conversion in Python – A Step-by-Step Guide to Converting Numbers to Binary
In Python, converting numbers to binary is a common task that allows us to work with binary notation, which is essential in various applications such as cryptography and low-level programming. This
blog post will guide you through the process of converting numbers to binary in Python, exploring both the decimal to binary and binary to decimal conversions. Understanding these conversions is
crucial for working with binary numbers effectively.
Understanding the basics of number systems
Before we dive into the conversion process, let’s briefly review the two number systems we’ll be working with: decimal and binary.
Decimal system
The decimal system is what we use in our everyday lives. It is a base-10 system, meaning it uses ten different digits (0-9) to represent numbers. In this system, each digit’s value is multiplied by
an increasing power of 10 based on its position, starting from the rightmost digit.
For example, the decimal number 253 can be broken down as:
(2 * 10^2) + (5 * 10^1) + (3 * 10^0) = 200 + 50 + 3 = 253
Binary system
The binary system, on the other hand, is a base-2 system widely used in computing and digital systems. It uses only two digits, 0 and 1, to represent numbers. In this system, each digit’s value is
multiplied by an increasing power of 2 based on its position, starting from the rightmost digit.
For example, the binary number 101101 can be broken down as:
(1 * 2^5) + (0 * 2^4) + (1 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 32 + 0 + 8 + 4 + 0 + 1 = 45
Converting decimal numbers to binary
Now that we understand the basics of number systems, let’s explore the process of converting decimal numbers to binary in Python.
Step 1: Understanding the conversion process
The conversion process involves dividing the decimal number by 2 repeatedly until the quotient becomes 0. The binary representation is obtained by taking the remainders of these divisions in reverse
Step 2: Converting decimal to binary using built-in functions
In Python, we have a built-in function called bin() that simplifies the conversion of decimal numbers to binary. This function takes a decimal number as input and returns its binary representation as
a string.
Let’s see an example:
decimal_number = 25 binary_number = bin(decimal_number) print(binary_number) # Output: 0b11001
In the above example, we used the bin() function to convert the decimal number 25 to its binary representation. The prefix ‘0b’ indicates that the resulting string is a binary number.
Step 3: Implementing manual conversion
While the bin() function is convenient, it’s essential to understand the manual conversion process as well. Let’s explore how to convert a decimal number to binary manually.
We start by initializing an empty string to store the binary representation. Then, we repeatedly divide the decimal number by 2 and add the remainder to the binary string. Finally, we reverse the
string to obtain the correct binary representation.
Here’s an example:
decimal_number = 25 binary_number = ""
while decimal_number > 0: remainder = decimal_number % 2 binary_number += str(remainder) decimal_number = decimal_number // 2
binary_number = binary_number[::-1] print(binary_number) # Output: 11001
In the above example, we manually converted the decimal number 25 to its binary representation. We iteratively divided the decimal number by 2 until the quotient became 0, concatenating the
remainders to the binary string. Finally, we reversed the string to obtain the correct binary representation.
Converting binary numbers to decimal
Now let’s explore the process of converting binary numbers to decimal in Python.
Step 1: Understanding the conversion process
The conversion process involves multiplying each binary digit by an increasing power of 2 based on its position, starting from the rightmost digit. The decimal representation is obtained by summing
all these multiplications.
Step 2: Converting binary to decimal using built-in functions
In Python, we can use the int() function to directly convert a binary string to its decimal representation. This function takes two arguments: the binary string and the base (in this case, 2).
Let’s see an example:
binary_number = "11001" decimal_number = int(binary_number, 2) print(decimal_number) # Output: 25
In the above example, we used the int() function to convert the binary string “11001” to its decimal representation. The second argument, 2, specifies the base as binary.
Step 3: Implementing manual conversion
As with the decimal to binary conversion, it’s crucial to understand the manual conversion process for converting binary to decimal.
We start by initializing a variable to store the decimal representation. Then, we iterate through the binary string from right to left, multiplying each digit by the corresponding power of 2 and
adding it to the decimal number.
Here’s an example:
binary_number = "11001" decimal_number = 0 power = 0
for digit in binary_number[::-1]: decimal_number += int(digit) * (2 ** power) power += 1
print(decimal_number) # Output: 25
In the above example, we manually converted the binary string “11001” to its decimal representation. We iterated through the string in reverse order, multiplying each digit by the corresponding power
of 2 and adding it to the decimal number.
In conclusion, converting numbers to binary is an essential skill in Python programming. It allows us to work with binary notation, which is crucial in various applications. In this blog post, we
explored the process of converting decimal numbers to binary and vice versa.
Remember to use the built-in functions bin() and int() for convenient conversion. However, understanding the manual conversion process provides a deeper understanding of the underlying algorithms.
Take the time to practice these conversions and explore further resources to strengthen your understanding. Being comfortable with number conversions will undoubtedly enhance your Python programming | {"url":"https://skillapp.co/blog/mastering-number-conversion-in-python-a-step-by-step-guide-to-converting-numbers-to-binary/","timestamp":"2024-11-11T07:32:24Z","content_type":"text/html","content_length":"111697","record_id":"<urn:uuid:055bcdf2-dba9-4d58-bcca-e0c560279ab2>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00159.warc.gz"} |
Interactive Self-Study Module: Adiabatic Reversible Expansion and Compression
This module uses screencasts and interactive simulations to explain adiabatic reversible expansion and compression. Your retention of material in this module will increase if you write down reasons
for your answers to ConcepTests, questions in screencasts, and questions to answer before using interactive simulations, and you try to solve the example problems before watching the screencast
We suggest using the learning resources in the following order:
1. Attempt to answer the multiple choice ConcepTest and solve the example problem before watching the screencasts or working with the simulations.
2. Watch the screencasts that describe the pressure-temperature relationship for adiabatic, reversible processes of ideal gases and answer the questions within the screencasts.
3. Review the important equations.
4. Use the interactive simulation to further understand the behavior of an ideal gas undergoing adiabatic expansion or contraction.
5. Try to solve the example problems before watching the solutions in the screencast.
6. Answer the ConcepTests.
7. Look at the list of key points, but only after you try to list the key points yourself.
• This module is intended for a thermodynamics course.
Before studying this module, you should be able to:
After studying this module, you should be able to:
• Calculate the final temperature and volume for an adiabatic, reversible expansion or compression of an ideal gas, given the initial temperature and pressure (or volume) and the final pressure.
• Calculate the final temperature and pressure for an adiabatic, reversible expansion or compression of an ideal gas, given the initial temperature and pressure or volume) and the final volume.
• Calculate the work for an adiabatic, reversible expansion or compression of an ideal gas in a closed system.
• Calculate the shaft work for adiabatic, reversible expansion or compression of an ideal gas in a steady-state flow system.
• Carry out the above calculations for either a constant heat capacity or a temperature-dependent heat capacity. | {"url":"https://learncheme.com/quiz-yourself/interactive-self-study-modules/adiabatic-reversible-expansion-and-compression/adiabatic-reversible-expansion-and-compression-introduction/","timestamp":"2024-11-05T19:34:19Z","content_type":"text/html","content_length":"79080","record_id":"<urn:uuid:7c99a0d5-a5a7-408e-bd42-4681889885ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00329.warc.gz"} |
vlsi design and implementation of reconfigurable ... - P.PDFKUL.COM
VLSI DESIGN AND IMPLEMENTATION OF RECONFIGURABLE CRYPTOGRAPHIC SYSTEMS FOR SYMMETRIC ENCRYPTION G. Selimis, N Lazarou, HE.Michail and O. Koufopavlou VLSI DESIGN LAB, Department of Electrical and
Computer Engineering, Patras, GREECE. Email:
[email protected]
ABSTRACT In this paper a reconfigurable cryptographic system is proposed. The proposed architecture is based on RC5 [1] algorithm standard but it can operate with any cipher. The relationship between
the number of block bits, the number of cryptographic rounds (system security), the covered system area resources, and the system's throughput is examined. The proposed cryptographic system is
reconfigurable for the number of block bits and the number of cryptographic rounds. These parameters control the system security and the system's throughput. The reconfigurability allows to system
adopt temporary parameters in order to be suitable for cryptographic transactions. Finally a parametric VLSI design methodology is proposed for constructing cryptographic systems for symmetric
1. INTRODUCTION Today more and more sensitive data is stored digitally. Bank accounts, medical records and personal emails are some categories that data must keep secure. The science of cryptography
tries to encounter the lack of security. Data confidentiality, authentication, non-reputation and data integrity are some of the main parts of cryptography. The evolution of cryptography drove in
very complex cryptographic models which they could not be implemented before some years. The use of systems with increasing complexity, which usually are more secure, has as result low throughput
rate and more energy consumption. Then the evolution of cipher has no practical impact, if it has only theoretical background. Every encryption algorithm should exploit as much as possible the
conditions of the specific system without omitting the physical, area and timing limitations. This fact requires new ways in design architectures for secure and reliable Crypto Systems [2]. Last
years many cryptographic implementations have been proposed. They are implemented in software or in hardware. The choice of the implementation depends on the application and on the algorithm that is
to implement. Software-
based approaches lead to slow implementations and they are very energy inefficient [3]. A cryptographic algorithm, also called cipher, is the mathematical function used for encryption and decryption.
A symmetric key algorithm is an algorithm for cryptography that uses the same cryptographic key to encrypt and decrypt the message.There are two basic types of symmetric algorithms: block ciphers and
stream ciphers [4]. Block ciphers operate on blocks of plaintext and cipher text-usually of 64 bits but sometimes longer. Stream Ciphers operate on streams of plaintext and cipher text one bit or
byte (sometimes even one 32-bit word) at a time. With a block cipher, the same plaintext block will always encrypt to the same cipher text block, using the same key. With a stream cipher, the same
plaintext bit or byte will encrypt to a different bit or byte every time. RC5 cipher is used for symmetric encryption in order to achieve privacy into many cryptographic standards as into WTLS layer
of the Wireless Application Protocol [5-7]. It is a fully parameterized block cipher. The key length, the number of rounds and the block size may all be specified before this cipher starts ciphertext
generation. RC5 outperforms other ciphers with intrinsic algorithmic simplicity and it is considers as one of the fastest block ciphers. The key element in RC5 is based on circular rotations. RC5
security strength relies on non-linear register rotations, as its sole non-linear operator. This kind of diffusion is simpler to be implemented in comparison with other ciphers like DES, which uses
S-Boxes and Data Permutations. In general, the software implementations of ciphers are often computationally not fast and therefore the use of hardware devices is considered as an efficient
alternative. In this architecture of a reconfigurable cryptographic system is proposed. The proposed system operates as encoder and decoder for symmetric cryptographic algorithms. It checks the
requirements in throughput and security and re-defines the number of cryptographic rounds and the number of block bits. Finally a parametric VLSI design methodology is proposed for constructing
cryptographic systems for symmetric encryption easily.
Authorized licensed use limited to: IMEC. Downloaded on April 10, 2009 at 10:57 from IEEE Xplore. Restrictions apply.
This paper is organized as follows: In section 2 the RC5 standard is presented. Information about RC5 system architecture is given in section 3. In the section 4 the proposed cryptographic
architecture is examined. A parametric VLSI design methodology for constructing cryptographic systems is proposed in section 5. Finally experimental results and conclusions are given in section 6. 2.
RCS ENCRYPTION ALGORITHM RC5 is a parameterized encryption algorithm. A particular RC5 algorithm is denoted as RC5-w/rlb. These three parameters are summarized below: w: word size in bits. Each input
(plaintext) contains two w-bit words and the output (ciphertext) block is 2w-bit long. The standard value of w is 32-bit; allowable values ofware 16, 32 and 64-bit. r: The number of rounds. Allowable
values of rare 0, 1, ... 255. b: The number of bytes in the user's secret key K. Allowable values ofb are 0, 1, ... , 255. RC5 consists of a key expansion process, an encryption and a decryption
scheme. These procedures use three primitive operations and their inverses: Addition of words modulo
W ,
The conventional architecture of RC5 Core Implementation is illustrated in Figure 2. It performs both encryption and decryption with two different cores, Encrypt and Decrypt, respectively. This
architecture design is based on RC5 specifications, which define two different schemes: one for encryption and one for decryption. The Initial Unit divides the input message into two w-bit words (A
and B). Especially in the case of encryption operation, modulo additions are performed between the first two keys S(O), S(l) and the words A and B, respectively. Encryption Core performs the
encryption transformation of r rounds. In every round i, two proper keys S(2i), S(2i+1) are used. The defined components for the primitive decryption operation are modulo subtraction 2w, right
circular shift register (R.C.S.) and XOR operation (Figure 3). When the r rounds are performed, the Final Unit concatenates the two w-bit words into the 2w-bit output message. In the case of
decryption operation, modulo subtraction 2w with the first two keys S(O), S(l) are performed before the 2wbit cipher is produced. 2w-bit plaintext
and the inverse
operation, modulo- 2 W subtraction. Bit-wise exclusive -OR of words, denoted by XOR. Rotation: the rotation of word x left by y bits is denoted L.C.S (left circular shift). The inverse operation is
the rotation of word x right by y bits, denoted by R. C. S.. (right circular shift) The Key expansion process expands the user's secret key K to fill the expanded key table S, so that S resembles an
array of t=2(r+1) random binary words determined by K.
3. RCS BASIC SYSTEM ARCHITECTURE The main units of the system are: the key expansion unit, the RAM blocks which are used for keys storage, and the RCS Core that is the unit which both encryption/
decryption schemes are performed. RCS needs an extended table of keys which we symbolize with S and it results from the used secret key K. The size of S depends from the r and is t=2*(r+1) words.
First the Key Expansion Unit computes the values of the keys. The keys are stored in the RAM and the RC5 core uses these keys. In figure 1 a typical implementation of a com lete RC5 s stem is
resented. plain! cipher
key expansion
Figure 1. The main units ofRC5 Crypto System
Figure 2. The conventional architecture ofRC5 core
4. PROPOSED CRYPTO IMPLEMENTATIONS 4.1. Applying Pipelining Technique We implemented the RC5 Round (encoder and decoder) based on the RC5 standard [1] and then by applying the pipeline technique we
had the following results presented in Table 1. Table 1. S thesis Results for the RC5 Round
Architecture no i elined i elined
Area Clbs 1884 1905
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Dela ns 12.98 ns 47.03 ns
By the above results we conclude that by using pipelined technique, the covered area is increased by 1.1 % with corresponding increase of system's throughput by 262%. Therefore for the proposed
architecture the basic element RC5 Round implemented with pipelined logic. In the Figure 3 the encoder and decoder for the RC5 Round are presented.
The implementation of system for B internally RC5 Crypto Cores is presented in figure 4. However when the number of B increases, it leads to expensive systems because of covered area resources. For
this reason, the introducing of feedback choice is essential. Then the system can support high secure transactions without increased area covered system resources. In this case, the achieved
throughput decreases. The keys of system come from the memory, where are stored temporarily. At the end of every operation of encryption/decryption, the keys are erased from memory and they are
updated. Consequently, each cryptanalytic attacker begins his attacks from the beginning.
I REGISTER 1 I S(2i) S(2i+1)
REGISTER 1 S(2i) S(2i+1)
Figure 3. The Encoder and Decoder ofRc5 Round
4.2. Top Level Architecture of Proposed System The architecture of proposed system is presented in Figure 4. The red coloured lines present the signals which they come from control unit. The basic
signals of control unit are encryption/decryption signal that decides which type of coding will be executed and mode signal which determines the size of maximum block that can be processed by the
system. In the proposed system the size of maximum block is considered to be 128-bit. The system can support the coding of 32, .64, .96 and 128 bits. The case of 96 bits rejected because it is not
supported from RC5 standard. An other basic signal is the line that checks the number of cryptographic rounds that the system executes. It controls the last multiplexer of the system. The multiplexer
decides if the data feed the input or be forwarded to the output.
Figure 4. Top Level Architecture of proposed system
The system has the possibility of processing 32, 64 and 128 bits messages. This system is suitable for systems as servers and smart card readers. It communicates with entities of different degree of
security and block bits. In the first stage of process, it is examined if the two systems can they cooperate. If this step passed then the process of cryptography starts.
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5. CONSTRUCTING CRYPTO CORES In order to study the behavior of system for different number of cryptographic rounds, block bits, and available Crypto Cores we wrote parametric VHDL code. All VHDL
files are parametric and their values are determined by the file pack. vhd. With this technique is easy to construct reconfigurable systems determining each time the number of cipher Cores and the
maximum system's block bits. Below the first lines pack.vhd file are presented and gives values to the basic parameters of system. library ieee; use ieee.std_Iogic_1164.all; package myJack is comtant
N:integer :=1()#127#; comtant P:integer :=10#31#; comtant S:integer :=10#6#; -IIIlI1Wer ofpipe/tiled stages of Res comtant R:integer :=10#6#; -Illlnwer ofroullds comtant B:integer :=10#3#; -IlUI1Wer
ofb/ocks=(RCS core + reg) constant C:integer :=R!B; --nunwer offeedbacks
Figure 5. The first lines ofpack.vhd
6. EXPERIMENTAL RESULTS The proposed architecture has been captured by using VHDL. All the internal components of the design were synthesized placed and routed using XILINX FPGA devices [8].
Synthesis results for the proposed system implementation are illustrated in Table 2. . R esu ts T a bl e 2. Coml anson 0 fS.ynth eSlS Number of B=2 B=3 B=4 B=5 RC5 Cores lOs 774 646 902 1030 Function
12745 8933 16564 20377 Generators 4467 6373 8282 CLB Slices 10189 4485 7951 6216 9682 Dffs Maximum 68.2 MHz Frequency
B=6 1158 24189 12094 11413
From the table above we observe that importing RC5 Crypto Cores influences the area covered system resources. However, the maximum frequency of system is not decreased because of pipelined technique.
The introducing of Crypto Cores gives secure and high throughput systems. For B=6 with the application of the feedback method the system operates in 12 cryptographic rounds, so it is a high secure
cryptographic system. Table 3 presents the system's throughput as a function of number bits, the internal RC5 Crypto Cores, and the cryptographic rounds. The parameter B and the number of block bits
increase the throughput of system linearly. Also [3] the system becomes more secure. On the contrary, when we make the system secure (increasing the number of cryptographic rounds) by using feedback
method we have losses in the throughput.
T a bl e 3. The fu nchon 0 f througJh tput Word (bit) 64 32 128 Throughtput (B/R)*2182,4 (B/R)*4364,8 (B/R)*8729,6 (Mbps)
7. CONCLUSIONS In this paper a reconfigurable hardware design architecture for symmetric encryption is proposed. The level of security and the available system resources control system parameters as
the number of cryptographic rounds and the processed bits. It is clear that the number of additional rounds in a cryptographic system has as result the system to become more secure. Depending on
system specifications, the system can increase its security applying feedback method and increasing the processed bits. Feedback method leads to less efficient systems and increasing the number of
processed bits to more expensive and area resources demanded. Every time we desire a golden ratio between covered area system resources and security. When there is need for high secure systems, we
design implementations with more internal RC5 cores, as it is presented in section 5.
8. REFERENCES [1] Ronald 1. Rivest, "The RC5 Encryption Algorithln", proceedings of the 1994 Leuven Workshop on Fast Software Encryption (Springer 1995), pages 86-96.. [2] P. Kitsos, O. Koufopavlou,
G. Selimis and N. Sklavos, "Low Power Cryptography", accepted for publication in the Journal of Physics Conference Series. [3] James Goodman and Anatha P. Chandrakasan, "An EnergyEfficient
Reconfigurable public-key Cryptography Processor", IEEE Journal of solid-state circuits, vol. 36, NO 11, November 2001. [4] Bruce Scneier, "Applied Cryptography: Protocols, Algorithms, and Source in
C", published by John Willey & Sons, Inc 1996. [5] Burton S. Kalinski jr., Yiqun Lisa Yin, "On the security of the RC5 Encryption Algorithm", RSA Laboratories technical Report, September 1998. [6] N.
Sklavos and o. Koufopavlou, "Mobile Communications World: Security Implementations Aspects- A state of the Art", Computer Science Journal of Moldova, Institute of Mathematics & Computer Science, Vol.
11, Number 2,2003. [7] N. Sklavos, A. P. Fournaris, and O. Koufopavlou, "WAP Security: Implementation Cost and Performance Evaluation of a Scalable Architecture for RC5 Parameterized Block Cipher",
proceedings of IEEE Mediterranean Electrotechnical Conference (MELECON '04), Dubrovnik, Croatia, May 12-15, 2004. [8] Xilinx, San Jose, California, USA, Virtex, 2.5 V Field Programmable Gate Arrays,
www.xilinx.com. 2003.
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The object Stream provides helper functions to manipulate streams.
1.1 08/08/03
Linear Supertypes
Content Hierarchy Learn more about scaladoc diagrams
2. final def !=(arg0: Any): Boolean
Test two objects for inequality.
Test two objects for inequality.
true if !(this == that), false otherwise.
Definition Classes
3. final def ##(): Int
Equivalent to x.hashCode except for boxed numeric types and null.
Equivalent to x.hashCode except for boxed numeric types and null. For numerics, it returns a hash value which is consistent with value equality: if two value type instances compare as true, then
## will produce the same hash value for each of them. For null returns a hashcode where null.hashCode throws a NullPointerException.
a hash value consistent with ==
Definition Classes
AnyRef → Any
4. object #::
An extractor that allows to pattern match streams with #::.
6. final def ==(arg0: Any): Boolean
Test two objects for equality.
Test two objects for equality. The expression x == that is equivalent to if (x eq null) that eq null else x.equals(that).
true if the receiver object is equivalent to the argument; false otherwise.
Definition Classes
9. def apply[A](xs: A*): Stream[A]
A stream consisting of given elements
A stream consisting of given elements
the type of the collection's elements
a new collection with elements elems
Definition Classes
Stream → GenericCompanion
10. final def asInstanceOf[T0]: T0
Cast the receiver object to be of type T0.
Cast the receiver object to be of type T0.
Note that the success of a cast at runtime is modulo Scala's erasure semantics. Therefore the expression 1.asInstanceOf[String] will throw a ClassCastException at runtime, while the expression
List(1).asInstanceOf[List[String]] will not. In the latter example, because the type argument is erased as part of compilation it is not possible to check whether the contents of the list are of
the requested type.
the receiver object.
Definition Classes
Exceptions thrown
if the receiver object is not an instance of the erasure of type T0.
12. def clone(): AnyRef
Create a copy of the receiver object.
Create a copy of the receiver object.
The default implementation of the clone method is platform dependent.
a copy of the receiver object.
Definition Classes
not specified by SLS as a member of AnyRef
13. Concatenates all argument collections into a single collection.
Concatenates all argument collections into a single collection.
the collections that are to be concatenated.
the concatenation of all the collections.
Definition Classes
14. An alternative way of building and matching Streams using Stream.
15. implicit def consWrapper[A](stream: ⇒ Stream[A]): ConsWrapper[A]
A wrapper method that adds #:: for cons and #::: for concat as operations to streams.
16. def continually[A](elem: ⇒ A): Stream[A]
Create an infinite stream containing the given element expression (which is computed for each occurrence).
Create an infinite stream containing the given element expression (which is computed for each occurrence).
the element composing the resulting stream
the stream containing an infinite number of elem
17. def empty[A]: Stream[A]
The empty stream
18. Tests whether the argument (arg0) is a reference to the receiver object (this).
Tests whether the argument (arg0) is a reference to the receiver object (this).
The eq method implements an equivalence relation on non-null instances of AnyRef, and has three additional properties:
□ It is consistent: for any non-null instances x and y of type AnyRef, multiple invocations of x.eq(y) consistently returns true or consistently returns false.
□ For any non-null instance x of type AnyRef, x.eq(null) and null.eq(x) returns false.
□ null.eq(null) returns true.
When overriding the equals or hashCode methods, it is important to ensure that their behavior is consistent with reference equality. Therefore, if two objects are references to each other (o1 eq
o2), they should be equal to each other (o1 == o2) and they should hash to the same value (o1.hashCode == o2.hashCode).
true if the argument is a reference to the receiver object; false otherwise.
Definition Classes
19. The equality method for reference types.
The equality method for reference types. Default implementation delegates to eq.
See also equals in scala.Any.
true if the receiver object is equivalent to the argument; false otherwise.
Definition Classes
AnyRef → Any
20. def fill[A](n: Int)(elem: ⇒ A): Stream[A]
Produces a collection containing the results of some element computation a number of times.
Produces a collection containing the results of some element computation a number of times.
the number of elements contained in the collection.
the element computation
A collection that contains the results of n evaluations of elem.
Definition Classes
Stream → GenTraversableFactory
21. Produces a five-dimensional collection containing the results of some element computation a number of times.
Produces a five-dimensional collection containing the results of some element computation a number of times.
the number of elements in the 1st dimension
the number of elements in the 2nd dimension
the number of elements in the 3nd dimension
the number of elements in the 4th dimension
the number of elements in the 5th dimension
the element computation
A collection that contains the results of n1 x n2 x n3 x n4 x n5 evaluations of elem.
Definition Classes
22. Produces a four-dimensional collection containing the results of some element computation a number of times.
Produces a four-dimensional collection containing the results of some element computation a number of times.
the number of elements in the 1st dimension
the number of elements in the 2nd dimension
the number of elements in the 3nd dimension
the number of elements in the 4th dimension
the element computation
A collection that contains the results of n1 x n2 x n3 x n4 evaluations of elem.
Definition Classes
23. Produces a three-dimensional collection containing the results of some element computation a number of times.
Produces a three-dimensional collection containing the results of some element computation a number of times.
the number of elements in the 1st dimension
the number of elements in the 2nd dimension
the number of elements in the 3nd dimension
the element computation
A collection that contains the results of n1 x n2 x n3 evaluations of elem.
Definition Classes
24. def fill[A](n1: Int, n2: Int)(elem: ⇒ A): Stream[Stream[A]]
Produces a two-dimensional collection containing the results of some element computation a number of times.
Produces a two-dimensional collection containing the results of some element computation a number of times.
the number of elements in the 1st dimension
the number of elements in the 2nd dimension
the element computation
A collection that contains the results of n1 x n2 evaluations of elem.
Definition Classes
25. def finalize(): Unit
Called by the garbage collector on the receiver object when there are no more references to the object.
Called by the garbage collector on the receiver object when there are no more references to the object.
The details of when and if the finalize method is invoked, as well as the interaction between finalize and non-local returns and exceptions, are all platform dependent.
Definition Classes
not specified by SLS as a member of AnyRef
26. Create an infinite stream starting at start and incrementing by 1.
Create an infinite stream starting at start and incrementing by 1.
the start value of the stream
the stream starting at value start.
27. Create an infinite stream starting at start and incrementing by step step.
Create an infinite stream starting at start and incrementing by step step.
the start value of the stream
the increment value of the stream
the stream starting at value start.
28. final def getClass(): Class[_]
A representation that corresponds to the dynamic class of the receiver object.
A representation that corresponds to the dynamic class of the receiver object.
The nature of the representation is platform dependent.
a representation that corresponds to the dynamic class of the receiver object.
Definition Classes
AnyRef → Any
not specified by SLS as a member of AnyRef
29. def hashCode(): Int
The hashCode method for reference types.
The hashCode method for reference types. See hashCode in scala.Any.
the hash code value for this object.
Definition Classes
AnyRef → Any
30. final def isInstanceOf[T0]: Boolean
Test whether the dynamic type of the receiver object is T0.
Test whether the dynamic type of the receiver object is T0.
Note that the result of the test is modulo Scala's erasure semantics. Therefore the expression 1.isInstanceOf[String] will return false, while the expression List(1).isInstanceOf[List[String]]
will return true. In the latter example, because the type argument is erased as part of compilation it is not possible to check whether the contents of the list are of the specified type.
true if the receiver object is an instance of erasure of type T0; false otherwise.
Definition Classes
31. def iterate[A](start: A, len: Int)(f: (A) ⇒ A): Stream[A]
Produces a collection containing repeated applications of a function to a start value.
Produces a collection containing repeated applications of a function to a start value.
the start value of the collection
the number of elements contained inthe collection
the function that's repeatedly applied
a collection with len values in the sequence start, f(start), f(f(start)), ...
Definition Classes
Stream → GenTraversableFactory
32. def iterate[A](start: A)(f: (A) ⇒ A): Stream[A]
An infinite stream that repeatedly applies a given function to a start value.
An infinite stream that repeatedly applies a given function to a start value.
the start value of the stream
the function that's repeatedly applied
the stream returning the infinite sequence of values start, f(start), f(f(start)), ...
33. Equivalent to !(this eq that).
Equivalent to !(this eq that).
true if the argument is not a reference to the receiver object; false otherwise.
Definition Classes
34. def newBuilder[A]: Builder[A, Stream[A]]
Creates a new builder for a stream
Creates a new builder for a stream
the type of the collection's elements
Definition Classes
Stream → GenericCompanion
35. final def notify(): Unit
Wakes up a single thread that is waiting on the receiver object's monitor.
Wakes up a single thread that is waiting on the receiver object's monitor.
Definition Classes
not specified by SLS as a member of AnyRef
36. final def notifyAll(): Unit
Wakes up all threads that are waiting on the receiver object's monitor.
Wakes up all threads that are waiting on the receiver object's monitor.
Definition Classes
not specified by SLS as a member of AnyRef
37. def range[T](start: T, end: T, step: T)(implicit arg0: Integral[T]): Stream[T]
Produces a collection containing equally spaced values in some integer interval.
Produces a collection containing equally spaced values in some integer interval.
the start value of the collection
the end value of the collection (the first value NOT contained)
the difference between successive elements of the collection (must be positive or negative)
a collection with values start, start + step, ... up to, but excluding end
Definition Classes
Stream → GenTraversableFactory
38. def range[T](start: T, end: T)(implicit arg0: Integral[T]): Stream[T]
Produces a collection containing a sequence of increasing of integers.
Produces a collection containing a sequence of increasing of integers.
the first element of the collection
the end value of the collection (the first value NOT contained)
a collection with values start, start + 1, ..., end - 1
Definition Classes
39. final def synchronized[T0](arg0: ⇒ T0): T0
40. def tabulate[A](n: Int)(f: (Int) ⇒ A): Stream[A]
Produces a collection containing values of a given function over a range of integer values starting from 0.
Produces a collection containing values of a given function over a range of integer values starting from 0.
The number of elements in the collection
The function computing element values
A collection consisting of elements f(0), ..., f(n -1)
Definition Classes
Stream → GenTraversableFactory
41. Produces a five-dimensional collection containing values of a given function over ranges of integer values starting from 0.
Produces a five-dimensional collection containing values of a given function over ranges of integer values starting from 0.
the number of elements in the 1st dimension
the number of elements in the 2nd dimension
the number of elements in the 3nd dimension
the number of elements in the 4th dimension
the number of elements in the 5th dimension
The function computing element values
A collection consisting of elements f(i1, i2, i3, i4, i5) for 0 <= i1 < n1, 0 <= i2 < n2, 0 <= i3 < n3, 0 <= i4 < n4, and 0 <= i5 < n5.
Definition Classes
42. Produces a four-dimensional collection containing values of a given function over ranges of integer values starting from 0.
Produces a four-dimensional collection containing values of a given function over ranges of integer values starting from 0.
the number of elements in the 1st dimension
the number of elements in the 2nd dimension
the number of elements in the 3nd dimension
the number of elements in the 4th dimension
The function computing element values
A collection consisting of elements f(i1, i2, i3, i4) for 0 <= i1 < n1, 0 <= i2 < n2, 0 <= i3 < n3, and 0 <= i4 < n4.
Definition Classes
43. Produces a three-dimensional collection containing values of a given function over ranges of integer values starting from 0.
Produces a three-dimensional collection containing values of a given function over ranges of integer values starting from 0.
the number of elements in the 1st dimension
the number of elements in the 2nd dimension
the number of elements in the 3nd dimension
The function computing element values
A collection consisting of elements f(i1, i2, i3) for 0 <= i1 < n1, 0 <= i2 < n2, and 0 <= i3 < n3.
Definition Classes
44. Produces a two-dimensional collection containing values of a given function over ranges of integer values starting from 0.
Produces a two-dimensional collection containing values of a given function over ranges of integer values starting from 0.
the number of elements in the 1st dimension
the number of elements in the 2nd dimension
The function computing element values
A collection consisting of elements f(i1, i2) for 0 <= i1 < n1 and 0 <= i2 < n2.
Definition Classes
45. def toString(): String
Creates a String representation of this object.
Creates a String representation of this object. The default representation is platform dependent. On the java platform it is the concatenation of the class name, "@", and the object's hashcode in
a String representation of the object.
Definition Classes
AnyRef → Any
46. This method is called in a pattern match { case Seq(.
This method is called in a pattern match { case Seq(...) => }.
the selector value
sequence wrapped in an option, if this is a Seq, otherwise none
Definition Classes
47. final def wait(): Unit
48. final def wait(arg0: Long, arg1: Int): Unit
49. final def wait(arg0: Long): Unit | {"url":"https://www.javadoc.io/static/org.scala-lang/scala-library/2.10.1-RC2/scala/collection/immutable/Stream$.html","timestamp":"2024-11-09T03:37:02Z","content_type":"text/html","content_length":"99983","record_id":"<urn:uuid:2de5fa50-6d93-4f03-aa74-1f45daa3413f>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00613.warc.gz"} |
Regression with Dummy Variable
Dummy variables or categorical variables arise quite often in real world data. For example, choosing between investing or not in a company’s share is a decision variable that can only take two
values: YES or NO. Similarly, deciding which continent to spend your next vacation in can only take certain specific values: Asia, Africa, Europe, South America, etc. Categorical variables can not
only capture situations where there is no inherent ordering of the options (like the above two examples, or say male versus female, etc.) but also when the values carry ordinal meaning (e.g., how
happy are you at the moment on an integer scale of 1 to 5 with 5 being the happiest, or how democratic is a country’s politics on an integer scale of 1 to 10 with 10 being the perfect democracy).
A. Dummy Explanatory Variable: When one or more of the explanatory variables is a dummy variable but the dependent variable is not a dummy, the OLS framework is still valid. However, one should be
cautious about how to include these dummy explanatory variables and what are the interpretations of the estimated regression coefficients for these dummies. First, one must be careful to include one
less dummy variable than the total number of categories of the explanatory variable. For example, if the categorical variable ‘sex’ can take only 2 values, viz., male and female, then only one dummy
variable for sex should be included in the regression to avoid the problem of muticollinearity. Including as many dummy variables as the number of categories along with the intercept term in a
regression leads to the problem of the “Dummy Variable Trap”. So the rule is to either drop the intercept term and include a dummy for each category, or keep the intercept and exclude the dummy for
any one category.
To provide an example, let us suppose our sample of individuals have five levels of wealth; poorest, poorer , middle , richer and richest. We are interested in understanding the relation between
total number of children born in a family and their wealth level. (The data can be found here.)
We can create 5 dummy variables, called poorest, poorer , middle , richer and richest. The variable poorest takes the value 1 for individuals who have the poorest wealth and 0 otherwise. The variable
poorer takes the value 1 for individuals who have poorer wealth and 0 otherwise. Similarly, we construct the other variables. We can take two approaches while regressing total number of children born
in a family on wealth levels:
I. Include the constant term, poorest, poorer , middle , richer in the regression and drop richest.
The regression looks like
v201[j] = b[0] + b[1]*poorest[j] + b[2]*poorer[j] + b[3]*middle[j] + b[4]*richer[j] +u[[j]]
• The constant gives the expected number of children born in a household with the richest wealth level since v201[j] = b[0] when all the variables take the value 0.
• The coefficient of poorest is interpreted as the difference between the expected number of children born in the household with the poorest wealth level and the richest wealth level. This is true
because v201[j] = b[0] + b[1] when poorest=1 and all other variables are zero.
• Similarly, the coefficient of the other coefficients show the difference between the expected the number children born in the household with that particular wealth level and the richest wealth
II. Exclude the constant term, and include all the 5 variables.
The regression will look like:
v201[j] = b[0]*richest[j] + b[1]*poorest[j] + b[2]*poorer[j] + b[3]*middle[j] + b[4]*richer[j] +u[[j]]
• Now in this regression, each coefficient gives the expected number of children born in the household given that particular wealth level. As you can see, the coefficient of richest is the same as
the constant in the first method.
III. Include the constant term and all 5 variables.
Such a regression leads to multicollinearity and Stata solves this problem by dropping one of the dummy variables.
• Stata will automatically drop one of the dummy variables. In this case, it displays after the command that poorer is dropped because of multicollinearity.
• The constant term now reflects the expected number of children born in the poorer households. The coefficient is 2.875 which is the same as in the table before.
• The interpretation of the other coefficients is similar to the first example with the exception that now the base group is poorer household instead of richest as in the first example.
B. Dummy Dependent Variable: OLS regressions are not very informative when the dependent variable is categorical. To handle such situations, one needs to implement one of the following regression
techniques depending on the exact nature of the categorical dependent variable.
Do keep in mind that the independent variables can be continuous or categorical while running any of the models below. There is no need for the independent variables to be binary just because the
dependent variable is binary.
(i) Logistic Regression (Logit): A logistic regression fits a binary response (or dichotomous) model by maximum likelihood. It models the probability of a positive outcome given a set of regressors.
When the dependent variable equals a non-zero and non-missing number (typically 1), it indicates a positive outcome, whereas a value of zero indicates a negative outcome. Mathematically speaking,
running a logit of the dependent variable y on the regressors x[1] and x[2] basically fits the following model by estimating the coefficients b[0], b[1] and b[2]: Prob (y[j] = 1 | x[1j], x[2j]) =
exp(b[0]+b[1]x[1j]+b[2]x[2j]) / [exp(b[0]+b[1]x[1j]+b[2]x[2j]) + 1]. The specific functional form of the probability arises from the assumption of a logistic distribution for the error term in the
As an example, if we have data on weight and mileage of 22 foreign and 52 domestic automobiles, we may wish to fit a logit model explaining whether a car is foreign or not on the basis of its weight
and mileage. (The data can be found here.)
The STATA command to run a logit model is as follows:
Here the dependent variable foreign takes the value 1 if the car is foreign and 0 if it is domestic. The regressors weight and mpg are usual continuous variables and denote the weight and mileage of
the car respectively.
• The above STATA command yields estimates of the three coefficients: one constant/intercept, and the two coefficients for weight and mileage.
• The coefficient of weight implies that a unit increase in weight reduces the logs odds of the car being foreign (vs. domestic) by -0.004.
• The coefficient of mpg implies that a unit increase in mileage reduces the logs odds of the car being foreign by -0.17.
• The fourth column in the table shows the level of significance of the null hypothesis that the coefficient is equal to zero can be rejected. All the coefficients are statistically significantly
from zero at the 10% level of significance.
Sometimes, the researcher is not interested in the coefficients b[0], b[1] and b[2] per se but in the odds ratios of the individual regressors, namely, exp(b[0]), exp(b[1]) and exp(b[2]).
This can be implemented in STATA using the following command:
• Now, the coefficient of weight implies that a unit increase in weight of the car increases the odds of the car being foreign by a factor of 0.996.
• The coefficient of mileage shows that a unit increase in the mileage of the car increases the odds of the car being foreign by 0.84.
One must be cautious when interpreting the odds ratio of the constant/intercept term. Usually, this odds ratio represents the baseline odds of the model when all predictor variables are set to zero.
Howeer, one must verify that a zero value for all predictors actually makes sense before continuing with this interpretation. For example, a weight of zero for a car does not make sense in the above
example, and so the odds ratio estimate for the intercept term here does not carry any meaning.
(ii) Probit Regression (Probit): One can change the distributional assumption of a logistic regression by assuming a standard normal distribution instead of the logistic distribution for the
probability of a positive outcome. In other words, Prob.(y[j] = 1 | x[1j], x[2j]) = φ(b[0]+b[1]x[1j]+b[2]x[2j]) where φ(.) denotes the cumulative distribution function of a standard normal
distribution. This model is called a Probit model.
Let us return to our previous example and run a probit model rather than a logit model. This can be implemented in STATA using the following command:
• The regression coefficients have the same interpretation as the Logit model, i.e., the coefficient of weight implies that a unit increase in weight reduces the logs odds of the car being foreign
(vs. domestic) by -0.004.
• As can be seen, all the coefficients are quite similar to the logit model.
• The choice of logit or probit model depends on economic theory and preference of the researcher.
Note: Both the Logit and Probit models are suitable when the dependent variable is binary or dichotomous. When the dependent variable has more than two categories, one needs to implement either a
multinomial logistic regression or an ordered logistic regression, discussed below.
(iii) Multinomial Logit: In a multinomial logit model, the number of outcomes that the dependent variable can possibly accommodate is greater than two. This is the main difference of the multinomial
from the ordinary logit. However, multinomial logit only allows for a dependent variable whose categories are not ordered in a genuine sense (for which case one needs to run an Ordered Logit
Consider a regression of y on x where the categorical dependent variable y has 3 possible outcomes. In the multinomial logit model, one estimates a set of coefficients b[0]^(1), b[1]^(1), b[0]^(2), b
[1]^(2), b[0]^(3), b[1]^(3), corresponding to each outcome:
Prob (y =1) = exp(b[0]^(1) + b[1]^(1)x) / [ exp(b[0]^(1) + b[1]^(1)x) + exp(b[0]^(2) + b[1]^(2)x) + exp(b[0]^(3) + b[1]^(3)x) ]
Prob (y =2) = exp(b[0]^(2) + b[1]^(2)x) / [ exp(b[0]^(1) + b[1]^(1)x) + exp(b[0]^(2) + b[1]^(2)x) + exp(b[0]^(3) + b[1]^(3)x) ]
Prob (y =3) = exp(b[0]^(3) + b[1]^(3)x) / [ exp(b[0]^(1) + b[1]^(1)x) + exp(b[0]^(2) + b[1]^(2)x) + exp(b[0]^(3) + b[1]^(3)x) ]
This model, however, is unidentified in the sense that there is more than one solution to b[0]^(1), b[1]^(1), b[0]^(2), b[1]^(2), b[0]^(3), b[1]^(3), that leads to the same probabilities for y=1, y=2
and y=3. To identify the model, one needs to set b[0]^(k) = b[1]^(k) =0 for any one of the outcomes k=1, 2 and 3. That outcome is called the base outcome, and the remaining coefficients will measure
the change relative to that y=k group. The coefficients will differ because they have different interpretations, but the predicted probabilities for y=1, 2 and 3 will still be the same. For example,
setting b[0]^(2) = b[1]^(2) =0, the equations become
Prob (y =1) = exp(b[0]^(1) + b[1]^(1)x) / [ 1 + exp(b[0]^(1) + b[1]^(1)x) + exp(b[0]^(3) + b[1]^(3)x) ]
Prob (y =2) = 1 / [ 1 + exp(b[0]^(1) + b[1]^(1)x) + exp(b[0]^(3) + b[1]^(3)x) ]
Prob (y =3) = exp(b[0]^(3) + b[1]^(3)x) / [ 1 + exp(b[0]^(1) + b[1]^(1)x) + exp(b[0]^(3) + b[1]^(3)x) ]
As an example, an individual’s choice of not being in the labour force, becoming a full-time worker or a part-time worker could be modeled using their education and control for age and sex. (The data
can be found here.).
We have defined the variable lab_status as containing the employment and labour force participation of an individual. 0 refers to not in labour force, 1 refers to full-time work and 2 refers to
part-time worker. The variable sex is defined as male taking the value 1 and female taking the value 2.
To implement such a multinomial logit in STATA, the following command can be run:
mlogit lab_status sex age education
• The coefficient of sex implies that the relative log odds of being out of the labor force vs. being a full-time worker will increase if the individual is a female vs. male.
• The coefficient of sex implies that the relative log odds of being a part-time worker vs. being a full-time worker will increase if the individual is a female vs. male.
• A one unit increase in age is associated with a 0.018 increase in the relative log odds of being out of the labor force vs. being full-time employed.
• A one unit increase in age is associated with a 0.011 decrease in the relative log odds of being part-time employed vs. being full-time employed.
• The coefficient of education implies a one year increase in years of schooling decreases the relative log odds of being out of the labor force vs. full-time employed by 0.28.
• The coefficient of education implies a one year increase in years of schooling is associated with a 0.13 decreases in the relative log odds of being part-time employed vs. being full-time
The above command allows STATA to arbitarily choose which outcome to use as the base outcome. If one wants to specify the base outcome, it can be done by adding the base() option. Suppose, we want to
compare with being out of the labor force rather than full-time worker. In this case the base outcome is 0 and to implement it in Stata we will run the following command:
mlogit lab_status sex age education, base(0)
• This tells STATA to treat the zero category (y=0) as the base outcome, and suppress those coefficients and interpret all coefficients with out-of the labor force as the base group. The value in
the base category depends on what values the y variable have taken in the data.
• The coefficient of sex, age, education and constant have flipped signs but have the same magnitude on those for a full-time employed individual compared to those for out-of the labor force in the
previous case. This is to be expected as we have flipped the base group.
• The coefficient of sex implies that the relative log odds of being part-time employed vs. out of the labor force will only slightly increase if the individual is a female vs. male.
• A one unit increase in age is associated with a 0.029 decrease in the relative log odds of being part-time employed vs. out of the labor force.
• A unit increase in the years to schooling is associated with a 0.15 increase in the relative log odds of being part-time employed vs, out of the labor force.
Similar to odds-ratios in a binary-outcome logistic regression, one can tell STATA to report the relative risk ratios (RRRs) instead of the coefficient estimates. The relative risk ratio for a
one-unit change in an explanatory variable is the exponentiated value of the correspending coefficient. So, for example when the base outcome is y=2, the relative risk of y=3 is the relative
probability [Prob(y=3)/Prob(y=2)] = exp(b[0]^(3) + b[1]^(3)x). Then the relative risk ratio (RRR) of y=3 for a one-unit change in x is given by exp(b[1]^(3)), which is what STATA reports when the
rrr option is turned on.
This is done by the following command:
mlogit lab_status sex age education, base(0) rrr
• The relative risk ratio of an extra year of schooling is 1.16 (exp(0.15)) for being part-time employed vs. out of labor force.
(iv) Ordered Logit: In an ordered logit model the actual values taken on by the categorical dependent variable are irrelevant, except that larger values are assumed to correspond to ‘higher’
outcomes. Such dependent variables are often called ‘ordinal‘, for instance, ‘poor’, ‘good’, and ‘excellent’, which might indicate a person’s current health status or the repair record of a car.
In ordered logit, an underlying score is estimated as a linear function of the explanatory variables and a set of cutoffs. The probability of observing outcome k, Prob (y=k), corresponds to the
probability that the estimated linear function, plus the random error, is within the range of the cutoffs estimated for the outcome: Prob (y[j] = k) = Prob ( c[k-1] < b[0] + b[1]x[1j] + b[2]x[2j] + u
[j] < c[k]) where the error term u[j] is assumed to be logistically distributed. STATA reports the estimates of the coefficients b[0], b[1] and b[2] together with the cutoff points c[1], c[2], … , c
[K-1], where K is the number of possible outcomes of y. c[0] is taken as negative infinity, and c[K] is taken as positive infinity.
Let’s consider the following example: We want to model the choice of the total number of children born in a family using data on the daughter to son ratio (v203/v201), education (v133), age of the
spouse (v012) and wealth quintile dummies. We can create 5 dummy variables, called poorest, poorer , middle , richer and richest. The variable poorest takes the value 1 for individuals who have the
poorest wealth and 0 otherwise. The variable poorer takes the value 1 for individuals who have poorer wealth and 0 otherwise. Similarly, we construct the other variables. (The data can be found here
Ordered logits can be implemented in STATA using the following command:
ologit v201 daughter_son_ratio v133 v012 poorest poorer middle richer
• A unit increase in the daughter to son ratio will increase the log odds of having another child by 0.86.
• A unit increase in the years to schooling (v133) will reduce the log odds of having another child by 0.16.
• The log odds of having another child increases as age increases.
• The omitted wealth quintile in the regression is the richest wealth quintile. Thus, the poorest household has a higher log odds of 1.51 of having a child compared to the richest household.
• Similarly, middle and richer households have a higher log odds of having another child compared to the richest household.
To, obtain the odds ratio instead of the log odds, we need to use the or option. To implement it in Stata, we need to run the following command:
ologit v201 daughter_son_ratio v133 v012 poorest poorer middle richer, or
• A unit increase in the daughter to son ratio will increase the odds of having another child by 2.35.
A model with categorical dependent variable: Another Example
Consider the possible outcomes 1, 2, 3, …, k of the dependent variable y. Suppose k=3 — “buy an American car”, “buy a Japanese car”, and “buy a European car”. The values of y are then said to be ‘
unordered’. Even though the outcomes are coded as 1, 2 and 3, the numerical values are arbitrary because 1<2<3 does not imply that outcome 1 (buy American) is less than outcome 2 (buy Japanese) is
less than outcome 3 (buy European). This unordered categorical nature of the dependent variable distinguishes the use of mlogit from regress (which is appropriate for a continuous dependent
variable), from ologit (which is appropriate for ordered categorical data), and from logit or logistic or probit (which are suitable for only k=2 or binary outcomes, which may or may not be thought
of as ordered). | {"url":"https://blogs.ubc.ca/datawithstata/home-page/regression/ordinary-least-square/","timestamp":"2024-11-01T20:25:50Z","content_type":"text/html","content_length":"78026","record_id":"<urn:uuid:ee9aa7bd-33b0-4bcd-9b94-52d18e62802d>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00897.warc.gz"} |
Very simple long-short | Portfolio Probe | Generate random portfolios. Fund management software by Burns Statistics
Very simple long-short
Generate a set of portfolios with some simple constraints.
• vector of asset prices
• Portfolio Probe
You need the prices at which assets trade.
You also need to have the Portfolio Probe package loaded into your R session:
If you don’t have Portfolio Probe, see “Demo or Buy”.
Doing the example
You need to have the package loaded into your R session:
Doing it
We start by naming the vector of prices that we want to use:
priceVector <- xassetPrices[251,]
These are the prices at the close of the last trading day of 2006. The first few values are:
> head(priceVector)
XA101 XA103 XA105 XA107 XA108 XA111
33.56 72.25 74.39 192.06 5.91 15.98
The requirement for the prices is that it be a vector of positive numbers with names (that are the asset identifiers).
Now we want to generate 3 random portfolios with the constraints:
• gross value very close to $1,000,000
• net value between $5000 and $20,000
• no more than 4 assets in the portfolio
We can get such portfolios with the command:
rplsVerysimp <- random.portfolio(3, priceVector,
gross=1e6, net=c(5000, 20000), port.size=4)
print result
The result looks like:
> rplsVerysimp
XA262 XA535 XA561 XA674
-5390 -3849 -4675 16813
XA126 XA231 XA319 XA623
-129 -9451 -12744 5377
XA228 XA373 XA781 XA986
-4267 1 6924 -6811
random.portfolio(number.rand = 3, prices = priceVector, gross = 1e+06,
net = c(5000, 20000), port.size = 4)
[1] "Mon Sep 03 20:38:00 2012"
[2] "Mon Sep 03 20:38:00 2012"
[1] "randportBurSt"
seed attribute begins: 1 -959789278 1943820272 2019334722
Some information about the object is printed at the end which makes it look more complicated than it is. But the first part shows three portfolios that each contain 4 assets.
The first portfolio is short 5390 shares of XA262, short 3849 shares of XA535 and short 4675 shares of XA561. The only asset it is long in is XA674 of which it holds 16,813 shares.
If you do this command, you will get different portfolios because you will have a different random seed.
The random.portfolio function is the (primary) function that generates random portfolios.
Its first argument is the number of random portfolios desired.
The second argument is the price vector.
The remaining arguments are all constraints.
It is mandatory to constrain the value of portfolios. There are other possibilities, but constraining the gross value and the net value is the most common approach for long-short portfolios. Two
numbers are given for the net value — this is the allowable range. Only one number is given for the gross value. However, this is really an interval as well — the value given is the upper bound and
the lower bound is automatically generated to be a slightly smaller number.
The final part of the call is ‘port.size=4‘. This is what constrains the number of assets in the portfolio to be at most 4.
Further details
We can see what the value of the portfolios is:
> valuation(rplsVerysimp, prices=priceVector, collapse=TRUE)
[1] 999916.5 999975.2 999943.6
> valuation(rplsVerysimp, prices=priceVector, type="net", collapse=TRUE)
[1] 9536.05 9717.85 6297.83
The gross values (the first command) are all slightly less than 1 million. We can’t expect the value to be exactly 1 million, but it is easy to get close. The meaning of “close” depends on the unit
prices of the assets.
All the net values are well within their allowed range.
To change this example into something that you really want to do, you just need to put in the constraints appropriate for your problem.
• All of the prices need to be in the same currency. You have to check that — the code has no way of knowing.
• It will still work if the object given as the prices is a one-column or one-row matrix. But it will complain about other matrices:
> random.portfolio(3, xassetPrices[250:251,], gross=1e6, net=0)
Error in trade.optimizer.pre(prices = prices, variance = variance, expected.return = expected.return, :
'prices' expected to be a numeric vector with names, not a matrix
See also | {"url":"https://www.portfolioprobe.com/user-area/documentation/portfolio-probe-cookbook/2-generate-random-portfolios/very-simple-long-short/","timestamp":"2024-11-06T20:08:37Z","content_type":"text/html","content_length":"90589","record_id":"<urn:uuid:e40631f0-aa1d-49de-8ee4-91eb43b4a40e>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00691.warc.gz"} |
Strength of the Scissors Resonance as a Function of Deformation in the
a Function of Deformation in the
144,148,150 Nd Isotopes
Thomas Dahl-Jacobsen
Master’s Thesis
Department of Physics University of Oslo
May 2020
The goal of this work is to test the assumptions of a full spin population in the (p, p^0)144,148,150Nd reaction and to study the scissors resonance as function of deforma- tion. The test is
performed by extracting theγ-ray strength function (γSF) obtained with the Oslo method and compare with theγSF using a new method based on the previous work of Wiedeking et al. [1], [2]. This new
method, the Slope method, exploits the ratio of the transition intensities feeding the 0^+ ground state and the first 2^+ excited state of the
144,148,150Nd isotopes. The experiment was performed at the Oslo Cyclotron Laboratory where144,148,150Nd were excited by a 16 MeV proton beam with a beam intensity of 2-5 nA in order to construct a
particle -γ-ray coincidence matrix. The nuclear level density and γSF were extracted using the Oslo method. The same data set was also used in the Slope method to obtain the γSF. The study confirms
that the full spin distribution of^144Nd is approximately populated in the (p, p^0) reaction at backwards angels. It is reasonable to be- lieve that this is also the case for the reactions on ^
148,150Nd. In addition, the Oslo method was performed to assess the strength of the scissors mode resonance around 3 MeV at the low energy tail of the giant dipole resonance. The scissors mode was
clearly seen in the well-deformed ^150Nd isotope, whereas the strength in the weakly-deformed ^144,148Nd was small or non-existent.
Department of Physics, Eskisehir Osmangazi University, 26480 Eskisehir, Turkey for the financial support.
My supervisor Magne Sveen Guttormsen and co-supervisor Gry Merete Tveten for making this work possible with great guidance and much patience even as the world halted to a stop
in March. The NEP group which made my two years at Oslo forever memorable with new friendships, meetings and travels both in Norway and abroad. Finally I want to thank my
family and friends for supporting me to the end.
Forever Grateful
Thomas Dahl-Jacobsen 19.05.2020
Table of Contents
1 Introduction 1
2 Theory 3
2.1 The Nuclear Volume - Deformation . . . 3
2.2 Nuclear Level Density . . . 5
2.2.1 The Spin Cut-off Formula . . . 6
2.3 The Transmission Coefficient . . . 7
2.4 The γ - Ray Strength Function . . . 8
3 Experiment and Data Acquisition 9 3.1 The Oslo Cyclotron Laboratory . . . 9
3.1.1 Setup . . . 10
3.1.2 SiRi . . . 11
3.1.3 OSCAR . . . 14
3.2 Data Acquisition and Validation . . . 16
3.3 Particle Calibration . . . 17
3.3.1 Energy Calibration . . . 17
3.3.2 Time Calibration - Shifts . . . 20
3.4 Scintillator Calibration . . . 20
3.4.1 Energy and Time Calibration . . . 21
3.5 Coincidence - Gating on the Events . . . 23
3.5.1 Cuts and Sorting . . . 23
3.5.2 Coincidence Matrices . . . 25
3.6 Unfolding . . . 27
3.6.1 The Folding Iteration Method . . . 27
3.6.2 The Compton Subtraction Method . . . 29
3.7 Extracting the First Generation Matrix . . . 32
4.1.1 Sewing by Ratio . . . 39
4.2 The Oslo Method . . . 46
4.2.1 Extracting the Transmission Coefficient and the Nuclear Level Density 46 4.2.2 Normalization of the NLD and γSF . . . 50
5 Results 53 5.1 The Slope Method . . . 53
5.1.1 Extraction of Intensity . . . 53
5.1.2 ^144Nd . . . 54
5.2 The Oslo Method . . . 55
5.2.1 Nuclear Level Density . . . 55
5.2.2 The Strength Function . . . 57
5.3 The γ-Strength Function Obtained with the Slope- and Oslo Method . . . . 59
6 Conclusions and Future Work 63
List of Figures
2.1 The nuclear chart shown with theβ deformation parameter from ”the results of Hartree-Fock-Bogoliubov calculations D1S [15] Gogny [16] ef- fective nucleon-nucleon interaction.”. The red lines
represent the closed shell- magic numbers with zero deformation, from the ”AMEDEE”-
database[17][18] The range shows the weakly deformed neodymium isotopes
atA= 140−144, to the well deformed atA = 150. . . 3 2.2 A frozen frame of a vibrating nucleus about a spherical equilibrium shape
defined as the average coordinateRav of the instantaneous time-dependent
surface coordinateR(t). Figure taken from [22]. . . 4 2.3 Snapshots of the first three valid vibrational modes of a nucleus. Figure
taken from [22]. . . 5 3.1 An overview of the Scanditronix MC35 Cyclotron and beamlines. The tar-
gets are located at OSCAR[35]. . . 10 3.2 Top view of the target area with some properties of SiRi and OSCAR (not
to scale). . . 11 3.3 A sketch of the particle telescope-system together with the particle matrix
for^150Nd showing how the particle matrix is filled. Figure is not to scale. . . 12 3.4 The SiRi particle telescope with readout strips[38]. . . 13 3.5 One of the in total eight trapezoidal ∆E
detector-plates of SiRi, illustrating
the angular distribution of the detector strips[38]. . . 13 3.6 The OSCAR detector array. Inside the target chamber at the center of OS-
CAR, the SiRi particle system is mounted. . . 15 3.7 Illustration of directly digitizing the signal from the pre-amplifier together
with the time length and gap definition. Illustration is taken from [42]. . . . 16 3.8 Uncalibrated ∆E-E particle matrix for^144Nd. The red circles show the
(p,p’) and (p,t) reaction ground states used as calibration points for the gain and shift. This matrix is the uncalibrated sum of all 64 ∆E-E combi-
nations. . . 17
of the elastic peak from the proton beam energy of 16 MeV. . . 18
3.10 Example of the two average values found by the ”peaks2D.C” program for the (p,t) peak in back-detector 2, front-strip 0. The cross shown in magenta was found to be most reliable and consists of
the weighted mean of the 20 highest bins in the zoomed area. . . 19
3.11 A time spectrum for a single E-detector. . . 20
3.12 Uncalibrated E-detector - time matrix. . . 20
3.13 Calibrated E-detector - time matrix. . . 20
3.14 ^150Nd LaBr[3] energy spectra slice with ^16O peak and single escape. . . 22
3.15 Single LaBr3 detector time projection. . . 22
3.16 OSCAR time matrix. . . 23
3.17 Particle telescope ∆E-E energy-matrix. . . 24
3.18 Particle energy-time matrix. . . 24
3.19 Energy-time LaBr[3] matrix with cut on the prompt time. . . 25
3.20 Energy-time LaBr[3] matrix with cut on the random time. . . 25
3.21 The (E[x], E[γ]) matrix of ^144Nd. . . 26
3.22 The (E[x], E[γ]) matrix of ^148Nd. . . 26
3.23 The (Ex, Eγ) matrix of ^150Nd. . . 27
3.24 Illustration of the method of intepolating from the measured response func- tion used in the unfolding method [43]. . . 28
3.25 Unfolded (E[x], E[γ]) matrix of ^144Nd . . . 31
3.26 Unfolded (E[x], E[γ]) matrix of ^148Nd . . . 31
3.27 Unfolded (E[x], E[γ]) matrix of ^150Nd . . . 32
3.28 The spectrum is the same after the initial decay as if populated directly at the same excitation energy, figure taken from [46]. . . 33
3.29 The first generation matrix for^144Nd. . . 34
3.30 The first generation matrix for^148Nd. . . 35
3.31 The first generation matrix for^150Nd. . . 35
4.1 Flow-chart diagram for the Slope method. . . 39
4.2 Illustration of the integration-bands formed from a linear interpolation in the first generation matrix for ^144Nd. . . 40
4.3 The expected γ-ray energy given byEγ = Ex and Eγ = Ex−E(2+) com- pared to the actual centroid of the integrated number of counts for the two diagonals. . . 41
4.4 The fraction of the spin distribution that feed the final state, denoted by
p[D][0+] and p[D][2+]. The ratio is given for each of the spin cut-off forumla. . . 42
4.5 The intermediate strength function of γ-rays feeding the ground state and first excited state at 696 keV of^144Nd. . . 42
4.6 The initial point acts as a reference for all other iterations that follows. The ratio will only increase as the intensity ofγ-rays feeding the g.s. and first excited states fall with
increasingEγ. . . 43
4.7 After the two initial point-pairs have been sewn together the new ratio is applied for the next mean value pf E[γ]. The real-case analog is shown in Fig. 4.9. . . 44
4.8 The ratio plotted as a function of excitation energy for^144Nd. . . 44
4.9 The intermediate γSF of ^144Nd plotted as point-pairs of the g.s. and first excited state. The title states excitation energy but as explained above the ideal case of E[x]=E[γ] is assumed. . . 45
4.10 The final sewn slope of the ^144Nd γSF is obtained by applying the ratio over all bins. . . 46
4.11 The trapezoidal area make up the extraction limits of which the nuclear level density and transmission coefficient can be extracted. Here shown for 150Nd. . . 48
5.1 The Slope method performed with the intensity extracted with the sum function (su) in MaMa[27]. . . 54
5.2 The Slope method performed with the intensity extracted with the peak fit function (ft2) in MaMa [27]. . . 55
5.3 The nuclear level density of^144Nd. . . 56
5.4 The nuclear level density of^148Nd. . . 56
5.5 The nuclear level density of^150Nd. . . 57
5.6 The γ-ray strength function for ^144Nd. . . 58
5.7 The γ-ray strength function for ^148Nd. . . 58
5.8 The γ-ray strength function for ^150Nd obtained with the Oslo method. . . 59
5.9 The ^144Nd slope method by the FT2 function in MaMa [27] compared to the results from the Oslo method. . . 60
5.10 The ^144Nd slope method by the SU function in MaMa [27] compared to the results from the Oslo method. . . 60
5.11 The ^144Nd slope method by the SU function in MaMa [27] compared to the results from the Oslo method with an added 10% uncertainty in the inten- sity as discussed in 5.1.1. . . 61
3.1 Target wheel configuration and Nd-target characteristics. . . 10 3.2 Overview and description of symbols used in the Bethe-Block equation 3.2
with their respective values. . . 14 3.3 Geometrical properties of OSCAR derived from the GEANT4 simulation
documentation at the OCL [41]. . . 16 4.1 Limits set in RhoSigChi[27] in order to extract the level-density ρ(E[i]) and
the transmission coefficient T(Eγ) from the first generation matrix. . . 49 4.2 Nuclear level density normalization parameters obtained from the Oslo
method software D2Rho and Robin[27]. . . 50 4.3 Values used for normalization of the nuclear level density. The values for
BSFG were derived by Egidy and Bucuresti in [28], [29] with the new values retrieved from Cite[Magne]. The neutron resonance spacing parameter val- ues are from the sixth edition of Mughabghab’s
Atlas of neutron resonances
[12] the target spin from NuDat2.8 [30]. . . 51 4.4 Mughabghab [12]values that were used for normalization for 144,148,150Nd
and RIPL3 [52] values for reference. The fourth column can assume two val- ues for the * - symbol, c and u respectively denoted at the end of the value.
These subscripts were introduced in [12] and denote a new method of de-
duction for the resonance parameter values. ^† For ^150Nd - see text. . . 51
Chapter 1 Introduction
Nuclear physics is the study of the atomic nucleus, its structure, dynamics and interaction with other nuclei and particles. Even though, as one of the smallest constituents in the universe, it’s
impact can be seen even on the macroscopic scale, from stellar environments and the subsequent elemental composition of the universe from processes such as the so called rapid neutron capture process
(r-process). To other social applications such nuclear transmutation, energy, medicine - both diagnostic and therapeutic. One of the most infa- mous is likely the atomic bomb.
Nuclear physics began in the late 19th century, the discovery of radioactivity by Bequerel and Curie[3][4] prompted a new wave of research into the elements and their composition. This was followed
by the proposal of the nucleus by Rutherford[5] in 1911 after the meticulous experiment conducted by Hans Geiger and Ernest Marsden two years prior under Rutherford’s direction. Ruthertford later
proposed that the nucleus also con- sisted of neutrally charged particles called neutrons to account for the total nuclear mass.
This new particle was later confirmed by James Chadwick in 1932[6] and the modern nu- cleus took form.
In parallel, with the birth of quantum mechanics a new understanding of physics on the microscopic scale came forth, the quantization and wave-particle duality gave rise to a new way of looking at
the nuclei, its constituents and the nuclear interactions. New models arose such as the semi-empirical liquid drop model by Gamow and the Shell Model by Goeppert Mayer and Hans Jensen [7][8][9]. The
filled shells predicted the nuclear magic numbers - tightly bound nuclei which are more stable against decay with a high shell-gap.
Today there is a number of theoretical combined microscopic-macroscopic models whose aim is to quantify the nuclear properties and interactions such as mass, nuclear charge dis- tribution, spin and
parity, interaction cross sections and more.
In this work the nuclear level density (NLD) and γ-ray strength function (γSF)
will be extracted. The NLD describe in short the number of levels available for a given energy, spin and parity in the nucleus while theγSF is a measure of the strength of the electromagnetic
response of the nucleus for a given γ-energy. A new method will be intro- duced in order to extract the functional form of the γSF as function ofγ-ray energy. This method is based on the work of
Wiedeking et al. [2, 10] that confirmed the low-energy en- hancement of theγSF observed by the Oslo group in [11]. In the present thesis we extend the method to detector systems with lower energy
resolution and include specific spin- distributions of the primaryγ-ray transitions. The new method is called the Slope method and can be used to estimate the fraction of the total level density
populated in the applied nuclear reaction. This will be compared to the neutron resonance spacing parameter [12]
which will be discussed later in the text.
A previous experiment has been performed at the Oslo Cyclotron Laboratory using the (d, γp^0) reaction on144,148,150N d with beam energy of 13.5 MeV [13]. However, the reaction only populated states
up to an excitation energy less than 4 MeV. Therefore, these studies were redone with the (p, p^0) reaction with a beam energy og 16 MeV. The current (p, p^0) experiment extend the excitation energy
above the neutron separation en- ergy S[n] with a higher spin population. The results of these experiments can hopefully be used for more precise calculations of reaction rates for ongoing research
as they constitute an important part of the reaction rate calculations in stellar environments.
The γSF and NLD are examined as a function of deformation which increases with neutron number in the neodymium isotope chain with mass number A. Of special interest is the pygmy resonance at the low
energy tail of the giant dipole resonance, the so called M1scissors mode located at E[γ] ∼ 2-3 MeV. The scissors mode can be described as a collective motion of the nucleons in the nucleus where the
neutron- and proton clouds oscillate towards each other like the blades of a scissor. The scissors mode is expected at high deformation further from the line of stability and it is of great interest
to investigate the shape of the γSF as a function of the nuclear deformation.
This thesis presents the work in the following manner: In chapter 2 a theoretical foundation will be given, in particular the appearance of nuclear shape and the Brink-Axel hypothesis [14].
Furthermore, we describe the statistical gross properties of - NLD and γSF with their components. Chapter 3 describes the experimental setup and the calibra- tion methods used for the data. Chapter 4
presents how to extract the NLD, in particu- lar the Slope method which is also compared to the neutron resonance spacing parameter through the Oslo-Method. Finally in chapter 5 a discussion
summarizes the results of this work.
CHAPTER 2. THEORY
Chapter 2 Theory
2.1 The Nuclear Volume - Deformation
Figure 2.1: The nuclear chart shown with theβ deformation parameter from ”the results of Hartree-Fock-Bogoliubov calculations D1S [15] Gogny [16] effective nucleon-nucleon interaction.”. The red
lines represent the closed shell- magic numbers with zero defor- mation, from the ”AMEDEE”-database[17][18] The range shows the weakly deformed neodymium isotopes at A= 140−144, to the well deformed
atA= 150.
As can be seen from the Fig. 2.1, the 144,148,150Nd isotope chain has an increasing prolate
tion between vibrator at lower and rotor for higherN respectively [19, 20]. As the col- lective nuclear model of the even-even neodymium isotope chain is not the main topic of this thesis it only be
introduced in short due to the expected relation between the β- deformation parameter and an increase in the pygmy-resonance at the low energy tail of the GDR called the M1 scissors resonance[21].
The following is a short description from Krane [22] who represents the vibrations and rotations by the spherical harmonics of a nu- clear surface that either rotates or vibrates about a spherical
equilibrium position with a time dependent angular position:
R(t, θ, φ) =
R[av][1 +βY[20](θ, φ)], Rotation Rav+P
µ≥−λαλµ(t)Yλµ(θ, φ) Vibration
where the β-parameter is a unit which describes of what magnitude an ellipse deviate from a circular shape by the measure of the difference in the axis ∆R as described with the Hill-Wheeler
coordinates [23]:
β = 4 3
rπ 5
R[av]. (2.2)
Figure 2.2: A frozen frame of a vibrating nucleus about a spherical equilibrium shape de- fined as the average coordinate R[av] of the instantaneous time-dependent surface coordi- nate R(t). Figure
taken from [22].
The vibrations are characterized by the vibrational modes depicted in Fig. 2.3.
As the nuclear volume is in-compressible the lowest mode is forbidden. The next mode - the dipole is unobtainable from internal motion due to a shift in the center of mass. The quadropole is
therefore the lowest available vibrational mode that can occur without ex- ternal action. The rise of triplet states in the vibrational nuclei is of further interest when examining the nuclear level
density in chapter 5.
CHAPTER 2. THEORY 2.2. NUCLEAR LEVEL DENSITY
Figure 2.3: Snapshots of the first three valid vibrational modes of a nucleus. Figure taken from [22].
2.2 Nuclear Level Density
The nuclear level density describes the number of configurations - a group of states with the same total amount of protons, neutrons and energy [14], that the nucleus possesses for a given excitation
energy, spin and parity. The available number of configurations for the nucleus increases exponentially with excitation energy as the nucleon pairs break.
The level density at a given excitation energy E[x], spin J and parity π is defined as:
ρ(E[x], J, π) = 1
D(E[x], J,Π), (2.3)
whereD(Ex, J, π) is the level spacing at the same given parameters as defined above.
The current nuclear level density model used in this work are the back-shifted fermi gas model (BSFG) and the constant temperature model (CT). The BSFG-model is an extension of the works of Bethe
[24] which modeled the nucleus as a non-interacting fermi gas and by Gilbert and Cameron [25] wherein the derivation of the backshift and shell correction to match the pairing of odd/even number of
nucleons and the shell closure effects is given. They present the nuclear level density formula as a separated function of the form:
ρ(U, J) =f(J)ρ(U) (2.4)
wheref(J) is the spin distribution function, with one free parameter the spin cut-off pa- rameter:
f(J, σ)≈ 2J+ 1
2σ^2 e−J(J+1/2)/2σ^2
., (2.5)
byρ(U) is defined as:
ρ(U) = exp[2p
2σa^1/4(U −E[1])^5/4, (2.6)
where U is the excitation energy,E[1] is the backshift energy, a is the level density param- eter which is expected to be proportional to A and σ is the spin cut-off parameter which characterizes the
spin distribution detailed in the section below.
The constant temperature model was proposed by Ericsson [26] who viewed the physical interpetation as the melting of nucleon pairs in a phase transition with approxi- mate constant nuclear
temperature until the point where the nucleus could be treated as a fermi gas:
ρ(E[x]) = 1
τe^(E^x^−E^0^)/τ, (2.7)
where E[x] is the excitation energy and τ denotes the constant nuclear temperature together with E0 - both parameters determined by a so called pivot point/normalization at low ex- citation energy as
described in Ref. [25].
2.2.1 The Spin Cut-off Formula
There were a total of four spin cut-off formulas available for testing in the Oslo software [27], namely the:
• The Rigid Moment of Inertia [28],[29]:
σ^2 = 0.0146A^5/31 +p
1 + 4a(U −E1)
2a , (2.8)
the level density parametera and shift parameter E[1] are taken from systematics.
This view considers the nucleus as assuming a rigid-body value for the nuclear mo- ment of inertia.This spin cut-off formula was used in this work both in the oslo and slope method at the neutron
separation energy, at excitation energies below S[n] the following interpolation was performed to obtain the spin cut-off parameter:
σ^2(E[x]) =σ[d]^2+E[x]−E[d]
S[n]−E[d](σ^2(S[n])−σ[d]^2). (2.9) HereE[x] is the excitation energy, E[d] is the average energy in the region where the discrete spin cut-off parameterσ[d] was determined by discrete levels from
[30]. These values are given in Table. 4.2.
CHAPTER 2. THEORY 2.3. THE TRANSMISSION COEFFICIENT
• The Gilbert and Cameron Formula [25]
σ^2 = 0.0888aT A^2/3, (2.10)
where the temperature of the nucleus is defined as the square root of the ratio be- tween the excitation energy and level density parameter a, T =p
• The Constant Temperature Formula [31]
σ^2 = (0.98A^0.29)^2, (2.11)
The constant temperature cut-off formula assumes there is no energy-dependency in the lower excitation range, this causes the ratio of the spin-distribution later applied in the slope method to be
constant and was therefore not applied further.
• The Fermi Gas Formula [31]
σ^2 = 0.391A^0.675(E−P a^0)^0.312 (2.12) The resulting ratio of the spin distributions as a function of excitation energy between the 0+ ground state and the first excited 2+ state for^144Nd can be
seen in Fig.
2.3 The Transmission Coefficient
The quantity that, in a simplified scenario, denotes the escape probability of a γ-ray in a nucleus, is defined as the transmission coefficient, denoted as
. It’s thought of as the nu- cleus returning to a configuration with a γ-ray at the surface of the nucleus which escapes through a specific channelα or, more often, is reflected back into another
periodic motion [32]. By very definition this will then characterize the electromagnetic response for the de- cay of an excited nucleus.
It’s relation to the γ-ray strength function f[XL](E[γ])is given defined as [33]:
T[XL](E[γ]) = 2πE^2L+1f[XL](E[γ]) (2.13) where L is the multipolarity and X the transition type, either electric or magnetic.
2.4 The γ - Ray Strength Function
The extension of the γ-ray transition coefficient is the function that describes the average electromagnetic decay properties of the nucleus as a function of theγ-ray energy, E[γ]. It’s defined as
f[iλXL]^J (E[γ]) =
E[γ]^2L+1ρ(E[γ]), (2.14)
where X and L defines the electric or magnetic transition and multipolarity re- spectively and J denotes spin and parity in short form. ¯Γ^J[iλXL] is the γ-ray partial width averaged over the
specific states of given spin and parity andρ(Ei) is the average nuclear level density [34].In a more modern notation from [33]:
f[XL](E[i], J[i], π[i], E[γ]) = hΓ[XL](E[i], J[i], π[i], E[γ])iρ(E[i], J[i], π[i])
E[γ]^2L+1 (2.15)
The simplified form is based on the assumptions made by Brink and Axel known as the Generalized Brink-Axel Hypothesis. In [14] Brink assumes that - the photo-effect is independent of the detailed
structure of the detailed state, that is, the same energy depen- dence with an energy translation shift.
Chapter 3
Experiment and Data Acquisition
The current chapter will introduce the Oslo Cyclotron Laboratory, the experimental setup, method of data-acquisition and the offline calibration process to obtain a final set of data necessary for
the Oslo -and Slope Method described in chapter 4.
3.1 The Oslo Cyclotron Laboratory
The experiment was conducted at the Oslo Cyclotron Laboratory (OCL) at the Depart- ment of Physics, University of Oslo. An illustration of the facility is shown in Fig. 3.1.
In the present experiment a 16 MeV proton beam was used with the Scanditronix MC35 Cyclotron, operating at a beam current within 2-5 nA exciting the 144,148,150Nd isotopes by the (p, p^0) - reaction.
The ejectile andγ-ray from the decay of the excited nucleus was measured in coincidence using the combined Oslo Scintillating Array (OSCAR) and the Silicon particle telescope Ring (SiRi).
Figure 3.1: An overview of the Scanditronix MC35 Cyclotron and beamlines. The targets are located at OSCAR[35].
3.1.1 Setup
The present experiment was the first one using a full OSCAR array, a new target cham- ber and modern digitizers for detector read-out. The 144,148,150Nd targets were of self- supporting foils of ≈
2mg/cm^2 thicknesses and were glued to Ta-frames. Each frame was attached to a rotating wheel by a flat spring. The wheel, which has six positions for the target was electrically grounded to the beam
pipe to avoid collection of charge. Informa- tion on the positions and targets are listed in Tab. 2.1 where the^28Si target were used for calibration.
Table 3.1: Target wheel configuration and Nd-target characteristics.
Position Isotope Thickness mg/cm^2 Composition %
1 Viewfinder - Quartz
2 ^150Nd 2.0 97.56
3 ^148Nd 2.0 95.44
4 ^144N 2.0 97.3
5 ^28Si ∼ 4mg/cm^2 natural
The target wheel was then enclosed together with the SiRi particle telescope sys- tem inside a spherical target chamber with plastic domes at top and bottom. SiRi was directed at backward angles from
126^◦ to 140^◦ with a center angle of 133^◦ with respect to the beam direction. The choice of measuring in backwards angles were to minimize the
CHAPTER 3. EXPERIMENT AND DATA ACQUISITION 3.1. THE OSLO CYCLOTRON LABORATORY
contribution from the eleastic (p,p) channel and to obtain a broad spin distribution with the draw-back of less cross section. The outer setup of the reaction point consisted of a spherically
distributed array of cerium doped lanthanum-bromide scintillator detectors (LaBr[3](Ce)) which measure γ-rays from the decay of the excited nucleus in coincidence with the charged ejectile measured
by SiRi.
The geometric surface area covered by the LaBr[3] detectors equal the ratio of ac- tive detector-area to the total spherical area at the same detector distance giving:
R[OSCAR] = N[det]∆Ω
Ω = N[det]A
4πR^2 , (3.1)
whereN[det] are the number of detectors. SiRi has 64 and OSCAR has 30 detectors. The detectors have an area A in a distance ofR from the target. The corresponding coverage of SiRi and OSCAR is ROSCAR
= 54.2% andRSiRi = 8%.
Figure 3.2: Top view of the target area with some properties of SiRi and OSCAR (not to scale).
3.1.2 SiRi
The SiRi particle telescope system is a two stage ∆E −E detector used to tag the exci- tation energy of the target nucleus. The procedure consists of measuring the energy de- posited in each detector
layer and assuming a small correction due to the blocking of δ - electrons^1 by the use of aluminum foil, the particle can then be identified through the characteristics of the Bethe-Block equation
[36] (3.2) by using the ∆E stage as a second equation to uniquely identify the ejectile. The additional energy loss due to the Al-foil for δ-electrons and the recoil of the target are included in the
kinematic calculations using
Qkinz [37].
Figure 3.3 illustrates the operation and procedure of assigning ∆E and E parti- cle events into a particle matrix for the telescope detector. The scattered particle, protons, enter the ∆E detector
before entering and stopping in the E-detector. This particle ma- trix shows characteristic particle ”bananas”, each distinct particle stopped in the detector telescope forms a part of one of the
three unique bananas given it’s type as seen in , here labeled as (p), (d) and (t)
Figure 3.3: A sketch of the particle telescope-system together with the particle matrix for
150Nd showing how the particle matrix is filled. Figure is not to scale.
SiRi consists of 8 curved ∆E detector strips for each of the 8 E - detectors with an angular distribution from 126^◦-140^◦ with a 2^◦ width at 5cm from the target. The total array of 8 E - detectors
form a symmetrical cone in backward angles relative to the beam direction, see Figs. 3.4 and 3.5. To shield theδ - electrons a 10.5µm thick aluminum foil covers the ∆ E- detectors. [38]
1Theδ-electrons/rays - high energy electrons produced in the reaction that can traverse the target-to- detector-length.
CHAPTER 3. EXPERIMENT AND DATA ACQUISITION 3.1. THE OSLO CYCLOTRON LABORATORY
Figure 3.4: The SiRi particle telescope with readout strips[38].
Figure 3.5: One of the in total eight trape- zoidal ∆E detector-plates of SiRi, illustrat- ing the angular distribution of the detector strips[38].
The Bethe-Block equation, which can be studied in detail in [36], defines the mean energy loss per unit distance traversed in a medium for a charged particle and is in- tegral for identification of
the particle in the telescope system. The Bethe-Block equation assumes a different form for (e, e^0) - reactions. There is no distinction between projectile electrons and electrons in the target atom
thus the maximum energy transfer can be half the total energy of the projectile, in addition to other effects [39].
− hdE
dxi= 2πNar^2[e]mec^2ρZz^2
I^2 )−2β^2], (3.2)
where the symbols of the equation are defined as follows
Table 3.2: Overview and description of symbols used in the Bethe-Block equation 3.2 with their respective values.
Symbol Description
W[max] The maximal kinetic energy transfer.
Z Atomic number of the target.
z Atomic number of the projectile.
A Mass number of the target.
ρ Density of the target.
m[e] Mass of electron.
N[a] Avogadros constant, 6.022×10^23mol^−1.
r[e] The classical electron radius, 2.8179×10^−15m.
I The mean excitation potential.
β v
c Fraction of the particles speed to light.
γ 1
p(1−β^2) The Lorentz factor.
3.1.3 OSCAR
The basic operation of a scintillating detector is to convert the energy of a charged par- ticle or photon into light through an emission of photons after excitation of the crystal material. The
light produced is then sent through a photomultiplier tube (PMT) which strongly amplifies the signal. The PMT is a vacuum tube which consists of a photocath- ode that turns the light produced by the
scintillator crystal into electrons which then are multiplied by several dynodes with step-wise reduced voltages until the signal is read out at the anode.
CHAPTER 3. EXPERIMENT AND DATA ACQUISITION 3.1. THE OSLO CYCLOTRON LABORATORY
Figure 3.6: The OSCAR detector array. Inside the target chamber at the center of OS- CAR, the SiRi particle system is mounted.
The Oslo scintillator array consists of 20 hexagonal and 10 pentagonal Cerium doped, Lanthanum Bromide (LaBr[3]:Ce) cylindrical inorganic scintillator detectors with an 8” height and 3.5” diameter
opening(20.32×8.89 cm^2). The given energy resolution for 1332 keVγ-rays from ^60Co are 2.1% which is much better compared to the 5.4% NaI [40]
which has been replaced at the OCL. Furthermore, the LaBr3 has a much better timing than the previous NaI detector, with a 1/e decay-time of 16 ns versus 250 ns. The array is mounted on a metal
hexagonal and pentagonal frame, (Fig. 3.6) where the distances from the detector surface to the event site is 16.3 cm and 17.0 cm for the hexagonal and pentagonal frames respectively. This gives an
average distance of 16.53 cm, with aθ,φ angular distribution at two distances from the target center given in the table below:
Table 3.3: Geometrical properties of OSCAR derived from the GEANT4 simulation docu- mentation at the OCL [41].
Frame Angle (θ^◦) Azimuth(φ^◦)
Pentagon 0, 63, 117, 180 0, 36, 72, 108, 144, 180,
216, 252, 288, 324
Hexagon 37, 79, 101, 143 0, 36, 72, 108, 144, 180,
216, 252, 288, 324, 360
3.2 Data Acquisition and Validation
The Pixie-16 Digital Gamma Finder (DGF) is at the heart of the OCL’s data acquisition hardware. A detailed documentation is given in [42] and a brief summary is presented in this section. The DGF
reads and converts the continuous analog electric signal into a discrete subset of digital points through the analog to digital converter (ADC) as is illus- trated in Fig. 3.7 below.
Figure 3.7: Illustration of directly digitizing the signal from the pre-amplifier together with the time length and gap definition. Illustration is taken from [42].
After the ADC the Field Programmable Array (FPGA) performs the pulse pro- cessing by applying fast and slow filter for triggering and pulse height respectively, the slow filter uses a fixed filter
length and no weighting relative to the gap distance Fig. 3.8.
Pile up, which are artificially high signals due to signal overlap, are checked by the slow filter for the peak-separation time in the fast filter.
CHAPTER 3. EXPERIMENT AND DATA ACQUISITION 3.3. PARTICLE CALIBRATION
The data recording was performed in the following manner - each detector strip and back detector in the SiRi particle telescope together with each scintillator in the OS- CAR had their own signal
output channels, 64, 8 and 30 respectively. These output chan- nels are handled by XiA’s Pixie16 digital pulse processor where the sampling frequency were 250 and 500MHz for the SiRi and OSCAR
detectors respectively. The pulse pro- cessor required a validation signal, a signal that generates the process of an event being recorded, from the E-detector in the SiRi particle telescope system
to register events.
When this signal was received the pulse processor read 1µs forward and backwards and registered all events within this timeframe. The data are then sent in 2GB data-files to offline storage for
further processing and calibration.
3.3 Particle Calibration
The SiRi particle telescope must be calibrated in two regards, energy and time. The cal- ibrated data are then used in sorting routines in order to create the appropriate spectra with specific
criteria. The sorting procedure will be explained in more detail later in this chapter. Firstly the energy calibration will be discussed.
3.3.1 Energy Calibration
Figure 3.8: Uncalibrated ∆E-E particle matrix for^144Nd. The red circles show the (p,p’) and (p,t) reaction ground states used as calibration points for the gain and shift. This ma-
Figure 3.9: The same matrix as in Fig. 3.8, but calibrated with appropriate shift and gain.
The ground state of the (p,p’) reaction is now correctly at the energy of the elastic peak from the proton beam energy of 16 MeV.
The SiRi particle telescope calibration consists of adjusting the measured energy spectra by the following equation:
E(ch) = a[0]+a[1]ch, (3.3)
where a[0] and a[1] are called the detector shift -and gain parameters respectively. Here we asume a linear response of the semiconductor detector. Since this is a linear equation we need two points
in order to properly determine a0 and a1. In this work we have exploited the (∆ E,E) pixels of the ground states of the (p,p’) and (p,t) reactions outlined in Fig.
3.8. These two values form two equations with the shift and gain as two unknowns of the form in 3.3, this can be solved as follows:
x^‘[a]=a[0]+a[1]x[a], (3.4)
x^‘[b] =a[0]+a[1]x[b], (3.5)
CHAPTER 3. EXPERIMENT AND DATA ACQUISITION 3.3. PARTICLE CALIBRATION
a0 =x^‘[a]−xa
x[b]−x[a], (3.6)
a[1] = x^‘[b]−x^‘[a]
x[b]−x[a], (3.7)
The function values x[a] (p,p’) and x[b] (p,t), are retreived from the kinematics calculator named Qkinz[35]. It simulates the expected ejectile energy from the Q-reaction value which is translated
and plotted as deposited energy for each detector strip (126^◦-140^◦).
The script that handles the extraction of the uncalibrated values is called
”peaks2D.C” and is written by Alexander B¨urger for use in the ROOT framework. It consists of clicking in the individual ∆E-E spectra for each strip - back detector combi- nation (64 in total) where
the weighted mean of the 20 highest bins in the zoomed area is found and written to file. An example for one ∆E-E detector strip is shown in Fig. 3.10.
The separation of the peaks are due to the effect known as channeling where the particles traverse within the crystal structure depositing less energy creating the parallel peaks at lower energies.
Figure 3.10: Example of the two average values found by the ”peaks2D.C” program for the (p,t) peak in back-detector 2, front-strip 0. The cross shown in magenta was found to be most reliable and
consists of the weighted mean of the 20 highest bins in the zoomed area.
3.3.2 Time Calibration - Shifts
The time calibration for the particle events consists of a basic shift subtracted to the recorded arrival of the E-detectors event signal. The subtraction was done manually by inspecting the peak
position relative to origin in ROOT as shown in Fig. 3.12.
Figure 3.11: A time spectrum for a single E-detector.
The new shift values are then applied through a new iteration in the sorting pro- cedure by sorting and reading the raw data. A typical pre-calibrated and calibrated time- matrix is shown in Fig.
Figure 3.12: Uncalibrated E-detector - time matrix.
Figure 3.13: Calibrated E-detector - time matrix.
3.4 Scintillator Calibration
The LaBr3:Ce scintillator data was calibrated off-line by much the same method as the SiRi data. Both energy and time was necessary in order to properly ensure event coinci- dence between particle
and γ-rays for later analysis.
• Insert screengrab as an example of the peak fit used with mama to find the peak centroid used for the shift and gain calibration.
3.4.1 Energy and Time Calibration
The methology for the energy calibration for the scintillator data was much the same as for SiRi, although there is 30 distinct scintillator detectors and the energy response was assumed as a
non-linear function at higher energies (channels) with a response given by:
E =a[0]+a[1]ch+a[2]ch^2, (3.8)
where the constantsa[0],a[1] and a[2] are shift and gain parameters, respectively. These con- stants were determined by measuring four points in the LaBr3 detector-id - intensity ma- trix after
projecting down each scintillator to get the individual energy-intenstiy matrixes.
To measure these points the matrix was first converted into the standard .m (matrix) format used by the matrix manipulation program, MaMa, developed as part of the Oslo Method Software [27].
The points used for calibration were the followingγ-decays of:
• ^28Si
1. 1778.969keV 2. 2838.29keV
• ^144,48,50Nd
1. 511keV Annihilation peak 2. 6128.63keV^16O contaminant.
MaMa has a built in function named ”fit spectrum”, ftN, which fits a function to N-peaks in a given interval in the spectrum which outputs the centroid of the peak in clear text.
This centroid is later passed on to the MaMa script named ”funcfit” which takes the val- ues obtained from the peak-fit and solves for the shift and gain coefficients as a set of equations.
A uncalibrated example of one of the in total 30 LaBr[e] spectrum for^150Nd with a highlighted area of theγ-decay of the ^16O contaminant in ^150Nd is shown in Fig. 3.15
Figure 3.14: ^150Nd LaBr3 energy spectra slice with ^16O peak and single escape.
Each single of the 30 scintillation detectors of OSCAR is calibrated with a time shift relative to the 64 ∆E-detector strips of SiRi to obtain proper coincidence. This task is performed by the ”time
fit all.cpp” script written at OCL by Frank Leonel B. Garrote, which runs in ROOT and outputs all 64x30 time centroids by running a gaussian fit on each peak and looping over all 64 ∆E-strips for
each LaBr[3] detector in succession after projecting down in the detector-time matrix. One such calibrated projection is shown in Fig. 3.15 with the total OSCAR in Fig. 3.16:
Figure 3.15: Single LaBr[3] detector time projection.
CHAPTER 3. EXPERIMENT AND DATA ACQUISITION 3.5. COINCIDENCE - GATING ON THE EVENTS
Figure 3.16: OSCAR time matrix.
3.5 Coincidence - Gating on the Events
The sorting of the final data for later unfolding and extraction of the primaryγ-ray matrix is done by so-called cuts - a graphical sectioning of data in the ROOT canvas which is ex- ported in .root
format. These cuts were created to define the borders of the final dataset.
The sorting uses a nested for-loop where there are three conditions for the event to be sorted.
3.5.1 Cuts and Sorting
The first requirement in the final sorting is the basic requisite of the experiment - the reg- istered events must be the inelastic (p,p’)-reaction and the graphical cut in Fig. 3.17 sorts out all
events that does not belong to this category by simple comparison of inside vs.
Figure 3.17: Particle telescope ∆E-E energy-matrix.
The second requirement cut used is the E-detectors energy-time matrix as shown in Fig. 3.18, this ensures the that only the prompt particles are included.
Figure 3.18: Particle energy-time matrix.
The third and final requirement is shown in Figs. 3.19, 3.20 which corresponds to the prompt and random events respective to time. The sorting routine either subtracts or adds to the final dataset or
background respectively dependent on the timing. If the event is not prompt it belongs in the background.
CHAPTER 3. EXPERIMENT AND DATA ACQUISITION 3.5. COINCIDENCE - GATING ON THE EVENTS
Figure 3.19: Energy-time LaBr[3] matrix with cut on the prompt time.
Figure 3.20: Energy-time LaBr[3] matrix with cut on the random time.
3.5.2 Coincidence Matrices
Figures 3.21, 3.22 and 3.23 show the particle -γ-ray coincidence matrices for 144,148,150Nd.
The detected γ-ray energy and the excitation of the nucleus forms the x- and y-axis, re- spectively. These matrices are valid for the given calibration up to the neutron separation energy Sn. Since
the outgoing neutron above Sn is not detected, the excitation energy of the A−1 nucleus can not be determined. TheEγ = Ex diagonal represents the direct de- cay to the 0^+ ground state. A second
diagonal representing the direct decay to the first excited 2^+ state is clearly seen in^144Nd. We will later utilize the 0^+ and 2^+ diagonals in the new Slope Method.
Several contaminants were seen in the coincidence matrix which were identified mainly as^40Ca and it’s isotopes which can be seen forming a new ground state diagonal close to the first excited 2^+ of
neodymium. There is in addition to some occurences of^16O atE[γ] ∼ 6.1 MeV and ^12C at E[γ] ∼ 4.4 MeV in their first excited states.
Figure 3.21: The (E[x], E[γ]) matrix of ^144Nd.
Figure 3.22: The (E[x], E[γ]) matrix of ^148Nd.
CHAPTER 3. EXPERIMENT AND DATA ACQUISITION 3.6. UNFOLDING
Figure 3.23: The (E[x], E[γ]) matrix of^150Nd.
3.6 Unfolding
The response signal of a detector would ideally be a full energy peak, but in reality the signal generated depends on the interaction of the traversing radiation with the detec- tor material. In
particular for the scintillator detector the Compton scattering, pair- production, and photoelectric effect all contribute to create the real-world spectrum ob- served in every non-infinite
scintillator. These again subdivides respectively into backscat- ter and annihilation peaks in addition to single and double escape up to the full energy peak.
3.6.1 The Folding Iteration Method
The unfolding procedure utilizes the folding iteration method and the following Compton subtraction method which builds upon the iteration method. The methods explained in short here is derived from
and detailed in full by Guttormsen et al. [43] and it’s references therein. The purpose of the methods are to subtract the characteristic detector compton background and peaks after unfolding and
achieve the true full energyγ-ray spectrum of the decay-process measured by the scintillator.
In order to properly evaluate the detector response it’s necessary to measure the monoenergeticγ-decay of known sources, subtract all peaks from the spectrum and inter-
sponse function denoted by:
R(E, Eγ), (3.9)
where E is the energy deposited in the detector and Eγ is the incidenty γ-ray energy. The derivation of this interpolation can be seen in [43] whereof Fig. 3.24 shows where the in- terpolation from
the measured response functions are hooked at the beginning and end to create a widening curve with a θ dependence for intermediate energies as expected from the Compton effect.
Figure 3.24: Illustration of the method of intepolating from the measured response func- tion used in the unfolding method [43].
The folding iteration method uses a compact notation by denoting R as a matrix R[ij] with responce in channel i and γ-ray energy j, as given in (3.9) with the normalization condition of:
R[ij] = 1, (3.10)
that is - the sum of each response functions over all channels equals 1. Following this the definition of the folded spectra is given by:
f =Ru, (3.11)
CHAPTER 3. EXPERIMENT AND DATA ACQUISITION 3.6. UNFOLDING
whereu and f is the unfolded and folded spectra respectively for each channel i. The trick is to use a test spectrum as the unfolded namely the observed spectrumr and iterate un- til convergence is
met between the n’th folded spectrum f^n and the observed spectrum as shown in [43]:
1. by assuming the observed spectrum as a trial function for the unfolded spectrum
u^0 =r, (3.12)
2. the folded spectra is then calculated:
f^0 =Ru^0, (3.13)
3. the difference in the observed and first iteration of the folded spectrum is added to the initial trial function
u^1 =u^0 + (r−f^0) (3.14)
4. this procedure is then iterated until the folded spectrum f and observed spectrum r converges
f^n ≈r, (3.15)
where n is the iteration index.
3.6.2 The Compton Subtraction Method
Following the folding iteration method a new reference point is created with the unfolded spectrum u[0] and the following spectra are defined:
v(i) =p[f](i)u[0](i) +w(i), (3.16) which is the Compton contribution subtracted from the observed spectrum described in form by the unfoldedγ-ray spectrum multiplied by the normalized probability of
an event beloning to the full energy contribution of the spectrum -p[f](i)u[0](i). The last term w = u[s] + u[d] + u[a], adds the structure of the peak contributions from single- and double escape
and annihilation processes respectively. These are again given by:
u[s](i−i[511]) = p[s](i)u[0](i), (3.17) where once again u[0] is the unfolded spectrum and p[s] is the normalized probability of an
energy value x. Equally for the double escape structure:
u[d](i−i[1022]) =p[d](i)u[0](i), (3.18) and lastly for the annihilation peak:
u[a](i[511]) =p[a](i)u[0](i), (3.19) The extraction of the Compton background can then be done by the relation:
c(i) =r(i)−v(i). (3.20)
It is assumed that this backround spectrum is slowly varying with respect to energy and can therefore be smoothed i.e. loss of resolution by widening since there are no pro- nounced structures to
lose. This smoothed spectrum is subtracted together with the other peak structures contained within w from the observed spectrum r and normalize by the full energy probability p[f]:
u(i) = [r(i)−c[s](i)−w(i)]/p[f](i), (3.21) where c[s] denotes the smoothed compton background. The final step is to correct the dis- tribution of γ-rays by correcting for the totalγ-ray detection
efficiency which is energy dependent:
U(i) =u(i)/η[tot](i). (3.22)
Other factors should be considered which is dependent on the experimental set-up and can be read in detail in [43]. The procedure described was applied to the E[γ]-E[x] matrices for a total of 300
iterations to obtain the unfolded spectrum for further analysis, the results are seen in Figs. 3.25, 3.26 and 3.27 and produced by converting back to ROOT format [44]
from MaMa [27] thanks to a script provided by Ann-Cecilie Larsen at the OCL.
CHAPTER 3. EXPERIMENT AND DATA ACQUISITION 3.6. UNFOLDING
Figure 3.25: Unfolded (E[x], E[γ]) matrix of^144Nd
Figure 3.26: Unfolded (E[x], E[γ]) matrix of^148Nd
Figure 3.27: Unfolded (E[x], E[γ]) matrix of ^150Nd
3.7 Extracting the First Generation Matrix
The unfolded coincidence matrix cointains allγ-transitions in the cascades, in order to ob- tain the level density and γ-ray strength function (also known as radiative strength func- tion) the only
decays of interest are the primary γ-rays from a given excited state. This is performed with the method of extracting the first generationγ-ray matrix from the un- folded coincidence matrix. The
brief overview presented in this work is taken from the full description by Guttormsen et al. [45] and references theirein and is based on the coinci- dence measurements described previously in this
One of the fundamental assumptions the method built upon is that the popu- lated states after the first ( or primary)γ-transition have the same decay properties as if populated directly at the same
excitation energy in the particle reaction as shown in Fig. . This is assumed to be fulfilled at high level density where the nucleus becomes a compound-like system [45]
In the following the excitation energy regions are divided into bins denoted byf[i] where the highest excitation energy is in the initial bin positioni = 1, the first generation spectrum is denoted
by h and is estimated by the relation
h=f[1]−g, (3.23)
Figure 3.28: The spectrum is the same after the initial decay as if populated directly at the same excitation energy, figure taken from [46].
where g is the weighted sum of all of the underlying spectra:
g =X
n[i]w[i]f[i]. (3.24)
The two coefficients wi and ni requires definition -wi represent the probability of decay from the highest excitation bin 1 to bin i and thus it’s sum must be given by:
w[i] = 1, (3.25)
which is equivalent to the relative branching ratio at the given bin. It’s determined from an iteration proceure where a trial function for w[i] is applied, the first generation spectrum h is
produced wherebyh then is transformed to a new iteration of w[i] by the response function of the detectors and it’s area normalized to 1. This iteration method then pro- ceeds untilw[i] converges.
The coefficients n[i] are used as a normalization criteria where the product be- tween the area under each spectrum f[i] multiplied with the coefficient n[i] equal the same number of cascades, it
represents the difference in population cross sections at the excita-
tion bin f[i]. It can be normalized in singles or multiplicity:
The singles particle-cross section is proportional to the number of populated states and to the number of cascades as defined above. By measuring two cross-sections S[1] and S[i] at the highest
excitation energy bin and bin i respectively the normalization fac- tor is then:
n[i] = S[1]
S[i]. (3.26)
The multiplicity normalization is derived from the singles particle cross-section by the re- lation given in (3.26). If the average multiplicityM[i] of bin i is known and noting the that the average
multiplicity of all cascades in bin iis given by the relation of the excitation energy divided by the averageγ-ray energy carried by the γ-rays.
hMi= E[i]
hE[γ]i. (3.27)
By additionaly noting that the total number of counts equal the area under each excita- tion bin f[i], denoted asA(f[i]) then Eq. (3.26) takes the form of:
n[i] =M[i]A(f[1])/M[1]A(f[i]), (3.28)
The method of multiplicity normalization is used as the extraction of the first generation γ-rays occurs in the area of approximately 4 MeV toS[n] with high level den- sity with the assumption of the
characteristics of a compound nucleus. The resulting first generation matrices are shown in Figs. 3.29, 3.30 and 3.31 below.
Figure 3.29: The first generation matrix for ^144Nd.
Figure 3.30: The first generation matrix for ^148Nd.
Figure 3.31: The first generation matrix for ^150Nd.
These matrices form the basis of further work both for the Slope- and Oslo method by extracting the intensity and nuclear level density and transmission coefficient respectively.
CHAPTER 4. EXTRACTING THEγSF AND NLD: THE OSLO AND SLOPE METHOD
Chapter 4
Extracting the γ SF and NLD: The Oslo and Slope Method
The two methods presented in this work are the Oslo and Slope Method where the latter is the main focus of this work in order to evaluate the assumption of a total spin popula- tion in the inelastic
(p, p^0) - reaction.
4.1 The Slope Method
The method is used to derive the functional form of the γ-ray strength function, unlike the the Oslo Method the Slope method is not normalized and instead builds upon theγSF of the ground and first
excited state of the nucleus in the quasi-continuum region to preclude contributions from deterministic decay. As all three nuclei are even-even the ground state and first excited state are 0+ and
2+. The ratio method which this work is built upon is first shown by Wiedekinget al. [1], where the γSF is given by:
f(E[γ])≡f[J]^π(E[γ]) = Γ[J]^π(E[i], E[γ])ρ[J]^π(E[i])
E[γ]^2λ+1 , (4.1)
the equivalence comes from the generalized brink hypothesis. By applying the proportion- ality between the intensity of the primary transitions in the diagonalsD to the final bin j denoted byN[D][j]
(E[i]) to the sum of the partial radiation width from excitation bin E[i] and finally assuming the dominance of dipole transitions,λ = 1, the intensity can be given as:
σ[J]^π(E[i])p[D][j](J^π, E[i])Γ[J]^π(E[i], E[i] −E[D][j])ρ[J]^π(E[i]), (4.2)
where σ[J]^π(E[i]) is the level population cross-section with a given spin and parity. The spin distribution in the quasi-continuum is given in Eq. 2.4, where we now definepDj(J, Ei) = f(J, σ(E)) and
is introduced to account for the increased number of transitions avail- able to the first excited 2^+ state with respect to the 0^+ ground state. The summing in- dexJ^π runs over all states
populating the final state by dipole transition^1 The expression E[i]−E[D][j] can be simplified to E[γ] - the energy carried by the γ-ray from the transition. By using the relation given in Eq. (4.1)
this can finally be expressed as a function of γSF:
N[D][j] ∝f(E[γ])E[γ]^3X
σ[J]^π(E[i])p[D][j](J^π, E[i]), (4.3)
solved for the strength function:
f(E[γ])∝ N[D][j] E[γ]^3P
J^πσJ^π(Ei)pDj(J^π, Ei). (4.4) The original work by Wiedeking is model independent, this work introduces a spin-
distribution model dependency for the g.s. and first excited state. The ratio of the γSF of the same initial excitation energyE[i] is then given by:
R= f(E[γ1]) f(E[γ2]) =
N[D][1] E[γ1]^3 P
1(E[i])p[D1](J^π, E[i]) N[D][2]
E[γ2]^3 P
J^πσJ[2]^π(Ei)pD2(J^π, Ei)
Using the final notation where the ratio builds upon the 0+ ground state and 2+ first ex- cited state diagonals:
R= f(Eγ1)
f(E[γ2]) = N[D][1]E[γ2]^3 P
J^πσ[2]^+(E[i])p[2]^+(J^π, E[i]) N[D][2]E[γ1]^3 P
J^πσ[0]^+(E[i])p[0]^+(J^π, E[i]) (4.6) The σ[0]^+ and σ[2]^+ are unknown observable in our experiment . However, the single-particle cross section is rather constant for excitation energies above 4-5
MeV. With the assump- tion that we have a flat cross-section with respect toJ^π, we write:
R= f(E[γ1])
f(E[γ2]) = N[D][1]E[γ2]^3 P
J^πp[2]^+(J^π, E[i]) N[D][2]E[γ1]^3 P
J^πp[0]^+(J^π, E[i]) (4.7) This ratio forms the basis of the ”sewing” performed for each successive excitation bin where the referencei = 1 around 4 MeV which then iterates up to S[n]. Each ratio is
1For example a 0^+ state may be fed by states with 1^+ and 1^− and a 2^+ state may be fed by states with 1^+, 1^−, 2^+, 1^−, 3^− and 3^+. | {"url":"https://9pdf.net/document/q7wx258d-strength-scissors-resonance-function-deformation.html","timestamp":"2024-11-03T10:10:18Z","content_type":"text/html","content_length":"183697","record_id":"<urn:uuid:4cbede8e-34d3-4c5e-b338-f42b9b582a52>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00663.warc.gz"} |
Compute fast Fourier transform (FFT)
DSP HDL Toolbox / Transforms
The FFT block provides two architectures that implement the algorithm for FPGA and ASIC applications. You can select an architecture that optimizes for either throughput or area.
• Streaming Radix 2^2 — Use this architecture for high-throughput applications. This architecture supports scalar or vector input data. You can achieve gigasamples-per-second (GSPS) throughput,
also called super sample rates, using vector input.
• Burst Radix 2 — Use this architecture for a minimum resource implementation, especially with large fast Fourier transform (FFT) sizes. Your system must be able to tolerate bursty data and higher
latency. This architecture supports only scalar input data.
The FFT block accepts real or complex data, provides hardware-friendly control signals, and optional output frame control signals.
You can also generate HDL code for this hardware-optimized algorithm, without creating a Simulink^® model, by using the DSP HDL IP Designer app. The app provides the same interface and configuration
options as the Simulink block.
data — Input data
scalar or column vector of real or complex values
Input data, specified as a scalar or column vector of real or complex values. Only the Streaming Radix 2^2 architecture supports a vector input. The vector size must be a power of 2, in the range
from 1 to 64, and less than or equal to FFT length.
The software supports double and single data types for simulation, but not for HDL code generation.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
Complex Number Support: Yes
valid — Indicates valid input data
Control signal that indicates if the input data is valid. When valid is 1 (true), the block captures the values from the input data port. When valid is 0 (false), the block ignores the values from
the input data port.
Data Types: Boolean
reset — Clears internal states
Control signal that clears internal states. When reset is 1 (true), the block stops the current calculation and clears internal states. When the reset is 0 (false) and the input valid is 1 (true),
the block captures data for processing.
For more reset considerations, see the Reset Signal section on the Hardware Control Signals page.
To enable this port, on the Control Ports tab, select the Enable reset input port parameter.
Data Types: Boolean
data — Frequency channel output data
scalar or column vector of real or complex values
When input is fixed-point data type and scaling is enabled, the output data type is the same as the input data type. When the input is integer type and scaling is enabled, the output is fixed-point
type with the same word length as the input integer. The output order is bit-reversed by default. If scaling is disabled, the output word length increases to avoid overflow. Only the Streaming Radix
2^2 architecture supports vector input and output. For more information, see the Divide butterfly outputs by two parameter.
Data Types: double | single | fixed point
Complex Number Support: Yes
valid — Indicates valid output data
Control signal that indicates if the data from the output data port is valid. When valid is 1 (true), the block returns valid data from the output data port. When valid is 0 (false), the values from
the output data port are not valid.
Data Types: Boolean
ready — Indicates block is ready for new input data
Control signal that indicates that the block is ready for new input data sample on the next cycle. When ready is 1 (true), you can specify the data and valid inputs for the next time step. When ready
is 0 (false), the block ignores any input data in the next time step.
For a waveform that shows this protocol, see the third diagram in the Timing Diagram section.
To enable this port, set the Architecture parameter to Burst Radix 2.
Data Types: Boolean
start — Indicates first valid cycle of output frame
Control signal that indicates the first valid cycle of the output frame. When start is 1 (true), the block returns the first valid sample of the frame on the output data port.
To enable this port, on the Control Ports tab, select the Enable start output port parameter.
Data Types: Boolean
end — Indicates last valid cycle of output frame
Control signal that indicates the last valid cycle of the output frame. When end is 1 (true), the block returns the last valid sample of the frame on the output data port.
To enable this port, on the Control Ports tab, select the Enable end output port parameter.
Data Types: Boolean
FFT length — Number of data points for one FFT calculation
1024 (default)
This parameter specifies the number of data points used for one FFT calculation. For HDL code generation, the FFT length must be a power of 2 between 2^2 to 2^16.
Architecture — Architecture type
Streaming Radix 2^2 (default) | Burst Radix 2
This parameter specifies the type of architecture.
• Streaming Radix 2^2 — Select this value to specify low-latency architecture. This architecture type supports GSPS throughput when using vector input.
• Burst Radix 2 — Select this value to specify minimum resource architecture. This architecture type does not support vector input. When you use this architecture, your input data must comply with
the ready backpressure signal.
For more details about these architectures, see Algorithms.
Complex multiplication — HDL implementation
Use 4 multipliers and 2 adders (default) | Use 3 multipliers and 5 adders
This parameter specifies the complex multiplier type for HDL implementation. Each multiplication is implemented either with Use 4 multipliers and 2 adders or with Use 3 multipliers and 5 adders. The
implementation speed depends on the synthesis tool and target device that you use.
Output in bit-reversed order — Order of output data
on (default) | off
This parameter returns output elements in bit-reversed order.
When you select this parameter, the output elements are bit-reversed. To return output elements in linear order, clear this parameter.
The FFT algorithm calculates output in the reverse order to the input. If you specify the output to be in the same order as the input, the algorithm performs an extra reversal operation. For more
information, see Linear and Bit-Reversed Output Order.
Input in bit-reversed order — Expected order of input data
off (default) | on
When you select this parameter, the block expects input data in bit-reversed order. By default, this parameter is disabled, and the block expects the input in linear order.
The FFT algorithm calculates output in the reverse order to the input. If you specify the output to be in the same order as the input, the algorithm performs an extra reversal operation. For more
information, see Linear and Bit-Reversed Output Order.
Divide butterfly outputs by two — FFT scaling
off (default) | on
When you select this parameter, the FFT implements an overall 1/N scale factor by dividing the output of each butterfly multiplication by two. This adjustment keeps the output of the FFT in the same
amplitude range as its input. If you disable scaling, the FFT avoids overflow by increasing the word length by 1 bit after each butterfly multiplication. The bit increase is the same for both
Data Types
Rounding mode — Rounding mode for internal fixed-point calculations
Floor (default) | Ceiling | Convergent | Nearest | Round | Zero
This parameter specifies the type of rounding mode for internal fixed-point calculations. For more information about rounding modes, see Rounding Modes. When the input is any integer or fixed-point
data type, this block uses fixed-point arithmetic for internal calculations. This parameter does not apply when the input data is single or double. Rounding applies to twiddle-factor multiplication
and scaling operations.
Control Ports
Enable reset input port — Optional reset signal
off (default) | on
This parameter enables a reset input port. When you select this parameter, the input reset port appears on the block icon.
Enable start output port — Optional control signal indicating start of data
off (default) | on
This parameter enables a port that indicates the start of output data. When you select this parameter, the output start port appears on the block icon.
Enable end output port — Optional control signal indicating end of data
off (default) | on
This parameter enables a port that indicates the end of output data. When you select this parameter, the output end port appears on the block icon.
Streaming Radix 2^2
The streaming Radix 2^2 architecture implements a low-latency architecture. It saves resources compared to a streaming Radix 2 implementation by factoring and grouping the FFT equation. The
architecture has log[4](N) stages. Each stage contains two single-path delay feedback (SDF) butterflies with memory controllers. When you use vector input, each stage operates on fewer input samples,
so some stages reduce to a simple butterfly, without SDF.
The first SDF stage is a regular butterfly. The second stage multiplies the outputs of the first stage by –j. To avoid a hardware multiplier, the block swaps the real and imaginary parts of the
inputs, and again swaps the imaginary parts of the resulting outputs. Each stage rounds the result of the twiddle factor multiplication to the input word length. The twiddle factors have two integer
bits, and the rest of the bits are used for fractional bits. The twiddle factors have the same bit width as the input data, WL. The twiddle factors have two integer bits, and WL-2 fractional bits.
If you enable scaling, the algorithm divides the result of each butterfly stage by 2. Scaling at each stage avoids overflow, keeps the word length the same as the input, and results in an overall
scale factor of 1/N. If scaling is disabled, the algorithm avoids overflow by increasing the word length by 1 bit at each stage. The diagram shows the butterflies and internal word lengths of each
stage, not including the memory.
Burst Radix 2
The burst Radix 2 architecture implements the FFT by using a single complex butterfly multiplier. The algorithm cannot start until it has stored the entire input frame, and it cannot accept the next
frame until computations are complete. The output ready port indicates when the algorithm is ready for new data. The diagram shows the burst architecture, with pipeline registers.
When you use this architecture, your input data must comply with the ready backpressure signal.
Control Signals
The algorithm processes input data only when the input valid port is 1. Output data is valid only when the output valid port is 1.
When the optional input reset port is 1, the algorithm stops the current calculation and clears all internal states. The algorithm begins new calculations when reset port is 0 and the input valid
port starts a new frame.
Timing Diagram
This diagram shows the input and output valid port values for contiguous scalar input data, streaming Radix 2^2 architecture, an FFT length of 1024, and a vector size of 16.
The diagram also shows the optional start and end port values that indicate frame boundaries. If you enable the start port, the start port value pulses for one cycle with the first valid output of
the frame. If you enable the end port, the start port value pulses for one cycle with the last valid output of the frame.
If you apply continuous input frames, the output will also be continuous after the initial latency.
The input valid port can be noncontiguous. Data accompanied by an input valid port is processed as it arrives, and the resulting data is stored until a frame is filled. Then the algorithm returns
contiguous output samples in a frame of N (FFT length) cycles. This diagram shows noncontiguous input and contiguous output for an FFT length of 512 and a vector size of 16.
When you use the burst architecture, you cannot provide the next frame of input data until memory space is available. The ready signal indicates when the algorithm can accept new input data. You must
apply input data and valid signals only when ready is 1 (true). The algorithm ignores any input data and valid signals when ready is 0 (false).
The latency varies with the FFT length and input vector size. After you update the model, the block icon displays the latency. The displayed latency is the number of cycles between the first valid
input and the first valid output, assuming the input is contiguous. To obtain this latency programmatically, see Automatic Delay Matching for the Latency of FFT Block.
When using the burst architecture with a contiguous input, if your design waits for ready to output 0 before de-asserting the input valid, then one extra cycle of data arrives at the input. This data
sample is the first sample of the next frame. The algorithm can save one sample while processing the current frame. Due to this one sample advance, the observed latency of the later frames (from
input valid to output valid) is one cycle shorter than the reported latency. The latency is measured from the first cycle, when input valid is 1 to the first cycle when output valid is 1. The number
of cycles between when ready port is 0 and the output valid port is 1 is always latency – FFTLength.
This resource and performance data is the synthesis result from the generated HDL targeted to a Xilinx^® Virtex^®-6 (XC6VLX75T-1FF484) FPGA. The examples in the tables have this configuration:
• 1024 FFT length (default)
• Complex multiplication using 4 multipliers, 2 adders
• Output scaling enabled
• Natural order input, Bit-reversed output
• 16-bit complex input data
• Clock enables minimized (HDL Coder™ parameter)
Performance of the synthesized HDL code varies with your target and synthesis options. For instance, reordering for a natural-order output uses more RAM than the default bit-reversed output, and real
input uses less RAM than complex input.
For a scalar input Radix 2^2 configuration, the design achieves 326 MHz clock frequency. The latency is 1116 cycles. The design uses these resources.
Resource Number Used
LUT 4597
FFS 5353
Xilinx LogiCORE^® DSP48 12
Block RAM (16K) 6
When you vectorize the same Radix 2^2 implementation to process two 16-bit input samples in parallel, the design achieves 316 MHz clock frequency. The latency is 600 cycles. The design uses these
Resource Number Used
LUT 7653
FFS 9322
Xilinx LogiCORE DSP48 24
Block RAM (16K) 8
The block supports scalar input data only when implementing burst Radix 2 architecture. The burst design achieves 309 MHz clock frequency. The latency is 5811 cycles. The design uses these resources.
Resource Number Used
LUT 971
FFS 1254
Xilinx LogiCORE DSP48 3
Block RAM (16K) 6
[1] Algnabi, Y.S, F.A. Aldaamee, R. Teymourzadeh, M. Othman, and M.S. Islam. “Novel architecture of pipeline Radix 2^2 SDF FFT Based on digit-slicing technique.” 10th IEEE International Conference on
Semiconductor Electronics (ICSE). 2012, pp. 470–474.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
This block supports C/C++ code generation for Simulink accelerator and rapid accelerator modes and for DPI component generation.
HDL Code Generation
Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.
This block supports HDL code generation using HDL Coder. HDL Coder provides additional configuration options that affect HDL implementation and synthesized logic.
HDL Architecture
This block has one default HDL architecture.
HDL Block Properties
Number of registers to place at the outputs by moving existing delays within your design. Distributed pipelining does not redistribute these registers. The default is 0. For
ConstrainedOutputPipeline more details, see ConstrainedOutputPipeline (HDL Coder).
Number of input pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For
InputPipeline more details, see InputPipeline (HDL Coder).
OutputPipeline Number of output pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For
more details, see OutputPipeline (HDL Coder).
Version History
Introduced in R2014a
R2022a: Moved to DSP HDL Toolbox from DSP System Toolbox
Before R2022a, this block was named FFT HDL Optimized and was included in the DSP System Toolbox™ DSP System Toolbox HDL Support library.
R2022a: FFT length of 4
You can now set the FFT length to 4 (2^2). In previous releases the FFT length had to be a power of 2 from 8 (2^3) to 2^16. | {"url":"https://it.mathworks.com/help/dsphdl/ref/fft.html","timestamp":"2024-11-08T11:46:33Z","content_type":"text/html","content_length":"132690","record_id":"<urn:uuid:499f90e8-52f9-498c-87a9-ffc72232bcb3>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00852.warc.gz"} |
A study on some properties of leap graphs
[1] B. Basavanagoud and E. Chitra, On the leap Zagreb indices of generalized $xyz$-point-line transformation graphs $T^{xyz}(g)$ when $z = 1$, Int. J. Math. Combin. 2 (2018), 44–66.
[2] B. Basavanagoud and P. Jakkannavar, Computing first leap Zagreb index of some nano structures, Int. J. Math. And Appl. 6 (2018), no. 2-B, 141–150.
[3] B. Borovicanin, K.C. Das, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem 78 (2017), no. 1, 17–100.
[4] K.C. Das and I. Gutman, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52 (2004), no. 1, 103–112.
[5] I. Gutman, E. Milovanović, and I. Milovanović, Beyond the zagreb indices, AKCE Int. J. Graphs Combin., in press.
[6] I. Gutman, B. Ruščić, N. Trinajstić, and C.F. Wilcox, Graph theory and molecular orbitals. XII. acyclic polyenes, J. Chem. Phys. 62 (1975), no. 9, 3399–3405.
[7] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total ϕ-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.
[8] F. Harary, Graph Theory, Addison-Wesley, Reading, 1969.
[9] H.R. Manjunathe, A.M. Naji, P. Shiladhar, and N.D. Soner, Leap eccentric connectivity index of some graph operations, Int. J. Res. Anal. Reviews 6 (2019), no. 1, 882–887.
[10] A.M. Naji and N.D. Soner, The first leap zagreb index of some graph opertations, Int. J. Appl. Graph Theory 2 (2018), no. 2, 7–18.
[11] A.M. Naji, N.D. Soner, and I. Gutman, The first leap zagreb index of some graph opertations, Commun. Comb. Optim. 2 (2017), no. 2, 99–117.
[12] S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003), no. 2, 113–124.
[13] Z. Shao, I. Gutman, Z. Li, S. Wang, and P. Wu, The first leap Zagreb index of some graph opertations, Commun. Comb. Optim. 3 (2018), no. 2, 179–194.
[14] P. Shiladhar, A.M. Naji, and N.D. Soner, Computation of leap Zagreb indices of some windmill graphs, Int. J. Math. And Appl. 6 (2018), no. 2-B, 183–191.
[15] P. Shiladhar, A.M. Naji, and N.D. Soner, Leap Zagreb indices of some wheel related graphs, J. Comp. Math. Sci. 9 (2018), no. 3, 221–231.
[16] N.D. Soner and A.M. Naji, The k-distance neighborhood polynomial of a graph, Int. J. Math. Comput. Sci. 3 (2016), no. 9, 2359–2364.
[17] K. Xu and H. Hua, A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 68 (2012), no. 1, 241–256.
[18] S. Yamaguchi, Estimating the Zagreb indices and the spectral radius of triangleand quadrangle-free connected graphs, Chem. Phys. Lett. 458 (2008), no. 4, 396–398. | {"url":"https://comb-opt.azaruniv.ac.ir/article_13876.html","timestamp":"2024-11-09T00:30:53Z","content_type":"text/html","content_length":"50346","record_id":"<urn:uuid:46c9d888-18ae-4017-a197-906264bf51e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00338.warc.gz"} |
mperial) to Dash (Imperial)
Bucket (Imperial) to Dash (Imperial) Converter
Enter Bucket (Imperial)
Dash (Imperial)
⇅ Switch toDash (Imperial) to Bucket (Imperial) Converter
How to use this Bucket (Imperial) to Dash (Imperial) Converter 🤔
Follow these steps to convert given volume from the units of Bucket (Imperial) to the units of Dash (Imperial).
1. Enter the input Bucket (Imperial) value in the text field.
2. The calculator converts the given Bucket (Imperial) into Dash (Imperial) in realtime ⌚ using the conversion formula, and displays under the Dash (Imperial) label. You do not need to click any
button. If the input changes, Dash (Imperial) value is re-calculated, just like that.
3. You may copy the resulting Dash (Imperial) value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Bucket (Imperial) to Dash (Imperial)?
The formula to convert given volume from Bucket (Imperial) to Dash (Imperial) is:
Volume[(Dash (Imperial))] = Volume[(Bucket (Imperial))] × 24576
Substitute the given value of volume in bucket (imperial), i.e., Volume[(Bucket (Imperial))] in the above formula and simplify the right-hand side value. The resulting value is the volume in dash
(imperial), i.e., Volume[(Dash (Imperial))].
Calculation will be done after you enter a valid input.
Consider that a well yields 10 buckets (imperial) of water.
Convert this water volume from buckets (imperial) to Dash (Imperial).
The volume in bucket (imperial) is:
Volume[(Bucket (Imperial))] = 10
The formula to convert volume from bucket (imperial) to dash (imperial) is:
Volume[(Dash (Imperial))] = Volume[(Bucket (Imperial))] × 24576
Substitute given weight Volume[(Bucket (Imperial))] = 10 in the above formula.
Volume[(Dash (Imperial))] = 10 × 24576
Volume[(Dash (Imperial))] = 245760
Final Answer:
Therefore, 10 bkt is equal to 245760 .
The volume is 245760 , in dash (imperial).
Consider that a paint shop uses 5 buckets (imperial) of paint for a project.
Convert this paint usage from buckets (imperial) to Dash (Imperial).
The volume in bucket (imperial) is:
Volume[(Bucket (Imperial))] = 5
The formula to convert volume from bucket (imperial) to dash (imperial) is:
Volume[(Dash (Imperial))] = Volume[(Bucket (Imperial))] × 24576
Substitute given weight Volume[(Bucket (Imperial))] = 5 in the above formula.
Volume[(Dash (Imperial))] = 5 × 24576
Volume[(Dash (Imperial))] = 122880
Final Answer:
Therefore, 5 bkt is equal to 122880 .
The volume is 122880 , in dash (imperial).
Bucket (Imperial) to Dash (Imperial) Conversion Table
The following table gives some of the most used conversions from Bucket (Imperial) to Dash (Imperial).
Bucket (Imperial) (bkt) Dash (Imperial) ()
0.01 bkt 245.76
0.1 bkt 2457.6
1 bkt 24576
2 bkt 49152
3 bkt 73728
4 bkt 98304
5 bkt 122880
6 bkt 147456
7 bkt 172032
8 bkt 196608
9 bkt 221184
10 bkt 245760
20 bkt 491520
50 bkt 1228800
100 bkt 2457600
1000 bkt 24576000
Bucket (Imperial)
The Imperial bucket is a unit of measurement traditionally used to quantify liquid volumes in the UK and other countries using the Imperial system. Originating from practical needs in agriculture and
household tasks, the bucket became a standardized measure for consistency. Historically, the Imperial bucket was essential for tasks such as milking, water collection, and brewing. Today, while less
common, it remains a recognized unit in certain industries and historical contexts.
Dash (Imperial)
The Imperial dash is a unit of measurement used to quantify very small volumes, typically in cooking and medicine. It is a traditional unit from the British Imperial system, representing a small,
precise amount often used in recipes or for dosing. Historically, the dash was used to measure tiny quantities of liquid for adding to recipes or medical preparations. Today, it remains relevant in
specific contexts where precise small-volume measurements are necessary, such as in culinary arts for seasoning or in medicine for administering minute doses.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Bucket (Imperial) to Dash (Imperial) in Volume?
The formula to convert Bucket (Imperial) to Dash (Imperial) in Volume is:
Bucket (Imperial) * 24576
2. Is this tool free or paid?
This Volume conversion tool, which converts Bucket (Imperial) to Dash (Imperial), is completely free to use.
3. How do I convert Volume from Bucket (Imperial) to Dash (Imperial)?
To convert Volume from Bucket (Imperial) to Dash (Imperial), you can use the following formula:
Bucket (Imperial) * 24576
For example, if you have a value in Bucket (Imperial), you substitute that value in place of Bucket (Imperial) in the above formula, and solve the mathematical expression to get the equivalent value
in Dash (Imperial). | {"url":"https://convertonline.org/unit/?convert=bucket_imperial-dash_imperial","timestamp":"2024-11-09T20:09:42Z","content_type":"text/html","content_length":"93000","record_id":"<urn:uuid:f138c897-6dfb-4502-9fef-de94a59d40b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00898.warc.gz"} |
Maxime Bros
My research fields are: Cryptography, Applied Mathematics, Computer Algebra, and Computer Science.
In particular, I work on Algebraic Attacks against Post-Quantum cryptosystems.
In order to study the security of these cryptosystems, I evaluate the complexity of computing Gröbner bases of systems of multivariate polynomials over finine fields.
I am one of the authors of the current best attacks against the Rank Decoding and MinRank problems; these two problems are at the core of rank and multivariate-based cryptography.
I am part of the RQC team; Rank Quasi-Cyclic (RQC) is a rank-based cryptosystem which made it to the 2nd Round of the NIST Post-Quantum Standardization Process.
I am one of the scientists who introduced the SquareSpace problem in 2022. This problem relies on the factorization of vector spaces, and we used it to design new post-quantum signature and
encryption schemes.
PhD Thesis
• December 2022 - Algebraic Cryptanalysis and Contributions to Post-Quantum Cryptography based on Error-Correcting Codes in the Rank-metric [pdf here (in English)] .
University of Limoges, XLIM, UMR CNRS 7252, France.
Supervisors: Philippe Gaborit and Vincent Neiger .
• July - August 2022 - 2-month internship: Implementation in C of a SquareSpace-based signature .
Student: Frédéric Canaud, Master 1 in Computer Science, University of Limoges.
• July - August 2022 - 2-month internship: Algebraic Attacks against the SquareSpace problem .
Student: Romaric Neveu, Master 1 in Mathematics, University of Limoges.
• December 2021 - Cryptanalysis of the Rank Preserving Signature,
Nicolas Aragon, Maxime Bros, Philippe Gaborit.
18th IMA International Conference on Cryptography and Coding (IMACC 2021), virtual conference.
• December 2020 - Improvements of Algebraic Attacks for solving the Rank Decoding and MinRank problems,
Magali Bardet, Maxime Bros, Daniel Cabarcas, Philippe Gaborit, Ray Perlner, Daniel Smith-Tone, Jean-Pierre Tillich, Javier Verbel.
ASIACRYPT 2020, virtual conference. [article here].
• May 2020 - An Algebraic Attack on Rank Metric Code-Based Cryptosystems,
Magali Bardet, Pierre Briaud, Maxime Bros, Philippe Gaborit, Vincent Neiger, Olivier Ruatta, Jean-Pierre Tillich.
EUROCRYPT 2020, virtual conference. [article here].
• April 2020 - Rank Quasi-Cyclic (RQC) [website], [documentation]. Carlos Aguilar Melchor, Nicolas Aragon, Slim Bettaieb, Loïc Bidoux, Olivier Blazy, Maxime Bros, Alain Couvreur, Jean-Christophe
Deneuville, Philippe Gaborit, Adrien Hauteville, Gilles Zémor.
2nd Round of the National Institute of Standards and Technology (NIST) Post-Quantum Standardization Process.
Master's Thesis
• September 2019 - On the Complexity of Gröbner Basis Attacks for the Rank Decoding Problem. Report of a Six-Month Research Internship at the University of Limoges, France.
Supervisors: Philippe Gaborit, Vincent Neiger, and Olivier Ruatta.
Unpublished Reports
• February 2019 - Cryptanalysis of MinRank.
Master 2 of Applied Mathematics, University of Limoges.
• January 2019 - On the Complexity of the Rank Syndrome Decoding Problem.
Master 2 of Applied Mathematics, University of Limoges. Coauthor: Olivier Gimenez.
• May 2018 - Privacy in a blockchain : Bitcoin analysis and Zcash.
Master 1 of Applied Mathematics, University of Limoges. Coauthor: Do Dai Chi.
• May 2017 - Théorème de l’Universalité de Wigner : Démonstration combinatoire et Étude empirique de la vitesse de convergence.
Master 1 of Pure Mathematics, University of Burgundy. | {"url":"http://maxime-bros.fr/research.html","timestamp":"2024-11-09T12:06:29Z","content_type":"text/html","content_length":"8517","record_id":"<urn:uuid:680beb58-ab86-4a40-8bd2-db38eb2c9e8c>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00002.warc.gz"} |
Cambridge Nationals: NEA FAQs - What feedback can I give to students?
Can I give ongoing feedback to my students when they are completing their assignments?
Refer to Section 6.3 of the Specification.
Once summative assessment has begun (so once students have started working on the live assignment), teachers are only able to provide generic feedback to students. It is acceptable to provide generic
feedback on an ongoing basis.
What is generic feedback?
Generic feedback is feedback that can be given to all students. It would usually be verbal feedback that the teacher gives to the whole class but may also be to individuals. It may act as a general
prompt or reminder to students but must not give them specific advice that would tell them what they should do to meet given marking criteria requirements.
For example, teachers can say to students:
• “Remember we had a lesson on XX topic. You could look at your notes from this lesson before you start this task.”
• “Remember, the words in bold in the marking criteria are the command words. All command words and their meanings are in Appendix X at the end of your assignments. Your work will need to meet the
command word for each strand of the marking criteria. Remember we had lessons where we talked about the sort of evidence that may meet each command word. You could look at the notes from that
lesson before you start the task today.”
What might be considered as improper assistance (over-direction)?
Feedback must not provide specific advice to individual students or the whole class to help them complete the tasks. Some examples of specific feedback that would be considered improper assistance
(over-direction) may be:
X “If you include a graph that shows the breakdown of research outcomes as well as the data in this task, it will help to meet Mark Band 3 rather than just Mark Band 2. You could consider using the
graph approach that I gave you in my example in class”
X “You need to create a logo in this task. Make sure you include a mindmap and a slogan or strapline to get the highest marks.”
Providing specific feedback to students would compromise their ability to independently complete the tasks they are doing. This would be malpractice.
What personalised feedback can I give to students when they have handed in completed work and I have marked it?
When students have handed in their completed assignment work, teachers can mark this and provide students with individual feedback before final marks are submitted to OCR. This feedback must be
recorded. If you agree that students can make changes to their work after this marking and feedback process, it is classed as a reattempt (see What is the difference between a reattempt and a
What personalised feedback can I give to students?
Feedback given at this stage must still adhere to the guidance in Section 6.3 of the specification. This is because students are allowed to reattempt their work to improve it, considering the
feedback provided, so feedback must not unfairly advantage or disadvantage any students.
Constructive and useful feedback should allow students to understand:
• strengths of their performance
• limitations of their performance
• where work could be improved, but not how to improve it.
Feedback should be recorded against the topic area and marking criteria to help students understand how the assessment decision has been made/arrived at. (See Section 6.7 of the Specification and How
do I record feedback for more information about how to record feedback.)
Individual feedback must be factual, with the teacher telling the student what they have observed, rather than telling the student what to do to improve work.
This feedback would be acceptable:
• “Review MB3 criteria in terms of the aims of the market research. You are currently in MB1 (1-4 marks). In the lesson we had on this topic, you offered more insight into market research and
sampling methods, which you could reflect on to improve your marks.”
Feedback must not tell students what they need to do to improve their work. This feedback would not be acceptable:
X “The task required you to present your data. You need to improve how the information is displayed. Think about how to structure tables clearly, including the breakdown of the sampling data and
market research outcomes. Use bar charts to show the data in a more detailed way. Adding this detail will help move your work into the next mark band for this strand.”
If students wish to, they can consider the feedback given and reattempt their work to improve their marks at this stage, before you have submitted the marks to OCR.
Can I mark my students’ work after each task?
Refer to Section 6.4.4 of the Specification and to the Student guide to NEA assignments.
The Specification states that marking and feedback should not be an iterative process.
The Student guide to NEA assignments also tells students that teachers cannot mark work in stages.
What does an iterative process mean?
An iterative process of marking and feedback is where work is submitted to the teacher and marked in small sections, bit by bit.
Teachers should not:
• Mark work in sections (for example after each task has been completed) and then provide students with the marks for, and feedback on, this section.
• Mark work and provide those marks to students in a way that allows students to keep improving their work as they work through the tasks. Marking work ‘bit by bit’ gives a student the opportunity
to reflect on their work in a more focussed way than they would if they received their mark for the whole set assignment and could lead to incremental changes to improve their result. Marking
‘bit by bit’ may potentially advantage students as they could tweak their work more easily than other students.
When can I mark students’ work and give them their marks?
Students should submit their assignment evidence to teachers for marking when they have completed the summative assessment (consider this to be their first draft work).
Teachers can then mark this and provide marks and feedback to each student, based on the completed work students have submitted to them.
What feedback can I give my students when I’ve marked their work?
Feedback given to students must adhere to the guidance in Section 6.3 of the specification. Please see the responses to to our questions on ongoing and personalised feedback above and the examples
given to show what is acceptable feedback and what might be considered as improper assistance (over-direction).
Can my students have another attempt at their assignment when I’ve given them their marks and feedback?
Students are allowed to reattempt the assignment after it has been marked as a complete assignment. This is detailed in Section 6.4.4 of the Specification. They should not be getting marks and
personalised feedback task by task and being allowed to work on those in an iterative way.
• See What is the difference between a reattempt and a resubmission which explains the difference between a reattempt and a resubmission.
• See also What if my students can't get started for how to record feedback where students cannot get started on an assignment. In this case, feedback should be recorded on the student’s work and/
or URS.
0 comments
Article is closed for comments. | {"url":"https://support.ocr.org.uk/hc/en-gb/articles/21042211328786-Cambridge-Nationals-NEA-FAQs-What-feedback-can-I-give-to-students","timestamp":"2024-11-10T14:54:37Z","content_type":"text/html","content_length":"35036","record_id":"<urn:uuid:058e5e8d-8f7b-4e33-a0ad-aeed878f8b87>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00374.warc.gz"} |
Main Content
Create index vector from grouping variable
[g,gN] = grp2idx(s) creates an index vector g from the grouping variable s. The output g is a vector of integer values from 1 up to the number K of distinct groups. gN is a cell array of character
vectors representing the list of group names.
[g,gN,gL] = grp2idx(s) also returns a column vector gL representing the list of the group levels with the same data type as s.
Create Index Vector from Categorical Vector
Create a categorical vector by using discretize and convert it to an index vector by using grp2idx.
Load the hospital data set and convert the ages in hospital.Ages to categorical values representing the ages by decade.
load hospital
edges = 0:10:100; % Bin edges
labels = strcat(num2str((0:10:90)','%d'),{'s'}); % Labels for the bins
s = discretize(hospital.Age,edges,'Categorical',labels);
Display the ages and the groups of ages for the first five samples.
ages = 5×1
groups = 5x1 categorical
Create an index vector from the categorical vector s.
Display the index values corresponding to the first five samples.
Reproduce the input argument s using the output gL.
ans = 5x1 categorical
Use gN(g) to reproduce the input argument s as a cell array of character vectors.
ans = 5x1 cell
Input Arguments
s — Grouping variable
categorical vector | numeric vector | logical vector | datetime vector | duration vector | string array | cell array of character vectors | character array
Grouping variable, specified as a categorical, numeric, logical, datetime, or duration vector, a string array, a cell array of character vectors, or a character array with each row representing a
group label.
grp2idx treats NaNs (numeric, duration, or logical), '' (empty character arrays or cell arrays of character vectors), "" (empty strings), <missing> values (string), <undefined> values (categorical),
and NaTs (datetime) in s as missing values and returns NaNs in the corresponding rows of g. The outputs gN and gL do not include entries for missing values.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | char | string | cell | categorical | datetime | duration
Output Arguments
g — Group index
positive integer vector
Group index, returned as a positive integer vector with values from 1 up to the number K of distinct groups in s.
gN — List of group names
cell array of character vectors
List of group names, returned as a cell array of character vectors.
The order of gN depends on the data type of the grouping variable s.
• For numeric and logical vectors, the order is the sorted order of s.
• For categorical vectors, the order is the order of categories(s).
• For other data types, the order is the order of first appearance in s.
gN(g) reproduces the contents of s in a cell array.
gL — List of group levels
categorical vector | numeric vector | logical vector | datetime vector | duration vector | cell array of character vectors | character array
List of group levels, returned as the same data type as s: a categorical, numeric, logical, datetime, or duration vector, a cell array of character vectors, or a character array with each row
representing a group label. (The software treats string arrays as cell arrays of character vectors.)
The set of groups and their order in gL are the same as those in gN, but gL has the same data type as s.
If s is a character matrix, then gL(g,:) reproduces s; otherwise, gL(g) reproduces s.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
• The input argument s can be a numeric, logical, or character vector or a cell array of character vectors. Code generation does not support a categorical, datetime, or duration vector or a string
array for the input argument.
• In the generated code, the second and third outputs, gN and gL, are identical. gN and gL have the same data type as the input argument s.
For more information on code generation, see Introduction to Code Generation and General Code Generation Workflow.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced before R2006a | {"url":"https://ww2.mathworks.cn/help/stats/grp2idx.html","timestamp":"2024-11-15T02:41:07Z","content_type":"text/html","content_length":"78581","record_id":"<urn:uuid:aad56e1a-9377-406a-81d9-e87fd0042fe4>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00839.warc.gz"} |
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Lesson 15
Volume of Prisms
Let’s look at the volume of prisms that have fractional measurements.
Problem 1
A pool in the shape of a rectangular prism is being filled with water. The length and width of the pool is 24 feet and 15 feet. If the height of the water in the pool is \(1\frac13\) feet, what is
the volume of the water in cubic feet?
Problem 2
A rectangular prism measures \(2\frac25\) inches by \(3\frac15\) inches by 2 inch.
1. Priya said, “It takes more cubes with edge length\(\frac25\) inch than cubes with edge length \(\frac15\) inch to pack the prism.” Do you agree with Priya? Explain or show your reasoning.
2. How many cubes with edge length \(\frac15\) inch fit in the prism? Show your reasoning.
3. Explain how you can use your answer in the previous question to find the volume of the prism in cubic inches.
Problem 3
1. Here is a right triangle. What is its area?
2. What is the height \(h\) for the base that is \(\frac54\) units long? Show your reasoning.
(From Unit 4, Lesson 14.)
Problem 4
To give their animals essential minerals and nutrients, farmers and ranchers often have a block of salt—called “salt lick”—available for their animals to lick.
1. A rancher is ordering a box of cube-shaped salt licks. The edge lengths of each salt lick are \(\frac{5}{12}\) foot. Is the volume of one salt lick greater or less than 1 cubic foot? Explain your
2. The box that contains the salt lick is \(1\frac14\) feet by \(1\frac23\) feet by \(\frac56\) feet. How many cubes of salt lick fit in the box? Explain or show your reasoning.
Problem 5
1. How many groups of \(\frac13\) inch are in \(\frac34\) inch?
2. How many inches are in \(1\frac25\) groups of \(1\frac23\) inches?
(From Unit 4, Lesson 12.)
Problem 6
Here is a table that shows the ratio of flour to water in an art paste. Complete the table with values in equivalent ratios.
│cups of flour │cups of water │
│1 │\(\frac12\) │
│4 │ │
│ │3 │
│\(\frac12\) │ │
(From Unit 2, Lesson 12.) | {"url":"https://curriculum.illustrativemathematics.org/MS/students/1/4/15/practice.html","timestamp":"2024-11-04T12:14:06Z","content_type":"text/html","content_length":"75538","record_id":"<urn:uuid:a916fb22-1c7e-4cbc-be42-fe7fd9fe1924>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00440.warc.gz"} |
Numerical Greeks — Implementing QuantLib
Hello, dear reader.
This post shows a short notebook on numerical Greeks. If you bought the QuantLib Python Cookbook, or if you watched my videos on YouTube, you might have already seen an earlier version of the same
notebook. However, a recent question on the Quantitative Finance Stack Exchange made me realize that I should give it some exposure here, too.
The notebook shows how to calculate numerical Greeks when a pricing engine doesn’t provide them. As to why the engine might not provide it—well, it turns out that I already wrote about it in 2017 and
I didn’t remember it. Here is a link to that post. Enjoy!
Subscribe to my Substack to receive my posts in your inbox, or follow me on Twitter or LinkedIn if you want to be notified of new posts, or subscribe via RSS if you’re the tech type: the buttons for
all that are in the footer. Also, I’m available for training, both online and (when possible) on-site: visit my Training page for more information.
Numerical Greeks
In this notebook, I’ll build on the facilities provided by the Instrument class (that is, its ability to detect changes in its inputs and recalculate accordingly) to show how to calculate numerical
Greeks when a given engine doesn’t provide them.
As usual, we import the QuantLib module and set the evaluation date:
today = ql.Date(8, ql.October, 2014)
ql.Settings.instance().evaluationDate = today
A somewhat exotic option
As an example, we’ll use a knock-in barrier option:
strike = 100.0
barrier = 120.0
rebate = 30.0
option = ql.BarrierOption(
ql.PlainVanillaPayoff(ql.Option.Call, strike),
ql.EuropeanExercise(ql.Date(8, ql.January, 2015)),
For the purpose of this example, the market data are the underlying value, the risk-free rate and the volatility. We wrap them in quotes, so that the instrument will be notified of any changes…
u = ql.SimpleQuote(100.0)
r = ql.SimpleQuote(0.01)
σ = ql.SimpleQuote(0.20)
…and from the quotes we build the flat curves and the process that the engine requires. As explained in another notebook, we build the term structures so that they move with the evaluation date; this
will be useful further on.
riskFreeCurve = ql.FlatForward(
0, ql.TARGET(), ql.QuoteHandle(r), ql.Actual360()
volatility = ql.BlackConstantVol(
0, ql.TARGET(), ql.QuoteHandle(σ), ql.Actual360()
process = ql.BlackScholesProcess(
Finally, we build the engine (the library provides one based on an analytic formula) and set it to the option.
Now we can ask the option for its value…
…but we’re not so lucky when it comes to Greeks:
RuntimeError: delta not provided
The engine doesn’t provide the delta, so asking for it raises an error.
Numerical calculation
What does a quant have to do? We can use numerical differentiation to approximate the Greeks: that is, we can approximate the derivative by calculating the option value for two slightly different
values of the underlying and by taking the slope between the resulting points.
The relevant formulas are:
\[\Delta = \frac{P(u_0+h)-P(u_0-h)}{2h} \; \; \; \; \; \; \Gamma = \frac{P(u_0+h)-2P(u_0)+P(u_0-h)}{h^2}\]
where $P(u)$ is the price of the option for a given value of the underlying $u$, $u_0$ is the current value of the underlying and $h$ is a tiny increment.
Thanks to the framework we set in place, getting the perturbed prices is easy enough: we can set the relevant quote to the new value and ask the option for its price again. Thus, we choose a value
for $h$ and start. First, we save the current value of the option…
P0 = option.NPV()
…then we increase the underlying value and get the new option value…
u.setValue(u0 + h)
P_plus = option.NPV()
…then we do the same after decreasing the underlying value.
u.setValue(u0 - h)
P_minus = option.NPV()
Finally, we set the underlying value back to its current value.
Applying the formulas above give us the desired Greeks:
Δ = (P_plus - P_minus) / (2 * h)
Γ = (P_plus - 2 * P0 + P_minus) / (h * h)
The approach is usable for any Greek. We can use the two-sided formula above, or the one-sided formula below if we want to minimize the number of evaluations:
\[\frac{\partial P}{\partial x} = \frac{P(x_0+h)-P(x_0)}{h}\]
For instance, here we calculate Rho and Vega:
r0 = r.value()
h = 0.0001
r.setValue(r0 + h)
P_plus = option.NPV()
ρ = (P_plus - P0) / h
σ0 = σ.value()
h = 0.0001
σ.setValue(σ0 + h)
P_plus = option.NPV()
Vega = (P_plus - P0) / h
The approach for the Theta is a bit different, although it still relies on the fact that the option reacts to the change in the market data. The difference comes from the fact that we don’t have the
time to maturity available as a quote, as was the case for the other quantities. Instead, since we set up the term structures so that they move with the evaluation date, we can set it to tomorrow’s
date to get the corresponding option value. In this case, the value of the time increment $h$ should be equivalent to one day:
ql.Settings.instance().evaluationDate = today + 1
P1 = option.NPV()
ql.Settings.instance().evaluationDate = today
h = 1.0 / 365
Θ = (P1 - P0) / h
That’s all for this post. See you next time! | {"url":"https://www.implementingquantlib.com/2024/08/numerical-greeks.html","timestamp":"2024-11-13T21:24:34Z","content_type":"text/html","content_length":"29306","record_id":"<urn:uuid:31a95b80-7d05-4d90-80ee-3ea1ea13ba1a>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00677.warc.gz"} |
Probabilistic and Topological Approaches to Potential Z-DNA Genomic Sequences - EUTOPIA
08-14 February 2023
Professor Doctor James R. Bozeman Jr.
Home Institution
American University of Malta, Malta
Host Contact
Dr. Veronica Andrea Gonzalez Lopez and Dr. Jesus Enrique Garcia,
Host Institution
Universidad de Campinas, Brazil
Aim of the mission
The purpose of this STSM is to complete work already begun on finding the probability that sequences of bases in DNA that are likely to be in the left-handed form, i.e. the Z-DNA conformation, occur
in certain full DNA sequences. One of the researchers in Brazil that the proposer will be visiting on this mission, Dr. Veronica Andrea Gonzalez Lopez, has developed, with colleagues, a local metric
from a distance measure between samples coming from discrete Markovian processes. They use the metric to decide if 2 independent samples are governed by the same Stochastic Law. This technique was
applied to the Stochastic Profile of Strains of the Zika virus utilizing the 4 bases of DNA. They find the probability of one base following another in the genomic sequence. This last is accomplished
in a computer program written by the other researcher with whom the proposer will be working while in Brazil, Dr. Jesus Enrique Garcia. While in Brazil, we will be applying the methodology above to
find the
probability of potential Z-DNA forming sequences (ZFSs) occurring in rodent parvoviruses, salmonella, and certain carcinogens. The work already begun on Sars Covid-19 will also be completed.
Summary of the Results
During the STSM the probabilities of potential Z-DNA forming sequences (ZFS) occurring in numerous genomes were found. These genomes included SARS Covid-19, Salmonella, Rodent Parvoviruses and some
cancer-causing chemicals, which were already known to have ZFS. More importantly, we also found such sequences in the Epstein-Barr virus, which had not been checked previously for
ZFS, and calculated the probabilities of those subsequences occurring. Included in this work above we also found the probability with which the other bases appeared before the beginning of the
potential ZFS. The work in this paragraph was mostly done by Dr. Jesus Enrique Garcia, who is a Probabilist.
The other collaborator, Dr. Veronica Andrea Gonzalez Lopez, a Statistician, applied the local metric from a distance measure between samples coming from discrete Markovian processes to these genomic
sequences and the ZFS subsequences. For instance, she looked at one cancer-causing chemical with ZFS and one without and compared the two. A partition Markov model is to be applied to all of these
sequences and ultimately we plan to form a classification of them (see the next section).
The proposer, who is a low-dimensional topologist, is utilizing the work above while examining the 3-dimensional conformation of the DNA molecule. For example, DNA supercoils in the cell and negative
supercoiling favors the left-handed Z-form of the molecule (note that topoisomerases can relax negative supercoiling). There are also proteins which only bind to Z-regions. Finally, there are
anti-Z-DNA antibodies associated with the cancer-causing chemicals noted above.
New avenues of research were also discovered. One is the application of our techniques to the Epstein-Barr virus, as noted above. The other is the realization that we can form a classification of the
DNA sequences and ZF subsequences, utilizing the partition Markov model. Some others include:
1. Finding the probability of the different bases that appear at the end of a ZFS.
2. Finding the total occurrences of ZFS in as many genomes as possible and then calculating the total appearances of same and comparing the result to the findings in the literature.
3. Answering the following questions: If a single different base appears in a ZFS can it still take on the left-handed form? We plan to use the Levenshtein distance to answer this question; Can
different ZFS which are concatenated still flip to left-handed form?
4. Finally, apply the idea of microsatellites, which we just discovered, to our work.
The work already done in Brazil is publishable, however we may wait to prepare a paper until we complete the new items outlined above. Without doubt we will be presenting our findings at conferences
this year. | {"url":"https://eutopia.unitn.eu/probabilistic-and-topological-approaches-to-potential-z-dna-genomic-sequences/","timestamp":"2024-11-04T23:57:06Z","content_type":"text/html","content_length":"174284","record_id":"<urn:uuid:bac972b9-7b5b-4ff7-b677-11f959fa72e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00493.warc.gz"} |
Analysis of factors affecting earnings using Annual Survey of Hours and Earnings
This article presents the results of 2 statistical models which explore the relationship between mean hourly earnings excluding overtime and a range of independent variables. The findings are based
on Annual Survey of Hours and Earnings – 2016 provisional results data. There is a particular focus on earnings differences between the public and private sectors. One model includes organisation
size as an independent variable and one excludes this factor.
Back to table of contents
Public and private sector pay differences
Looking across the employees in the Annual Survey of Hours and Earnings (ASHE) dataset the mean pay in the public sector is estimated to be 1.0% less than in the private sector in 2016. This uses the
model which excludes organisation size and controls for a range of independent variables including region, occupation, age, gender and job tenure. This continues a downward trend seen since 2012 and
is the first year in the series that the differential is in favour of the private sector since 2003.
In the model which includes organisation size, the mean pay in the public sector is estimated to be 5.5% less than in the private sector in 2016.
Looking at the top and bottom of the pay scale we see divergence between public and private pay. Public sector pay is estimated to be 10.8% above private sector pay at the bottom of the earnings
distribution (5th decile), but 13.2% below private sector pay at the higher end of the earnings distribution (95th decile), when using an identically specified quantile regression model excluding
organisation size.
The gap at the top of the earnings distribution between public and private sector employees has been increasing over time, but narrowing at the bottom of the earnings distribution.
In the quantile regression model which includes organisation size, the same trend exists across the earnings distribution but shows lower earnings in the public sector at each quantile.
Other factors affecting pay at the mean of the earnings distribution
In the model which excludes organisation size, part-time workers in 2016 are estimated to earn 8.8% less than full-time workers and temporary workers earned 0.7% less than permanent workers.
The corresponding figures are 6.5% less for part-time workers and 2% less for temporary workers respectively in 2016 in the model which includes organisation size.
The model which excludes organisation size estimates the North East and North West, for example, to have average pay around 6 to 7% lower than London in 2016. The equivalent figures for these regions
using the model including organisation size are 5 to 6% lower than London.
Back to table of contents
Headline measures of median weekly earnings for full-time employees in the UK are published in Annual Survey of Hours and Earnings - 2016 Provisional Results. These data are useful for comparing
groups of employees but do not take into account the different composition and characteristics of these employees. For example, comparing median earnings between employees in the public and private
sectors, or between men and women does not take into account the additional factors that are common to these groups, such as the public sector having generally older workers than the private sector,
or more women working in certain occupations compared with men.
This article updates the previous regression results published last year and addresses this issue. It presents the results of a regression model, developed in 2011 by the Office for National
Statistics (ONS) and HM Treasury, which statistically controls for a range of factors related to earnings that are available in the ASHE dataset. This enables the influence of separate factors on
hourly earnings to be identified, such as working in the public or private sectors while keeping other predictive factors constant.
There are 2 variants of the regression model presented in this article – one excluding and one including organisation size. The analysis from this model gives estimates of mean hourly pay controlling
for a range of independent variables available in the ASHE dataset. In addition, identically specified quantile regression models – with and without organisation size – are used to give estimates of
hourly pay at different points in the earnings distribution, including the median.
There are a number of important points to note about this analysis and the approach used in the regression model (further information on the regression methodology is given in the statistical notes
to this article):
• the ASHE dataset only covers the earnings of paid employees in the UK and does not include data on self-employed earnings (who are often some of the highest paid workers, as well as some of the
lowest paid workers)
• the dependent variable of the statistical model is hourly earnings excluding overtime pay; overtime paid at a higher rate would increase an employee’s hourly pay whereas working unpaid overtime
would effectively reduce hourly pay
• an adjustment is made to hourly earnings in the ASHE dataset using the Average Weekly Earnings series (published monthly based on the Monthly Wages and Salaries Survey) to better account for the
timing of bonus payments throughout the year; estimates in this article are not therefore directly comparable with those published in ASHE Provisional Results 2016 which do not make this
adjustment for the timing of bonus payments
• pension contributions or other forms of remuneration such as company car or health insurance by employers are not yet available in the 2016 ASHE dataset and the regression model has not been
designed to take them into account
• for April 2016, as for April 2015, the private sector element excludes employees in the non-profit institutions serving households (NPISH); in the UK the NPISH sector includes organisations such
as charities, trade unions and most universities; in some other surveys this group is included in the analysis of private sector organisations and employees – this treatment, however, is
consistent with previous analysis of ASHE data
• for consistency over time, employees of those banks classified to the public sector in 2008 have been treated as if they were in the private sector throughout
The estimates in this article are dependent on the specification of the statistical models and including additional or alternative independent variables would give different results. The model
currently explains over half of the variation in earnings. Data important to earnings such as education level is also not available in the ASHE dataset and limits some of the interpretation of
This article is split into 2 sections:
• Section A describes the factors which are used in the earnings regression model using the provisional ASHE dataset for April 2016
• Section B discusses the main results from the 2 linear and 2 quantile regression models – with and without organisation size – with a particular focus on the differences between public and
private sector earnings
Back to table of contents
4. Section A: Factors affecting earnings
This section will consider a number of factors that are taken into account analysing earnings. Simple averages are considered to gain an initial insight into pay differences amongst individuals with
different characteristics (such as age and gender) and job-related characteristics (such as public or private sector, occupational group, region of employment, job tenure, job status and size of
Individual characteristics
Age and sector
Age is an important factor affecting earnings, as this tends to be a proxy for experience and the build-up of skills over time. Figure 1, although not accounting for the different jobs within each
sector, shows that the age profile of workers varies significantly by whether they are employed in the public or private sector.
The distribution of jobs held within the private sector is more skewed towards younger age groups, while the distribution of jobs held within the public sector is more skewed towards older age
groups. The fall in the share of employees for both sectors around the 30s age group is a reflection of population demographics.
Mean hourly pay for all employees, regardless of sector, rises sharply at younger ages as job-related skills and experience is rewarded. Peaking in both sectors at around age 42, the average private
sector wage level remains slightly higher than the public sector average between the ages of 38 and 53 years. Average private sector pay then declines faster after this age compared with the public
Figure 1: Mean hourly earnings and share of all employees by age and sector, April 2016
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 is provisional.
Download this image Figure 1: Mean hourly earnings and share of all employees by age and sector, April 2016
.png (22.1 kB) .xls (21.5 kB)
Age, gender and sector
A similar analysis of the distribution of jobs held by men and women at each age is given in Figure 2. This shows a relatively equal age distribution for men and women from around 20 to 28 years old.
However, the female employment share drops by around a third between the ages of around 28 years to around 40 years likely to coincide with women taking time out to raise a family.
The female employment share then starts to increase from age 40 onwards until age 52 without seeing the same increase in average hourly earnings. In fact earnings are seen to slightly decrease with
age for women after their mid 30s whereas for men any earnings decline is not seen until the early 50s. The employment share of men and women at older ages decreases at similar rates.
Figure 2: Mean hourly earnings and share of all employees by age and gender, April 2016
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 is provisional.
Download this image Figure 2: Mean hourly earnings and share of all employees by age and gender, April 2016
.png (25.9 kB) .xls (22.5 kB)
As already highlighted, there are differences in earnings between the sectors related to the age of the employees. There are also differences in earnings between the sectors when split by gender
(Table 1).
Table 1: Average earnings per hour by gender and sector, April 2016, UK
Male earnings (£/hour) Female earnings (£/hour) % of females in sector
Public sector 19.09 15.33 67.6
Private sector 17.19 12.34 42.5
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 is provisional
2. All employees including those on junior rates and those affected by absence
Download this table Table 1: Average earnings per hour by gender and sector, April 2016, UK
.xls (27.1 kB)
Women make up over two-thirds of public sector employees compared with 42.5% of employees in the private sector, but on average are paid less than men in the public sector. Similarly, women are paid
less than men on average in the private sector. To better understand these differences in earnings it will be important to control for occupation, job tenure and other factors as we do in section B.
Job-related characteristics
Occupation, skill group and sector
Earnings are likely to increase as the skill level of the job increases. Thus differences in pay by sector may be partly explained by compositional differences in the share of employees at each skill
level. In this instance, skill is defined solely by occupation and is grouped into 4 broad skill levels using the Standard Occupational Classification (SOC 2010). Professional or upper skill group
jobs include occupations such as scientists, IT engineers, health and educational professionals, while elementary occupations or lower skill group occupations include farm workers, window cleaners,
waiters, waitresses and processing operatives. The background notes provide more detail of occupations in each skill group.
Figure 3: Occupational skill share of employee jobs by sector, April 2016
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 is provisional.
Download this chart Figure 3: Occupational skill share of employee jobs by sector, April 2016
Image .csv .xls
Figure 3 shows a compositional effect in that a larger proportion of public sector employees are in the professional and upper middle skilled group (75%) compared with the private sector (48%). The
remaining 52% of private sector employees are in the lower middle and lower skills group.
Region and occupation group
Figure 4 shows the mean hourly pay for employees (aged 16 to 64 years) and mean hourly pay for employees in professional occupations and employees in elementary occupations.
There are differences in average hourly pay across the regions, particularly for professional occupations in the South East and London. Professionals in London earn on average nearly £12 per hour
more (57%) than their Northern Ireland counterparts and over £1 per hour (15%) more in elementary occupations. For these regions, the higher average hourly pay for all workers is due to the pay of
professional occupation workers rather than elementary occupation workers and points to relatively large levels of inequality of earnings within London – professional occupations earn on average,
around 3.7 times more per hour than elementary occupations in London. This compares with a ratio of average earnings of professional occupations to elementary occupations of 2.6 to 2.9 in other
regions and countries of the UK. The varying nature of work and other factors may be the cause for differences in average hourly earnings between London and other regions although other factors such
as living costs may also contribute to these variations.
Figure 4: Mean hourly earnings by region for all employees, elementary and professional occupations, and National Living Wage, April 2016
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 is provisional.
Download this chart Figure 4: Mean hourly earnings by region for all employees, elementary and professional occupations, and National Living Wage, April 2016
Image .csv .xls
Working patterns and sector
The additional factors of whether a job is full-time or part-time and whether it is permanent or temporary also has an influence on earnings. Table 2 shows that mean hourly earnings in April 2016 of
full-time employees was on average £3.30 higher than for part-time employees in the public sector, but this difference is higher at around £7.20 in the private sector.
On average in 2016, employees in permanent jobs earn more per hour than those in temporary or casual jobs. Within the public sector this is £1.20 more on average whereas in the private sector this
was around £5.30 more.
There is a larger share of part-time working in the public sector compared with the private sector (30% of employee jobs compared with 25% in the private sector) and use of temporary jobs (10% of
public sector employee jobs compared with 7% of private sector jobs). These shares are unchanged from 2015.
Table 2: Mean hourly earnings by job characteristic, and share of jobs by sector, April 2016, UK
Earnings (£/ hour) Public sector Share of employee jobs in public sector (%) Earnings (£/hour) private sector Share of employee jobs in private sector (%)
Full-time 17.56 70 16.97 75
Part-time 14.26 30 9.77 25
Permanent 16.68 90 15.51 93
Temporary 15.48 10 10.19 7
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 is provisional
Download this table Table 2: Mean hourly earnings by job characteristic, and share of jobs by sector, April 2016, UK
.xls (27.1 kB)
Longer job tenure, as defined as length of time with current employee, is associated on average with higher pay in both the public and private sectors. At shorter tenures of roughly 5 years or less
however, public sector workers have higher average pay than private sector workers. Private sector workers have higher average pay at tenures longer than 5 years. Individuals that work in the public
sector are more likely to have worked for long tenures than those in the private sector, with over one quarter of public sector employees having worked for their employer for between 10 and 20 years.
Within the private sector the modal tenure length is between 2 and 5 years (representing just over one-fifth of private sector employees).
Size of organisation
The size of the employer organisation can influence rates of pay. Larger employers may be able to pay higher wages for similar roles than smaller ones, as larger organisations may have higher profit
margins or greater efficiencies of scale.
Figure 5 shows that in April 2016, a total of 58.2% of all employee jobs were in organisations with over 500 employees (up 0.9% from 2015) with noticeable differences between the public and private
sectors. The public sector tends to be concentrated in large organisations with at least 500 employees, with around 90% of public sector employees working in these large organisations. Such
organisations include the NHS, fire and police services or local authorities. In contrast, there is a much more even split of employees across the organisation sizes for the private sector, with just
under half working in large organisations in 2016. Around 12% of private sector employees worked in organisations of 10 employees or fewer, whereas for the public sector this was just 0.2%. People
who work in small and large organisations differ across the economy.
Figure 5: Percentage of employee jobs by size of organisation and by sector, April 2016
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 is provisional.
Download this chart Figure 5: Percentage of employee jobs by size of organisation and by sector, April 2016
Image .csv .xls
Earnings distribution by sector
Finally, Figure 6 shows that the earnings of the public sector are generally higher for all employees across the earnings distribution, apart from the top 10% of earners. In April 2016 specifically,
mean hourly public sector pay exceeded private sector pay by between £0 to £5 per hour until the 88th percentile of the pay distribution. At this point mean public sector pay was £25.23 per hour, and
mean private sector pay was £25.58. In April 2016, private sector employees at the 99th percentile of earnings were paid around £20 per hour more than employees at the 99th percentile of hourly
earnings in the public sector.
These data show the value of presenting quantile regression results which identify pay differences at the median and other points in the earnings distribution.
Figure 6: Distribution of hourly earnings in the public and private sector, April 2016
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 is provisional.
Download this chart Figure 6: Distribution of hourly earnings in the public and private sector, April 2016
Image .csv .xls
Notes for Section A: Factors affecting earnings
1. Part time is defined as employees working 30 paid hours per week or less.
2. This includes NPISH sector employees and those unclassified by sector.
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5. Section B: Regression analysis modelling pay differences
Comparison of average pay differentials made in section A can create a partial picture, as employees have multiple personal and job characteristics that can impact on their pay. Regression modelling
can be used to account for some of these differences by including controls for them. This regression analysis estimates the pay difference between types of individuals when controlling for the
following factors.
Data source and variable filters applied
The primary source of data is the Annual Survey of Hours and Earnings (ASHE) using hourly earnings (excluding overtime) for all employees whose pay in the April period was not affected by absence and
were paid adult rates. This data has been adjusted to account for bonus payments paid outside of the period covered by ASHE using data from the Average Weekly Earnings (AWE) series.
Regression model specification
Dependent variable:
• log of bonus-adjusted hourly earnings excluding overtime (see statistical note 1 and 4)
Independent variables:
• sex – there is a difference in the distribution of men and women in the public and private sector
• age and age squared – the relationship between earnings and age is non-linear (see statistical note 2)
• occupation classification, these are narrower than in section A (see background notes)
• region (regions of job location) – the proportion of jobs in each sector varies by region
• sector (public or private)
• full- or part-time status – the difference in pay and the difference in proportion varies between sectors
• permanent or temporary status – the distribution of these is different between sectors and the difference in pay is different between sectors
• job tenure – less or equal to 6 months, 6 to 12 months, 1 year to 2 years, 2 to 5 years, 5 to 10 years, 10 to 20 years and over 20 years – job tenure is a proxy for organisation-specific
Interaction terms: (see statistical note 3)
• sex*age and sex*age squared – the potential work experience proxied by age for males and females is different, that is, women experience more career interruptions than males
• occupation*age and occupation*age squared – the return to work experience may be different for different occupations
• occupation*region – industry and labour market structures that impact on wages differ between regions
In the model controlling for organisation size^1 we also include as explanatory variables:
• organisation size, categorised into 6 bands by number of workers: less than or equal to 10, 11 to 25, 26 to 50, 51 to 250, 251 to 500, 501 and over
• organisation size*occupation – the effect of organisation size on pay may differ for different occupation classifications
The Analysis of factors affecting earnings using the Annual Survey of Hours and Earnings linear regression dataset gives the coefficients generated by the linear regression model and provides
metadata for the variable names and model specifications. The statistical notes in this section explain how to interpret these results.
Differences in private and public sector pay
Figure 7: Average difference in mean hourly pay (excluding overtime) between public and private sector workers expressed as a percentage of private pay, April 2002 to April 2016
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. No data is available for Northern Ireland for 2002 to 2003, estimates for these years refer to Great Britain.
2. Data for 2016 is provisional.
Download this image Figure 7: Average difference in mean hourly pay (excluding overtime) between public and private sector workers expressed as a percentage of private pay, April 2002 to April 2016
.png (50.6 kB) .xls (31.2 kB)
Comparing sectors and controlling for the variables set out earlier in this section, Figure 7 shows the average difference and 95% confidence intervals in hourly pay between public and private sector
workers expressed as a percentage of private sector pay between April 2002 and 2016 (see statistical note 5 for its calculation method). A positive number indicates a pay gap in favour of the public
sector and a negative number indicates a pay gap in favour of the private sector.
The model results, controlling for the factors listed in section A, provide the opposite result to that described in section A (where it was found hourly earnings in the public sector were greater
than in the private sector). The model results excluding organisation size show 2016 is the first year since 2003 in which the public sector earns less than the private sector; by 1%. There has been
a steady reduction in the pay premium for the public sector since 2010, the premium moving in favour of the private sector in 2016.
The model controlling for organisation size shows that public sector employees earn on average 5.5% less than employees in the private sector in 2016. This gap has widened from 2015 when it was 3.4%.
In this model, the pay differential in favour of the private sector has been increasing since 2010 (increasingly negative public sector coefficients).
It is therefore implied in both cases – with and without organisation size included in the regression model – that growth in mean hourly earnings for employees in the private sector has grown faster
(all other factors being equal) than in the public sector between 2015 and 2016. This may be a reflection of restrictions on public sector pay increases for 2015 to 2016 announced in Budget 2013 and
generally faster growth in pay in the private sector during this time. For example, between April 2015 and April 2016, pay growth was 3.3% in the private sector compared with 1.8% for the public
sector (Average Weekly Earnings, EARN02 dataset).
Quantile regression results
The Analysis of factors affecting earnings using the Annual Survey of Hours and Earnings quantile regression dataset gives the coefficients generated by the quantile regression models and provides
metadata for the variable names and model specifications. The statistical notes in this section explain how to interpret these results.
The linear regression model considers the difference in the mean pay of public and private sector workers. This does not take account of the fact that the distribution of pay tends to be narrower in
the public sector than the private sector and so does not give a complete picture.
It is possible to use quantile regression methods to estimate the difference in the median pay of public and private sector workers as well as the difference at each percentile, for example, the 5th
or 10th percentile. This is useful as it indicates if the pay gap is different at different points of the pay distribution, effects which cannot be captured by mean regressions.
It should be noted that estimates across different quantiles of the income distribution compare the average hourly pay for a certain distribution of the public sector workforce to the average pay for
a certain distribution of the private sector workforce. For example, if we observe a positive public sector premium at the lower end of the distribution, this does not necessarily imply that if an
individual in this part of the income distribution working in the public sector was to move to the private sector, they would earn a lower hourly pay. Instead it implies that individuals in the lower
end of the public sector income distribution – conditional on observed characteristics – earn an hourly premium compared with the individuals in the lower end of the private sector income
The pay gap between private and public sector workers has been estimated for the 2nd, 5th and 10th percentiles, the median and the 90th, 95th and 98th percentile for 2012 to 2016, using the quantile
regression model both excluding and including organisation size.
Table 3 shows the difference in hourly pay between public and private sector workers expressed as a percentage of private pay by percentile for April 2012 to 2016, where a positive number indicates
the pay gap is in favour of the public sector.
Table 3: Quantile and mean regression results for differences in public sector pay as a percentage of private sector pay, 2012 to 2016
Year 2^nd 5^th 10^th 50^th 90^th 95^th 98^th Mean
Without organizational size
2012 12.9 13.1 12.4 6.3 -3.3 -5.9 -9.9 3.1
2013 12.3 12.6 11.9 4.7 -3.2 -6.4 -10.6 2.3
2014 10.7 11.6 11.0 2.6 -4.8 -7.4 -10.4 0.4
2015 13.1 12.9 11.4 1.1 -9.9 -11.2 -13.9 1
2016 10.9 10.8 10.0 0.4 -10.9 -13.2 -17.2 -1
With organizational size
2012 9.2 9.5 9.0 2.5 -7.2 -9.9 -13.9 -1
2013 8.0 8.4 8.3 1.1 -7.4 -10.9 -15.3 -2.1
2014 6.6 7.5 7.2 -0.7 -9.2 -11.7 -14.2 -3.8
2015 8.1 7.7 7.1 -3.4 -14.1 -16.0 -19.0 -3.4
2016 6.8 6.8 5.8 -4.3 -15.4 -17.7 -21.5 -5.5
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 is provisional
Download this table Table 3: Quantile and mean regression results for differences in public sector pay as a percentage of private sector pay, 2012 to 2016
.xls (28.2 kB)
In 2016, both the model controlling for and not controlling for organisation size indicate that the pay gap is wider at the top end of the distribution than the bottom. There have also been changes
at the 50th percentile (median) where the size of the gap has been decreasing.
For instance at the 5th percentile, public sector workers earned 6.8% (10.8% in the model without organisation size) more on average than private sector workers in 2016. At the other end of the pay
distribution, at the 95th percentile, public sector workers earned 17.7% (13.2% in the model without organisation size) less on average than private sector workers in 2016. From 2012, the pay gap on
average has been narrowing at the lower end of the distribution and increasing at the higher end.
The pay gap in favour of the public sector at the bottom of the distribution may be due to the private sector having more jobs paid close to the National Minimum Wage than the public sector. In
comparison, the pay gap in favour of the private sector at the top of the distribution can be explained by the fact that the public sector, in general, does not have the very high wages at the top of
the wage distribution as seen in the private sector.
Analysis of other factors
In addition to sector, the linear regression model in 2016 found almost all other variables within the model were statistically significant at the 1% level.
Working pattern
One of the largest impacts on earnings, with all other factors being equal, comes from an employee’s job tenure – how long an employee has been in their current job. Figure 8 compares the pay
differential of the job tenure groups with those who have been in their job for 6 months or less and suggests that the longer a person is in a job, the more they earn, allowing for all other
individual and job-related factors such as gender, occupation and region.
At the same time the model suggests this relationship holds with each subsequent group earning more than the previous group, up to those in the same job for 20 years or more earning 22.3% more than
those newly started in a job.
Figure 8: Average difference in mean hourly pay (excluding overtime) by job tenure group, aged 16 to 64, 2016
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 is provisional.
2. This percentage is the difference in pay compared with the reference group. .
3. The reference group is those that have been employed for less than 6 months.
4. Results are significant at a 1% level.
Download this chart Figure 8: Average difference in mean hourly pay (excluding overtime) by job tenure group, aged 16 to 64, 2016
Image .csv .xls
Controlling for all other factors, including public and private sectors, in the model which excludes organisation size, part-time workers in 2016 earned 8.8% less than full-time workers when
controlling for other factors and temporary workers earned 0.7% less than permanent workers. In the model which includes organisation size in 2016, the figures are 6.5% less for part-time workers and
2% less for temporary workers respectively.
Organisation size
The model including organisation size observes the effect of how large a firm someone is employed in, on their earnings. In 2016, relative to employees in organisations with more than 500 employees,
those employed in all breakdowns of smaller firms earn relatively less on average. Specifically, employees in organisations with between 11 and 55 employees earn between 8 to 9% less than
organisations with more than 500 employees. For larger organisations (with between 50 and 499 employees) the earnings penalty is 4 to 6% on average. The pay differentials are not as regressive as
other variables observed, with the impact being much more uniform amongst firms of differing size.
Regional variation
Table 4: Average difference in mean hourly pay (excluding overtime) between London employees and other regions of England and the devolved administrations of the UK
Percentage difference in average hourly pay to London
Region Excluding organisation size Including organisation size
North East -7.04 -5.91
North West -5.68 -4.88
Yorkshire and The Humber -6.18 -5.20
East Midlands -5.09 -4.38
West Midlands -5.40 -4.68
South West -4.34 -3.58
East -5.40 -4.52
South East
Wales -7.22 -6.07
Scotland -2.37
Northern Ireland -10.32 -8.50
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 are provisional
2. All results are significant at the 1% level other than for the South eaast which shows no statistical difference
Download this table Table 4: Average difference in mean hourly pay (excluding overtime) between London employees and other regions of England and the devolved administrations of the UK
.xls (28.2 kB)
Table 4 shows the difference in pay between London and the rest of the UK for the linear regression model including andexcluding organisation size. The model which excludes organisation size
estimates the North East and North West, for example, have average pay around 6 to 7% lower than London in 2016 when controlling for other factors. The equivalent figures for these regions using the
model including organisation size are 5 to 6% lower than London.
The estimates presented in this article are by no means a definitive measure of earnings. In fact the model explains 55% of the difference in earnings between people. A different model containing
additional or alternative independent variables may explain more of the variation. However, it is not possible to include all factors that affect pay, such as employee ability or motivation, as
information on these factors is not available. In addition these are estimates based on a sample such that different samples would give different results.
Notes for Section B: Modelling pay differentials in the public and private sector:
1. The logic for a creating model that includes organisation size and another that does not is as follows. Firstly, if the view is held that public sector employees should earn the same as private
sector employees irrespective of organisation size, then it would be useful to see the results without organisation size included. From a statistical point of view, as over 90% of those working
in the public sector are also in large organisations thus the inclusion of organisation size can lead to issues of collinearity and have an impact on the precision of the estimate of the public
and private sector pay differential. The model with organisation size, however, still remains statistically valid.
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1. The dependent variable is expressed in log form. If the distribution of a variable has a positive skew, taking a natural logarithm of the variable sometimes helps fitting the variable into a
model. Also, when a change in the dependent variable is related with percentage change in an independent variable, or vice versa, the relationship is better modelled by taking the natural log of
either or both of the variables.
2. When accounting for the age of employees in the regression model, we have incorporated a variable for both age and age squared; this is due to the Taylor series approximations. Taylor series
approximations tell us that for many smooth functions, they can be approximated by a polynomial, so including terms like Χ2 or Χ3 let us estimate the coefficients for the approximation for a
known or unknown non-linear function of Χ, or in this case age.
3. As well as the suite of independent variables observed in the model, a number of interaction terms are included. These are added to account for the assumption that some characteristics interact
with one another.
The presence of a significant interaction indicates that the effect of the first independent variable (α1) on the dependent variable (Ω) is different at different values of a second independent
variable (α2). It is tested by adding a term to the model in which the 2 independent variables are multiplied.
Ω = A + β1*α1 + β2*α2 + β3*α1*α2
Adding an interaction term to a model drastically changes the interpretation of all of the coefficients. If there were no interaction term, β1 would be interpreted as the unique effect on the Ω
(in this case pay). But the interaction means that the effect of α1 Ωon is different for different values of α2. So the unique effect of 1 is not limited to β1, but also depends on the values of
β3 and α2.
The unique effect of 1 is represented by everything that is multiplied by α1 in the model:
β1 + β3* α2. β1 is now interpreted as the unique effect of α1 on Ω only when α2= 0.
4. When interpreting the outputs of the model, care needs to be taken with the coefficients of variables. While the independent variables are in their original state, the dependent variable is in
its log-transformed state, therefore, the coefficient (β) for the independent variables do not simply reflect the percentage change but (100*β)% for a one unit increase in the independent
variable, with all other variable in the model held constant.
5. Taking into consideration the statistical notes, the calculations conducted to estimate the pay gap between the public and private sector follow algebraically
In the final model, public or private sector is coded as:
The base category is the private sector and there is a dummy variable that codes for public sector called “_l_banks_rec_1” in the regression output.
The variable “_l_banks_rec_2” measures the unclassified pay premium relative to the private sector. This sometimes gets dropped in the regression estimation. This is important for estimation but
not the focus of the analysis.
Taking model 6 and deriving the formula for the public sector pay premium. Assume other characteristics the same in Δ:
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Survey details
The Annual Survey of Hours and Earnings (ASHE) is based on a 1% sample of employee jobs taken from HM Revenue and Customs Pay-as-you-earn (PAYE) records. Information on earnings and hours is obtained
from employers and treated confidentially. ASHE does not cover the self-employed nor does it cover employees not paid during the reference period.
Quality and Methodology Information
The Annual Survey of Hours and Earnings Quality and Methodology Information document contains important information on:
• the strengths and limitations of the data
• the quality of the output: including the accuracy of the data and how it compares with related data
• uses and users
• how the output was created
Classification of SOC 2010
The Standard Occupational Classification 2010 (SOC2010) separates the labour market into 9 major groups, based on criteria such as the qualifications, skills and experience associated with each job.
These 9 major groups can be combined further into 4 skill groups (levels 1 through 4, where level 1 indicates relatively low skill requirements and level 4 indicates relatively high skill
requirements). Table 4 describes some of the important features of each skill group:
Table 5: SOC 2010 classification of skill groups and share of employees by skill group, April 2016
Skill group Proportion of men, ASHE 2016 Proportion of women, ASHE 2016 Typical occupations
1 (low) 12 11 Labourers (e.g. agriculture, construction), cleaners and basic admin workers
2 (lower-mid) 26 46 Secretaries, carers, hairdressers, cashiers, machine operatives, transport drivers
3 (upper-mid) 31 16 Skilled trade workers, associate professionals and technical occupations
4 (upper) 31 27 Professionals (e.g. teachers, doctors, scientists, engineers, managers, directors)
Source: Annual Survey of Hours and Earnings (ASHE) - Office for National Statistics
1. Data for 2016 is provisional
Download this table Table 5: SOC 2010 classification of skill groups and share of employees by skill group, April 2016
.xls (28.2 kB)
In March 2012, the 2011 ASHE estimates were published on a Standard Occupational Classification (SOC) 2010 basis (they had previously been published on a SOC 2000 basis). Since the SOC forms part of
the methodology by which ASHE data are weighted to produce estimates for the UK, this release marked the start of a new time series and therefore care should be taken when making comparisons with
earlier years.
Similarly, methodological changes in 2004 and 2006 also resulted in discontinuities in the ASHE time series. On 28 February 2014 we published a methodological note explaining the impact of the change
in Standard Occupational Classification on the estimates of public and private sector pay.
The earnings information in ASHE relates to gross pay before tax, National Insurance or other deductions and excludes payments in kind. With the exception of annual earnings, the results are
restricted to earnings relating to the survey pay period and so exclude payments of arrears from another period made during the survey period; any payments due as a result of a pay settlement but not
yet paid at the time of the survey will also be excluded.
Most of the published ASHE analyses (that is, excluding annual earnings) relate to employees on adult rates whose earnings for the survey pay period were not affected by absence. They do not include
the earnings of those who did not work a full week and whose earnings were reduced for other reasons, such as sickness. Also, they do not include the earnings of employees not on adult rates of pay,
most of whom will be young people. More information on the earnings of young people and part-time employees is available in the main survey results. Full-time employees are defined as those who work
more than 30 paid hours per week or those in teaching professions working 25 paid hours or more per week.
Sampling error
ASHE aims to provide high quality statistics on the structure of earnings for various industrial, geographical, occupational and age-related breakdowns. However, the quality of these statistics
varies depending on various sources of error.
Sampling error results from differences between a target population and a sample of that population. Sampling error varies partly according to the sample size for any particular breakdown or
Non-sampling error
ASHE statistics are also subject to non-sampling errors. For example, there are known differences between the coverage of the ASHE sample and the target population (that is, all employee jobs). Jobs
that are not registered on PAYE schemes are not surveyed. These jobs are known to be different to the PAYE population in the sense that they typically have low levels of pay. Consequently, ASHE
estimates of average pay are likely to be biased upwards with respect to the actual average pay of the employee population. Non-response bias may also affect ASHE estimates. This may happen if the
jobs for which respondents do not provide information are different to the jobs for which respondents do provide information. For ASHE, this is likely to be a downward bias on earnings estimates
since non-response is known to affect high-paying occupations more than low-paying occupations.
Further information about the quality of ASHE, including a more detailed discussion of coverage and non-response errors, is available on our archive website.
Re-weighting of the Labour Force Survey
Returned data from ASHE are weighted to UK population totals from the Labour Force Survey (LFS). The LFS itself has recently been reweighted, using revised UK and subnational population estimates
consistent with the 2011 Census and updated population projections. We have found there to be negligible impact of this on the ASHE results. Further information on the LFS reweighting can be found on
our LFS user guidance pages
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Captivate blog
Review the output tutorial published in this post. The two first use cases were about a knowledge check question where the learner needs to create physics formulas, by clicking the terms in the
formula. Those terms are multistate objects, used as buttons. In a previous post we created one shared action to get the 'State number' which is stored in an associated variable for each multistate
object. That same post also explained how to choose the sequence of the states.
The first use case has 3 terms in the formula, the second one has 4 terms. The terms are identical multistate objects, at the start of the exercise they show the same Normal state.
In this article youi'll see how to create the action for the Submit button, and the On Enter action which is needed to reset the slide and thus allow multiple visits to the slides.
Submit actions
Those are advanced action triggered by the Success event of the Submit buttons.
Formula with 3 terms
To the left of the equal sign the term should be 'Force', which is state 3 in the multistate object.
To the right of the equal sign we need 'Pressure' and 'Area', sequence is not important. Those are states 1 and 2. The easiest way is to calculate the sum of the two state numbers, which should be 3.
To avoid confusion, create another user variable v_right to calculate the sum. The condition will be very simple: v_1 (state number to the left of the equal sign) needs to be equal to v_right. To
show the result, another multistate object SS_Feedback was created with these three states:
This feedback shape is on the slide (not hidden) but the Normal state is invisible to the learner. I used the same feedback for the next slide (4 term formula) by timing it for the rest of the
project. It will be hidden after the two formula slides.
The Submit action doesn't need much explanation; I used the Delay command to mimick the default time a feedback is shown, allowing a new attempt:
Formula with 4 terms
Due to the sequence of the states, a correct formula will also have the same sum to the left and to the right of the equal sign. Of course we need another user variable v_left to store the sum to
the left. However, checking the sums will not be sufficient in this case. If the learner chooses exactly the same states on both sides, the sums will also be the same. Example: if only the states for
the 'oval 'are chose on both sides. To exclude that situation, I use another expression with a multiplication of the state numbers. For the result of the multiplication two extra user variables a; re
defined: v_leftM and v_rightM
Example: if only 'oval' states are used sum will be: 5 = 2 + 3. However the multiplication will result in 6 = 2 * 3. Whereas for the drum the sum is 5 = 1 + 4 but the multiplcation will result in 4
= 1 * 4. Look at the Submit action in this case:
Enter actions (reset)
Those are advanced actions triggered by the On Enter event of the slides, to reset the situation.
Formula with 3 terms
The 3 variables storing the state numbers need to be reset to their default value (1: corresponding with the Normal state). That is not necessary for the v_right, because it is always calculated. The
option 'Retain state on revisit slide' is not activated, which means that on revisiting the slide the Normal state is reset automatically. The variable v_states needs to be redefined.
Since the multistate object SS_Feedback is timed for more than one slide, and can be hidden by actions on later slides, it is important to show this object and put it back in the Normal state. The
automatic reset to the Normal state is not happening due to the timing. The Next button needs to be hidden.
Formula with 4 terms
The 4 variables storing the state numbers need to be reset to their default value (1: corresponding with the Normal state). That is not necessary for v_right, v_left, v_leftM and v_rightM because
they will be calculated. The option 'Retain state on revisit slide' is not activated, which means that on revisiting the slide the Normal state is reset automatically. The variable v_states needs to
be redefined.
The resetting for the feedback object is for the same reason as in the other Formula slide.
I used a similar workflow in this small project, which is meant as exercise for the math's class about Multiplication tables.
It is a responsie project, uses the same shared action I have used in the use case I described here. The Submit actions are a little bit more complicated. Your challenge: try to replicate this
project. This is the link:
In this post you can find a Captivate tutorial, illustrating 4 use cases where the state number was tracked and used to find a solution.
This post will explain the setup of one Shared action to track the state number, when a multistate object is clicked to advance to the next state. To explain I will use the first two (physics) use
cases. The first one has 3 identical multistate objects, with 3 states, and two possible correct answers:
The second one has 4 identical multistate objects with 4 states, multiple correct answers are possible:
You can find one of the correct answers in those screenshots. The workflow will be described step-by-step.
Both use cases have:
1. Success event of the multistate objects which are used as buttons. These events will be used to advance to the next state and to fill a variable with the tracking number. The index start with 1,
which means that the Normal state is state 1. Being this the only goal of all the multistate objects in both use cases, the use of a shared action is recommended over duplicated and edited
advanced actions. The shared action will be labeled StateNumberAct.
2. The Success event of the Submit button. The action triggered by this even will have two goals: first to check the answer, second to show the Next button if the answer is correct. For a wrong
answer the learner can have another attempt at changing the states and submitting the new situation. Those actions are advanced actions, labeled SubmitAct_3 and SubmitAct_4. They are similar but
not identical.
3. The Success event of the Next button when it appears which is the Default 'Go to Next Slide'.
4. The On Enter event of the slides which is a Reset action for the slide. These advanced actions are similar but not identical, and they are labeled EnterThreeStates and EnterFourStates.
Each multistate object needs an associated variable to store the tracking number. Since that number will be used only on the same slide, the variables can be reused. They will be reset with the On
Enter actions. The four variables are labeled v_1, v_2, v_3 and v_4 (only needed for the second use case). If you are a fan of my blog, you may remember that I have those variables in a shared action
in my general library to be used as external library. See Creation of Variables.
For the shared action StateNumber Act an extra variable v_states is used to store the number of states in the multistate object. The value of this variable will be defined by the On Enter action of
the slide.
StateNumberAct (shared action)
This action will be triggered by the Success event of each multistate object used as button:
It is a two-decision action. The value of the tracking variable (used as parameter 2) is defined by incrementing, while the command 'Go to Next State' is done on the multistate object. The second
conditional decision checks if the maximum number of states is reached (stored in v_states) and resets the tracking variable to 1 in that case. The first parameter is the multistate object used as
button, which is a compulsory parameter contrary to the variable which is a candidate parameter, needs to be indicated to be a parameter.
Sequence of States
The state numbers will be used to check for a correct answer in combination with mathematical operators. For that reason there is no arbitrary choice:
Use case with 3 states
It has two correct answers, left side of the equation sign needs always to be 'Force'. I choose this to be state 3.
To the right side of the equation sign it can be 'Area x Pressure' or 'Pressure x Area'. Those are in state 1 and 2, sequence less important. Reason: check for correctness will calculate the sum of
the tracking variables to the right of the equation sign, which needs to be 3. That lead to a very simple condition. Try to figure it out, or wait for the next blog post.
Use case with 4 states
This is bit more complicated. Similar to the 3 state case, we can calculate the sum of the state numbers to the left and the right. This leads to a sequence of state numbers as shown in the
screenshot at the start. In all cases that sum will result in '5', because it will be either '2+3' or '1+4', in all correct answers symbolized in this list:
• 2+3 = 1+4
• 3+2 = 1+4
• 2+3 = 4+1
• 3+2 = 4+1
• 1+4 = 2+3
• 4+1 = 2+3
• 1+4 = 3+2
• 4+1 = 2+3
However there is a 'booby trap' here! Because the condition based on the sum will also result in correct when the states on the left and the right side are identical. Example: 2+3 = 2+3 which is not
a correct answer. More is needed than the simple sum condition. Any idea? Find the solution in the next blog post.
Multistate objects did change my workflows in Captivate a lot when they were introduced. Use the tag 'multistate object' or 'states' in my blog to see this proved. However I often miss the
possibility to follow up states. There is no system variable for that purpose. Partially due to some questions on the forums, I set up some use cases to see the usefulness of state tracking. In this
introduction you'll find a Captivate published output of 4 use cases and a short summary of the basic setup I used.
Check the tutorial below. These are the four use cases:
1. Three identical multistate objects are used, and learner needs to reconstruct a physical formula by clicking through the states. Two solutions are correct:
2. Four identical multistate objects, and another formula (related with the previous one). A lot more combinations are possible here, both sides need to result in the same Force.
3. One multistate object, being a Carousel of 10 shapes (fSVG text). Some requests (forum question):
Flag number needs to be shown
Back button only visible when in second state
Forward button only hidden when in last state
Next button on slide may appear only when all states have been visited
4. Slot machine with 3 identical multistate objects having 8 states.
You can reach the rescalable version of this example using this link, or watch the embedded one below.
For the design I used some slides from the Quick Start Project 'Diverse'.
Basics for Setup
Tracking variable
For each multistate object a user variable is defined. It will store the tracking number. I choose to have numbers identifying the state, with Normal being state 1, or in programming language: index
starts with 1. None of the multistate objects in the example project has an 'empty' Normal state. If that were the case, the index could have been 0 which is the common approach for most system
variables. However the system variable cpInfoCurrentSlide also has an index starting with 1.
For the use cases the state commands 'Go to Next State' and/or 'Go to Previous State' are used. So, it is necessary to reset the variable when either the last state or the first state is reached, to
get a correct value in all situations.
Example for the first formula: once the third state is reached, the variable needs to be rest to 1 for the next click.
This is more of a general tip when using advanced/shared actions. It is easier to set up a workflow if you can spread the actions over multiple events.
For the first two use cases (formula): the multistate objects are interactive which leads to the use of a shared action because the check of the answer is done by clicking a button. This is also more
user friendly: control over submitting the answer is given to the learner.
For the Carousel the multistate object is not interactive. Only the success events of the Forward and Back button are used. You'll see in the videos that this makes the action more complicated
because all situations need to be cared for.
The slot machine is using JS for a random number and a While command, and can be triggered by the multistate object which is interactive. I added a Check button, to avoid a very complicated action.
It is also the only example where I provided a Reset button (see previous blog).
On Enter event is used to reset variables and the situation on a slide. This is necessary when you allow to revisit the slides, but also because I tend to reuse variables.
State Choice
In the specific use cases of the formulas, the state tracking variable is used with mathematical operators to check for the correct answer. In that case it is important to choose a correct sequence
for the states.
Example for the first use case: 'Force' is in the third state, because the used check function is to create the sum of the tracking variables for the first and the second state (1+2) which needs to
be equal to 3 (third state).
I will post workflows for the use cases, including (interactive) video in future posts.
This is a short blog post, answering a software simulation (assessment mode) question on the forum. If you are interested, here is the link.
You probably know that in such an assessment scores are attached to one interactive object on each slide: either a click box or a Text Entry Box.It offten happens that the same action can be done by
clicking more than one 'target' on the slide, which will need customizing that slide to add at least one extra click box, which will also be scored. However that will lead to a results slide where
the acquired percentage is not correct. Reason is that all scored objects are included in the total score. You can verify that easily in the Advanced Interaction panel, to be opened from the Project
menu or with the shortcut key F9. Here is an example. Slide 59 has two target click boxes, both are scored.
The OP wants to show the correct percentage to the learner in the project, but for the reporting to the LMS Pass/Fail reporting is sufficient which can be done by basing it on the required pass
□ A while ago this blog post described using the system variables used in the Results slide: Quizzing System Variables. For this blog post the relevant system variables are:
cpQuizInfoTotalQuizPoints: will have the value you see at the top of the Advanced Interaction panel, and is not correct in this case because it includes the scores of the extra target click
cpQuizInfoPointsscored: will be correct because on each quiz slide only one of the targets can be clicked.
cpInfoPercentage: is calculated from both previous variables and will be lower than expected for the learner.
cpQuizInfoPointsPerQuestionSlide: this system variable is not behaving as expected for slides with multiple scored objects. It will NOT show the total score of the slide, but the score of ONE
scored object. I will presume that in case of a slide with two target click boxes, both of them will have the same score. This is also valid if you prefer shape buttons instead of click boxes.
Since there is no need to transfer the percentage to the LMS, it is possible to use user variables. Create these variables:
v_TotalScore: will replace the system variable cpQuizInfoTotalQuizPoints
v_TooMuch: will store the total score of the extra click boxes. I assume that click boxes on the same slide have the same score.
v_Percentage: will store the result of the calculation from v_TotalScore and cpQuizInfoPointsscored
The easiest way would be to count the number of extra targets. If all the click boxes have the same score of 1 point you could calculate the correct percentage with this advanced action:
You can trigger this advanced action with the On Enter event of the custom Score slide. It will allow you to display the correct Percentage.
Situation becomes bit more complicated if not dual targets (click box/shape button) have the same score (of 1 point). In that case you can take advantage of the weird behavior of the quizzing system
variable cpQuizInfoPointsPerQuestionSlide (see above). I used this shared action, triggered by the Success action of the targets, and the 'Last Attempt' action which is used here (Success action on
a background Shape covering the complete slide). I assumed that the Last Attempt action of the targets is set to No Action. That typical workflow was used by the OP, and explained in this blog post.
You see that this shared action has no parameters, you could also create an identical advanced action, if you still fear shared actions. My main reasons for this choice:
• Shared action can have a small description
• Shared action appears in the Library, which make it easier to check the Usage
• This shared action has no parameters, so it takes the same time as the advanced action to be applied to all the necessary events.
• It is much easier to transfer this shared action to another Project.
There is a small change to the action to be triggered On Enter for the Results slide:
On the Results slide you can insert most of the default fields, an overview:
• You scored: the system variable cpQuizInfoPointsscored is correct
• Maximum score: you need to replace the system variable by the user variable v_TotalScore
• Correct questions: will be OK, it is the system variable cpQuizInfoTotalCorrectAnswers
• Total Questions: will not be OK, you could add a supplementary command to the shared action similar to replace the system variable cpQuizInfoTotalQuestionsPerProject by a decremented user
• Accuracy: you need to replace the system variable cpInfoPercentage by the user variable v_Percentage
• Attempts: will be OK, it is the system variable cpQuizInfoAttempts
Coincidence because this blog, answer to another forum question, is an alternative workflow to my most recent blog post about Software Simulation Assessments.
Summary of the request:
• Learner needs to click a specific region, two attempts are allowed.
• On Success the Success message appears and the project advances to the next slide.
• On First failure a Hint message appears and an arrow indicates where to click. Moreover a Retry button appears.
• When the Retry button is clicked the Hint text and arrow should disappear.
• If second attempt is successful, Success message appears and the project advances to the next slide.
• If the second attempt is failed, a Sorry message appears with a Continue button to go to the next slide.
Contrary to the previous article, the slide is created manually (no click boxes are added automatically as in a simulation). For that reason I preferred shape buttons over click boxes. A shape button
can be made invisible by setting Opacity and Stroke to zero.
The same idea is at the origin of the workflow: at the bottom of the Timeline, a shape button (invisible to the learner) covering the full slide is added. Have a look at this screenshot of the
Timeline of the first slide (Flags):
The Success event of this big shape button will trigger the FailureAct action to show the Failure messages.
Sample project
Watch a scalable version using this link. The fixed resolution version is below:
The Timeline screenshot of the first slide (Flags) shows the objects on this slide, from bottom to top:
• The shape button covering the slide SB_Back. This button will trigger an advanced action 'FailureAct' with the Success event!
• Gr_Flags groups the four flag images
• Title which function as instruction
• A multistate shape SS_Failure. The normal state is invisible (same as the shape buttons), it has a state for the two Failure situations. The Hint message is stored in a user variable v_hint.
Reason: this allows to time this message shape for the rest of the project. The workflow is needed in a sequence of similar slides, message can be hidden when no longer needed.
• Shape button SB_Next, also a multistate object. I took out the Rollover and Down states, but it has a custom state 'Second' where the label changed to 'Continue'.
• Shape button SB_Flags, multistate object where the custom state 'Indicator' adds the arrow pointing to the correct location. This shape button is 'invisible' and over the Belgian flag on the
first slide, over Portugal in the second slide (see screenshot belos). One of the advantages of shape buttons over click buttons: they can have custom states, and you can edit the form, needs not
to be a rectangle:
• The Text caption is added to show the state of the variable v_attempt. It is part of the debugging workflow, and I forgot to hide it before publishing.
Two user variables are needed:
1. v_attempt: has a start value of 0 and will be incremented on each failure (which is a click on the button SB_Backx).
2. v_hint: will store the text to appear on the Hint message (first failure). Careful: you will need to increase the length of the inserted variable because 50 will not be sufficient in most cases.
FailureAct (Advanced Action)
This action is triggered by the Success event of the shape button SB_Back. The OP preferred to avoid Shared actions. You can assign this action to that shape button on the first slide, but you'll
need a duplicate advanced action for each slide, because the color marked items need to be changed:
Action triggered by the Success event of the shape button SB_Next. Similar to the previous action, you'll need a duplicate action for each slide. The to be edited items are marked in color.
EnterAct (Shared Action)
It is a very simple shared action, with only one parameter. The text of the Hint is the parameter (literal). The reset of the other items has been done or is automatically happening because the
option 'Retain State on Slide Revisit' is unchecked for the multistate objects which is the default.
The following screenshot shows how the '{P}' button is clicked and opens the Parameters dialog box where you enter the parameter:
For the Success situation I just kept the default Success message. This is the Actions tab for the shape button SB_Flags and similar:
Two messages are allowed, Success 'caption' (is a shape) is activated, and the Last Attempt action is set to 'No action'.
The Advanced Interaction panel shows which actions are attached to which events:
Thanks to a question on the Adobe forum: ‘How to have two different Failure messages for click boxes in a software simulation, assessment mode?’, I created a sample project showing the solution, and
will explain the workflow step-by-step. It has become a pretty long article, be prepared.
At this moment multiple Failure messages (up to 3) depending on the number of the attempt, are only possible for question slides. A click box nor a button have that functionality. Moreover
interactive objects, question slides, Drag&Drop slides have only two events to trigger an action: Success and Last Attempt. This means that you cannot use the first attempt to change a situation at
all. Of course you could add events using JavaScript or InfoSemantics CpExtra widget. I accepted the challenge to try to figure this out only with Captivate’s features.
Example project
Watch this project either with the embedded fixed resolution version below, or by clicking this link for a rescalable one (recommended).
Step-by-step workflow
Slide setup
Have a look at the Timeline of the first slide with a scored click box:
From bottom to top you have
• A shape button, labeled ‘SB_Back2’ (because this was slide 2 in the project). That interactive object covers the complete slide and needs to be at the bottom, behind the original click box. This
means that such a workflow is impossible in a fluid box where stacking is prohibited. It may seem better to create a SVG (used as button) which has a ‘hole’ corresponding with the original click
box. However that would be a lot of work, since that click box will be in another location on all slides. Only the Success event of this shape button will be used, and you can leave the number of
attempts at Infinite or 1, doesn’t matter.
Check the screenshot of the Properties, Actions tab: the Success message which appears when the Back button is clicked will act as First Failure message. You may call this ‘cheating’.
This back button is triggering a shared action labeled ‘BackAct’.
• The shape ‘SS_Hint’ has no role in the workflow, was in this assessment as help.
• CB_Assess is the original click box inserted in the software sim, assessment mode. The OP provided 2 attempts, but that has to be changed to 1 (see further). I deactivated the Failure message,
and kept only the Success message. You see the Properties of this click box to the right. I didn’t use the Failure message, but replaced it by a multistate shape.
The action On Success is to proceed to the next slide with the next action to be done by the learner. Since there is only one attempt, the Last Attempt action is a Failure action. It is an
advanced action ‘FailAct’.
It is impossible to have states for the default feedback messages of interactive objects. You can see in the Timeline that this multistate shape is labeled SS_Failure and that it is displayed for
the rest of the project, always on top. Its visibility is triggered by state changes.
• The Failure message, which is the second message, is labeled ‘SS_Failure’ and you see the two states in this screenshot. The Normal state is completely transparent and has not text. The state
‘Second’ has the failure message, and the multistate object will be hidden until the ‘second’ Failure attempt happens.
FailAct (Advanced action)
This action is triggered by the Last Attempt event of the Click box.
The Failure shape is shown, and will have the Second state in place (see below in the shared action ‘BackAct’. Since it is not a default feedback message, it will not appear automatically nor leave
the slide paused for 3 seconds as will be the case for the Success message. That is the reason for the Delay command, followed by navigation to the next slide.
You see that this advanced action will be the same for all the click boxes, because of the Timing of the SS_Failure message for the rest of the project.
BackAct (Shared action)
This action is triggered by the Success event of the shape button SB_Back2.
Because there are so many instances of that shape button, I largely prefer using a shared action over duplicated edited advanced actions. In the next part (other slides) you’ll see how to avoid
having to enter the parameters on each slide. Here is the screenshot of that action, with the indication of its parameters:
The state of the Failure message will be changed to show the ‘second’ message when the click box is not correctly clicked. To allow such a failure you need to hide the back shape button so that a
click outside of the click box is possible. But since you don’t want the Failure message to appear right away, you need to hide it. As you could see it will be made visible with the Last Attempt
advanced action. Sequence of the actions is very important here: you need to change the state before hiding.
ResetAct (Shared action)
This action will be triggered On Enter for each slide with a scored click box. Since the OP provided multiple attempts on quiz level, I needed also to take into account that the slide will be
revisited and custom objects have to be reset in their original status. Custom objects are the shape Back button (SB_Backx) and the multistate shape used for the second Failure message (SS_Failure).
That explains why I preferred a shared action with these three parameters:
This is the screenshot of the action with parameters filled in for the first slide:
Slides setup (after first slide)
1. Slides with a Click Box
• SS_Failure: since this shape is timed for the rest of the project, you don’t have to do anything on slides with a click box.
• SB_Backx: the shape button covering the slide. I recommend to copy/paste SB_Back2 from the slide with the first click box, having the Shared action ‘FailAct’ applied to it. Take the time to
rename the new copy which will have a generic name starting with SB_Back. I prefer to just change the number to the slide number. It is very important that you send this new instance to the Back,
using either the right-click menu, Arrange or the Align toolbar, Arrange options.
The attached shared action will be edited automatically to use the correct Back button for the slide.
• EnterAct: needs to be applied manually to the slide. There is no workaround, you’ll need to choose the correct parameters. Only the first parameter (SB_Backs) is different, both the failure
message and its state are always the same.
2. Slides with Text Entry Boxes, FMR or Video Demo slides, Static slides, Results slide
The two failure messages were only needed for the click boxes. You’ll need a simple On Enter action to make sure that the Failure shape has been hidden. With static slides I mean the very short
slides which sometimes appear during capture and eventually intermediate explanation slides. You can use the Advanced Interaction panel to help with this setup. Here is an example of this panel for
some slides:
Watch the labeling of the Shape back buttons on slides with Click boxes, and the applied actions to the Back button, the Click box and On Enter.
Slide 7 has a Text Entry box and shows the simple State action to hide the Failure message.
This project had 26 interactive objects, Click boxes and Text Entry Boxes. Each of them has a score of 1pt, which leads to the total score showing at the top. It is that score which is stored in the
quizzing system variable cpQuizInfoTotalProjectPoints/cpQuizInfoTotalQuizPoints and is used for calculating the displayed fields on the Results slide.
My apologies for this rather long blog post, where I tried to explain the full workflow in detail. A simple question, however with not a real simple answer, agreed? Questions are always welcome. Post
them in the comment.
Another post meant as answer to a more complicated question in the Adobe forums. You can find the (long) thread under this link. Title 'Play different audio on slide revisit' is not covering the
final required situations. I will explain the end situation in short terms.
• Project has a menu slide with 4 buttons. Each of the buttons jumps to a chapter of the course. There is narration on this slide, which needs CC and that means it has to be slide audio. That slide
has staggered items synchronizing with the narration, and also several effects applied to objects. The learner can click each button multiple times. Buttons are all active.
• At the end of each chapter the learner returns to the menu slide. On return the playhead should skip immediately towards the end of the slide, when all the objects are in place and narration has
finished. Except...
• When the four buttons have been clicked at least once another audio clip needs to be played. That clip doesn't need CC.
The solution is less easy than I thought, hence this blog post.
Example project
I created a barebone project for testing, using some slides from the Quick Start Project Safety. Have a look, to understand better what I explained about the request.
The course has 7 slides:
• Slide 1: Title (poster image is based on this first slide)
• Slide 2 = menu slide
• Slides 3-6: one slide for each chapter
• Slide 7: End slide
The menu slide should appear in 3 possible states:
1. First visit: default state, play head will start from the first frame and pause at the pausing point, narration (slide audio) and effects will be visible.
2. Later visits, when not all buttons have been clicked at least once: audio shouldn't play and no need to wait for the staggered objects and their effects. The easiest way to realize both is using
the shared action explained in this blog post:
Force first view and/or play audio only on first visit
3. Visit when all buttons have been clicked at least once: situation like in 2 BUT another audio clip needs to play. At the end of that clip something needs to happen. I choose to jump to the end
slide, but this can be replaced easily by another command.
Have a look at the Timeline of the menu slide in the Example project:
You see the staggered objects including an entrance effect. Those are the white numbered shape buttons. The meaning of the top most button (SB_Done) will be explained below but you see that it has
audio attached (object audio).
Step-by-step workflow
Create the user variables shown in this screenshot. The tracking variables (one per chapter) v_1 - v_4 are Booleans, default value =0. The variable v_skip will be populated with the duration of the
narration audio clip in seconds. The variable v_audio gets the duration of the final audio clip. As indicated in its description v_buttons is meant for the sum of the tracking variables, which will
be calculated whenever the menu slide is revisited.
Button SB_Done
Since it proved to be impossible to use Play Audio for the final audio clip in the project, I opted to use Object Audio. That type of audio plays when the object appears, or is made visible. The
shape button which I use is invisible because both Opacity (fill) and Stroke are set to 0. The eye button in its Properties is marked 'Invisible in Output'. The audio clip is added in the Options
tab. The action nor the attempts matter, since the Jump to the last slide will not be done by clicking this invisible button. Have a look at the Style and Options tab in these screenshots:
ReturnAct (shared action)
This action is triggered by the success event of the Return buttons at the end of each chapter, it is pretty simple to understand from the Preview screenshot:
EnterAct (advanced action)
This action is triggered by the On Enter event of the menu slide, takes care of the three possible situations (see Analysis above). Both conditional decisions are mutually exclusive because they are
based on the value of the sum variable v_Buttons.
I could have used JS to calculate the value of v_buttons to avoid the sequence of the 3 Expression commands.
If you want to reuse this action for other projects, I would recommend to convert it in a shared action. Be careful with the two literals (duration of both audio clips). They can be tricky, and it
may be better to replace them by variables getting their value in the Variables dialog box.
Overview: Advanced Interaction
This screenshot shows all events and actions in the example project.
This course offers a detailed (video) workflow how to create and use for Drag&Drop objects.
The video is also available as standalone on YouTube, can be skipped in the interactive Captivate tutorial. The tutorial has also a Table of Content in Overlay, to allow complete free navigation.
The mastered skills are used for two games, which also take advantage of the available InBuilt states for Drag&Drop objects. More detailed workflow for the games will appear as blog posts. I used
Ready-to-go slides from the Quick Start Project 'Business', non-responsive version. Some licensed images from Adobe Stock were used as well.
You can watch the tutorial as scalable HTML output from this link (recommended- or watch the embedded fixed resolution output below:
This is the fifth blog post about default Quizzing slides. Five of the compulsory master slides in any theme are linked with Quizzing. For this article the type of project is important:
non-responsive project, responsive project with Fluid Boxes, Responsive project with Breakpoints. Same master slides are used for Pretest and Knowledge Check slides.
With CP2019 Overlay Knowledge Check slides are available in an Interactive Video. Those slides also use the master slides in the chosen theme. The Quiz slides in a VR project however cannot be
formatted using master slides at this moment. You have to accept the default styling. It is possible to edit the text containers (mostly shapes) but that has to be done on each slide individually. I
sincerely hope that formatting functionality will be extended in next versions.
As for the object styles, for master slides editing depends on the type of project: non-responsive (can be published as scalable HTML output), responsive with Fluid boxes, responsive with
Breakpoints. Between those types master slides are not interchangeable.
Non-responsive project
In the previous posts Terminology and Submit Process I explained the meaning of Embedded Objects on the quizzing master slides as being objects without an individual Timeline. You find them on the
four master slides for Quizzes. They appear as placeholder on inserted Quiz slides. Each of them is using a dedicated Object style from the Quizzing Category (see Object Style Manager). As I wrote
in Preferences it is a bit confusing that for the included Captivate themes, the default feedback messages use shapes as container, not captions which means you cannot change them directly for the
Default Labels dialog box where only caption styles are available.
The background of the quizzing master slides can be set up like any other master slide. Since they have a lot of placeholder objects (mostly shapes) which can be styled individually you will mostly
see either:
• Master Slide Background: this means that the background of the Main Master slide is inherited. Often that will be a solid color, a gradient or eventually a not invasive texture image.
• Custom: if the Main Master slide background is not suited, you need to use this option where you’ll find again a solid color, a gradient or a texture.
However feel free to use an image as background. Or you can override the background on individual quiz slides by inserting an image. Since the embedded objects are always on top of the stack, that
image will automatically be below the quiz objects.
1. Use Guides (see Guides Rule) if you want to edit the layout of the placeholders, if you need a Matching, Hotspot and/or Likert question for your course to have a consistent design with the more
common MCQ,T/F… questions sharing the same master slide.
2. If you have a dark background, you can have issues with Matching slides: the link lines between the two columns are black, not possible to edit that color. Moreover since you probably would want
a light color for text, the dropdown list will be unreadable. Solution is to create a duplicate master slide, where you put a light shape under Answer Area, and change font color to a dark color
(new object style).
3. If you expect to have multiline answers, the Answer area can be too small. It is not possible to edit the individual answer shapes n the master slides (which is a pity), that has to be done on
the quiz slides. Increasing the height of the Answer area will make the workflow easier. Decrease the height of the other placeholders. That is possible for the Question title, the Feedback
messages. Be careful with the Question placeholder: questions can also need more than one line. Eventually you can move the Progress indicator next to the Question title, to free up more space
for the Answer Area.
4. If you want some quiz slides including an image, create another duplicate of the master slide. Decrease the width of the Answer area and insert an image placeholder as shown in this screenshot:
5. In any custom theme I create, I will always edit the results master slide: drag the Retake button over the Review button. It will avoid that the learner is confused when quiz is set up with
multiple attempts, and Review is enabled. Design of Captivate’s quiz means that all attempts are considered to be exhausted if the learner clicks the Review button. Moving that button will not
cause any problems. If there is only one attempt, the Retake button is not appearing, and the Review button is visible immediately. If there is no Review functionality, no problem neither.
If you want to override the design, and allow Review before a Retake, but without showing the correct answers, you could use the workflow described in Review before Retake.
Responsive project – Fluid Boxes
An in-depth exploration of the quizzing master slides in the Fluid Boxes themes is described in this article.
Every embedded object is in its (normal) fluid box, with the exception of the feedback messages sharing a static fluid box. That is necessary to save space, they are stacked which is not possible in
a normal fluid box.
Some of the tips mentioned for non-responsive projects are also valid for this type of project:
1. Use of Guides is strongly recommended. For responsive projects rulers are in %. I know that CP2019 allows to use Position Properties panel for fluid boxes, but I find setting up a grid with
Guides much quicker.
2. For Matching questions using a dark background: fill the fluid box of the Answer area with a lighter (semi-transparent) color, and change the font color to avoid the mentioned problems. Do this
on a duplicate master slide (MCA, T/F…).
3. In the article mentioned above, I described the workflow for long answers in the Answer area. Instead of resizing the placeholders you need to resize the fluid boxes.
4. For the duplicate master slide allowing adding an image, you’ll have create two fluid boxes in the present fluid box for the Answer area. That means taking out first the content by unlocking from
Fluid box, dragging it into the scratch area. Create two horizontal child fluid boxes. Relock the answer area to FB and redrag it into one of the new child fluid boxes. You can insert a
placeholder in the second child fluid box.
5. The easy workflow described for a non-responsive project is not possible. The buttons are in a normal fluid box, stacking is not possible. Here a real tweak is needed. Tweaks and special
situations will be explained in later blog posts.
Responsive project with Breakpoints
I have to confess that the toughest part of this type of projects is getting the quiz slides behave properly on all mobile devices. Although in some cases I prefer this type of responsive project for
content slides because of the real freedom of design allowing to reduce layouts for small screens to the bare minimum by replacing items, quiz slides are a real pain.
It is very important to set up the object styles properly, especially those where fonts are included. Full explanation is to be found in ‘Object styles for Responsive Projects’.
The problem has become bigger, because Captivate no longer has included themes for Breakpoints as was the case until CP9. I still continue to use those themes, especially for the Quizzing master
slides. At least they provide a start for setting up the responsiveness. For the rest a lot of testing (trial and error) is needed.
Most of the tips mentioned for non-responsive projects are fully valid, taken into account that you need to use the Position Properties to make all items responsive. No limitations however as for
Fluid boxes: stacking is allowed, grouping is allowed etc… You can drag the Review button under the Retake button.
This fifth post was supposed to be the last one in the sequence about default quiz slides. However I will add another one explaining partial scores and penalties, after seeing several questions in
the social media. You should now have a pretty complete overview now. Next articles will be about tweaking the default design (submit process, use of feedback images, use of audio…) and special
situations like Branch aware, Pretest setup, Remediation. custom quiz slides, …
I also strongly recommend to get acquainted with the category 'Quizzing' of the system variables.
In previous articles I talked about these topics:
• Terminology of Quizzes, including the different types of quiz slides within Captivate
• Submit Process, including an overview of the Embedded objects
• Attempts and Scores: for normal or random quiz slides
Differences between Knowledge Check slides and Quiz slides are fully explored in this article.
This part has its focus on the Quiz Preferences. some of them have to be set up preferably before inserting any quiz slide, or creating a question pool. The same is the case for editing the design of
the quizzing master slides which will be the topic of the 5th post in the series. Up till now the described content was valid for non-responsive projects, responsive projects with Fluid Boxes and
responsive projects with Breakpoints. That will also be the case for the Preferences. Sorry for this very lengthy article, sometimes bit boring but I didn't find an in-depth explanation anywhere
Preferences to set up BEFORE inserting quiz slides
It is also important to set up the design of quiz slides in the master slides before inserting questions, but in case you want to edit later on, it is possible. However this part of the Preferences
needs to be done imperatively before inserting quiz slides. It concerns the default messages to appear, the labels on buttons etc. Too bad, not everything can be found in the same location in
Preferences. And the amount of items to be edited can be different depending on the type of ‘quiz’ slides. Will try to structure everything as completely as possible for each type.
Default labels
This is the most intuitive location in Quiz Preferences. The number of fields showing up can be different. This is a screenshot of the dialog box for normal quiz slides:
The first part of the dialog box shows the text for all the messages. I changed the text for the Correct and Incorrect Message. That change will NOT show up in the Master slides! That is the reason I
insist on making those changes before inserting quiz slides. Resetting the Master slide cannot be used on existing quiz slides to revert to the changed labels. Since I have to create courses i
multiple languages, I have exported Preferences with those edited labels for each language.
Very confusing IMO is the indication of the Style used for the messages. Look at the Correct message, which tells that the ‘Default Success Caption Style would be used for that message. That is NOT
true in the default setup of Captivate, the default style used will be the Default Success Shape Style. Look at the Properties panel, I had the Success message selected on this slide. Reason is that
in Preferences, Defaults normally the option to use Captions instead of Shapes for SFH is unchecked. However, this also means that you cannot switch to another shape style in the dialog box shown
above. You have to switch to the Object Style Manager if you don’t like the used style for this message. You can find more details about the OSM in previous blog posts.
Second part of the dialog box shows the labels on the buttons of the quiz slides. You may already know that only the old type of buttons can be used (Text, Image or Transparent), not Shape buttons.
For those buttons the label has to be entered in the Properties panel, not on the stage by double-clicking the button as is the case for shape buttons. Change the labels here in the Defaults to avoid
having to do it on each quizzing slide. Change will not be transferred to the quizzing master slide neither. Alternative could be to change labels on all the quizzing master slides. Contrary to the
messages, you can use the dropdown lists for the button styles to change the style if wanted.
The progress indicator is also in the dialog box (was on my pleading for translation reasons). It is a weird object, neither caption nor shape, just text. You can edit the style in the Object Style
manager however, under Quizzing Objects.
Defaults under Settings
Quiz Results messages
Under Quiz Preferences, Settings more default messages can be found. Let us start with the Quiz Result messages, which will appear in the so-called ‘Review Area’ on the score slide:
In that dialog box you can edit the Pass and Failure messages (which do not appear on the Results master slide!) and eventually indicate which fields you want to show (but that can be done later).
The legacy indicator for an email button is not valid anymore since many versions.
Question Review messages
Under settings you also find the button for the Quiz Review messages which only is activated when Review is allowed. Except for the Incomplete message, most of the messages will only be used for
disabled learners. Since quite a lot of versions no text message appears for other learners, only checkmarks.
The Incomplete message will appear in the Review area (same name as on the score slide)? If you don’t need the Accessibility messages and do force the user to answer each question (see later), it is
perfectly possible to make that review area placeholder much smaller on the quizzing master slides. However do not do that for the Review area on the results master slide, it is needed for the Pass/
Fail messages mentioned before.
Submit All
When you activate Submit All functionality, this button gets activated. The label on the Submit All button which replaces the normal Submit button, has been set already under Defaults in Quiz
Preferences? But the popup dialog box that appears when that button is clicked has several defaults as well, which appear in this dialog box (I jsut corected Ok to OK).
You’ll see that contrary to the other Defaults, there is no way to set styles here at all . This default dialog popup box will take on the style of the theme. If you dont like that style, go into the
Object Style Manager, and find the style ‘Runtime Dialog’. It is listed independently from Standard Objects and Quizzing objects, since it can occur for both normal slides and quizzing slides.
All dialog boxes shown in this part are valid for quiz slides, pool questions and KC slides. They are the same for responsive and non-responsive projects. Screenshots were taken in a responsive
project with fluid boxes.
More Preferences
The other preferences can be set up whenever you want although I would personally prefer to activate the Submit All if you want that feature, since it has consequences for the default settings for
that option. Similar for the Review functionality if you need the Incomplete message.
I will not go into details for all the settings here because it would be almost impossible since a lot has to be tested out with the LMS you are using in your courses. If that LMS accepts only
packaging using Reporting (SCO’s) you need to activate Reporting. Only a few LMS’s have a dedicated setting in the LMS dropdown list. Choose the protocol to be used, in many cases that will be SCORM
1.2, AICC is really outdated, and not every LMS supports the more powerful SCORM 2004.
Just a few tips about the marked parts in this screenshot:
Template: is the SCORM template to be used. With the default template the reporting data are sent after each slide, you can choose to send it only at the end if the course proves to be very slow and
many users are taking it on the same network. However that means that when the connection is interrupted, the learner will have to retake the quiz slides.
Slide Views: be careful with using this instead or together with Quiz requirement. I often see problems when clients use a requirement of 100% of slide views. If you need to use that requirement,
set the percentage bit lower at least. Much better is to use at least one scored object at the end slide, that needs to be clicked by the learner and which you can use for a score requirement.
Data to Report: in general most LMS’s prefer percentages, not points. Especially for SCORM 1.2 if the total score is higher than 100points.
Never Send Resume Data: under the Advanced button. That (negative) option is unchecked by default, which means that bookmarking is turned on. On a second session learner can return to the slide
where he left in a previous session. If you check that option, learner will always be returned to the first slide. Setting up the alternative, bookmarking by the TOC (Self-paced learning) makes no
sense when using a LMS.
Other Settings
In a previous article about Embedded objects some of the Settings about activating/deactivating buttons was already indicated Above I already mentioned several options under Quiz Preferences,
Settings. In this new screenshot I marked some other choices:µ
Required: in almost all cases it is recommended to keep the default choice which is Optional. If you prefer one of the other choices it can lead to problems, where learners get stuck. There are
better ways to achieve ‘Answer all’ or ‘Learner must take the quiz’. Set up the navigation in such a way that the learner cannot skip the quiz (do not use totally free navigation with the scrubbing
bar in the playbar, or with the TOC). Once the learner is on the first question, with the default button setup he can only continue when he has answered the question. If you use remediation (go to
content slide) you cannot use the more strict options neither.
Interaction ID Prefix: necessary when using same question pool in more courses to be reported to the same LMS.
Review mode navigation buttons: has to be activated if you allow Review (bit lower). If you forget those buttons, learner will be stuck on the first question slide during Review, unless you provide
another navigation possibility.
Branch aware: may I point to a recent article where I explore in-depth this functionality which makes the quizzing system variables dynamic?
Show Progress: either Relative (Question x of y), or Absolute (Question x) allows to turn on/off the Progress indicator.
Allow Backward movement: it may seem strange that I allow this, whereas the Back button is unchecked on quiz slides (and playbar is hidden? Reason: if you want remediation, where the learner is sent
to a content slide after failing a question, it should be possible to do so if that content slide is before the quiz.
Hide Playbar in Quiz: unless you have “Submit all” activated, you should hide the playbar at least on quizzes (if you have one in the course). Reason: it makes no sense because when going back to a
previous quiz slide or content slide by playbar the learner will not be able to change the answer at all which makes it a frustrating experience.
Pass or Fail
Herre a screenshot of that easy dialog box (easy compared with the more complicated previous ones):
The passing score (cpQuizInfoQuizPassPercent or cpQuizInfoQuizPassPoints) can be set. In some cases it will prove necessary to check if same is used in the Manifest file. Again: prefer percentage to
If you have more than one attempt on Quiz level, you should activate the Retake button. It will appear on the socre slide as mentioned in the previous Quizzing post. In that post I also explained
that the actions on Passing and Failing (after last attempt) only occur after reaching the last frame of the score slide.
This was a pretty boring article, hope you found some interesting documentation.
I didn’t yet talk about Pretest quizzes here, nor about the quiz slides in a VR project or 360 slides, nor about the KC slides in an interactive video. Articles are already pretty long. Will be in a
later post.
Quizzing master slides have been mentioned several times. There is already one article about their setup for Fluid Boxes.
And some of you will be looking forward to possible ‘tweaks’? I have been creating so many cheating workflows, expect more in the future. | {"url":"https://blog.lilybiri.com/?page=2","timestamp":"2024-11-13T10:53:10Z","content_type":"text/html","content_length":"196967","record_id":"<urn:uuid:2c786f74-756b-4175-a35f-1012f9fea841>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00187.warc.gz"} |
Need help in very interesting OA problem - Codeforces
Many of my friends asked this interesting OA problem to me:
Given an array of strings $$$A$$$ of size $$$N$$$ and two integers $$$B$$$ and $$$C$$$.
Let $$$D$$$ be the number of strings of length $$$C$$$ that contains exactly $$$B$$$ of the strings in $$$A$$$ as substring.
Return $$$D$$$ mod $$$10^9+9$$$.
$$$1 \le N \le 6$$$
$$$1 \le |A[i]| \le 50$$$
All the $$$N$$$ strings are distinct
$$$0 \le B \le N$$$
$$$1 \le C \le 50.$$$
Note that if a substring belonging to set $$$A$$$ occurs multiple times in my resulting string, it is counted only once.
My approach:
Let $$$Z$$$ be the size of the alphabet $$$(26)$$$.
Let $$$dp[i][j][k][m]$$$ denote the number of strings satisfying the constraints:
1) It is of length $$$i$$$.
2) The longest suffix present in it which is a proper prefix of some string belonging to set $$$A$$$ is the substring $$$[k...i]$$$ and the string whose proper prefix is the $$$j^{th}$$$ string in
set $$$A$$$, in case of multiple such strings in $$$A$$$ choose the one with longest length. In case no such suffix exists, we can put a fake "empty string" at index $$$0$$$ in set $$$A$$$ (rest of
the strings are numbered from $$$1$$$ to $$$N$$$) and assume that substring is $$$[i+1 , i]$$$.
3) The substrings (belonging to set $$$A$$$) which has already been occurred is denoted by mask $$$m$$$, more formally, if the $$$w^{th}$$$ string in $$$A$$$ has already occurred, then $$$w^{th}$$$
bit of $$$m$$$ is set, otherwise its not.
I'll write transitions via forward style dp, if I'm adding the $$$(i+1)^{th}$$$ character, then, it might "complete" some substrings, by this I mean, some suffix which was a proper prefix of some
string in $$$A$$$ before adding character will now be a complete string belonging to set $$$A$$$. Note that all such strings will be the suffix of that longest suffix.
So, some new bits in mask $$$m$$$ will be set, All this can be calculated, since we already know the longest suffix, in fact lets precalculate, $$$info[i][j][k]$$$ which gives a tuple $$$(bitmask, L,
idx)$$$. If we consider the prefix of length $$$j$$$ of $$$i^{th}$$$ string in set $$$A$$$ and add character $$$k$$$ at the end, $$$w^{th}$$$ bit in bitmask is set iff, entire $$$w^{th}$$$ string in
$$$A$$$ occurs as a substring in that prefix after adding character $$$k$$$, $$$L$$$ denotes the length of the longest suffix in resulting string (after adding character $$$k$$$) that is a proper
prefix of $$$idx^{th}$$$ string in set $$$A$$$, this precomputation can be done naively in $$$O(N*C*Z*N)$$$.
So, after adding $$$(i+1)^{th}$$$ character (denote it by $$$c$$$), new mask is $$$ (m | info[j][i-k+1][c][0])$$$, new length of longest suffix is $$$info[j][i-k+1][c][1]$$$ so, add its contribution
towards state $$$dp[i+1][info[j][i-k+1][c][2]][i+2-info[j][i-k+1][c][1]][(m | info[j][i-k+1][c][0])]$$$.
I think the complexity would be $$$O(C*N*C*2^{N}*Z)$$$, which might pass. However, I think I'm overkilling it and there has to be simpler solution. I'm not even sure whether my solution is correct or
» 14 months ago, # |
» 14 months ago, # |
» 14 months ago, # |
← Rev. 2 → +3
Edit: incorrect
• » 14 months ago, # ^ |
» +3
Geothermal How do you ensure that we count all strings that occur? For example, suppose that you expect the strings to appear in the order 1, 2, 3, 4, 5, 6 and are counting occurrences where four
of the strings appear. How are you ensuring that string 5 never appears before any of the first four strings?
□ » 14 months ago, # ^ |
» +3
I do not, thank you for correcting me.
14 months ago, # |
A simpler solution (in the sense of requiring less original thought after applying a standard algorithm; the underlying idea is probably basically the same) that has the same time
» complexity as yours is to build an Aho-Corasick automaton on the $$$N$$$ input strings. For each node in the automaton, create a bitmask indicating which of the $$$N$$$ input strings could
end in this node.
Then, run a DP where your state is the number of characters used, a mask indicating which of the $$$N$$$ strings have appeared in the string you're building, and the node of the
Aho-Corasick automaton corresponding to the string you've built so far. To transition, iterate over the next letter in the string, transition to the corresponding Aho-Corasick node, and
add any input strings appearing in that node to your bitmask.
There are $$$O(C2^N N |A[i]|)$$$ states and $$$26$$$ transitions per state, giving the same complexity as your solution.
• » 14 months ago, # ^ |
» 0
ShivanshJ Thanks! I don't know Aho Corasick, time for me to learn this!
• » 13 months ago, # ^ |
» 0
bicsi It's literally the same solution though, is it not?
14 months ago, # |
← Rev. 3 → 0
I am little confused in understanding the task.
(Edit: I got it now , so we need to find A_i as substring of assumed Good string S not in other way. I thought we need to find number of substring in A_i or in total A)
yup N <= 6, DP bitmask seems the good intuitive way here. | {"url":"https://mirror.codeforces.com/blog/entry/120928","timestamp":"2024-11-13T22:02:39Z","content_type":"text/html","content_length":"118102","record_id":"<urn:uuid:4abc10a7-a7be-4574-914c-e28cb191ba13>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00570.warc.gz"} |
Executing Arithmetic: Enhancing Large Language Models' Computational Abilities by Emulating Turing Machines
Bibliographic Information:
Lai, J., Xu, J., Yang, Y., Huang, Y., Cao, C., & Xu, J. (2024). Executing Arithmetic: Fine-Tuning Large Language Models as Turing Machines. arXiv preprint arXiv:2410.07896.
Research Objective:
This paper aims to address the limitations of large language models (LLMs) in performing arithmetic operations by proposing a novel framework that enables LLMs to learn and execute computational
logic, similar to Turing machines.
The researchers developed a Composable Arithmetic Execution Framework (CAEF) that consists of two main components: executors and aligners. Executors are responsible for performing step-by-step
computations based on the underlying logic of arithmetic operators, while aligners translate between the original arithmetic expressions and the internal representation used by the executors. The
framework utilizes LoRA adapters to fine-tune a pre-trained LLaMA 3.1-8B model on datasets generated by simulating Turing machine executions for various arithmetic operations.
Key Findings:
CAEF achieved high accuracy across seven common mathematical operations (addition, subtraction, multiplication, division, greater than, less than, and equal to) on the LLaMA 3.1-8B model. The
framework demonstrated superior performance compared to baseline models, including GPT-4o and LLaMA 3.1-8B Instruct, particularly when handling operands with up to 100 digits. The evaluation revealed
that the executor component effectively learned the arithmetic logic, achieving over 99% accuracy in most cases.
Main Conclusions:
The study demonstrates that fine-tuning LLMs to emulate Turing machines can significantly improve their ability to perform arithmetic operations by enabling them to learn and execute computational
logic rather than relying on memorization. The proposed CAEF framework, with its composable nature, offers a promising approach to enhance LLMs' capabilities in handling complex mathematical tasks.
This research contributes to the field of machine learning by addressing a key limitation of LLMs in arithmetic reasoning. The proposed framework has the potential to enhance the applicability of
LLMs in domains requiring accurate and efficient numerical computations.
Limitations and Future Research:
The study acknowledges limitations in handling repeated digit patterns and computational efficiency. Future research could explore methods to improve the robustness of the framework in dealing with
such patterns and optimize the utilization of KV cache to enhance computational speed. Additionally, automating the generation of Turing machine prototypes for different operators could further
streamline the framework's implementation.
How might CAEF be adapted to handle more complex mathematical concepts beyond basic arithmetic, such as algebra or calculus?
Adapting CAEF to handle complex mathematical concepts like algebra or calculus presents exciting challenges and opportunities. Here's a breakdown of potential approaches: 1. Expanding the
Representation: Symbolic Manipulation: CAEF currently uses a digit-based representation. To handle algebra, the representation needs to incorporate variables, coefficients, and operations like
addition, subtraction, multiplication, division, and exponentiation. For calculus, we'd need to represent functions, derivatives, integrals, and limits. Tree Structures: Shifting from linear
sequences to tree structures can represent mathematical expressions more naturally. For instance, an expression like "(2x + 3) * (x - 5)" can be represented as a tree, making it easier for the LLM to
parse and manipulate. 2. Enhancing Executors and Composers: Algebraic Executors: Develop specialized executors for algebraic operations. For example, an executor for solving linear equations could
perform Gaussian elimination step-by-step. Calculus Executors: Create executors for differentiation and integration rules. These executors would need to understand the structure of functions and
apply the appropriate rules. Recursive Composition: Enable more sophisticated composition of executors. For instance, an executor for solving quadratic equations might call upon executors for
addition, multiplication, square root, and conditional logic. 3. Leveraging External Knowledge: Mathematical Libraries: Integrate CAEF with external mathematical libraries or symbolic computation
engines. This would allow the LLM to offload complex computations and focus on high-level reasoning. Formal Verification: Employ formal verification techniques to ensure the correctness of the LLM's
symbolic manipulations, especially in high-stakes applications. Example: Consider solving a simple algebraic equation like "2x + 5 = 11". A CAEF-based approach might involve: Representation:
Represent the equation as a tree structure. Executors: Utilize executors for subtraction (11 - 5), division (6 / 2), and variable isolation. Composer: The composer would orchestrate the executors to
first subtract 5 from both sides, then divide both sides by 2, resulting in the solution x = 3. Challenges: Scalability: Handling complex mathematical expressions efficiently will require careful
design of the representation and executors. Generalization: Training LLMs to generalize across a wide range of mathematical concepts and problem types remains a significant challenge.
Could the reliance on Turing machine emulation limit the framework's ability to discover novel or more efficient computational strategies?
Yes, the reliance on Turing machine emulation could potentially limit CAEF's ability to discover novel or more efficient computational strategies. Here's why: Inherent Bias: By design, CAEF guides
the LLM to learn computational logic through the lens of Turing machines, which operate on a step-by-step, sequential model of computation. This could bias the LLM towards solutions that resemble
traditional algorithms. Exploration Constraints: While Turing machines are theoretically capable of universal computation, the specific implementation within CAEF might constrain the LLM's
exploration of alternative computational pathways. The predefined states, transitions, and commands could limit the LLM's ability to discover strategies that deviate from the provided framework.
Potential for Innovation: However, it's important to note that CAEF's current limitations don't entirely preclude the possibility of discovering novel strategies. Emergent Behavior: LLMs are known to
exhibit emergent behavior, meaning they can sometimes learn to perform tasks in ways that were not explicitly programmed. It's conceivable that with sufficient training data and architectural
modifications, a CAEF-based LLM could develop novel computational shortcuts or optimizations within the confines of the Turing machine framework. Hybrid Approaches: Combining CAEF with other
techniques, such as reinforcement learning or evolutionary algorithms, could enable the exploration of a wider range of computational strategies. These methods could help overcome the limitations of
the Turing machine model and potentially lead to the discovery of more efficient solutions. Example: Imagine a CAEF-based LLM tasked with multiplying large numbers. While it might initially learn to
perform multiplication through repeated addition (as guided by the Turing machine model), with enough data and exploration, it's conceivable that the LLM could discover and internalize more efficient
algorithms like Karatsuba multiplication, even if these algorithms are not explicitly encoded in the framework. Future Directions: To foster innovation, future research could explore: Relaxing
Constraints: Investigate ways to make CAEF's representation and execution model more flexible, allowing the LLM greater freedom to explore alternative computational paths. Rewarding Efficiency:
Incorporate reward signals during training that encourage the LLM to discover and utilize more efficient computational strategies.
What are the ethical implications of developing LLMs with enhanced mathematical abilities, particularly in contexts where such skills could be misused?
Developing LLMs with enhanced mathematical abilities raises significant ethical concerns, especially considering the potential for misuse. Here are some key areas of concern: 1. Bias and
Discrimination: Data Bias: If the training data used to develop these LLMs contains biases related to mathematical aptitude or stereotypes associated with certain demographics, the resulting models
could perpetuate and even amplify these biases. This could lead to unfair or discriminatory outcomes in applications like educational assessment, loan approvals, or job recruitment. Algorithmic
Transparency: The complexity of LLMs makes it challenging to understand how they arrive at their solutions. This lack of transparency could make it difficult to detect and mitigate bias, potentially
leading to unintentional harm. 2. Misinformation and Manipulation: Generating False Evidence: LLMs with advanced mathematical skills could be used to generate fabricated data, statistics, or
scientific findings that appear credible but are ultimately false. This could have serious consequences in areas like public health, climate change, or political discourse. Personalized Persuasion:
By leveraging their mathematical prowess, LLMs could be used to create highly targeted and persuasive arguments or propaganda tailored to exploit an individual's vulnerabilities or biases. 3.
Economic Disparity and Job Displacement: Automating Skilled Labor: As LLMs become more adept at mathematical tasks, they could potentially automate jobs that currently require human expertise, such
as data analysis, financial modeling, or scientific research. This could exacerbate economic inequality and lead to job displacement. Access and Affordability: Access to powerful LLMs with enhanced
mathematical abilities could be concentrated among well-resourced organizations or individuals, creating a digital divide and further disadvantaging those who lack access. 4. Security Risks:
Adversarial Attacks: LLMs are vulnerable to adversarial attacks, where small, carefully crafted changes to the input can lead to significant changes in the output. In the context of mathematics, this
could be exploited to manipulate financial markets, disrupt critical infrastructure, or compromise sensitive data. Autonomous Weapon Systems: While speculative, there's a concern that LLMs with
advanced mathematical abilities could be used to develop more sophisticated and potentially autonomous weapon systems, raising ethical questions about accountability and the potential for unintended
consequences. Mitigating Ethical Risks: Addressing these ethical implications requires a multi-faceted approach: Responsible Data Practices: Ensure that training data is diverse, representative, and
free from harmful biases. Algorithmic Transparency and Explainability: Develop techniques to make LLM decision-making processes more transparent and understandable. Regulation and Oversight:
Establish clear guidelines and regulations for the development and deployment of LLMs with enhanced mathematical abilities. Public Education and Awareness: Promote public understanding of the
capabilities, limitations, and potential risks associated with these technologies. By proactively addressing these ethical concerns, we can work towards harnessing the power of LLMs with enhanced
mathematical abilities for societal benefit while mitigating the risks of misuse. | {"url":"https://linnk.ai/insight/machine-learning/executing-arithmetic-enhancing-large-language-models-computational-abilities-by-emulating-turing-machines-eXj40okO/","timestamp":"2024-11-02T06:15:27Z","content_type":"text/html","content_length":"340834","record_id":"<urn:uuid:6ed4e444-9499-4ef3-a08c-6e394dc63466>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00664.warc.gz"} |
Introduction to Bayesian Data Analysis
The listed learning units belong to the course Introduction to Bayesian Data Analysis. Do you want to access all course content?
Week 0 - Initial Setup
Please install the latest versions of R and RStudio, rstan, brms, and other necessary packages in R. In order to get the most out of this course, please read the textbook chapters 1-4 (the textbook
link is provided below) as the course progresses. Each chapter belongs to the corresponding week in this course.
Week 1 - Introduction
Learn the foundational ideas about random variables and probability distributions. Reading: Chapter 1 of the textbook.
Week 2 - Bayesian data analysis
Understand Bayes' rule, derive the posterior using Bayes' rule; visualize the prior, likelihood, and posterior; distinguish between the prior, likelihood, and posterior; incorporate prior knowledge
into the analysis. Reading: Chapter 2.
Week 3 - Computational Bayesian data analysis
Derive the posterior through sampling; build a simple linear regression model using brms; visualize prior predictive distributions, perform sensitivity analysis and posterior predictive checks.
Reading: Chapter 3.
Week 4 - Bayesian regression and hierarchical models
Perform simple linear regressions using the normal and binomial likelihoods to answer the following research questions: (i) Does attentional load affect pupil size? (ii) Does trial id affect response
times? (iii) Does set size affect recall accuracy? Take a brief look-ahead at linear mixed models. Reading: Chapter 4 and up to section 5.3 of Chapter 5.
I like, I wish
Please provide your feedback on the course. | {"url":"https://open.hpi.de/courses/bayesian-statistics2023/overview","timestamp":"2024-11-03T23:19:37Z","content_type":"text/html","content_length":"65483","record_id":"<urn:uuid:74ee9aa0-1d25-40cc-8697-f50b6324d091>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00811.warc.gz"} |
Blog: Logs for the Monero Research Lab Meeting Held on 2017-11-13
Logs for the Monero Research Lab Meeting Held on 2017-11-13
November 13, 2017
<suraeNoether> Hi everyone! fluffypony, knaccc, luigi1111, anonimal, binaryFate, dEBRUYNE, endogenic, gingeropolous, hyc, JollyMort[m], moneromooo, nioc, smooth, stoffu, vtnerd… anyone else I'm
forgetting to ping?
<hyc> hola
<suraeNoether> 2) I want someone to volunteer to put up meeting logs on either the monero-project github or my personal github (which would end up on monero-project after a few weeks) as well as
maybe throwing up a vCal as sarang suggested. Anyone want to make this their "thing"? :P
<suraeNoether> I can download my logs and upload all the past meetings tomorrow morning after I get home
<hyc> I won't volunteer because I usually have a conflicting meeting in this time slot
<suraeNoether> but I want someone to make it their thing to improve reliability. It's easy for me to de-prioritize stuff like that in favor of other stuff
<suraeNoether> hyc that's fine, we could just try to all remind each other instead of it being one person's responsibility
<suraeNoether> any other announcements/administrivia?
<suraeNoether> oh, I'm going to add a point to the agenda: 4) educational outreach
<sarang> yes
<suraeNoether> enough is moving on that front for us to need to discuss it
<sarang> goog idea
<suraeNoether> bab execution?
<andytoshi> sarang: (sorry to interrupt the meeting) the dot product is a scalar op but you still need to do a multiexp to compute a vector pedersen commitment corresponding to each vector, to check
that the commitments are correct
<sarang> yup, true
<suraeNoether> np andytoshi
<suraeNoether> these aren't super formal. if you can't tell
<sarang> We can chat more afterward about it if you want
<suraeNoether> office hours are probably a better term for it. :P
<suraeNoether> okay, 3) Projects. Sarang is clearly on bulletproofs…
<sarang> Yeah, gonna code up linear and log versions
<silur> sorry I missed this bulletproof stuff
<sarang> we have andytoshi present, which is very helpful
<silur> a quick overview on that line?
<sarang> Proposal in a recent paper about efficient and small range proofs
<silur> sounds nice
<suraeNoether> nice. i put spectre down 2 weeks ago so i can work on the multisig paper and code vetting. thigns are looking good. I'm pretty confident I will be ready to send it off somewhere (the
void) by the end of the month
<sarang> Our current proofs are linear in size with the number of bits in the range
<sarang> Bulletproofs have a linear version but the main contribution is a logarithmic version
<sarang> We're gonna more carefully examine the verification complexity, which would likely be a deciding factor
<suraeNoether> why do they bother with the linear version? simply because they can? (anything worth proving is worth proving twice?)
<sarang> The logarithmic version uses a more complex argument, but you can basically drop it in to replace the linear part
<andytoshi> suraeNoether: the log version is a very clean modification of the linear version, which is simpler to understand
<sarang> it's a good buildup
<andytoshi> to go from linear to log it's literally "in the final step replace these two vectors with a log-sized proof that their dot product is this"
<suraeNoether> oh, nice~!
<suraeNoether> that's elegant
<sarang> It's a cool idea
<hyc> in the final step, magic
<andytoshi> and this also lets them do a MPC for multiparty rangeproofs where everyone passes around full vectors then a coordinator aggregates into one log-sized proof, which results in constant
number of rounds (3 or 4? unsure) vs log-many
<sarang> Yeah andytoshi y'all did an excellent job with very solid applications for aggregation
<andytoshi> thanks. the real magic was the stanford people, esp benedikt
<hyc> So MPC gets us multisig too?
<sarang> I'm super interested to see the efficiency comparisons when we finish them
<andytoshi> hyc: it gets aggregation of outputs which have different owners
<andytoshi> i'm not sure about multisig, that isn't covered
<hyc> ok
<andytoshi> good call, i should investigate this anyway for mimblewimble
<sarang> Yeah, I keep forgetting that it's also logarithmic in # of outputs
<suraeNoether> it's very helpful to have you here, andytoshi. I've been thinking about whether or not the bulletproof approach could apply as a ring signature by storing the signing keys in
polynomial coefficients somehow. do you have any thoughts in that regard?
<andytoshi> suraeNoether: no clue, i've thought about it a bit and i don't see any obvious way to do it
<suraeNoether> ok. on another note, hyc as far as MPC for multisig goes….
<suraeNoether> currently the coordinating signer fashions the range proof without input from threshold members… after all, the range proof isn't really part of "getting permission from our multisig
coalition members to send this transaction."
<silur> there is a method by peikert
<hyc> oh good point surae
<silur> I remember when I constructed the encrypted storage EIP for ethereum
<suraeNoether> i don't see a super good reason to distribute the range proof for the use cases we see in cryptocurrency right now
<silur> there is a "compression" algo by peikert for something like this
<andytoshi> if you have a coordinating signer that knows the opening of the commitments (which is what it sounds like if it can make a rangeproof) then you can keep doing that with bulletproofs
<suraeNoether> silur if you find a link i'd be curious
<andytoshi> suraeNoether: in MW the rangeproof keys are used for authentication, that's all
<andytoshi> but monero doesn't care about that
<suraeNoether> yeah, i'm very ignorant about MW, so I'll change my wording above to "use cases we see in Monero right now" as opposed to cryptocurrency in general
<suraeNoether> so, anyone else working on any research projects? After I'm done with multisig this month, I plan on going back to spectre and attempting some simulations on random networks
<suraeNoether> but that's low priority now that multisig is sooo close.
<suraeNoether> (the multisig paper, obv)
<sarang> I'm splitting time between proofs and outreach
<sarang> (which is #4 on agenda)
<suraeNoether> ok. knaccc, silur, anyone else?
<suraeNoether> well, let's talk about educational outreach then
<silur> still working on my quantum-shuffle
<silur> and possible usecase on PRNG
<suraeNoether> nice
<suraeNoether> i'm very interested in that
<silur> would love to have you as a reviewer :D
<suraeNoether> *on another note, btw, before education…* our H_p has been pointed out to me as …. non-standard a few times. can anyone give me any documentation at all on it? i think we chatted about
this early this week, but since i've been mainly working offline, I haven't been able to participate too much
<knaccc> sarang msged me some EC operation counts for bulletproofs
<suraeNoether> silur sure,just make sure you find at least three other reviewers too haha
<knaccc> i'll be looking at attempting a comparison
<knaccc> which should be fairly easy, and would not require coding
<suraeNoether> sarang, do you still need that done? i think you mentioned you had completed a timing analysis earlier?
<sarang> We know basic operation counts
<knaccc> actually we can pretty much state the answer now
<suraeNoether> oh ok, neat
<knaccc> so it's 5*64 scalarmultbase ops
<knaccc> which is the heaviest part by far, assuming 64 bit range proofs
<knaccc> for bulletproofs this is
<knaccc> i.e. 320 scalarmultbases
<knaccc> vs 128 doublescalarmultbases for existing range proofs
<knaccc> so we need to look up some performance stats to get that comparison
<andytoshi> for verification you can do much better than that
<andytoshi> you can do it in 128
<knaccc> ooh nice
<hyc> and relative cost of doublescalarmultbase vs single?
<knaccc> that's perfact then, bulletproofs win!
<sarang> Yeah I messaged some stuff for the log version knaccc
<suraeNoether> allrighty then… did we ever "make a decision" on sandy2x knaccc?
<andytoshi> oh, maybe 132 or something
<andytoshi> idk per numbers for ed25519. you probably want a multiscalarmultbase
<knaccc> suraeNoether sandy2x is only applicable if we move to curve25519, which we would only do if we had variable base scalarmults, which we don't
<suraeNoether> oh, i think i had something written down someplace incorrectly, then
<knaccc> i'm sure sandy2x can be modified to work on ed25519 though
<suraeNoether> oh, but
<knaccc> i'm looking up those performance stats hyc did now
<suraeNoether> the reason we were looking into sandy2x was to see if the variable base scalarmults from RuffCT could be made faster, right?
<andytoshi> anyway you have 64 G bases, 64 H bases, 6 L bases, 6 R bases, plus the standard bases and you can precompute scalars for every one of them and do one multiexp with 141 bases
<knaccc> suraeNoether yes
<andytoshi> that's the inner product anyway. i haven't worked out the rangeproof, it'll add another small constant
<suraeNoether> nice andytoshi
<knaccc> ed25519 monero 10k timings
<knaccc> ge_double_scalarmult_base_vartime 2.282629
<knaccc> ge_scalarmult 2.462409
<knaccc> ge_scalarmult_base 1.498808
<knaccc> all in seconds
<silur> noice
<suraeNoether> hyc knaccc so i guess what i'm asking is: have we gotten enough information about speeds in sandy2x to figure out whether we can actually use RuffCT in the near future? or has this
officially been shelved for a few hard forks?
<hyc> knaccc, any thoughts?
<knaccc> so 132 scalarmults for bulletproofs has the compute cost of 142 doublescalarmultbases that existing Monero range proofs use
<knaccc> so it's comparable
<sarang> mhmm that's in line with the counts I found as well
<knaccc> suraeNoether ruffct I think I remember as having similar compute costs to existing MLSAGs if we switch from ed25519 to curve25519
<sarang> What should be the next step for bulletproofs then, so as not to duplicate effort?
<knaccc> because ruffct was 2x as heavy, but curve25519 is 0.5x the cost for variable base scalarmults
<knaccc> i.e. ruffcy doesn't help with compute costs, only with storage costs
<knaccc> and would involve a lot of extra complexity to have curve25519 and ed25519 coexisting
<andytoshi> knaccc: sorry, it's 141 not 132. and that's just for the inner product, i think there are a couple more for the full rangeproof
<andytoshi> but the existing rangeproofs have basically no room to do multiexp (or even double-exp), so we expect it to be faster anyway
<andytoshi> on secp we've gotten far enough in the analysis to say "build it and benchmark" because they're pretty damn close
<knaccc> andytoshi ah thanks, ok then so bulletproofs are 19% more compute intensive to verify than Monero range proofs
<suraeNoether> ouch
<andytoshi> if you use a separate scalarmultbase for every one
<suraeNoether> 19% is huge with an exponential space-time tradeoff….
<knaccc> andytoshi i'm not familiar with the terms multiexp / double-exp
<andytoshi> "exp" means "scalarmultbase
<andytoshi> "
<knaccc> ok, i see, as in exponent when written in alternate notation
<andytoshi> "yeah
<knaccc> and multiexp is what?
<andytoshi> knaccc: if you are doing a*G + b*H + c*I + d*J you can compute that way faster than computing the individual terms and adding. in libsecp we explored three algorithms, bos-coster, strauss
and pippinger and found that strauss was fastest for small numbers of points while pippinger was fastest for large numbers. see https://github.com/bitcoin-core/secp256k1/pull/486
<sarang> andytoshi: what's the general scaling on that?
<andytoshi> N/log(N)
<sarang> OK nice
<suraeNoether> andytoshi i didn't know that… it makes sense, though… i need to look into that
<sarang> We do that what, 2n times in the inner product?
<suraeNoether> if that's the case, ruffct might could be sped up
<sarang> Er, 2log(n) times
<andytoshi> suraeNoether: https://github.com/bitcoin-core/secp256k1/pull/486
<knaccc> andytoshi ah interesting, are you familiar with the way that ed25519 makes multiexps faster than if curve25519 is used?
<andytoshi> sarang: you can precomute the scalars and do it just once. the paper does it 2log(n) times but this isn't necessary
<knaccc> i'm wondering if there is something special about in that regard
<andytoshi> knaccc: no, i think ed25519 has some specific speedups but the algos i know are curve-agnostic
<knaccc> sepcial about secp256k1 in that regard
<suraeNoether> andytoshi thanks for the link
<andytoshi> we do use our endomorphism to get a small speedup but otherwise everything we've done is curve-agnostic
<knaccc> because we know having investivated that multiexp is 2x faster (with 2 bases) on ed25519 than on curve25519, which is why Monero uses ed25519
<knaccc> it's helped by the twisted edwards curve for some reason
<andytoshi> yes, we don't get 2x for two points, we get something like a 30% speed improvement… but for thousands of points we were getting like a 7x improvement with some older code
<suraeNoether> knaccc: so verification time for sandy2x + curve25519 + ruffct's 2x-sized keys = (approximately) current implementation + ed25519 + our currently sized keys (per ring member, fo
<knaccc> wow ok thanks. Did you have to write your own optimizations to achieve that, using the published approaches you mentioned? or were libraries available for all that stuff for your curve?
<andytoshi> knaccc: we wrote our own code, the fastest library for our curve is ours :)
<knaccc> suraeNoether c8rrect
<knaccc> andytoshi whoa kudos!
<andytoshi> knaccc: but you can check out bos-coster, it requires a heap implementation but it's easy to understand and implement, and djb has python code you can basically translate into C
<knaccc> lol you're too smart for planet earth. you have a warped sense of "easy" :)
<andytoshi> i did this as a first try, which gave impressive numbers, but not as good as strauss for small # of bases and not as good as pippenger for large # of bases, so we dropped it. i forget the
exact numbers now
<suraeNoether> ok, i had no idea about these multiexp tricks andytoshi is describing… we might could make ruffct doable using those tricks, but i need to read the paper again. (I'm beginning to think
a proper name would be Bootle CT, but does Bootle really need more things named after him? lol)
<suraeNoether> knaccc the solutions are all trivial once you know them. :P heh
<knaccc> heh ok
<suraeNoether> okay, NOW that we've gotten some technicals a little more out of the way… sarang, you want to describe the bangalore thing?
<sarang> OK so maybe a good step forward is some comparative testing with code for bulletproofs, which can include proposed speed improvements as part of the test code?
<sarang> (just to finish this discussion)
<suraeNoether> (yeah, i don't want to rush us through it…)
<suraeNoether> i agree with you sarang
<sarang> So about outreach, if we wanna talk that
<sarang> I got an offer for one of us to do a two-week cryptocurrency course in Bangalore next May
<suraeNoether> and while we are doing that, we should keep an eye on Bootle's short accountable ring sigs (ruffct, RTRS CT, whatever you want to call it) to see if we can use multiexp
<suraeNoether> yes
<suraeNoether> outreach
<suraeNoether> one of us?
<sarang> Still need to discuss getting a second person too
<sarang> For sure the expenses are all paid for one instructor
<sarang> Unsure about a second one, depending on how they hire in India vs. elsewhere
<knaccc> suraeNoether maybe i've misunderstood, but i didn't think andytoshi was saying that the optimizations would apply unless you had fixed bases
<knaccc> and RuffCT does not have fixed bases
<knaccc> it's all variable base scalarmults
<suraeNoether> my impression was that the speed-up came from computing the sum in toto all at once instead of computing each term individually and summing them one at a time
<suraeNoether> andytoshi, want to clarify? :P
<knaccc> andytoshi i.e. when you said "a*G + b*H + c*I + d*J", were you implying that G,H,I,J were fixed bases that were amenable to precomputation?
<andytoshi> ah, no
<suraeNoether> or is it enough to simply have those stored locally?
<andytoshi> it's enough to have them stored locally
<sarang> cool
<knaccc> oh whoa, that's awesome, thanks andytoshi
<andytoshi> strauss does some precomputation on the fly, so if they _are_ fixed bases you can cache this and get even more speedup
<andytoshi> but you don't need it (and neither pippinger nor bos-coster have any precomp)
<sarang> OK good, so we should certainly test this w/ bulletproofs simultaneously
<sarang> Anyway suraeNoether, any more questions/comments about the outreach thing?
<knaccc> andytoshi oh wait, when you said "stored locally", you meant precomputed and stored locally?
<knaccc> i.e. GHIJ all still need to be precomputed? which is costly?
<andytoshi> no
<knaccc> what does stored locally mean then
<andytoshi> idk, i was just copying the phrase from suraeNoether
<andytoshi> i meant literally nothing by it
<sarang> "You know what the points are"?
<andytoshi> yeah, i guess
<sarang> I get it
<knaccc> lol well if you don't know what the points are, how can you do anything at all? i'm so confused :)
<sarang> knaccc the answer would be "doesn't matter, but if it's possible to also precompute, cool"
<sarang> that's all
<knaccc> ok cool
<knaccc> that's great news then
<andytoshi> i don't know :P. anyway we use these for aggregate signatures where the bases are public keys that we definitely can't precompute, and our numbers come from this
<suraeNoether> all i meant was
<suraeNoether> well
<suraeNoether> i'm going to shut my mouth for a bit
<suraeNoether> because i haven't read the bootle paper recently and i don't want to mis-speak
<suraeNoether> sarang for india? no more questions for now….
<suraeNoether> my e-mail to a local university has not yet received a reply, either, but I sent it on a Thursday evening… if I don't receive a contact by middle of this week, i'm going to try to
contact some of the other, more prestigious schools around me…
<sarang> Is there interest in getting two people there to teach?
<sarang> vs just one?
<suraeNoether> uhm
<sarang> If it's just one, there are no expenses for the community or other supporters
<suraeNoether> i wouldn't mind it, i think it would be fun
<suraeNoether> like, i know this is a weird thing to say but
<sarang> With two people, I don't know if they'd cover the second person
<suraeNoether> "wanna go to india and teach for 2 weeks?" "eh, i could take it or leave it. " hehe
<suraeNoether> i think it'd be fun
<suraeNoether> if it's politically costly or something in order to establish funding, i don't really have a dog in that fight
<sarang> Yes, and there are parts of the country (like India) where gifted students don't have great opportunities for this
<sarang> So we'd be doing the STEM world a big favor
<suraeNoether> maybe. india has more honors students than america has students
<sarang> Yes but in the U.S. it's easy to find challenging opportunities if you're gifted
<sarang> in India it is not
<sarang> and being labeled an "honors student" in your own school doesn't mean squat comparatively
<suraeNoether> ehrm
<suraeNoether> anyway, i see what you are saying
<suraeNoether> but also
<suraeNoether> on a non-teaching note
<sarang> My point is that doing this anywhere in the world is a good idea, but I think serving a place like India is a great idea
<suraeNoether> I was chatting with a friend who brought up the following idea, which I have been kicking around in my head for a few days. Monero should take some charge in producing friendly,
educational, currency-agnostic infographics and educational documents for people with varying levels of tech education
<suraeNoether> sure
<sarang> On cryptocurrency in general?
<sarang> Because CCs work in a lot of different ways
<suraeNoether> yep! thing is…
<suraeNoether> there's an analogy to compare against cars. every cryptocurrency (that isn't a shitcoin) seems to be doing something differently with very differetn design philosophies
<suraeNoether> but that's like
<suraeNoether> as if in 1860 the internal combustion engine was invented, attracted oodles of capital, and then every manufacturing company in the world created their own weird design of internal
<sarang> We have some really nice Monero topic videos
<suraeNoether> it's… kind of insane
<sarang> Things like those?
<suraeNoether> we do
<suraeNoether> uhm
<suraeNoether> the sorts of things I was thinking about
<suraeNoether> range from educational infographics and youtube videos that are short and designed for a 13 year old with a limited attention span… all the way up to a computer science student who is
just learning about these things for the first time but has some stuff under their belt…
<suraeNoether> all the way up to educational infographics for politicians and policy makers who are ignorant about tech in general
<sarang> How general though?
<suraeNoether> well, the short videos for younger people (or grand mothers) would be quite general
<sarang> You can lead a horse to crypto primitives, but can't make him put them together in every currency's way
<suraeNoether> hey, look, a saying that people should learn
<suraeNoether> well, actually that idiom is kinda terrible
<suraeNoether> but that's the point
<suraeNoether> but anyway
<suraeNoether> the problem with the idea is that it's so general and not well-specified
<suraeNoether> there needs to be a scope defined ahead of time
<sarang> Yes, I think scope is the biggest hurdle
<suraeNoether> but i like the idea of MRL kind of… providing non-tribalistic, educational descriptions of the cc space, for internet users that aren't necessarily computer scientists
<sarang> Because it gets really complex really fast
<suraeNoether> nod
<suraeNoether> anyway, that's all I have for discussion today
<suraeNoether> I'm a little impressed we actually took over an hour!
<suraeNoether> Does anyone have anything else they want to chat about?
<suraeNoether> okay, well
<suraeNoether> \/meeting
<suraeNoether> good job, everyone. :P
<hyc> more tribalism!!
<hyc> n/m
<suraeNoether> hehe
<suraeNoether> i mean, honestly
<suraeNoether> zcash's trusted setup and dash's masternodes
<suraeNoether> i wouldn't necessarily trust my life savings to a dash masternode
<suraeNoether> i certainly wouldn't trust my double-spend protection to a 6-person ceremony
<sarang> Different tools for different needs, I suppose
<suraeNoether> but these are solutions to cryptocurrency problems that come from different design philosophies with different traits and different security/threat models in mind
<suraeNoether> nod right
<sarang> I was listening to a radio news show that interviewed someone about the nature of Bitcoin
<sarang> He claimed that it was about "trusting the network"
<sarang> which I thought was utter BS
<suraeNoether> heh
<sarang> It's the exact opposite
<knaccc> suraeNoether they've figured out how to make it a 100,000 person ceremony now
<sarang> you trust _nobody_
<suraeNoether> knaccc yeah I saw that
<suraeNoether> well, maybe i need to come up wiht a more finite mission statement for this particular idea
<suraeNoether> anyway
<suraeNoether> okay, everyone
<suraeNoether> good meeting, good meeting, no one brought doughnuts, but hey
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"One loop tests of higher spin AdS/CFT" - Department of Physics and Astronomy
UNC-CH String Theory Seminar
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“One loop tests of higher spin AdS/CFT”
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Rocko Mama’s is considering three potential sites, A, B, and C, in which to build its latest state-of-the-art café. The aim is to locate a site with the minimum cost of production, taking into
account both fixed and variable costs. Each site has a different fixed annualized cost, with Site A costing $10,000,000, Site B costing $20,000,000, and Site C costing $25,000,000. The variable costs
per output produced also vary between the three sites, with Site A having a variable cost of $2,500 per output, Site B having a variable cost of $2,000, and Site C having a variable cost of $1,000.
Rocko Mama’s has estimated that the new facility will serve between 0 and 60,000 clients per year. Given the above information, it is essential to determine which site is optimal for which range of
volume, and whether Site C should be recommended at any point. To determine which site is optimal for a particular volume, we must consider the total cost of production at each site. The total
production cost for each site can be calculated as follows: Total Cost = Fixed Cost + (Variable Cost x Output Produced) For Site A, the total cost can be calculated as follows: Total Cost for Site A
= $10,000,000 + ($2,500 x Output Produced) For Site B, the total cost can be calculated as follows: Total Cost for Site B = $20,000,000 + ($2,000 x Output Produced) For Site C, the total cost can be
calculated as follows: Total Cost for Site C = $25,000,000 + ($1,000 x Output Produced) To determine the volume at which Site C would be recommended, we must compare the total cost of producing
various volumes of output for each site. Assuming that the café serves 60,000 clients per year, we can calculate the total cost of production for each site as follows: Total Cost for Site A =
$145,000,000 Total Cost for Site B = $140,000,000 Total Cost for Site C = $85,000,000 In this case, Site C is the clear winner, with the lowest total cost of production. However, it is essential to
note that Site C is only recommended when the café serves 60,000 clients per year. At lower levels of volume, the fixed cost makes Site C prohibitively expensive. To determine the volume at which
Site A is optimal, we can calculate the total cost of producing various volumes of output for each site. Assuming that the café serves 30,000 clients per year, we can calculate the total cost of
production for each site as follows: Total Cost for Site A = $80,000,000 Total Cost for Site B = $80,000,000 Total Cost for Site C = $55,000,000 In this case, Site A and Site B both have the same
total cost. However, it is essential to note that Site A is the preferred option due to its lower fixed cost. Lastly, to determine the range of volume over which Site B is optimal, we can calculate
the total cost of producing various volumes of output for each site. Assuming that the café serves between 0 and 60,000 clients per year, we can determine the range of volume at which Site B is
optimal using the graph below. [insert graph here] From the graph, it appears that Site B is the optimal choice for volumes between approximately 10,000 and 45,000 clients per year. While Site A and
Site C may be more cost-efficient at lower and higher levels of volume, respectively, Site B is the most economical for the majority of the range of volume being considered. In conclusion, when
considering where to build its latest state-of-the-art café, Rocko Mama’s should carefully evaluate the fixed and variable costs of each potential site and determine which option offers the lowest
total cost of production. Site C is only recommended when the café serves 60,000 clients per year, while Site A is optimal at lower levels of volume due to its lower fixed cost. Site B is the optimal
choice for most of the range of volume being considered. By taking these factors into account, Rocko Mama’s can ensure that it is making an informed decision when selecting the location for its new
by gpt-3.5-turbo
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Follow Class Ace for product announcements and AI tips & tricks: | {"url":"https://www.classace.io/answers/write-an-essay-about-22-of-the-three-sites-being-considered-by-rocko-mamas-a-b-and-c-at-which-to-build-its-latest-state-of-the-art-caf-the-goal-is-to-locate-at-a-minimum-cost-site-where-cost-is-measur","timestamp":"2024-11-05T23:32:05Z","content_type":"text/html","content_length":"74823","record_id":"<urn:uuid:ef7be5b7-5f4c-4de9-9fc4-4d9e45d70f0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00071.warc.gz"} |
2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)
Cite as
Joshua Ani, Michael Coulombe, Erik D. Demaine, Yevhenii Diomidov, Timothy Gomez, Dylan Hendrickson, and Jayson Lynch. Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets,
and Counter Machines. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 5:1-5:21, Schloss
Dagstuhl – Leibniz-Zentrum für Informatik (2023)
Copy BibTex To Clipboard
author = {Ani, Joshua and Coulombe, Michael and Demaine, Erik D. and Diomidov, Yevhenii and Gomez, Timothy and Hendrickson, Dylan and Lynch, Jayson},
title = {{Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets, and Counter Machines}},
booktitle = {2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)},
pages = {5:1--5:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-275-4},
ISSN = {1868-8969},
year = {2023},
volume = {257},
editor = {Doty, David and Spirakis, Paul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023.5},
URN = {urn:nbn:de:0030-drops-179414},
doi = {10.4230/LIPIcs.SAND.2023.5},
annote = {Keywords: Gadgets, robots, undecidability, Petri nets} | {"url":"https://drops.dagstuhl.de/entities/volume/LIPIcs-volume-257","timestamp":"2024-11-07T15:29:52Z","content_type":"text/html","content_length":"227919","record_id":"<urn:uuid:88b82d76-7f2c-47d0-9c7c-ff1c6cec3b9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00805.warc.gz"} |
How do you differentiate ycosx^2-y^2=xy? | HIX Tutor
How do you differentiate #ycosx^2-y^2=xy#?
Answer 1
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y \left(2 x \sin \left({x}^{2} + 1\right)\right)}{\cos \left({x}^{2}\right) - x - 2 y}$.
Firstly, let's apply the derivative operator to both sides, I'll try my best to colour code the use of the product rule:
#d/dx [color(red)(ycos(x^2)) color(teal)(- y^2)] = d/dx[color(purple)(x)color(orange)y]#
#color(red)(dy/dx * cos(x^2) + y * (-sin(x^2) * 2x)) color(teal)(- 2y * dy/dx) = color(purple)(1) * color(orange)(y) + color(purple)(x) * color(orange)(dy/dx)#
#dy/dx * cos(x^2) - 2y * dy/dx - x * dy/dx = 2xy * sin(x^2) + y #
#dy/dx (cos(x^2) - 2y - x) = 2xy * sin(x^2) + y#
#dy/dx = (2xy * sin(x^2) + y)/(cos(x^2) - 2y - x)#
#dy/dx = (y(2xsin(x^2 + 1)))/(cos(x^2) - x - 2y)#.
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Answer 2
To differentiate the equation ( y \cos(x^2) - y^2 = xy ) with respect to ( x ), you'll use the implicit differentiation technique.
First, differentiate each term of the equation with respect to ( x ), then solve for ( \frac{{dy}}{{dx}} ). The steps are as follows:
1. Differentiate ( y \cos(x^2) ) using the product rule.
2. Differentiate ( -y^2 ) with respect to ( x ).
3. Differentiate ( xy ) using the product rule.
4. Combine the results and solve for ( \frac{{dy}}{{dx}} ).
Applying these steps will give you the derivative of ( y ) with respect to ( x ), ( \frac{{dy}}{{dx}} ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Trade Margin Calculator - Calculate Trade Margin Easily Online
Trade margin
Did you know ? The rules that mathematical logic derives from the research of thinking are analogous to the rules of mathematical, especially algebraic, operations. The task of mathematical logic is
actually the same as the task of traditional logic, namely to develop procedures by means of which we could verify the correctness of our logical conclusions, or in other words, by means of which we
could distinguish valid conclusions from invalid ones. | {"url":"https://multimathcalculator.com/percentage-calculations/trade-margin","timestamp":"2024-11-15T02:23:10Z","content_type":"text/html","content_length":"21249","record_id":"<urn:uuid:e7d75cac-a677-431e-9ac4-4cb69ebfe664>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00014.warc.gz"} |
Ottoneu Auctioneering Series Part II: On Using PoTP to Assign Player Values
By: Brad Williamson
Topic: Ottoneu, Auction drafting, PoTP
In part one of this series we looked at different types of auctions, which types of auctions Ottoneu utilizes, how to prepare for an auction draft, and how to behave while the draft is happening. But
none of this matters if you don’t have a system in place to assign valuations to the players you are interested in. Our sytem, PoTP, or Percentage of Total Points, assigns these valuations. Here’s
how it works:
Points Leagues
Ottoneu actually offers four different scoring options; each falls into one of two types: rotisserie or points-based. Though this article will focus on Ottoneu, it is important to remember that PoTP
allows us to generalize and customize results for any other rotisserie, points-based, or head-to-head league, not only Ottoneu. Let’s begin with points-based leagues. Here is the scoring template for
Fangraphs Points leagues, one of the four scoring options available in Ottoneu:
Pretty self-explanatory, but what isn’t clear is how this alters players’ values. Before getting into our system of putting prices to players, I think it’s important to point out a few key things
this scoring format does. It immediately devalues steals, runs, and RBIs, and essentially equalizes all bullpen arms. A dynamite set-up man is now almost the same as a good closer, while a speedster
with no other skills is likely not worth putting on your team. Now, how should you properly assign values to your players? It’s actually rather simple, but requires diligence.
Assigning Precise Values and Understanding Draft Capital
Capital is succinctly defined here: “wealth in the form of money or other assets owned by a person or organization or available or contributed for a particular purpose such as starting a company or
Succeeding in Ottoneu requires a change of paradigm. You must consider your team an organization that is investing in assets. You must think like a GM, which is the entire premise of the game. Still,
many owners underestimate what this entails—don’t be that owner. By assigning values to a player you are declaring how much capital you believe that player to be worth. If you assign too little, you
will be outbid, but if you assign too much, you will run out of money. The key to Ottoneu is not loading up on stars; the key is to build a complete team with both production and value that exceed
capital at as many positions as possible. This is how you do that:
First, pick a player you’re interested in. I will pick Jose Ramirez for this exercise because I’d like to illustrate the wide gap between valuations. This is much different than typical drafts where
the top-15 or -20 players will have steady ADPs. Here, each player’s value depends directly on how much you think they’re worth in relation to your budget instead of how much value they’re worth in
relation to the round you’re picking in, a subtle but extraordinarily important point to understand. In a non-auction draft, the draft itself essentially limits your capital, each subsequent round
being worth slightly less than the round before it, with every owner given the opportunity to take the player of their choice without penalty during each round. Round one might be worth an average of
$44, round two $39, round three $34—the exact values make no difference in traditional drafts. But the key that you must understand is that this traditional type of draft automatically limits the
capital you have to spend. You don’t have to worry about managing that aspect of your team; in Ottoneu you must manually manage this capital. If you spend $44 on three different players you might get
three first-round picks, but this increases the chance that the rest of your team will be worthless. I cannot stress the importance of understanding this subtle but absolutely critical difference in
draft formats.
Next, make your best prediction of what you expect your player’s—in this case Ramirez—season to look like. Here’s mine for Jose:
625 Abs = -625
185 hits = 1,036
36 doubles = 104.4
6 triples = 34.2
34 homers = 319.6
90 BBs = 270
2 HBP = 6
32 SBs = 60.8
6 CS = –16.8
This gives us 1,189.2 points. Now decide how many points you’re trying to earn in your league. Let’s shoot high and say we want to assure victory with 20,000 points. Therefore I am predicting that
Ramirez will score 5.95% of my points. I talked in part one of this series about creating hard-cap values for players that you promise yourself not to exceed when bidding. At first glance it makes
sense to simply allocate 5.95% of my budget to Jose’s hard cap, but this doesn’t work. And here we’ve hit upon another one of the mainstays of Ottoneu: salary cap management.
If 20 players each earn 5% of your points and you draft them at 5% of your salary cap, then you will fill only half your roster, not to mention this would be illegal. But if you draft 40 players who
you can afford but only earn 2% of your points, you won’t field a good team. You have to find value–excess capital in other words–in every player you select, which is why this preparation is so
important. Even if Ramirez has a 50/50 season (which he won’t), buying him at auction for $80 is a bad purchase.
To ensure you’re getting good players at good values, find their percentage of total points (PoTP), calculate how much of your budget that amounts to, and set that player’s hard cap to 75% of this.
In our example Jose’s 5.95 PoTP comes to $23.8 of our $400 budget. We’ll be generous with our hard cap just this once and call it $24. Some simple math tells us that 75% of this is $18. You will not
under any circumstance bid more than $18 for Jose, except when using…
Your Two-Hitter-Two-Pitcher and Big Exceptions
So your hard cap on Jose is $18, but you really think he’s going to have a huge year and you don’t believe $18 will get him, even with aggressive, smart bidding. Once you’ve assigned values to all
the players you’re interested in, go back through the list and pick just a few of your favorites. On draft-day, depending how the auction goes, you can bid 100% of the PoTP (but this is truly a hard
cap, no more than a $24 bid for Jose, even if he’s one of your exceptions) on four players–two hitters and two pitchers. You don’t have to do this, but you can, and you cannot do more than four, or
more than two of each. Stick to the plan because it will work.
In addition to the two-hitter-two-pitcher exception, you can go big on two additional players. Think of these as your first- and second-round picks. Using this system won’t get you many of the
top-tier players, but it will give you a balanced and productive team that will compete for years to come; sometimes, however, it’s important to get that ace or top hitter despite the price. Pick one
batter and one pitcher you need and you may bid 150% of their PoTP on each of them. You could therefore bid up to $36 for Jose. In no circumstance should you win with bids of 100% or more of a pl
ayer’s PoTP more than these six times.
This ensures you are able to get several stars while managing your entire team and promises that you’ll have money to spend on the waiver wire later in the year. If you do declare an emergency and
spend more than the PoTP allots, then you must subtract the amount you go over from another player’s cost. Otherwise your entire budget will be skewed. Therefore if you are able to get a player
beneath their pre-exception PoTP price, then you can spend that saved money elsewhere, using it even on top of a big exception should you deem such a purchase responsible.
Rotisserie Leagues
Assigning your PoTP for rotisserie will work very similarly; you only need to make a couple small adjustments. Here are the 5X5 stats used in Ottoneu leagues:
The inclusion of average, ERA, and WHIP make this calculation slightly more difficult, but it’s still possible to assign proper values. Let’s assume a 15-team league, so with 10 categories there is a
maximum of 150 points. Whereas 20,000 points was our goal in the points league, let’s set our goal here at 100 points. To calculate how much production you need to reach this goal, simply aim for the
80th percentile at each position’s statistics and then calculate a comprehensive total. Reaching such a goal will give you a realistic chance to win any league. For 2019, among hitters with at least
200 ABs, qualified pitchers, and relievers, the 80th percentile values were as follows:
Assuming all your positions log approximately equal ABs, which they should if your team is balanced, then we can just add these totals to arrive at a team goal. We will also assume you’ve rostered
five starters, three good relievers, and one closer, though these might fluctuate. You’ll have to adjust this aspect of your calculations if you plan on using a different amount of pitchers. The
saves category is the main problem here, but you can assume you have at least one good closer who will provide ~35 saves in addition to the charted values. Here are your calculated team goals using
the 80th percentile marker from the 2019 season:
Calculating Precise Batter Values in Rotisserie
Let’s stick with Jose:
AVG = .296 = 10 points above team’s AVG goal
HR = 34/208 = 16.35% of team’s home run goal
RBI = 100/624 = 16.03% of team’s RBI goal
SB = 32/80 = 40% of team’s SB goal
R = 100/644 = 15.53% of team’s R goal
Since all five categories are equally valuable you can weight them equally. The PoTP system does this very easily by finding the player’s comprehensive percentage (CP) of the five stats; in this
scenario the CP comes to 17.58% [(HR%+RBI%+SB%+R%)/500]. Then, for every point above or below the AVG goal add or subtract a tenth of a percentage to adjust the calculation. We are predicting 10
points above goal, so we add 1% and get 18.58 PoTP. This comes to a whopping $74.32, or $74-$75. The only difference here with the PoTP system between Points leagues and rotisserie is that you will
set your hard cap to 50% of this value, so in this case we have Jose hard-capped at $37.
There is a huge gap between his value in points leagues ($18) and rotisserie leagues ($37), but this can easily be explained. We stated earlier that the points league devalues steals, runs, and RBIs,
but these stats are important in this rotisserie format. Different players will carry league-dependent capital, but don’t be wary of these disparities. Trust the system.
You should also be cautious when using your exceptions in rotisserie, because some players will be very expensive. When bidding on your six hard-cap-exempt players, use 75% of their PoTP price as
your new hard cap. In this example, if you want to use Jose as one of your two batters, his hard cap would increase from $37 to $56, or 75% of his $74 value. For your big exception, bid 100% of the
player’s value instead of the 150% that you used in the Fangraphs points league. In this example, Jose’s absolute maximum price will be $74 if you decide to use your big batter exception on him.
That is a huge bid. Although the system allows this bid, I suggest setting a limit that you’re willing to spend on any single player. I suggest not spending more than $50 on any one player, but you
can choose a different limit or even risk not having one, but remember that you need to field a full team and save some of your budget for the waiver wire.
Calculating Precise Pitcher Values in Rotisserie
Pitchers’ calculations will look very similar, but let’s do one for illustrative purposes. Let’s use Julio Teherán for this exercise. And remember, these are my rankings. You need to make your own.
Every owner will value every player differently, so don’t trust their ADPs, don’t trust what others are saying, make your own calculations!
My prediction for Julio’s 2020 campaign:
W = 14
SV = 0
ERA = 3.60
WHIP = 1.22
K = 220
And his percentages:
W = 14/95 = 14.74% of team’s W goal
SV = 0/59 = 0% of team’s SV goal
ERA = 3.60 > 3.15 = .45 above team’s ERA goal
WHIP = 1.22 > 1.06 = .16 above team’s WHIP goal
K = 220/1,503 = 14.64% of team’s K goal
Once again they are all weighted equally, so let’s find Julio’s CP [(W%+SV%+K%)/500]: 5.88%. But we still need to adjust for ERA and WHIP. For every hundredth point of ERA and WHIP in either
direction, adjust a hundredth of a percentage. For Julio this means we need to adjust by -.61 percentage points, giving us a 5.27 PoTP. This comes to $21.08, let’s call it $22 because he’s so
durable. Cut it in half and we get a hard cap of $11 on Julio. Remember, if you think he’ll earn more wins or log a higher ERA, your hard cap will be different from mine. That’s okay.
Some Warnings for Rotisserie Players Regarding Fluctuating PoTP Values
Your weighting system will only be accurate if you don’t exceed each individual category. Remember to balance your team. Once you have a group of players who reach your team’s goal in one category,
each subsequent intended starting player who adds to that category will only count for 50% in further weightings for that category. For example, let’s say you want a pitching staff of Verlander,
Cole, Strasburg, Scherzer, and Castillo. Nevermind that you would have no money to spend on any other player, you would be wasting stats. Here’s why:
Imagine you’re in the middle of your draft. You’ve selected Verlander, Cole, Stras, and Scherzer, who together hypothetically reach 95 wins, achieving your team goal. Then, when you go to look up
Castillo’s PoTP, you see he now earns only 50% credit for wins despite a predicted 15 victories. Once you’ve drafted players whose allocations reach your team goals, further additions of these stats
are superfluous and will be handicapped. If you counted all these stats, your team could easily end up wildly unbalanced.
Castillo’s value pre-draft would include his wins, of course, but this is something you must be extremely cognizant of during the actual draft. Despite a specific player’s theoretical PoTP, their
actual PoTP can fluctuate if you’ve already drafted similar players earlier in the draft. This can be confusing to track, especially if the draft is a fast draft, but you must try. Blundering this
aspect of the draft is what oftentimes results in highly unbalanced teams, and in Ottoneu more than in any other format, it’s difficult to rebalance your team mid-season.
If, however, the intended starting players you have drafted meet all your team goals, then you can once again begin fully crediting subsequent stats into their PoTP values, as these will be surplus
statistics. But this is only in the case that you have filled all your goals with other intended starting players.
Therefore, returning to the previous example, say you’ve selected Verlander, Cole, Stras, Scherzer, and several closers, and have met all three counting stats. If by some miracle Castillo is still on
the board, you can now select him at his pre-draft PoTP value, assuming you have the funds; his wins can now be factored into his value because you’ve met all goals and balanced your team. But if you
select him before you have your closers he will carry inherently less value since you still need saves while his wins are unnecessary.
While it is true that the value of statistical categories will fluctuate more than simply back-and-forth from full- to half-value, this is merely a system meant to assist you in balancing your team
during the draft. I do not claim that Castillo’s wins are actually worth exactly half of what they were worth in the given scenario, but assuming this helps you draft wisely and carefully.
Planning Your Draft Strategy Around PoTP
Of course you cannot simply target all the best players. That would be too expensive. If you only research the elite players, then you will not have a good draft. You need to have a well-rounded
pre-draft ranking of players. With your budget of $400 you need to field a 40-man roster with 22 starters and 18 reserves or prospects. This is why you can afford to have the 4-player exception and
two big exceptions, because at least 10 of your players are going to be low-cost minor leaguers, inexpensive relievers, or platoon bats. For players valued at $7 or less before their hard cap, feel
free to bid at 100% of their PoTP.
Not everyone is trying to fill an entire roster. Still, in dynasty leagues where you might only need a few players, the PoTP system is easily applicable. Instead of finding your PoTP values with a
$400 budget, simply use your available salary cap minus the amount you need for in-season waiver wire moves as your total percentage. And when setting goals for the picks you need to make, all you
must do is subtract the expected stats you already have on your team from your season’s goal, and what you have left over is what your new picks need to deliver. This works smoothly whether you’re in
a points or rotisserie league as long as you are thorough with your research and calculations.
It also helps if you plug players into categories, or tiers. You don’t have to do this, but it can certainly help you budget, though theoretically, if you follow the PoTP plan accordingly, you won’t
have budgeting concerns. The only issue becomes if you’re unable to get any of your players at the PoTP values you’ve calculated. To prepare for this eventuality, I recommend mock drafting if
possible, but even if mocks are unavailable to you, the following solutions can still help if you’re in multiple drafts and botch the first one. At least you can salvage the remaining drafts.
In this case you have two options: find new players you didn’t catch the first time around, because there are always undervalued players to be had, or re-examine the players you’ve already priced and
see if you can alter their hard caps through new analysis. But whatever you do, tempting as it might be, don’t ditch the PoTP strategy; it won’t work if there are leaks in your team because you’ve
illogically fudged numbers and statistics.
It’s easy to think you won’t be able to get enough good players, but this just isn’t true. You might not get everyone you want, but the people who do get these expensive players will regret it very
soon. By following the PoTP framework you will ensure your team is competitive this year while remaining fluid in its ability to make transactions both throughout the year and into next offseason and
How to Value Prospects and Bench Players
This is tough because you might predict their PoTP to be zero due to not getting a call this year, or maybe they’re valued at $2 because they won’t get much playing time, but their value is in the
future and in slow game days. If there are prospects you love and have unbiased reasons for loving, feel free to bid up to $20, or about the value of one good starter, of your budget on a select
group of players solely for your team’s future. The money spent here on nonproduction shouldn’t hinder you because you should have already met your primary goals with your intended starting players,
while these players will help ensure you have a healthy team into the future. Similarly, assuming you’ve filled your needs, feel free to obtain economical bats or pitching reclamation efforts, but do
so with PoTP in mind; simply because they only cost $3 according to your prediction is no reason to assume paying $4 is okay, and neither should you try to pay only $2.
By this point you should have the PoTP system mastered for your drafts, but there is one other key aspect of Ottoneu, and fantasy baseball in general, where PoTP will assist you. Read part three of
this series to find out what that is. | {"url":"https://www.centerfieldofgravity.com/2020/01/22/ottoneu-auctioneering-series-part-ii-on-using-potp-to-assign-player-values/","timestamp":"2024-11-12T19:10:14Z","content_type":"text/html","content_length":"242794","record_id":"<urn:uuid:71a53142-88a5-4dfd-9210-d5128a830b8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00019.warc.gz"} |
Degrees of Freedom - Kinetic Theory of Gases | Physics
The minimum number of independent coordinates needed to specify the position and configuration of a thermo-dynamical system in space is called the degree of freedom of the system.
1. A free particle moving along x-axis needs only one coordinate to specify it completely. So its degree of freedom is one.
2. Similarly a particle moving over a plane has two degrees of freedom.
3. A particle moving in space has three degrees of freedom.
Suppose if we have N number of gas molecules in the container, then the total number of degrees of freedom is f = 3N.
But, if the system has q number of constraints (restrictions in motion) then the degrees of freedom decreases and it is equal to f = 3N-q where N is the number of particles.
1. Monoatomic molecule
A monoatomic molecule by virtue of its nature has only three translational degrees of freedom.
Therefore f = 3
Example: Helium, Neon, Argon
2. Diatomic molecule
There are two cases.
1. At Normal temperature
A molecule of a diatomic gas consists of two atoms bound to each other by a force of attraction. Physically the molecule can be regarded as a system of two point masses fixed at the ends of a
massless elastic spring.
The center of mass lies in the center of the diatomic molecule. So, the motion of the center of mass requires three translational degrees of freedom (figure 9.5 a). In addition, the diatomic molecule
can rotate about three mutually perpendicular axes (figure 9.5 b).
But the moment of inertia about its own axis of rotation is negligible (about y axis in the figure 9.5). Therefore, it has only two rotational degrees of freedom (one rotation is about Z axis and
another rotation is about Y axis). Therefore totally there are five degrees of freedom.
f = 5
2. At High Temperature
At a very high temperature such as 5000 K, the diatomic molecules possess additional two degrees of freedom due to vibrational motion[one due to kinetic energy of vibration and the other is due to
potential energy] (Figure 9.5c). So totally there are seven degrees of freedom.
f = 7
Examples: Hydrogen, Nitrogen, Oxygen.
3. Triatomic molecules
There are two cases.
Linear triatomic molecule
In this type, two atoms lie on either side of the central atom as shown in the Figure 9.6
Linear triatomic molecule has three translational degrees of freedom. It has two rotational degrees of freedom because it is similar to diatomic molecule except there is an additional atom at the
center. At normal temperature, linear triatomic molecule will have five degrees of freedom. At high temperature it has two additional vibrational degrees of freedom.
So a linear triatomic molecule has seven degrees of freedom.
Example: Carbon dioxide.
Non-linear triatomic molecule
In this case, the three atoms lie at the vertices of a triangle as shown in the Figure 9.7
It has three translational degrees of freedom and three rotational degrees of freedom about three mutually orthogonal axes. The total degrees of freedom, f = 6
Example: Water, Sulphurdioxide. | {"url":"https://www.brainkart.com/article/Degrees-of-Freedom_36284/","timestamp":"2024-11-08T13:56:58Z","content_type":"text/html","content_length":"49293","record_id":"<urn:uuid:55312558-d4a5-4a89-989a-f8105d9cd18f>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00215.warc.gz"} |
dispersion.m - A MatLab script for phase angle and amplitude correction of pressure bar signals
dispersionFactors_PR29.mat (9.84 kB)
dispersion.m - A MatLab script for phase angle and amplitude correction of pressure bar signals
(A Python version of this script is also available)
In processing the signals from split Hopkinson pressure bar (SHPB) experiments it is often assumed that longitudinal stress waves in the pressure bars propagate one-dimensionally at a common velocity
c_0, and so measurements taken at the strain gauges are often simply translated to the end of the bar using a suitable time delay. In reality, stress waves propagate at a specific phase velocity,
c_p, which is a function of frequency and the bar's diameter, one-dimensional wave speed and Poisson's ratio. Phase velocity decreases as the frequency of a wave increases, leading to dispersion of a
signal as it propagates down the bar. Dispersion of the stress pulse is accompanied by a frequency-dependent variation in stress and strain across the bar cross-section, so that a signal recorded on
the surface of the bar at some distance from the specimen will not accurately describe the stresses the specimen was subjected to, and hence cannot be used to accurately determine the specimen
This script uses an implementation of the dispersion-correction method described by Tyas and Pope (2005) to ensure that the inferred measurements of axial stress and strain accurately represent the
specimen behaviour. In this method:
1. The time-domain strain signal is converted into the frequency domain using the fast Fourier transform (FFT)
2. A correction is applied to the phase angle of each frequency component to account for the dispersion over the distance between the strain gauge and the bar end, using Bancroft's equation.
3. A correction is applied to the amplitude of each frequency component using the factors M_1 and M_2, which account for the variation of strain and Young's modulus across the bar cross section,
respectively. These are derived from Davies' analysis of the radial effects in a cylindrical pressure bar.
4. The signal is transformed back into the time domain using the inverse FFT.
Dispersion.m uses a pre-calculated, normalised look-up table of phase velocity, M1 and M2 to improve calculation time. A lookup table for a Poisson's ratio of 0.29 has been provided in the .zip file,
and other tables can be constructed using the relationships defined in Tyas and Pope (2005) with the accompanying phasevelocity.m script.
Further information on the operation of the script is also available in Barr (2016), linked below.
• There is no personal data or any that requires ethical approval
• The data complies with the institution and funders' policies on access and sharing
Sharing and access restrictions
• The uploaded data can be shared openly
Data description
• The file formats are open or commonly used
Methodology, headings and units
• There is a file including methodology, headings and units, such as a readme.txt | {"url":"https://orda.shef.ac.uk/articles/software/dispersion_m_-_A_MatLab_script_for_phase_angle_and_amplitude_correction_of_pressure_bar_signals/3996876","timestamp":"2024-11-14T10:35:54Z","content_type":"text/html","content_length":"176969","record_id":"<urn:uuid:ed81cab9-3d63-4ffb-ac53-aab18e4ce191>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00309.warc.gz"} |
A(n) 72.6 kg astronaut becomes separated from the shuttle, while on a space walk. She finds herself 48.1 m away from the shuttle and moving - DocumenTVA(n) 72.6 kg astronaut becomes separated from the shuttle, while on a space walk. She finds herself 48.1 m away from the shuttle and moving
A(n) 72.6 kg astronaut becomes separated from the shuttle, while on a space walk. She finds herself 48.1 m away from the shuttle and moving
A(n) 72.6 kg astronaut becomes separated from the shuttle, while on a space walk. She finds herself 48.1 m away from the shuttle and moving with zero speed relative to the shuttle. She has a(n) 0.704
kg camera in her hand and decides to get back to the shuttle by throwing the camera at a speed of 12 m/s in the direction away from the shuttle.
(a) How long will it take for her to reach the shuttle?
in progress 0
Physics 3 years 2021-07-20T04:45:39+00:00 2021-07-20T04:45:39+00:00 1 Answers 26 views 0 | {"url":"https://documen.tv/question/a-n-72-6-kg-astronaut-becomes-separated-from-the-shuttle-while-on-a-space-walk-she-finds-herself-15714244-11/","timestamp":"2024-11-10T17:52:54Z","content_type":"text/html","content_length":"82468","record_id":"<urn:uuid:ec47c313-e54f-4f26-87ee-1657ed7f0a3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00379.warc.gz"} |
Home Assignment 1
Problem 4 was to find the general solution to $uu_{xy} = u_xu_y$. I used the hint and got $\frac{\partial}{\partial y} \ln u_x = \frac{\partial}{\partial y} \ln u$, but I'm not sure where to go from
Junjing, you are right but I am not sure if anyone but me would be able to read your solution | {"url":"https://forum.math.toronto.edu/index.php?topic=1667.0","timestamp":"2024-11-03T07:11:18Z","content_type":"application/xhtml+xml","content_length":"36035","record_id":"<urn:uuid:f1363c26-63c8-405d-a072-bbf7958c3b3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00468.warc.gz"} |
adding and subtracting fractions with like denominators worksheet
solving multiple variable polynomial equation
prealgebra solving equations by multiplying and dividing worksheet
how to solve multivariable algebra equation with fractions
worksheet on adding,subtracting,multiplying,dividing integers
Thank you for visiting our site! You landed on this page because you entered a search term similar to this:
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│ Math Objective │ To show students how least common multiples are achieved using two and three numbers. │
│ Skills/Outcomes │ • Using a formula with a constant │
│ │ • Filling a formula across a row │
│ Files to Use │ lcm.doc │
│ │ │
│ Download Info/Instructions │ lcm.xls │
│ Optional Resources │ calculators │
ClassroomDiscussion and Activities
Computer Lab Activities
ClassroomDiscussion and Activities
│ Teacher Note: This activity should not replace traditional paper and pencil methods for finding common multiples, especially for the numbers frequently found in denominators of fractions. │
When the purpose of teaching common multiples is to find the leastcommon multiple for fractions, the numbers used should stay withinthe range of fractions encountered in normal life. Students
shouldthen be working with denominators of 2, 3, 4, 5, 6, 8, 16, 12, and10.
It may be helpful to have students organize their searches forcommon multiples on a grid such as the one below. The top number isthe factor used to multiply the numbers in the problems. An exampleof
this kind of student worksheet is available in lcm.doc.
Find the least common multiple for 6 and 15.
This grid will provide a natural transition to the spreadsheetactivity. You can use calculators to have students complete exerciseslike the one above. Just multiply row and column numbers (e.g., 6
x2, 6 x 3) and fill in the grid. The next step is to look back overthe series and find the common number. In this case, it would be 30.This practice provides a foundation for common multiples.
To prepare the computer activity, give students a blankspreadsheet grid. These can be found in grid wksht. Have themwrite the formula in cells B2 and B3 as shown below, using thenumbers in the
problem (in this case, 6 and 15) as the constants.Tell them that in the lab, they will use the FILL RIGHT command tocopy the formula over as many places as needed. You may want studentsto write this
command in the cells to the right of the formulas sothat they remember.
ComputerLab Activities
│ Teacher Note: Here's a formatting tip. It is helpful to use colors and borders to help keep the different multiples separate from the factors. We have set up some sample exercises like this in │
│ lcm.xls. │ | {"url":"https://softmath.com/tutorials2/adding-and-subtracting-fractions-with-like-denominators-worksheet.html","timestamp":"2024-11-08T21:11:53Z","content_type":"text/html","content_length":"34528","record_id":"<urn:uuid:6c544948-6896-40f7-b8bf-eb6ac0263d2f>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00579.warc.gz"} |
57 research outputs found
We use density functional theory to calculate the electronic band structures, cohesive energies, phonon dispersions, and optical absorption spectra of two-dimensional In$_2$X$_2$ crystals, where X is
S, Se, or Te. We identify two crystalline phases (alpha and beta) of monolayers of hexagonal In$_2$X$_2$, and show that they are characterized by different sets of Raman-active phonon modes. We find
that these materials are indirect-band-gap semiconductors with a sombrero-shaped dispersion of holes near the valence-band edge. The latter feature results in a Lifshitz transition (a change in the
Fermi-surface topology of hole-doped In$_2$X$_2$) at hole concentrations $n_{\rm S}=6.86\times 10^{13}$ cm$^{-2}$, $n_{\rm Se}=6.20\times 10^{13}$ cm$^{-2}$, and $n_{\rm Te}=2.86\times 10^{13}$ cm$^
{-2}$ for X=S, Se, and Te, respectively, for alpha-In$_2$X$_2$ and $n_{\rm S}=8.32\times 10^{13}$ cm$^{-2}$, $n_{\rm Se}=6.00\times 10^{13}$ cm$^{-2}$, and $n_{\rm Te}=8.14\times 10^{13}$ cm$^{-2}$
for beta-In$_2$X$_2$.Comment: 9 pages. arXiv admin note: text overlap with arXiv:1302.606
We report calculations of the electronic structure of silicene and the stability of its weakly buckled honeycomb lattice in an external electric field oriented perpendicular to the monolayer of Si
atoms. We find that the electric field produces a tunable band gap in the Dirac-type electronic spectrum, the gap being suppressed by a factor of about eight by the high polarizability of the system.
At low electric fields, the interplay between this tunable band gap, which is specific to electrons on a honeycomb lattice, and the Kane-Mele spin-orbit coupling induces a transition from a
topological to a band insulator, whereas at much higher electric fields silicene becomes a semimetal
We use density functional theory to calculate the electronic band structures, cohesive energies, phonon dispersions, and optical absorption spectra of two-dimensional In2X2 crystals, where X is S,
Se, or Te. We identify two crystalline phases (alpha and beta) of monolayers of hexagonal In2X2, and show that they are characterized by different sets of Raman-active phonon modes. We find that
these materials are indirect-band-gap semiconductors with a sombrero-shaped dispersion of holes near the valence-band edge. The latter feature results in a Lifshitz transition (a change in the
Fermi-surface topology of hole-doped In2X2) at hole concentrations n(S) = 6.86 x 10(13) cm(-2), n(Se) = 6.20 x 10(13) cm(-2), and n(Te) = 2.86 x 10(13) cm(-2) for X= S, Se, and Te, respectively, for
alpha-In2X2 and n(S) = 8.32 x 10(13) cm(-2), n(Se) = 6.00 x 10(13) cm(-2), and n(Te) = 8.14 x 10(13) cm(-2) for beta-In2X2
Recently nanomechanical devices composed of a long stationary inner carbon nanotube and a shorter, slowly-rotating outer tube have been fabricated. In this Letter, we study the possibility of using
such devices as adiabatic quantum pumps. Using the Brouwer formula, we employ a Green's function technique to determine the pumped charge from one end of the inner tube to the other, driven by the
rotation of a chiral outer nanotube. We show that there is virtually no pumping if the chiral angle of the two nanotubes is the same, but for optimal chiralities the pumped charge can be a
significant fraction of a theoretical upper bound.Comment: 4 pages, 5 figure
We report on the radiative recombination of photo-excited carriers bound at native donors and acceptors in exfoliated nanoflakes of nominally undoped rhombohedral gamma-polytype InSe. The binding
energies of these states are found to increase with the decrease in flake thickness, L. We model their dependence on L using a two-dimensional hydrogenic model for impurities and show that they are
strongly sensitive to the position of the impurities within the nanolayer. (c) 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0
Unported License
Recent transport measurements [Churchill \textit{et al.} Nat. Phys. \textbf{5}, 321 (2009)] found a surprisingly large, 2-3 orders of magnitude larger than usual $^{13}$C hyperfine coupling (HFC) in
$^{13}$C enriched single-wall carbon nanotubes (SWCNTs). We formulate the theory of the nuclear relaxation time in the framework of the Tomonaga-Luttinger liquid theory to enable the determination of
the HFC from recent data by Ihara \textit{et al.} [Ihara \textit{et al.} EPL \textbf{90}, 17004 (2010)]. Though we find that $1/T_1$ is orders of magnitude enhanced with respect to a Fermi-liquid
behavior, the HFC has its usual, small value. Then, we reexamine the theoretical description used to extract the HFC from transport experiments and show that similar features could be obtained with
HFC-independent system parameters.Comment: 5 pages plus 2 supplementary material
This is the author accepted manuscript. The final version is available on open access from the Royal Society of Chemistry via the DOI in this recordIII-VI post-transition metal chalcogenides (InSe
and GaSe) are a new class of layered semiconductors, which feature a strong variation of size and type of their band gaps as a function of number of layers (N). Here, we investigate exfoliated layers
of InSe and GaSe ranging from bulk crystals down to monolayer, encapsulated in hexagonal boron nitride, using Raman spectroscopy. We present the N-dependence of both intralayer vibrations within each
atomic layer, as well as of the interlayer shear and layer breathing modes. A linear chain model can be used to describe the evolution of the peak positions as a function of N, consistent with first
principles calculationsNational Science Centre, PolandEngineering and Physical Sciences Research Council (EPSRC)Scientific and Technological Research Council of Turkey (TUBITAK)Royal SocietySamsung
Advanced Institute of Technology (SAIT)European Research Council (ERC | {"url":"https://core.ac.uk/search/?q=authors%3A(Zolyomi%2C%20V.)","timestamp":"2024-11-07T18:54:37Z","content_type":"text/html","content_length":"163031","record_id":"<urn:uuid:74bab314-ca94-4c9a-a977-faf78e05ce52>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00752.warc.gz"} |
559 Centimeter/Hour Squared to Inch/Minute/Second
Centimeter/Hour Squared [cm/h2] Output
559 centimeter/hour squared in meter/second squared is equal to 4.3132716049383e-7
559 centimeter/hour squared in attometer/second squared is equal to 431327160493.83
559 centimeter/hour squared in centimeter/second squared is equal to 0.000043132716049383
559 centimeter/hour squared in decimeter/second squared is equal to 0.0000043132716049383
559 centimeter/hour squared in dekameter/second squared is equal to 4.3132716049383e-8
559 centimeter/hour squared in femtometer/second squared is equal to 431327160.49
559 centimeter/hour squared in hectometer/second squared is equal to 4.3132716049383e-9
559 centimeter/hour squared in kilometer/second squared is equal to 4.3132716049383e-10
559 centimeter/hour squared in micrometer/second squared is equal to 0.43132716049383
559 centimeter/hour squared in millimeter/second squared is equal to 0.00043132716049383
559 centimeter/hour squared in nanometer/second squared is equal to 431.33
559 centimeter/hour squared in picometer/second squared is equal to 431327.16
559 centimeter/hour squared in meter/hour squared is equal to 5.59
559 centimeter/hour squared in millimeter/hour squared is equal to 5590
559 centimeter/hour squared in kilometer/hour squared is equal to 0.00559
559 centimeter/hour squared in meter/minute squared is equal to 0.0015527777777778
559 centimeter/hour squared in millimeter/minute squared is equal to 1.55
559 centimeter/hour squared in centimeter/minute squared is equal to 0.15527777777778
559 centimeter/hour squared in kilometer/minute squared is equal to 0.0000015527777777778
559 centimeter/hour squared in kilometer/hour/second is equal to 0.0000015527777777778
559 centimeter/hour squared in inch/hour/minute is equal to 3.67
559 centimeter/hour squared in inch/hour/second is equal to 0.061132983377078
559 centimeter/hour squared in inch/minute/second is equal to 0.0010188830562846
559 centimeter/hour squared in inch/hour squared is equal to 220.08
559 centimeter/hour squared in inch/minute squared is equal to 0.061132983377078
559 centimeter/hour squared in inch/second squared is equal to 0.000016981384271411
559 centimeter/hour squared in feet/hour/minute is equal to 0.30566491688539
559 centimeter/hour squared in feet/hour/second is equal to 0.0050944152814232
559 centimeter/hour squared in feet/minute/second is equal to 0.000084906921357053
559 centimeter/hour squared in feet/hour squared is equal to 18.34
559 centimeter/hour squared in feet/minute squared is equal to 0.0050944152814232
559 centimeter/hour squared in feet/second squared is equal to 0.0000014151153559509
559 centimeter/hour squared in knot/hour is equal to 0.0030183585430556
559 centimeter/hour squared in knot/minute is equal to 0.000050305975717593
559 centimeter/hour squared in knot/second is equal to 8.3843292862654e-7
559 centimeter/hour squared in knot/millisecond is equal to 8.3843292862654e-10
559 centimeter/hour squared in mile/hour/minute is equal to 0.000057891082743445
559 centimeter/hour squared in mile/hour/second is equal to 9.6485137905742e-7
559 centimeter/hour squared in mile/hour squared is equal to 0.0034734649646067
559 centimeter/hour squared in mile/minute squared is equal to 9.6485137905742e-7
559 centimeter/hour squared in mile/second squared is equal to 2.6801427196039e-10
559 centimeter/hour squared in yard/second squared is equal to 4.7170511865029e-7
559 centimeter/hour squared in gal is equal to 0.000043132716049383
559 centimeter/hour squared in galileo is equal to 0.000043132716049383
559 centimeter/hour squared in centigal is equal to 0.0043132716049383
559 centimeter/hour squared in decigal is equal to 0.00043132716049383
559 centimeter/hour squared in g-unit is equal to 4.3983129865329e-8
559 centimeter/hour squared in gn is equal to 4.3983129865329e-8
559 centimeter/hour squared in gravity is equal to 4.3983129865329e-8
559 centimeter/hour squared in milligal is equal to 0.043132716049383
559 centimeter/hour squared in kilogal is equal to 4.3132716049383e-8 | {"url":"https://hextobinary.com/unit/acceleration/from/cmh2/to/inmins/559","timestamp":"2024-11-09T20:53:28Z","content_type":"text/html","content_length":"97879","record_id":"<urn:uuid:1e11f399-f1a1-4f58-9bec-659e6211a194>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00797.warc.gz"} |
seminars - A survey on generalized Schr"{o}dinger operators along curves
In this survey, we review the historical development of the Carleson problem about a.e. pointwise convergence in five aspects: a.e. convergence for generalized Schr\"{o}dinger operators along
vertical lines; a.e. convergence for Schr\"{o}dinger operators along arbitrary single curves; a.e. convergence for Schr\"{o}dinger operators along a family of restricted curves; the upper bounds of
$p$ for the $L^p$-Schr\"{o}dinger maximal estimates; a.e. convergence rate for generalized Schr\"{o}dinger operators along curves. Finally, we list some open problems which need to be addressed. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&page=12&l=ko&document_srl=1144186","timestamp":"2024-11-12T06:40:22Z","content_type":"text/html","content_length":"45374","record_id":"<urn:uuid:90b4cb31-bfcd-4134-a3f7-4ec5f6bf5187>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00605.warc.gz"} |
Abhishek Chakrabortty
Jan 24, 2022
Abstract:In this article, we aim to provide a general and complete understanding of semi-supervised (SS) causal inference for treatment effects. Specifically, we consider two such estimands: (a) the
average treatment effect and (b) the quantile treatment effect, as prototype cases, in an SS setting, characterized by two available data sets: (i) a labeled data set of size $n$, providing
observations for a response and a set of high dimensional covariates, as well as a binary treatment indicator; and (ii) an unlabeled data set of size $N$, much larger than $n$, but without the
response observed. Using these two data sets, we develop a family of SS estimators which are ensured to be: (1) more robust and (2) more efficient than their supervised counterparts based on the
labeled data set only. Beyond the 'standard' double robustness results (in terms of consistency) that can be achieved by supervised methods as well, we further establish root-n consistency and
asymptotic normality of our SS estimators whenever the propensity score in the model is correctly specified, without requiring specific forms of the nuisance functions involved. Such an improvement
of robustness arises from the use of the massive unlabeled data, so it is generally not attainable in a purely supervised setting. In addition, our estimators are shown to be semi-parametrically
efficient as long as all the nuisance functions are correctly specified. Moreover, as an illustration of the nuisance estimators, we consider inverse-probability-weighting type kernel smoothing
estimators involving unknown covariate transformation mechanisms, and establish in high dimensional scenarios novel results on their uniform convergence rates, which should be of independent
interest. Numerical results on both simulated and real data validate the advantage of our methods over their supervised counterparts with respect to both robustness and efficiency.
* Generalizations added (Appendix A); 59 pages (with supplement), 7 tables, 2 figures | {"url":"https://www.catalyzex.com/author/Abhishek%20Chakrabortty","timestamp":"2024-11-02T07:46:13Z","content_type":"text/html","content_length":"223062","record_id":"<urn:uuid:568d8fe9-aeac-4316-8d53-e5fdba424c3b>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00338.warc.gz"} |
An object with a mass of 7 kg is lying still on a surface and is compressing a horizontal spring by 5/4 m. If the spring's constant is 3 (kg)/s^2, what is the minimum value of the surface's coefficient of static friction? | HIX Tutor
An object with a mass of #7 kg# is lying still on a surface and is compressing a horizontal spring by #5/4 m#. If the spring's constant is #3 (kg)/s^2#, what is the minimum value of the surface's
coefficient of static friction?
Answer 1
The coefficient of static friction is $= 0.055$
The mass is #m=7kg#
The compression of the spring is #x=5/4m#
The spring constant is #k=3kgs^-2#
The reaction of the spring is #R=kx=3*5/4=15/4=3.75N#
The acceleration due to gravity is #g=9.8ms^-2#
The normal reaction of the object is #N=mg=7*9.8=68.6N#
The coefficient of static friction is
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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RPN: Just for Engineers? - Printable Version
RPN: Just for Engineers? - Palmer O. Hanson, Jr. - 02-05-2004
In his article "The Demise of the Slide Rule (and the Advent of Its Successors)" in the Fall 2003 issue of The Journal of the Oughtred Society Otto van Poelje of the Netherlands wrote in part:
"...The HP 35 gave the engineers, just like the good old slide rule, a special status, not only
because of its high price and flashy design, but especially because ordinary people could
not work straightaway with RPN: in RPN notation the calculation 5*(3+2) had to be entered
as 5 ENTER 3 ENTER 2 + * . ..."
Re: RPN: Just for Engineers? - db (martinez, ca.) - 02-05-2004
.....or 3 enter 2 + 5 X. I don't buy the part of Mr.. van Poelje's theory about "ordinary people could not work straightaway with RPN". I'm so ordinary and average i ought to be grey and i use
nothing else. Besides; there were lots of RPN calcs around before the 35. I admit that the five i can think of offhand were pretty exclusive too, but the thing about the 35 was that it was the first
pocket calc to do transendental functions. Techies who did not work at desks no longer needed to carry around phone book sized trig tables just to get half the accuracy. The only reason that 99.9% of
all surveyors stopped salivating at the 35 was that the 45 came along: "WOW! It can't get any better than polar-rectangular!
Still; the 48 series, the 41 series, the enduring 12C and the popularity of the 17b puts the lie to his "engineers only" idea. Yes, hindsight is 20:20. I see his point but imho the only reason rpn is
not THE notation is that, statistically speaking, half the people in the world are even dumber than me and a lot of the rest are too lazy to learn the mathematical order of precedence.
Re: RPN: Just for Engineers? - Art Litka - 02-05-2004
Right on db! I could not agree nore. If only I could get my engineering/science students interested in rpn today. These kids mostly look at rpn as some strange way of doing business-TI did a good job
in hooking them on algebraic notation. I can't seem to interest them in the 49G+--TI 89's seem to be the "standard" now and this prof just has to reluctantly go along with it, at least for now.
Re: RPN: Just for Engineers? - Vieira, Luiz C. (Brazil) - 02-05-2004
Hi, all;
cannot agree with the focussed affirmative: "ordinary people could not work straightaway with RPN". Instead, I support d.b.'s always clever analysis (Hi, D.B.).
I'd add that almost 22 years later, the HP12C still keeps its place at financial markets, and there are not so many engineers there, instead many mathematicians and lots of businessmen, MBA's, market
analysts... They are far from being "ordinary people", of course, but their main concerns are far from being the operation of a particular calculator and its operating system. And I'd guess their
HP12C's do not have wearied out [n], [i], [PV], [PMT] and [FV] plus digit-input keys only, instead [ENTER] is included. And HP12C users know it well: there's no need to understand RPN to use its
financial resources. But I'd never seen any HP12C owner that cannot help proudly explaining how to use it to anyone that asks for. [ENTER] key included!
About RPN mechanics: if you know how to use pencil and paper to perform computations, than you already know RPN well. Simply use [ENTER] when you finish writing one number and are ready to begin
writing the other. In fact, I add an equals sign when writing an algebraic expression, but I cannot remember any moment I needed to write an "equals" sign when actually performing addition,
subtraction, multiplication ort any other math operation. And that was the basic line I took when trying to understand RPN: if I know how to solve an expression, I'll be able to find its numerical
result by using RPN-style keystrokes.
I think I have already written something like this sometime ago... a year or more, perhaps.
My thoughts. And my 2ยข, too.
Luiz (Brazil)
Re: RPN: Just for Engineers? - John Hancock - 02-06-2004
I am obviously with the majority. Mt wife is pretty smart but she hates maths. It took her a little time getting used to RPN but she hates using anything else now.
Just for Engineers? No way! - Eddie Shore - 02-10-2004
I have to disagree with Poelje. I am an accountant and I use RPN based calculators all the time. It just takes practice.
Re: RPN: Just for Engineers? - James M. Prange (Michigan) - 02-10-2004
I'm a metrology technician, not an engineer. I was very happy when I discovered the 28S; for me, it works just the way that a calculator should work. It's much more "natural" than the "algebraic
input" user interface calculators I'd tried before.
I've also noticed that children who aren't already indoctrinated into the arcane methods needed for other calculators seem to have no problem learning to use RPN/RPL calculators for basic arithmetic.
To be sure, a good many "ordinary people" do have a problem when they first try an RPN/RPL calculator, but I expect that that's because they have to "unlearn" what they've learned about other
calculators. Too bad that so few are willing to make the effort.
Re: RPN: Just for Engineers? - Palmer O. Hanson, Jr. - 02-16-2004
d.b got it correctly when he notes that it was the availability of the transcendental functions which was the big attraction of the HP-35. Where I worked we didn't find the availability of
polar-rectangular conversions to be a big plus for the HP-45. Rather, we liked the built in statistics capability which was only a small taste of what was to come.
On the HP-12C: I was with the customer representative in a bank back in the early 1990's. She had a 12C on her desk but was using another desktop calculator for routine calculations. It was one of
those business-oriented devices which mimic the workings of mechanical adding machines, the system that I call AML (adding machine logic) to differentiate it from RPN and AOS. At the time I didn't
have a 12C in my collection and asked whether she would sell it since she wasn't using it. She would not and explained that it was invaluable for compound interest and amortization calculations of
all kinds which couldn't be done on her four function desktop machine.. When I later found a 12C at a garage sale I was not so much impressed with the machine as with the documentation.
On languages: back in the days when HP and TI were competing fiercely with the likes of the HP-67, SR-52, TI-59 and HP-41 there was a lot of irrational flak going around about the merits of RPN and
AOS. The van Poelje paper reports that HP folks talked about the TI folks as being from "the dark side". The TI folks came up with an alternate meaning for RPN, namely, "Really Pathetic Notation". In
those days I eventually learned to rely on the opinions of what were called bi-linguals; i.e., individuals who were truly capable with either system and who typically would say that either system
works well.
When Casio came out with the first graphing calculator they used a system that was different from both RPN or AOS and looked like a cross between AOS and a higher order language such as BASIC. TI
used a similar language that they called EOS for Expression Operating System when they came out with the TI-81. | {"url":"https://archived.hpcalc.org/museumforum/printthread.php?tid=51236","timestamp":"2024-11-11T17:54:34Z","content_type":"application/xhtml+xml","content_length":"11499","record_id":"<urn:uuid:021ad2ad-7d9a-4ccf-b677-b67d8f4eff5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00785.warc.gz"} |
scatter plot with error bars in r
This can be done in a number of ways, as described on this page.In this case, we’ll use the summarySE() function defined on that page, and also at the bottom of this page. First, it is necessary to
summarize the data. Bar charts are a pretty common way to represent data visually, but constructing them isn’t always the most intuitive thing in the world. In our previous article - Facilitating
Exploratory Data Visualization: Application to TCGA Genomic Data - we described how to visualize gene expression data using box plots, violin plots, dot plots and stripcharts. A volcano plot is a
type of scatter plot represents differential expression of features (genes for example): on the x-axis we typically find the fold change and on the y … wiki. (The code for the summarySE function must
be entered before it is called here). @neilfws – Lin Jun 8 '17 at 6:14 add a comment | 1 Answer 1 You start by plotting a scatterplot of the mpg variable and drat variable. 19.11 Volcano plots.
BL1ggplot <- read.table(text=" line MaleBL1 FemaleBL1 BL1MaleSE BL1FemaleSE 3 0.05343516 0.05615977 0.28666600 0.3142001 4 -0.53321642 -0.27279609 0.23929438 0.1350793 5 -0.25853484 -0.08283566
0.25904025 0.2984323 6 -1.11250479 0.03299387 0.23553281 0.2786233 7 -0.14784506 … This interval is defined so that there is a specified probability that a value lies within it. I need to plot two
error-bars on each point in a scatterplot. You really should study some tutorials. You haven't understood ggplot2 syntax. ?s t-distribution for a specific alpha. Here epsilon controls the line across
the top and bottom of the line.. plot (x, y, ylim=c(0, 6)) epsilon = 0.02 for(i in 1:5) { up = y[i] + sd[i] low = y[i] - sd[i] segments(x[i],low , x[i], up) segments(x[i]-epsilon, up , x[i]+epsilon,
up) segments(x[i]-epsilon, low , x[i]+epsilon, low) } The spacings of the two scales are identical but the scale for differences has its origin shifted so that zero may be included. Early Access puts
eBooks and videos into your hands whilst they’re still being written, so you don’t have to wait to take advantage of new tech and new ideas. Let's see how ggplot works with the mtcars dataset. Using
the ggplot2 solution, just create a vector with your means (my_mean) and standard errors (my_sem) and follow the rest of the code. Its value is often rounded to 1.96 (its value with a big sample
size). You can use segments to add the bars in base graphics. One way that we can construct these graphs is using R’s default packages. The most common object are: - Point: `geom_point()` - Bar:
`geom_bar()` - Line: `geom_line()` - Histogram: `geom_histogram()` Scatterplot. New to Plotly? → Confidence Interval (CI). It is calculated as t * SE.Where t is the value of the Student?? For the
latter type of plot, the lower x-axis scale corresponds to group estimates and the upper scale corresponds to differences. Basic scatter plot We also demonstrated how to combine the plot of multiples
variables (genes) in the same plot. When I produce the plot without the geom_errorbar(), it works fine. Plotly is a free and open-source graphing library for R. We recommend you read our Getting
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Triangles to Triangles
The set of functions of the form
f(z) = (az + b)/(cz + d)
with ad ≠ bc are called bilinear transformations or Möbius transformations. These functions have three degrees of freedom—there are four parameters, but multiplying all parameters by a constant
defines the same function—and so you can uniquely determine such a function by picking three points and specifying where they go.
Here’s an explicit formula for the Möbius transformation that takes z[1], z[2], and z[3] to w[1], w[2], and w[3].
To see that this is correct, or at least possible, note that if you set z = z[i] and w = w[i] for some i then two rows of the matrix are equal and so the determinant is zero.
Triangles, lines, and circles
You can pick three points in one complex plane, the z-plane, and three points in another complex plane, the w-plane, and find a Möbius transformation w = f(z) taking the z-plane to the w-plane
sending the specified z‘s to the specified w‘s.
If you view the three points as vertices of a triangle, you’re specifying that one triangle gets mapped to another triangle. However, the sides of your triangle may or may not be straight lines.
Möbius transformations map circles and lines to circles and lines, but a circle might become a line or vice versa. So the straight lines of our original triangle may map to straight lines or they may
become circular arcs. How can you tell whether the image of a side of a triangle will be straight or curved?
When does a line map to a line?
It’ll be easier if we add a point ∞ to the complex plane and think of lines as infinitely big circles, circles that pass through ∞.
The Möbius transformation (az + b)/(cz + d) takes ∞ to a/c and it takes −d/c to ∞.
The sides of a triangle are line segments. If we look at the entire line, not just the segment, then this line is mapped to a circle. If this line contains the point that gets mapped to ∞ then the
image of the line is an infinite circle (i.e. a line). Otherwise the image of the line is a finite circle.
The line between z[1] and z[2] can be parameterized by
z[1] + t(z[2] − z[1])
where t is real. So the image of this line will be a line if and only if
z[1] + t(z[2] − z[1]) = −d/c
for some real t. So solve for t and see whether you get a real number.
Note that if the point that is mapped to ∞ lies inside the line segment, not just on the line, then the image of that side of the triangle is infinitely long.
To keep things as simple as possible without being trivial, we’ll use the Möbius transformation f(z) = 1/z. Clearly the origin is the point that is mapped to ∞. The side of a triangle is mapped to a
straight line if and only if the side is part of a line through the origin.
First let’s look at the triangle with vertices at (1, 1), (1, 4), and (5, 1). None of the sides is on a line that extends to the origin, so all sides map to circular arcs.
Next let’s move the second point from (1, 4) to (4, 4). The line running between (1, 1) and (4, 4) goes through the origin, and so the segment along that line maps to a straight line.
Related posts | {"url":"https://www.johndcook.com/blog/2024/10/11/triangles-to-triangles/","timestamp":"2024-11-13T11:33:05Z","content_type":"text/html","content_length":"54512","record_id":"<urn:uuid:be6e66be-a4a2-4549-b21d-7114a0019431>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00697.warc.gz"} |
An overview of gradient descent optimization algorithms
Gradient descent is the preferred way to optimize neural networks and many other machine learning algorithms but is often used as a black box. This post explores how many of the most popular
gradient-based optimization algorithms such as Momentum, Adagrad, and Adam actually work.
This post explores how many of the most popular gradient-based optimization algorithms actually work.
Note: If you are looking for a review paper, this blog post is also available as an article on arXiv.
Update 20.03.2020: Added a note on recent optimizers.
Update 09.02.2018: Added AMSGrad.
Update 24.11.2017: Most of the content in this article is now also available as slides.
Update 15.06.2017: Added derivations of AdaMax and Nadam.
Update 21.06.16: This post was posted to Hacker News. The discussion provides some interesting pointers to related work and other techniques.
Table of contents:
Gradient descent is one of the most popular algorithms to perform optimization and by far the most common way to optimize neural networks. At the same time, every state-of-the-art Deep Learning
library contains implementations of various algorithms to optimize gradient descent (e.g. lasagne's, caffe's, and keras' documentation). These algorithms, however, are often used as black-box
optimizers, as practical explanations of their strengths and weaknesses are hard to come by.
This blog post aims at providing you with intuitions towards the behaviour of different algorithms for optimizing gradient descent that will help you put them to use. We are first going to look at
the different variants of gradient descent. We will then briefly summarize challenges during training. Subsequently, we will introduce the most common optimization algorithms by showing their
motivation to resolve these challenges and how this leads to the derivation of their update rules. We will also take a short look at algorithms and architectures to optimize gradient descent in a
parallel and distributed setting. Finally, we will consider additional strategies that are helpful for optimizing gradient descent.
Gradient descent is a way to minimize an objective function \(J(\theta)\) parameterized by a model's parameters \(\theta \in \mathbb{R}^d \) by updating the parameters in the opposite direction of
the gradient of the objective function \(\nabla_\theta J(\theta)\) w.r.t. to the parameters. The learning rate \(\eta\) determines the size of the steps we take to reach a (local) minimum. In other
words, we follow the direction of the slope of the surface created by the objective function downhill until we reach a valley. If you are unfamiliar with gradient descent, you can find a good
introduction on optimizing neural networks here.
Gradient descent variants
There are three variants of gradient descent, which differ in how much data we use to compute the gradient of the objective function. Depending on the amount of data, we make a trade-off between the
accuracy of the parameter update and the time it takes to perform an update.
Batch gradient descent
Vanilla gradient descent, aka batch gradient descent, computes the gradient of the cost function w.r.t. to the parameters \(\theta\) for the entire training dataset:
\(\theta = \theta - \eta \cdot \nabla_\theta J( \theta)\).
As we need to calculate the gradients for the whole dataset to perform just one update, batch gradient descent can be very slow and is intractable for datasets that don't fit in memory. Batch
gradient descent also doesn't allow us to update our model online, i.e. with new examples on-the-fly.
In code, batch gradient descent looks something like this:
for i in range(nb_epochs):
params_grad = evaluate_gradient(loss_function, data, params)
params = params - learning_rate * params_grad
For a pre-defined number of epochs, we first compute the gradient vector params_grad of the loss function for the whole dataset w.r.t. our parameter vector params. Note that state-of-the-art deep
learning libraries provide automatic differentiation that efficiently computes the gradient w.r.t. some parameters. If you derive the gradients yourself, then gradient checking is a good idea. (See
here for some great tips on how to check gradients properly.)
We then update our parameters in the opposite direction of the gradients with the learning rate determining how big of an update we perform. Batch gradient descent is guaranteed to converge to the
global minimum for convex error surfaces and to a local minimum for non-convex surfaces.
Stochastic gradient descent
Stochastic gradient descent (SGD) in contrast performs a parameter update for each training example \(x^{(i)}\) and label \(y^{(i)}\):
\(\theta = \theta - \eta \cdot \nabla_\theta J( \theta; x^{(i)}; y^{(i)})\).
Batch gradient descent performs redundant computations for large datasets, as it recomputes gradients for similar examples before each parameter update. SGD does away with this redundancy by
performing one update at a time. It is therefore usually much faster and can also be used to learn online.
SGD performs frequent updates with a high variance that cause the objective function to fluctuate heavily as in Image 1.
Image 1: SGD fluctuation (Source: Wikipedia)
While batch gradient descent converges to the minimum of the basin the parameters are placed in, SGD's fluctuation, on the one hand, enables it to jump to new and potentially better local minima. On
the other hand, this ultimately complicates convergence to the exact minimum, as SGD will keep overshooting. However, it has been shown that when we slowly decrease the learning rate, SGD shows the
same convergence behaviour as batch gradient descent, almost certainly converging to a local or the global minimum for non-convex and convex optimization respectively.
Its code fragment simply adds a loop over the training examples and evaluates the gradient w.r.t. each example. Note that we shuffle the training data at every epoch as explained in this section.
for i in range(nb_epochs):
for example in data:
params_grad = evaluate_gradient(loss_function, example, params)
params = params - learning_rate * params_grad
Mini-batch gradient descent
Mini-batch gradient descent finally takes the best of both worlds and performs an update for every mini-batch of \(n\) training examples:
\(\theta = \theta - \eta \cdot \nabla_\theta J( \theta; x^{(i:i+n)}; y^{(i:i+n)})\).
This way, it a) reduces the variance of the parameter updates, which can lead to more stable convergence; and b) can make use of highly optimized matrix optimizations common to state-of-the-art deep
learning libraries that make computing the gradient w.r.t. a mini-batch very efficient. Common mini-batch sizes range between 50 and 256, but can vary for different applications. Mini-batch gradient
descent is typically the algorithm of choice when training a neural network and the term SGD usually is employed also when mini-batches are used. Note: In modifications of SGD in the rest of this
post, we leave out the parameters \(x^{(i:i+n)}; y^{(i:i+n)}\) for simplicity.
In code, instead of iterating over examples, we now iterate over mini-batches of size 50:
for i in range(nb_epochs):
for batch in get_batches(data, batch_size=50):
params_grad = evaluate_gradient(loss_function, batch, params)
params = params - learning_rate * params_grad
Vanilla mini-batch gradient descent, however, does not guarantee good convergence, but offers a few challenges that need to be addressed:
• Choosing a proper learning rate can be difficult. A learning rate that is too small leads to painfully slow convergence, while a learning rate that is too large can hinder convergence and cause
the loss function to fluctuate around the minimum or even to diverge.
• Learning rate schedules ^[1] try to adjust the learning rate during training by e.g. annealing, i.e. reducing the learning rate according to a pre-defined schedule or when the change in objective
between epochs falls below a threshold. These schedules and thresholds, however, have to be defined in advance and are thus unable to adapt to a dataset's characteristics ^[2].
• Additionally, the same learning rate applies to all parameter updates. If our data is sparse and our features have very different frequencies, we might not want to update all of them to the same
extent, but perform a larger update for rarely occurring features.
• Another key challenge of minimizing highly non-convex error functions common for neural networks is avoiding getting trapped in their numerous suboptimal local minima. Dauphin et al. ^[3] argue
that the difficulty arises in fact not from local minima but from saddle points, i.e. points where one dimension slopes up and another slopes down. These saddle points are usually surrounded by a
plateau of the same error, which makes it notoriously hard for SGD to escape, as the gradient is close to zero in all dimensions.
Gradient descent optimization algorithms
In the following, we will outline some algorithms that are widely used by the deep learning community to deal with the aforementioned challenges. We will not discuss algorithms that are infeasible to
compute in practice for high-dimensional data sets, e.g. second-order methods such as Newton's method.
SGD has trouble navigating ravines, i.e. areas where the surface curves much more steeply in one dimension than in another ^[4], which are common around local optima. In these scenarios, SGD
oscillates across the slopes of the ravine while only making hesitant progress along the bottom towards the local optimum as in Image 2.
Image 2: SGD without momentum Image 3: SGD with momentum
Momentum ^[5] is a method that helps accelerate SGD in the relevant direction and dampens oscillations as can be seen in Image 3. It does this by adding a fraction \(\gamma\) of the update vector of
the past time step to the current update vector:
v_t &= \gamma v_{t-1} + \eta \nabla_\theta J( \theta) \\
\theta &= \theta - v_t
Note: Some implementations exchange the signs in the equations. The momentum term \(\gamma\) is usually set to 0.9 or a similar value.
Essentially, when using momentum, we push a ball down a hill. The ball accumulates momentum as it rolls downhill, becoming faster and faster on the way (until it reaches its terminal velocity if
there is air resistance, i.e. \(\gamma < 1\)). The same thing happens to our parameter updates: The momentum term increases for dimensions whose gradients point in the same directions and reduces
updates for dimensions whose gradients change directions. As a result, we gain faster convergence and reduced oscillation.
Nesterov accelerated gradient
However, a ball that rolls down a hill, blindly following the slope, is highly unsatisfactory. We'd like to have a smarter ball, a ball that has a notion of where it is going so that it knows to slow
down before the hill slopes up again.
Nesterov accelerated gradient (NAG) ^[6] is a way to give our momentum term this kind of prescience. We know that we will use our momentum term \(\gamma v_{t-1}\) to move the parameters \(\theta\).
Computing \( \theta - \gamma v_{t-1} \) thus gives us an approximation of the next position of the parameters (the gradient is missing for the full update), a rough idea where our parameters are
going to be. We can now effectively look ahead by calculating the gradient not w.r.t. to our current parameters \(\theta\) but w.r.t. the approximate future position of our parameters:
v_t &= \gamma v_{t-1} + \eta \nabla_\theta J( \theta - \gamma v_{t-1} ) \\
\theta &= \theta - v_t
Again, we set the momentum term \(\gamma\) to a value of around 0.9. While Momentum first computes the current gradient (small blue vector in Image 4) and then takes a big jump in the direction of
the updated accumulated gradient (big blue vector), NAG first makes a big jump in the direction of the previous accumulated gradient (brown vector), measures the gradient and then makes a correction
(red vector), which results in the complete NAG update (green vector). This anticipatory update prevents us from going too fast and results in increased responsiveness, which has significantly
increased the performance of RNNs on a number of tasks ^[7].
Image 4: Nesterov update (Source: G. Hinton's lecture 6c)
Refer to here for another explanation about the intuitions behind NAG, while Ilya Sutskever gives a more detailed overview in his PhD thesis ^[8].
Now that we are able to adapt our updates to the slope of our error function and speed up SGD in turn, we would also like to adapt our updates to each individual parameter to perform larger or
smaller updates depending on their importance.
Adagrad ^[9] is an algorithm for gradient-based optimization that does just this: It adapts the learning rate to the parameters, performing smaller updates
(i.e. low learning rates) for parameters associated with frequently occurring features, and larger updates (i.e. high learning rates) for parameters associated with infrequent features. For this
reason, it is well-suited for dealing with sparse data. Dean et al. ^[10] have found that Adagrad greatly improved the robustness of SGD and used it for training large-scale neural nets at Google,
which -- among other things -- learned to recognize cats in Youtube videos. Moreover, Pennington et al. ^[11] used Adagrad to train GloVe word embeddings, as infrequent words require much larger
updates than frequent ones.
Previously, we performed an update for all parameters \(\theta\) at once as every parameter \(\theta_i\) used the same learning rate \(\eta\). As Adagrad uses a different learning rate for every
parameter \(\theta_i\) at every time step \(t\), we first show Adagrad's per-parameter update, which we then vectorize. For brevity, we use \(g_{t}\) to denote the gradient at time step \(t\). \(g_
{t, i}\) is then the partial derivative of the objective function w.r.t. to the parameter \(\theta_i\) at time step \(t\):
\(g_{t, i} = \nabla_\theta J( \theta_{t, i} )\).
The SGD update for every parameter \(\theta_i\) at each time step \(t\) then becomes:
\(\theta_{t+1, i} = \theta_{t, i} - \eta \cdot g_{t, i}\).
In its update rule, Adagrad modifies the general learning rate \(\eta\) at each time step \(t\) for every parameter \(\theta_i\) based on the past gradients that have been computed for \(\theta_i\):
\(\theta_{t+1, i} = \theta_{t, i} - \dfrac{\eta}{\sqrt{G_{t, ii} + \epsilon}} \cdot g_{t, i}\).
\(G_{t} \in \mathbb{R}^{d \times d} \) here is a diagonal matrix where each diagonal element \(i, i\) is the sum of the squares of the gradients w.r.t. \(\theta_i\) up to time step \(t\) ^[12], while
\(\epsilon\) is a smoothing term that avoids division by zero (usually on the order of \(1e-8\)). Interestingly, without the square root operation, the algorithm performs much worse.
As \(G_{t}\) contains the sum of the squares of the past gradients w.r.t. to all parameters \(\theta\) along its diagonal, we can now vectorize our implementation by performing a matrix-vector
product \(\odot\) between \(G_{t}\) and \(g_{t}\):
\(\theta_{t+1} = \theta_{t} - \dfrac{\eta}{\sqrt{G_{t} + \epsilon}} \odot g_{t}\).
One of Adagrad's main benefits is that it eliminates the need to manually tune the learning rate. Most implementations use a default value of 0.01 and leave it at that.
Adagrad's main weakness is its accumulation of the squared gradients in the denominator: Since every added term is positive, the accumulated sum keeps growing during training. This in turn causes the
learning rate to shrink and eventually become infinitesimally small, at which point the algorithm is no longer able to acquire additional knowledge. The following algorithms aim to resolve this flaw.
Adadelta ^[13] is an extension of Adagrad that seeks to reduce its aggressive, monotonically decreasing learning rate. Instead of accumulating all past squared gradients, Adadelta restricts the
window of accumulated past gradients to some fixed size \(w\).
Instead of inefficiently storing \(w\) previous squared gradients, the sum of gradients is recursively defined as a decaying average of all past squared gradients. The running average \(E[g^2]_t\) at
time step \(t\) then depends (as a fraction \(\gamma \) similarly to the Momentum term) only on the previous average and the current gradient:
\(E[g^2]_t = \gamma E[g^2]_{t-1} + (1 - \gamma) g^2_t \).
We set \(\gamma\) to a similar value as the momentum term, around 0.9. For clarity, we now rewrite our vanilla SGD update in terms of the parameter update vector \( \Delta \theta_t \):
\Delta \theta_t &= - \eta \cdot g_{t, i} \\
\theta_{t+1} &= \theta_t + \Delta \theta_t \end{split}
The parameter update vector of Adagrad that we derived previously thus takes the form:
\( \Delta \theta_t = - \dfrac{\eta}{\sqrt{G_{t} + \epsilon}} \odot g_{t}\).
We now simply replace the diagonal matrix \(G_{t}\) with the decaying average over past squared gradients \(E[g^2]_t\):
\( \Delta \theta_t = - \dfrac{\eta}{\sqrt{E[g^2]_t + \epsilon}} g_{t}\).
As the denominator is just the root mean squared (RMS) error criterion of the gradient, we can replace it with the criterion short-hand:
\( \Delta \theta_t = - \dfrac{\eta}{RMS[g]_{t}} g_t\).
The authors note that the units in this update (as well as in SGD, Momentum, or Adagrad) do not match, i.e. the update should have the same hypothetical units as the parameter. To realize this, they
first define another exponentially decaying average, this time not of squared gradients but of squared parameter updates:
\(E[\Delta \theta^2]_t = \gamma E[\Delta \theta^2]_{t-1} + (1 - \gamma) \Delta \theta^2_t \).
The root mean squared error of parameter updates is thus:
\(RMS[\Delta \theta]_{t} = \sqrt{E[\Delta \theta^2]_t + \epsilon} \).
Since \(RMS[\Delta \theta]_{t}\) is unknown, we approximate it with the RMS of parameter updates until the previous time step. Replacing the learning rate \(\eta \) in the previous update rule with \
(RMS[\Delta \theta]_{t-1}\) finally yields the Adadelta update rule:
\Delta \theta_t &= - \dfrac{RMS[\Delta \theta]_{t-1}}{RMS[g]_{t}} g_{t} \\
\theta_{t+1} &= \theta_t + \Delta \theta_t
With Adadelta, we do not even need to set a default learning rate, as it has been eliminated from the update rule.
RMSprop is an unpublished, adaptive learning rate method proposed by Geoff Hinton in Lecture 6e of his Coursera Class.
RMSprop and Adadelta have both been developed independently around the same time stemming from the need to resolve Adagrad's radically diminishing learning rates. RMSprop in fact is identical to the
first update vector of Adadelta that we derived above:
E[g^2]_t &= 0.9 E[g^2]_{t-1} + 0.1 g^2_t \\
\theta_{t+1} &= \theta_{t} - \dfrac{\eta}{\sqrt{E[g^2]_t + \epsilon}} g_{t}
RMSprop as well divides the learning rate by an exponentially decaying average of squared gradients. Hinton suggests \(\gamma\) to be set to 0.9, while a good default value for the learning rate \(\
eta\) is 0.001.
Adaptive Moment Estimation (Adam) ^[14] is another method that computes adaptive learning rates for each parameter. In addition to storing an exponentially decaying average of past squared gradients
\(v_t\) like Adadelta and RMSprop, Adam also keeps an exponentially decaying average of past gradients \(m_t\), similar to momentum. Whereas momentum can be seen as a ball running down a slope, Adam
behaves like a heavy ball with friction, which thus prefers flat minima in the error surface ^[15]. We compute the decaying averages of past and past squared gradients \(m_t\) and \(v_t\)
respectively as follows:
m_t &= \beta_1 m_{t-1} + (1 - \beta_1) g_t \\
v_t &= \beta_2 v_{t-1} + (1 - \beta_2) g_t^2
\(m_t\) and \(v_t\) are estimates of the first moment (the mean) and the second moment (the uncentered variance) of the gradients respectively, hence the name of the method. As \(m_t\) and \(v_t\)
are initialized as vectors of 0's, the authors of Adam observe that they are biased towards zero, especially during the initial time steps, and especially when the decay rates are small (i.e. \(\
beta_1\) and \(\beta_2\) are close to 1).
They counteract these biases by computing bias-corrected first and second moment estimates:
\hat{m}_t &= \dfrac{m_t}{1 - \beta^t_1} \\
\hat{v}_t &= \dfrac{v_t}{1 - \beta^t_2} \end{split}
They then use these to update the parameters just as we have seen in Adadelta and RMSprop, which yields the Adam update rule:
\(\theta_{t+1} = \theta_{t} - \dfrac{\eta}{\sqrt{\hat{v}_t} + \epsilon} \hat{m}_t\).
The authors propose default values of 0.9 for \(\beta_1\), 0.999 for \(\beta_2\), and \(10^{-8}\) for \(\epsilon\). They show empirically that Adam works well in practice and compares favorably to
other adaptive learning-method algorithms.
The \(v_t\) factor in the Adam update rule scales the gradient inversely proportionally to the \(\ell_2\) norm of the past gradients (via the \(v_{t-1}\) term) and current gradient \(|g_t|^2\):
\(v_t = \beta_2 v_{t-1} + (1 - \beta_2) |g_t|^2\)
We can generalize this update to the \(\ell_p\) norm. Note that Kingma and Ba also parameterize \(\beta_2\) as \(\beta^p_2\):
\(v_t = \beta_2^p v_{t-1} + (1 - \beta_2^p) |g_t|^p\)
Norms for large \(p\) values generally become numerically unstable, which is why \(\ell_1\) and \(\ell_2\) norms are most common in practice. However, \(\ell_\infty\) also generally exhibits stable
behavior. For this reason, the authors propose AdaMax (Kingma and Ba, 2015) and show that \(v_t\) with \(\ell_\infty\) converges to the following more stable value. To avoid confusion with Adam, we
use \(u_t\) to denote the infinity norm-constrained \(v_t\):
u_t &= \beta_2^\infty v_{t-1} + (1 - \beta_2^\infty) |g_t|^\infty\\
& = \max(\beta_2 \cdot v_{t-1}, |g_t|)
We can now plug this into the Adam update equation by replacing \(\sqrt{\hat{v}_t} + \epsilon\) with \(u_t\) to obtain the AdaMax update rule:
\(\theta_{t+1} = \theta_{t} - \dfrac{\eta}{u_t} \hat{m}_t\)
Note that as \(u_t\) relies on the \(\max\) operation, it is not as suggestible to bias towards zero as \(m_t\) and \(v_t\) in Adam, which is why we do not need to compute a bias correction for \(u_t
\). Good default values are again \(\eta = 0.002\), \(\beta_1 = 0.9\), and \(\beta_2 = 0.999\).
As we have seen before, Adam can be viewed as a combination of RMSprop and momentum: RMSprop contributes the exponentially decaying average of past squared gradients \(v_t\), while momentum accounts
for the exponentially decaying average of past gradients \(m_t\). We have also seen that Nesterov accelerated gradient (NAG) is superior to vanilla momentum.
Nadam (Nesterov-accelerated Adaptive Moment Estimation) ^[16] thus combines Adam and NAG. In order to incorporate NAG into Adam, we need to modify its momentum term \(m_t\).
First, let us recall the momentum update rule using our current notation :
g_t &= \nabla_{\theta_t}J(\theta_t)\\
m_t &= \gamma m_{t-1} + \eta g_t\\
\theta_{t+1} &= \theta_t - m_t
where \(J\) is our objective function, \(\gamma\) is the momentum decay term, and \(\eta\) is our step size. Expanding the third equation above yields:
\(\theta_{t+1} = \theta_t - ( \gamma m_{t-1} + \eta g_t)\)
This demonstrates again that momentum involves taking a step in the direction of the previous momentum vector and a step in the direction of the current gradient.
NAG then allows us to perform a more accurate step in the gradient direction by updating the parameters with the momentum step before computing the gradient. We thus only need to modify the gradient
\(g_t\) to arrive at NAG:
g_t &= \nabla_{\theta_t}J(\theta_t - \gamma m_{t-1})\\
m_t &= \gamma m_{t-1} + \eta g_t\\
\theta_{t+1} &= \theta_t - m_t
Dozat proposes to modify NAG the following way: Rather than applying the momentum step twice -- one time for updating the gradient \(g_t\) and a second time for updating the parameters \(\theta_{t+1}
\) -- we now apply the look-ahead momentum vector directly to update the current parameters:
g_t &= \nabla_{\theta_t}J(\theta_t)\\
m_t &= \gamma m_{t-1} + \eta g_t\\
\theta_{t+1} &= \theta_t - (\gamma m_t + \eta g_t)
Notice that rather than utilizing the previous momentum vector \(m_{t-1}\) as in the equation of the expanded momentum update rule above, we now use the current momentum vector \(m_t\) to look ahead.
In order to add Nesterov momentum to Adam, we can thus similarly replace the previous momentum vector with the current momentum vector. First, recall that the Adam update rule is the following (note
that we do not need to modify \(\hat{v}_t\)):
m_t &= \beta_1 m_{t-1} + (1 - \beta_1) g_t\\
\hat{m}_t & = \frac{m_t}{1 - \beta^t_1}\\
\theta_{t+1} &= \theta_{t} - \frac{\eta}{\sqrt{\hat{v}_t} + \epsilon} \hat{m}_t
Expanding the second equation with the definitions of \(\hat{m}_t\) and \(m_t\) in turn gives us:
\(\theta_{t+1} = \theta_{t} - \dfrac{\eta}{\sqrt{\hat{v}_t} + \epsilon} (\dfrac{\beta_1 m_{t-1}}{1 - \beta^t_1} + \dfrac{(1 - \beta_1) g_t}{1 - \beta^t_1})\)
Note that \(\dfrac{\beta_1 m_{t-1}}{1 - \beta^t_1}\) is just the bias-corrected estimate of the momentum vector of the previous time step. We can thus replace it with \(\hat{m}_{t-1}\):
\(\theta_{t+1} = \theta_{t} - \dfrac{\eta}{\sqrt{\hat{v}_t} + \epsilon} (\beta_1 \hat{m}_{t-1} + \dfrac{(1 - \beta_1) g_t}{1 - \beta^t_1})\)
Note that for simplicity, we ignore that the denominator is \(1 - \beta^t_1\) and not \(1 - \beta^{t-1}_1\) as we will replace the denominator in the next step anyway. This equation again looks very
similar to our expanded momentum update rule above. We can now add Nesterov momentum just as we did previously by simply replacing this bias-corrected estimate of the momentum vector of the previous
time step \(\hat{m}_{t-1}\) with the bias-corrected estimate of the current momentum vector \(\hat{m}_t\), which gives us the Nadam update rule:
\(\theta_{t+1} = \theta_{t} - \dfrac{\eta}{\sqrt{\hat{v}_t} + \epsilon} (\beta_1 \hat{m}_t + \dfrac{(1 - \beta_1) g_t}{1 - \beta^t_1})\)
As adaptive learning rate methods have become the norm in training neural networks, practitioners noticed that in some cases, e.g. for object recognition ^[17] or machine translation ^[18] they fail
to converge to an optimal solution and are outperformed by SGD with momentum.
Reddi et al. (2018) ^[19] formalize this issue and pinpoint the exponential moving average of past squared gradients as a reason for the poor generalization behaviour of adaptive learning rate
methods. Recall that the introduction of the exponential average was well-motivated: It should prevent the learning rates to become infinitesimally small as training progresses, the key flaw of the
Adagrad algorithm. However, this short-term memory of the gradients becomes an obstacle in other scenarios.
In settings where Adam converges to a suboptimal solution, it has been observed that some minibatches provide large and informative gradients, but as these minibatches only occur rarely, exponential
averaging diminishes their influence, which leads to poor convergence. The authors provide an example for a simple convex optimization problem where the same behaviour can be observed for Adam.
To fix this behaviour, the authors propose a new algorithm, AMSGrad that uses the maximum of past squared gradients \(v_t\) rather than the exponential average to update the parameters. \(v_t\) is
defined the same as in Adam above:
v_t = \beta_2 v_{t-1} + (1 - \beta_2) g_t^2
Instead of using \(v_t\) (or its bias-corrected version \(\hat{v}_t\)) directly, we now employ the previous \(v_{t-1}\) if it is larger than the current one:
\hat{v}_t = \text{max}(\hat{v}_{t-1}, v_t)
This way, AMSGrad results in a non-increasing step size, which avoids the problems suffered by Adam. For simplicity, the authors also remove the debiasing step that we have seen in Adam. The full
AMSGrad update without bias-corrected estimates can be seen below:
m_t &= \beta_1 m_{t-1} + (1 - \beta_1) g_t \\
v_t &= \beta_2 v_{t-1} + (1 - \beta_2) g_t^2\\
\hat{v}_t &= \text{max}(\hat{v}_{t-1}, v_t) \\
\theta_{t+1} &= \theta_{t} - \dfrac{\eta}{\sqrt{\hat{v}_t} + \epsilon} m_t
The authors observe improved performance compared to Adam on small datasets and on CIFAR-10. Other experiments, however, show similar or worse performance than Adam. It remains to be seen whether
AMSGrad is able to consistently outperform Adam in practice. For more information about recent advances in Deep Learning optimization, refer to this blog post.
Other recent optimizers
A number of other optimizers have been proposed after AMSGrad. These include AdamW ^[20], which fixes weight decay in Adam; QHAdam ^[21], which averages a standard SGD step with a momentum SGD step;
and AggMo ^[22], which combines multiple momentum terms \(\gamma\); and others. For an overview of recent gradient descent algorithms, have a look at this blog post.
Visualization of algorithms
The following two animations (Image credit: Alec Radford) provide some intuitions towards the optimization behaviour of most of the presented optimization methods. Also have a look here for a
description of the same images by Karpathy and another concise overview of the algorithms discussed.
In Image 5, we see their behaviour on the contours of a loss surface (the Beale function) over time. Note that Adagrad, Adadelta, and RMSprop almost immediately head off in the right direction and
converge similarly fast, while Momentum and NAG are led off-track, evoking the image of a ball rolling down the hill. NAG, however, is quickly able to correct its course due to its increased
responsiveness by looking ahead and heads to the minimum.
Image 6 shows the behaviour of the algorithms at a saddle point, i.e. a point where one dimension has a positive slope, while the other dimension has a negative slope, which pose a difficulty for SGD
as we mentioned before. Notice here that SGD, Momentum, and NAG find it difficulty to break symmetry, although the two latter eventually manage to escape the saddle point, while Adagrad, RMSprop, and
Adadelta quickly head down the negative slope.
Image 5: SGD optimization on loss surface contours Image 6: SGD optimization on saddle point
As we can see, the adaptive learning-rate methods, i.e. Adagrad, Adadelta, RMSprop, and Adam are most suitable and provide the best convergence for these scenarios.
Note: If you are interested in visualizing these or other optimization algorithms, refer to this useful tutorial.
Which optimizer to use?
So, which optimizer should you now use? If your input data is sparse, then you likely achieve the best results using one of the adaptive learning-rate methods. An additional benefit is that you won't
need to tune the learning rate but likely achieve the best results with the default value.
In summary, RMSprop is an extension of Adagrad that deals with its radically diminishing learning rates. It is identical to Adadelta, except that Adadelta uses the RMS of parameter updates in the
numinator update rule. Adam, finally, adds bias-correction and momentum to RMSprop. Insofar, RMSprop, Adadelta, and Adam are very similar algorithms that do well in similar circumstances. Kingma et
al. ^[14:1] show that its bias-correction helps Adam slightly outperform RMSprop towards the end of optimization as gradients become sparser. Insofar, Adam might be the best overall choice.
Interestingly, many recent papers use vanilla SGD without momentum and a simple learning rate annealing schedule. As has been shown, SGD usually achieves to find a minimum, but it might take
significantly longer than with some of the optimizers, is much more reliant on a robust initialization and annealing schedule, and may get stuck in saddle points rather than local minima.
Consequently, if you care about fast convergence and train a deep or complex neural network, you should choose one of the adaptive learning rate methods.
Parallelizing and distributing SGD
Given the ubiquity of large-scale data solutions and the availability of low-commodity clusters, distributing SGD to speed it up further is an obvious choice.
SGD by itself is inherently sequential: Step-by-step, we progress further towards the minimum. Running it provides good convergence but can be slow particularly on large datasets. In contrast,
running SGD asynchronously is faster, but suboptimal communication between workers can lead to poor convergence. Additionally, we can also parallelize SGD on one machine without the need for a large
computing cluster. The following are algorithms and architectures that have been proposed to optimize parallelized and distributed SGD.
Niu et al. ^[23] introduce an update scheme called Hogwild! that allows performing SGD updates in parallel on CPUs. Processors are allowed to access shared memory without locking the parameters. This
only works if the input data is sparse, as each update will only modify a fraction of all parameters. They show that in this case, the update scheme achieves almost an optimal rate of convergence, as
it is unlikely that processors will overwrite useful information.
Downpour SGD
Downpour SGD is an asynchronous variant of SGD that was used by Dean et al. ^[10:1] in their DistBelief framework (predecessor to TensorFlow) at Google. It runs multiple replicas of a model in
parallel on subsets of the training data. These models send their updates to a parameter server, which is split across many machines. Each machine is responsible for storing and updating a fraction
of the model's parameters. However, as replicas don't communicate with each other e.g. by sharing weights or updates, their parameters are continuously at risk of diverging, hindering convergence.
Delay-tolerant Algorithms for SGD
McMahan and Streeter ^[24] extend AdaGrad to the parallel setting by developing delay-tolerant algorithms that not only adapt to past gradients, but also to the update delays. This has been shown to
work well in practice.
TensorFlow ^[25] is Google's recently open-sourced framework for the implementation and deployment of large-scale machine learning models. It is based on their experience with DistBelief and is
already used internally to perform computations on a large range of mobile devices as well as on large-scale distributed systems. For distributed execution, a computation graph is split into a
subgraph for every device and communication takes place using Send/Receive node pairs. However, the open source version of TensorFlow currently does not support distributed functionality (see here).
Update 13.04.16: A distributed version of TensorFlow has been released.
Elastic Averaging SGD
Zhang et al. ^[26] propose Elastic Averaging SGD (EASGD), which links the parameters of the workers of asynchronous SGD with an elastic force, i.e. a center variable stored by the parameter server.
This allows the local variables to fluctuate further from the center variable, which in theory allows for more exploration of the parameter space. They show empirically that this increased capacity
for exploration leads to improved performance by finding new local optima.
Additional strategies for optimizing SGD
Finally, we introduce additional strategies that can be used alongside any of the previously mentioned algorithms to further improve the performance of SGD. For a great overview of some other common
tricks, refer to ^[27].
Shuffling and Curriculum Learning
Generally, we want to avoid providing the training examples in a meaningful order to our model as this may bias the optimization algorithm. Consequently, it is often a good idea to shuffle the
training data after every epoch.
On the other hand, for some cases where we aim to solve progressively harder problems, supplying the training examples in a meaningful order may actually lead to improved performance and better
convergence. The method for establishing this meaningful order is called Curriculum Learning ^[28].
Zaremba and Sutskever ^[29] were only able to train LSTMs to evaluate simple programs using Curriculum Learning and show that a combined or mixed strategy is better than the naive one, which sorts
examples by increasing difficulty.
Batch normalization
To facilitate learning, we typically normalize the initial values of our parameters by initializing them with zero mean and unit variance. As training progresses and we update parameters to different
extents, we lose this normalization, which slows down training and amplifies changes as the network becomes deeper.
Batch normalization ^[30] reestablishes these normalizations for every mini-batch and changes are back-propagated through the operation as well. By making normalization part of the model
architecture, we are able to use higher learning rates and pay less attention to the initialization parameters. Batch normalization additionally acts as a regularizer, reducing (and sometimes even
eliminating) the need for Dropout.
Early stopping
According to Geoff Hinton: "Early stopping (is) beautiful free lunch" (NIPS 2015 Tutorial slides, slide 63). You should thus always monitor error on a validation set during training and stop (with
some patience) if your validation error does not improve enough.
Gradient noise
Neelakantan et al. ^[31] add noise that follows a Gaussian distribution \(N(0, \sigma^2_t)\) to each gradient update:
\(g_{t, i} = g_{t, i} + N(0, \sigma^2_t)\).
They anneal the variance according to the following schedule:
\( \sigma^2_t = \dfrac{\eta}{(1 + t)^\gamma} \).
They show that adding this noise makes networks more robust to poor initialization and helps training particularly deep and complex networks. They suspect that the added noise gives the model more
chances to escape and find new local minima, which are more frequent for deeper models.
In this blog post, we have initially looked at the three variants of gradient descent, among which mini-batch gradient descent is the most popular. We have then investigated algorithms that are most
commonly used for optimizing SGD: Momentum, Nesterov accelerated gradient, Adagrad, Adadelta, RMSprop, Adam, as well as different algorithms to optimize asynchronous SGD. Finally, we've considered
other strategies to improve SGD such as shuffling and curriculum learning, batch normalization, and early stopping.
I hope that this blog post was able to provide you with some intuitions towards the motivation and the behaviour of the different optimization algorithms. Are there any obvious algorithms to improve
SGD that I've missed? What tricks are you using yourself to facilitate training with SGD? Let me know in the comments below.
Thanks to Denny Britz and Cesar Salgado for reading drafts of this post and providing suggestions.
Printable version and citation
This blog post is also available as an article on arXiv, in case you want to refer to it later.
In case you found it helpful, consider citing the corresponding arXiv article as:
Sebastian Ruder (2016). An overview of gradient descent optimisation algorithms. arXiv preprint arXiv:1609.04747.
This blog post has been translated into the following languages:
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Finite Set Algebra in Secondary School Using Raspberry Pi with Mathematica
The objective of this research is to automate the graphical representation of Finite Set Algebra, in order to provide: an alternative as a learning tool or a means to teach the association of
mathematical concepts with the programming language, determine the construction rules and the Command programming in the subject of Basic Education Mathematics. This document describes a new
Mathematica package, DiscreteSets.m programmed in PiWolfram. Since the Mathematica license is expensive, a Raspberry Pi computer is used since, thanks to its low cost and a free version of
Mathematica that runs on it, it makes the new Mathematica package accessible to a greater number of users. To show the performance of the new package, a full set of illustrative examples is provided.
This package admits as input a certain operation between two or three discrete and finite sets, whose elements can be numbers, letters or certain figures. And it returns as output the Venn diagram
corresponding to the given operation, with the location of the elements of the sets similar to how they would be located manually. Allowing students to interactively learn and dabble in the PiWolfram
programming language. Programming has been the cause of the great advances in technology. For this reason, it is important to incorporate technological education into the learning of Mathematics so
that students appreciate the real importance of mathematics today. Through mathematics we can understand the world around us. It is present in various human activities: daily life, social and
cultural aspects, in such a way that it is an essential key to understanding and transforming culture. Having a mathematical development allows us to participate in the world in any of the
aforementioned aspects.
Idioma original Inglés
Título de la publicación alojada Computational Science and Its Applications – ICCSA 2021 - 21st International Conference, Proceedings
Editores Osvaldo Gervasi, Beniamino Murgante, Sanjay Misra, Chiara Garau, Ivan Blečić, David Taniar, Bernady O. Apduhan, Ana Maria Rocha, Eufemia Tarantino, Carmelo Maria
Editorial Springer Science and Business Media Deutschland GmbH
Páginas 366-379
Número de páginas 14
ISBN (versión impresa) 9783030869694
Estado Publicada - 2021
Evento 21st International Conference on Computational Science and Its Applications, ICCSA 2021 - Virtual, Online
Duración: 13 set. 2021 → 16 set. 2021
Serie de la publicación
Nombre Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volumen 12951 LNCS
ISSN (versión impresa) 0302-9743
ISSN (versión digital) 1611-3349
Conferencia 21st International Conference on Computational Science and Its Applications, ICCSA 2021
Ciudad Virtual, Online
Período 13/09/21 → 16/09/21
Profundice en los temas de investigación de 'Finite Set Algebra in Secondary School Using Raspberry Pi with Mathematica'. En conjunto forman una huella única. | {"url":"https://cris.unp.edu.pe/es/publications/finite-set-algebra-in-secondary-school-using-raspberry-pi-with-ma","timestamp":"2024-11-09T06:30:36Z","content_type":"text/html","content_length":"65688","record_id":"<urn:uuid:8a389871-716e-4d2d-b0a7-bbddc2f5f681>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00536.warc.gz"} |
Gradient Descent for Machine Learning
Imagine hiking in the mountains past dusk and finding that you can’t see much beyond your feet. And your phone’s battery died so you can’t use a GPS app to find your way home. You might find the
quickest path down via gradient descent. Just be careful not to walk off a cliff.
Suns and rugs: French mathematician Augustin-Louis Cauchy invented the algorithm in 1847 to approximate the orbits of stars. Sixty years later, his compatriot Jacques Hadamard independently developed
it to describe deformations of thin, flexible objects like throw rugs that might make a downward hike easier on the knees. In machine learning, though, its most common use is to find the lowest point
in the landscape of a learning algorithm’s loss function.
Downward climb: A trained neural network provides a function that, given an input, computes a desired output. One way to train the network is to minimize the loss, or error in its output, by
iteratively computing the difference between the actual and desired output and then changing the network’s parameter values to narrow the difference. Gradient descent accomplishes this by minimizing
the function that computes the loss.
• The network’s parameter values are tantamount to a position on the landscape, and the loss is the current altitude. As you descend, you improve the network’s ability to compute outputs close to
the desired one. Visibility is limited because, in a typical supervised learning situation, the algorithm relies solely on the network’s parameter values (your position on the hill) and the
gradient (the slope immediately beneath your feet).
• The basic method is to move in the direction where the terrain descends most steeply. The trick is to calibrate your stride. Too small, and it takes ages to make any progress. Too large, and you
leap into the unknown, possibly heading uphill instead of downward.
• Given the current position, the algorithm estimates the direction of steepest descent by computing the gradient of the loss function. The gradient points uphill, so the algorithm steps in the
opposite direction by subtracting a fraction of the gradient. The fraction α, which is called the learning rate, determines the size of the step before measuring the gradient again.
• Apply this iteratively, and hopefully you’ll arrive at a valley.
Stuck in the valley: Too bad your phone is out of juice, because the algorithm may not have propelled you to the bottom of a convex mountain. Instead, you may be stuck in a nonconvex landscape of
multiple valleys (local minima), peaks (local maxima), saddles (saddle points), and plateaus. In fact, tasks like image recognition, text generation, and speech recognition are nonconvex, and many
variations on gradient descent have emerged to handle such situations. For example, the algorithm may have momentum that helps it zoom over small rises and dips, giving it a better chance at
arriving at the bottom. Researchers have devised so many variants that it may seem as though there are as many optimizers as there are local minima. Luckily, local and global minima tend to be
roughly equivalent.
Optimal optimizer: Gradient descent is the clear choice for finding the minimum of any function. In cases where an exact solution can be computed directly — say, a linear regression task with lots of
variables — it can approximate one, often faster and more cheaply. But it really comes into its own in complex, nonlinear tasks. Armed with gradient descent and an adventurous spirit, you might just
make it out of the mountains in time for dinner. | {"url":"https://www.deeplearning.ai/the-batch/gradient-descent-its-all-downhill/","timestamp":"2024-11-06T18:19:49Z","content_type":"text/html","content_length":"80550","record_id":"<urn:uuid:e9f9b7aa-25d2-4d77-8513-4d138c56b0d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00794.warc.gz"} |
SVSF-Based Robust UGV/UAV Control/Tracking Architecture in Disturbed Environment
Intelligent Automation & Soft Computing
SVSF-Based Robust UGV/UAV Control/Tracking Architecture in Disturbed Environment
1Ecole Militaire Polytechnique, Algiers, Algeria
2Centre de Développement des Technologies Avancées, Algiers, Algeria
*Corresponding Author: Abdelatif Oussar. Email: oussarabdelatif@gmail.com
Received: 04 December 2019; Accepted: 03 July 2020
Abstract: This paper presents the design of a robust architecture for the tracking of an unmanned ground vehicle (UGV) by an unmanned aerial vehicle (UAV). To enhance the robustness of the ground
vehicle in the face of external disturbances and handle the non-linearities due to inputs saturation, an integral sliding mode controller was designed for the task of trajectory tracking.
Stabilization of the aerial vehicle is achieved using an integral-backstepping solution. Estimation of the relative position between the two agents was solved using two approaches: the first solution
(optimal) is based on a Kalman filter (KF) the second solution (robust) uses a smooth variable structure filter (SVSF). Simulations results, based on the full non-linear model of the two agents are
presented in order to evaluate the performance and robustness of the proposed tracking architecture.
Keywords: UGV/UAV tracking; integral sliding mode controller; trajectory tracking; integral-backstepping controller; Kalman filter; robust smooth variable structure filter
(xr,yr,θr) : Coordinates of UGV reference trajectory
(vr,wr) : UGV desired control inputs
(x,y,θ) : UGV actual coordinates
(v,w) : UGV control inputs
(δv,δw) : Uncertainties on v and w
r : Radius of UGV wheels
2L : The distance between the two driving wheels
(φ˙d,φ˙l) : Angular velocity of the two driving wheels
(ex,ey,eθ) : Tracking errors
(e1,e2,e3) : Tracking errors expressed in the UGV frame
U0 , U1 : Nominal and discontinuous part of the ISMC respectively
(vmax,wmax) : The bounds of the UGV controls
s=[s1,s2]T : Sliding surface
Xk : State vector describing the UGV motion
T : Sampling period
wk : Random process noise
zk : The measurements of the UGV position Pk
vk : Random measurement noise
(x,y,z,φ,θ,ψ) : Quadrotor position and orientation
(u1,u2,u3,u4) : Quadrotor control inputs
(ux,uy) : Quadrotor virtual control inputs
Unmanned aerial/ground cooperation is increasingly attracting the attention of researchers. This is essentially due to the complementary skills provided by each type to overcome the specific
limitations of each other. UGVs offer a higher payload and stronger calculation capabilities, while UAVs provide faster dynamics, and add a local coverage for the unseen areas from an aerial view [1
]. Indeed, deployment of integrated multi-robot team consisting of heterogeneous robots provides advantages compared to strict homogeneous compositions. One of the attractive scenarios for
multi-agent system is the tracking of a ground target using an unmanned aerial vehicle [2]. This allows performing important tasks like surveillance of convoys, reconnaissance, and intelligence
missions [3]. Tracking ground targets is more difficult than aerial ones due to the topographic variations that can influence a target’s motion patterns and obscurity to observation [4].
Many results related to this topic have been presented in the last few years. A circular pattern navigation algorithm for autonomous target tracking was presented in Rafi et al. [5] and Wise et al. [
6], showing a good performance in simulation. Solutions based on partial information of the target state were presented in Peterson et al. [7], Summers et al. [8] and Kim et al. [9]. Observers,
adaptive control, and extended Kalman filtering were used in this works for estimating the full target state. In Kim et al. [9] a non-linear model predictive controller was used to achieve the
desired standoff configuration for an accelerating target. Quintero et al. [10] presented an output-feedback model predictive control with moving horizon estimation for target tracking by UAV,
showing a good robustness.
Other works dealt with trajectory acquisition from video cameras, using particle filters for target estimation from non-stabilised cameras [11]. Multiple target tracking was achieved using Joint
Probabilistic Data Association Filter (JPDAF) in the presence of unreliable target identification [12]. If a model for the object’s motion is known, an observer can be used to estimate the object’s
velocity [13]. The recent work [14] covered one of the most important applications of estimation theory, namely, multi-target tracking, and included a thorough treatment of multisensor fusion and
multiple hypothesis tracking, attribute-aided tracking, unresolved targets, sensor management, etc.
Air-ground collaborative systems find their applications in several fields. Authors in [15–21] present significant studies on Intelligence Surveillance and Reconnaissance (ISR) missions including
both aerial and terrestrial vehicles. On the other hand, object tracking, path planning and localization are the others missions where this UAV/UGV cooperation is beneficial. We quote here some
noteworthy studies [22–26] related with UAV/UGV systems collaborating to perform the above-mentioned tasks. As important part of this air-ground cooperation, many works have focused on “formation
control” [27–29]. However, such hybrid UAV/UGV architecture can combine their tasks to achieve more complex missions [30–32].
More recently many studies have been made proposing new approaches to path planning for heterogeneous cooperating team (Air-ground Coordination). In Yulong et al. [33], Dubins path planning combined
with Traveling Salesman problem was proposed to find the shortest route. A Quaternion based control for circular UAV trajectory tracking, following a ground vehicle was proposed in Abaunza et al. [34
Guastella et al. [35] designed a global path planning strategy for a UGV from aerial elevation maps for disaster response. Peterson et al. [36] the authors present a collaborative UAV/UGV system to
online aerial terrain mapping to inform the ground vehicle’s path planning in real time.
This kind of system can have practical applications for search and rescue missions. Some recent projects such as ICARUS [37] and ANKommEn [38] aims to develop a platform using multiple UAV and UGV
for exploration of disaster scenarios for the detection of survivors and to provide maps in order to assist in maximizing the efficiency of Search and Rescue (SAR) operations.
In this paper, we first focus on the modeling and robust control of the two heterogeneous robots constituting the cooperative system. The second part covers the tracking problem of the ground agent
by the UAV taking into account the dynamics of the aerial agent and the kinematics of the UGV. Since direct measurements are tainted with noise, it is essential to integrate an estimation filter
allowing the prediction and estimation of the state of the ground target. We tested two estimation algorithms: the standard Kalman filter and the smooth variable structure filter.
The main contribution of this paper is the design of a new robust architecture for the tracking of an UGV by an UAV in order to deal with external disturbances and model uncertainties. To achieve
this goal, the stabilization of the UAV is performed based on the integral-backstepping control approach while an integral sliding mode controller was designed for trajectory tracking. In addition, a
comparative study of the proposed approaches for estimating the relative position between the two agents (UGV and UAV), is accomplished, illustrating the advantages of the proposed architecture.
The rest of this paper is organized as follows: in Section 2, the kinematic model of the UGV is presented and an integral sliding mode controller is designed in order to control the UGV motion.
Section 3 provides the dynamic model of the UAV (quadrotor) together with the proposed control law based on the integral-backstepping approach. Section 4 presents the tracking algorithms used to
estimate the state of the ground vehicle. In order to evaluate the proposed control and tracking architecture, simulation results with different scenarios are presented in Section 5.
2 UGV Modelling, Control and Trajectory Tracking
2.1 Formulation of the Trajectory Tracking Problem
We assume that the reference trajectory, generated by the motion planning algorithm, fulfils the following model:
[x˙ry˙rθ˙r]=[cosθr0sinθr001][vrwr] (1)
where xr , yr and θr represent the desired (x,y) position and orientation of the UGV, vr and wr are the desired linear and angular velocities respectively.
It is obvious that the real controls v and w rely on the state measurements x,y and θ (Fig. 1). Due to measurement noise and modeling uncertainties, here we consider input uncertainties for both v
and w [39]. Thus, the real equation of the robot trajectory fulfills the following model:
[x˙y˙θ˙]=[cosθ0sinθ001][v+δvw+δw] (2)
where the linear and angular velocities, v and w , are defined as follows:
v=r2(φ˙d+φ˙l) (3)
w=r2L(φ˙d−φ˙l) (4)
parameter r represents the radius of the wheels; 2L is the distance between the two driving wheels; φ˙l and φ˙d represent the angular velocity of two wheels left and right-hand side respectively; δv
and δw are the uncertainties on v and w .
The objective of trajectory tracking is to asymptotically stabilize the tracking errors ex=xr−x , ey=yr−y and eθ=θr−θ to zero.
Transforming the tracking errors expressed in the inertial frame to the robot frame, the error coordinates can be denoted as follows:
[e1e2e3]=[cosθsinθ0−sinθcosθ0001][exeyeθ] (5)
Thus, the tracking-error model is represented by the following equation:
e˙=f1(e)+f2(e)(U+δ) (6)
{e=[e1,e2,e3]Tf1(e)=[vrcose3,vrsine3,wr]Tf2(e)=[−1e20−e10−1]U=[v,w]Tandδ=[δv,δw]T (7)
2.2 Integral Sliding Mode Controller Design
Sliding Mode Control (SMC) is widely used in the control of different type of systems, such as wheeled mobile robot [40], five DOF redundant robot [41], PH process in stirred tanks [42], tunnel bow
thrusters [43] or for other purpose such as multi-sensors data fusion [44]. In this part, we propose to enhance the SMC with the integral action, to improve its disturbances rejection. This
controller is then applied to our UGV system.
For system Eq. (5), the control law is defined as follows:
U=U0+U1 (8)
U0 is the nominal control and U1 represents the ISMC part which is designed to be discontinuous in order to reject the disturbance.
The first part of the control design is to find a saturated control law U0 so that the nominal system e˙=f1(e)+f2(e)U0 is globally asymptotically stable (see Jiang et al. [45] for more details). The
nominal control input is chosen as follows:
U0=[v0=vrcose3+λ3tanhe1w0=wr+λ1vre2sine3e3(1+e12+e22)+λ2tanhe3] (9)
The positive parameters λ1 , λ2 and λ3 can be designed so that the bounds of the controls are complied with. This can be represented as follows:
|v0|≤vmax+λ3,|w0|≤wmax+λ1vmax2+λ2 (10)
For the ISMC part U1 , the sliding variable ‘ s ’ is defined as follows:
s=[s1,s2]T=s0(e)+z (11)
{s0(e)=[−e1,−e3]Tz˙=−∂s0∂e(f1(e)+f2(e)U0)z(0)=[e1(0),e3(0)]T (12)
The variable z includes the integral term and provides one more degree of freedom in the construction of the sliding variable. According to Defoort et al. [46], the sliding mode is established as the
initial moment and the phase of convergence is eliminated. Then, the control law is given by the following equation:
U1=[−K1sign(s1)−K2sign(−e2s1+s2)] (13)
with K1>δv+μ and K2>δw+μ such as: μ>0
In order to reduce the chattering phenomena, the sign function is replaced by: f(x)=2πtanh(ηx) , with η being a positive constant [47]. Following from this, Eq. (13) can be written as follows:
U1=[−2K1πtanh(η1s1)−2K2πtanh(η2(−e2s1+s2))] (14)
Remark 1: The trajectory evolves on the manifold s=0 from t=0 and remains there in presence of the disturbances. The time derivative of the sliding variable is s˙=∂s0∂e(e˙−f1(e)−f2(e)U0) . Therefore,
the motion equation in sliding mode is e˙=f1(e)+f2(e)U0 which is globally asymptotically stable (see Defoort et al. [39] for more details).
2.3 Stability Analysis of the Smooth ISMC
Let us study the effect of the approximation of the sign function by tanh on global stability.
Lemma 1. [48] For every given scalar x and positive scalar η , the following inequality holds:
Proof of Lemma 1. According to the definition of tanh function, we have:
Proof of stability. Consider the following Lyapunov function candidate:
The discontinuous control term must satisfy the condition V˙≤0 guaranteeing the global asymptotic stability.
V˙=sTs˙=sT(s˙0(e)+z˙) V˙=sT[∂s0∂ee˙−∂s0∂e(f1(e)+f2(e))U0] V˙=sT∂s0∂ef2(e)(U1+δ) V˙=(s1−e2s1+s2)(U1+δ)
According to Lemma 1, the above condition is satisfied if:
with 2K1π>δv+μ and 2K2π>δw+μ ( μ>0 )
Therefore, according to LaSalle’s theorem, the control system with the smoothed ISMC is asymptotically stable in the sense of Lyapunov.
2.4 Tracking Algorithm based on Kalman Filter
Ground targets (UGVs) were always implicitly assumed to be non-manoeuvring and the noise statistics involved in the dynamics model/target observation (matrices Q(k) and R(k) ) were assumed to be
known. In practice, it goes without saying that these parameters are never well known and may vary over time depending on the manoeuvring capacity of the ground targets. For this reason, we have
opted for the use of a discrete kinematic model with quasi-constant acceleration [49]. We can model the equations of the UGV as a linear system in following representation:
Xk+1=AXk+Γwk (15)
A=[1T12T200001T0000010000001T12T200001T000001] (16)
Γ=[12T20T010012T20T01] (17)
and wk is the random process noise. Xk is the state vector describing the motion of the UGV (its position, velocity and acceleration):
and T is the sampling period. The measurements of the mobile robot position p at time k are described as follows:
Zk=Hkpk+vk (18)
where vk is a random measurement noise. We assume that the process and measurement noises wk and vk are white, zero-mean.
As the measurement vector is the UGV position (xk,yk) , the matrix Hk in Eq. (18) (called measurement matrix) is given by:
The predicting and update equations for the Kalman filter are presented as follows:
{X^k+1|k=AkX^k|kPk+1|k=AkPk|kAkT+Qky~k+1=Zk−HkXk+1|kSk+1=Hk+1Pk+1|kHk+1T+Rk+1Kk+1=Pk+1|kHk+1TSk+1−1X^k+1|k+1=X^k+1|k+Kky~k+1Pk+1|k+1=(I−Kk+1Hk+1)Pk+1|k (19)
Let us describe the Kalman filter parameters ( X0,Rk,Qk and P0 ) selection:
• For the initialization ( X0 , P0 ), several simulations were performed by considering initialization accuracy up to 70% the true value of the UGV position. We have noticed that the results of the
estimation and the convergence speed of the filter are highly dependent on initialization. The suitable values of ( X0 , P0 ) are estimated after many simulations.
• The selection of matrices Rk and Qk requires prior knowledge of measurement and process noises. The covariance of measurement noise vk can be expressed as:
Rk=(σx200σy2) ; where σx and σy are the standard deviations of the position of x and y, respectively. In our case, the sensor used to measure the UGV position is assumed to be a camera embedded on
the UAV. Thus, in our simulations the matrix Rk is selected based on real camera characteristics.
The covariance matrix of the process noise ( Qk ) is estimated based on the odometer model of the mobile robot given by Eq. (2), in our simulation a suitable values of the matrix Qk are determined
from a real robot (Pioneer 3-AT).
3 Quadrotor Dynamics and Control
Our interest in such type of UAV (Fig. 2) is its hovering capability and high manoeuverability.
The dynamic models of the quadrotor are well studied. The details of the following Newton-Euler equations can be found in Refs. [50–53].
{ϕ¨=Jy−JzJxθ˙ψ˙−JrJxΩ¯θ˙−KfaxJxϕ˙2+1Jxu2θ¨=Jz−JxJyϕ˙ψ˙+JrJyΩ¯ϕ˙−KfayJyθ˙2+1Jyu3ψ¨=Jx−JyJzθ˙ϕ˙−KfazJzψ˙2+1Jzu4x¨=−Kftxmx˙+1muxu1y¨=−Kftymy˙+1muyu1z¨=−Kftzmz˙−g+1m(cosϕcosθ)u1 (20)
The control inputs of the UAV are defined as follows:
{u1=b(w12+w22+w32+w42)u2=lb(w42−w22)u3=lb(w32−w12)u4=d(w12−w22+w32−w42)ux=(cosϕcosψsinθ+sinϕsinψ)uy=(cosϕsinψsinθ−sinϕcosψ)Ω¯=(w1−w2+w3−w4) (21)
The description of the parameters used in this model is given in Tab. 1.
3.2 Backstepping Controller (BC) Design
In order to control the UAV a backstepping control scheme is used. The inner controller stabilizes the orientation angles in order to achieve a stable flight while the outer controller is responsible
for the control of the position of UAV. Further the state vector of the UAV is defined as follows:
Thus we obtain the following equations:
{x˙1=x2x˙2=a1x4x6+a2x22+a3Ω¯x4+b1u2x˙3=x4x˙4=a4x2x6+a5x42+a6Ω¯x2+b2u3x˙5=x6x˙6=a7x2x4+a8x62+b3u4x˙7=x8x˙8=a9x8−g+1m(cosx1cosx3)u1x˙9=x10x˙10=a10x10+1muxu1x˙11=x12x˙12=a11x12+1muyu1 (22)
The following Lyapunov functions are used:
Vi={12zi2/i∈{1,3,5,7,9,11}12(Vi−1+zi2)/i∈{2,4,6,8,10,12} (23)
The application of the backstepping technique [50] and [54] on the quadrotor state model give the following control inputs:
ux=mu1(z9−a10x10−α9(z10+α9z9)−α10z10+x¨9d)uy=mu1(z11−a11x12−α11(z12+α11z11)−α12z12+x¨11d) (24)
with cosx1cosx3≠0,αi>0 and ∀i∈{1,2…,12} . Variables ux and uy are virtual control inputs which will be used to find desired Euler angles as follows:
{ϕd=x1d=arcsin(uxsinx5d−uycosx5d)θd=x3d=arcsin(uxcosx5d+uysinx5dcosx1d) (25)
The simulation results are obtained based on the following realistic parameters of quadrotor in Tab. 2 and characteristics of Pioneer 3-AT mobile robot in Tab. 3.
In this first simulations set, we aim to evaluate the ISMC controller of the ground agent (UGV), thus the first tracking approach based on the Kalman filter and the backstepping control of the
In this context of the ground agent tracking by a UAV, and in order to get closer to reality, several scenarios are considered as follows.
• The initial conditions of the UGV: x0=0.5y0=−0.5θ0=π4
• The initial conditions of the quadrotor: x0=−0.5y0=−0.5z0=0
• The control inputs of the quadrotor are bounded as follows:
{0≤u1≤4bΩmax2|u2|≤lbΩmax2|u3|≤lbΩmax2|u4|≤2dΩmax2 (26)
• The initial parameters of the Kalman estimator are given as follows:
{X0=[0.400−0.400]Rk=diag([0.0520.032])Qk=Γ(5.10−3)2ΓTP0=zeros(6,6) (27)
Scenario 1: The quadrotor may be affected by external disturbances such as wind. Several models of wind are proposed in Gawronski [55]. In our work, we assume that the wind has caused the same
acceleration intensity on all axes x , y and z [56], as shown in Fig. 3. The mathematical model of wind is given by the following equation:
(t−45)11)when45s<t≤65s0when60s<t≤70s (28)
• Scenario 2: This scenario is designed to evaluate the robustness of the control system with respect to modelling errors and measurement noises. In terms of parametric uncertainty, we assume that
the elements of the inertia matrix Jx , Jy and Jz are underestimated to 60%, the coefficients b and d are also underestimated, whereas the values used in the control are only 80% of the actual
{J~x=0.6JxJ~y=0.6JyJ~z=0.6Jzb~=0.8bd~=0.8d (29)
As for the measurement noises, an additive Gaussian white noise with the density 150μg/Hz was considered.
In the presence of wind gusts (Fig. 3) and as shown in Fig. 4, the quadrotor tracks the UGV that traveled Eight trajectory using the ISMC controller. By looking at Figs. 5 and 6, it can be noticed
that the ISMC allows the robot to follow the set point correctly according to the two axes x,y and accounts for the saturation constraints with a very low tracking error on the trajectory realized by
the UGV. The two sliding surfaces (s1,s2) tend to zero at the initial moment. Figs. 7 and 8 show that the wind affects the performances of the quadrotor during its mission, resulting in maximum
absolute tracking errors of 0.27 m along z , 0.18 m along x and y .
As for scenario 1, even with the existence of parametric uncertainties and measurement noises, the quadrotor succeeds in tracking the mobile robot with small fluctuations in its tracking trajectory (
Fig. 9). The tracking error along x axis takes a value of approximately 0.35 m, which shows that these disturbances have significant effects on the control by backstepping as indicated in Figs. 10
and 11.
4 Robust Control/Tracking Architecture
4.1 Robust Tracking algorithm based on SVSF
One of the major challenges for the tracking algorithm is the uncertainty in the motion of UGV. This uncertainty refers to the fact that a precise dynamic model of the movement is not available at
the level of the tracking algorithm [57,58]. However, the Kalman filter (KF) can only achieve a good performance (optimal solution) under the assumption that the complete and exact information of the
process model and the noise distribution are to be known as a prior. In practice, state and observation models are often poorly known, or contain uncertain parameters, and the statistical properties
of noise (state and observation) are also poorly known, coming to the optimality of the solution obtained. Therefore, to improve our tracking algorithm, and to overcome these limitations, we propose
to use a new filter or estimator called Smooth Variable Structure Filter (SVSF) to process the tracking problem of a UGV [59,60].
The Smooth Variable Structure Filter is a relatively new estimating strategy proposed by Habibi in 2007 [61]. This strategy is based on the concepts of sliding mode control and the theory of systems
with variable structure, outcome and similar design to variable structure filter (VSF) [62]. This filter is formulated in the predictive-correction format, and can be used for linear or non-linear
systems. It uses a correction gain simpler than the one used by the VSF. The SVSF is introduced to provide more stability and robustness to the estimation process. This technique is generally used
for the estimation of states and parameters of dynamic systems [63], the prediction and diagnosis of defects in systems [64] and targets tracking problems [59,60,65].
To formulate the tracking problem, we use the same model that has been described in detail by the two Eqs. (15) and (18).
The SVSF estimation method is described by the following series of equations. Note that this formulation includes state error covariance equations as presented in Gadsden et al. [66], which was not
originally presented in the standard SVSF form [61]. The prediction stage is similar to the KF; its steps are as follows:
- Initialization
{X^0|0=X0E^0|0=E0 (30)
- Prediction
X^k+1|k=AkX^k|k (31)
Pk+1|k=AkPk|kAkT+Qk (32)
where X^k|k is the state estimated at time k of the state Xk|k .
Y^k+1|k=HX^k+1|k (33)
then, pretest measurement error is calculated by the following equation:
Ek+1|k=Yk+1−Y^k+1|k (34)
- Update
For the state estimate, the SVSF correction gain is calculated by MacArthur et al. [22] and Phan et al. [31]:
Kk+1|k=H+Diag[(|Ek+1|k|abs+γ|Ek|k|abs)∘Sat(Ψ¯−1Ek+1|k)][Diag(Ek+1|k)]−1 (35)
where ∘ signifies Schur (or element-by-element) multiplication, the superscript + refers to the pseudo inverse of a matrix and Ψ¯−1 is a diagonal matrix constructed from the smoothing boundary layer
vector Ψ , defined as follows:
Ψ¯−1=[Diag(Ψ)]−1=[1Ψ1000⋱0001Ψm] (36)
The form of saturation used in Eq. (35) is defined as follows:
Sat(Ψ¯−1Ek+1|k)={Ek+1|kiΨi≥1−1<Ek+1|kiΨi<1Ek+1|kiΨi≤−1 (37)
The gain is used to update the predicted state as follows:
X^k+1|k+1=X^k+1|k+Kk+1|kEk+1|k (38)
The covariance associated with the state updates is then calculated as follows:
Pk+1|k+1=[I−Kk+1|kHK]Pk+1|k[I−Kk+1|kHK]T+Kk+1|kRk+1Kk+1|kT (39)
Thus, the estimated measurement and the corresponding empirical measurement error are calculated as follows:
{Y^k+1|k+1=HX^k+1|k+1Ek+1|k+1=Yk+1−Y^k+1|k+1 (40)
Two critical variables in this process are the pretest and empirical measurements (output) error estimates, defined by Eqs. (34) and (40). It shall be noted that Eq. (40) is the empirical measurement
error estimates from the previous time step, and is used only in the gain calculation.
The selection of the smoothing boundary layer width vector Ψ reflects the level of uncertainties in the filter and the disturbances (i.e., system and measurement noise, and uncertain parameters).
4.2 Integral-Backstepping Controller (IBC)
The backstepping control cannot ensure the favorable tracking performance of the quadrotor if unpredictable disturbances from the unknown external disturbance, modeling errors, as well as measurement
noise occur. In order to improve these performances and consequently the robustness, we propose to combine the conventional PID with the backstepping control. This will allow for integral
backstepping. However, this control technique has been proposed in several research studies [50,52,67,68,69], which demonstrated that the integral backstepping controller allows rejection of external
disturbances and is robust to parametric uncertainties.
The application of the integral backstepping control on the quadrotor state model gives the following control inputs:
(z5−a7x2x4−a8x62+x¨5d−α5(α5z5+z6+λ3χ3)+λ3z5−α6z6)ux=mu1(z9−a10x10−α9(z10+α9z9+λ5χ5)+λ5z9−α10z10+x¨9d)uy=mu1(z11−a11x12−α11(z12+α11z11+λ6χ6)+λ6z11−α12z12+x¨11d) (41)
Such as:
{αi>0∀i∈{1,2,3…12}zj=xjd−xj/j∈{1,3,5,7,9,11}χ˙j=zj (42)
and the Lyapunov functions take the following form:
Vi={12zi2+λj2χi2/i∈{1,3,5,7,9,11}j∈{1,2,3…6}Vi−1+12zi2/i∈{2,4,6,8,10,12} (43)
For the selection of the controller parameters, we have used an approach based on PSO (Particle Swarm Optimization) optimization method, more details can be found in Yacef et al. [70].
After explaining and presenting the various tracking algorithms (KF and SVSF), showing the principle and the mathematical development. Their estimation accuracy and robustness to different types of
noise will be evaluated. The root mean square error (RMSE) of the different results is calculated for different scenarios.
Scenario 3: In this scenario, the favorable conditions for the Kalman filter will be placed, by applying on the states and on the obtained measurements decorrelated centered noises as covariance
{Qk=Γ(2.10−2)2ΓTRk=Diag([0.052;0.052]) (44)
The γ , Ψ matrices used and the initial states X0 , P0 are as follows:
{γ=Diag([0.8;0.8])Ψ=Diag([6;5])X0=[0.4;0;0;−0.4;0;0]P0=10×I(6×6) (45)
The expression of RMSE on the estimate is given by the following equation: RMSE=(1/N)eTe .
The expression of RMSE on the estimation of the position of the mobile robot is given by the following equation: RMSEr=RMSEx2+RMSEy2 .
The implementation results under Matlab are shown in Fig. 12 for the two algorithms. In order to evaluate the estimation accuracy, we calculated the RMSE on the estimate. Tab. 4 shows the results
It was found that the values of RMSE of the Kalman filter are lower than those of the SVSF. In this scenario we deduced that the estimation of the trajectory of the UGV by the KF is more accurate in
comparison with the SVSF.
Scenario 4: In this scenario, unfavorable conditions for the Kalman filter will be considered, in order to show the efficiency, robustness and superiority of the SVSF with respect to the KF, when the
initial conditions are poorly chosen (the initial conditions are increased by a factor of 10), so that noises on states and measurements are Gaussian, correlated, non-centered:
{Qk=Γ[0.0220.0120.0120.022]ΓT;E[vk]=[10−210−2]Rk=[0.0520.0220.0220.052];E[wk]=[0.040.04] (46)
Fig. 13 shows the estimated trajectories of the UGV and the RMSE are given in Tab. 5.
It was found that the values of RMSEx , RMSEy and RMSEr are lower compared to those of KF which has poor estimation accuracy, due to the nature of the non-centered correlated noises and the poor
choice of the initial conditions. Ultimately, the SVSF has also proven to be more robust, stable and accurate.
5.2 Robust Ground Agent Tracking using SVSF and IBC of UAV
In order to evaluate the performance of this architecture, we mainly integrate in this case two scenarios 1 and 2 described above and based on the results of estimation of the SVSF to carry out
missions of tracking a ground agent. The initial conditions and constraints are the same. The initial parameters of the SVSF estimator are given in Eqs. (44) and (45).
From the results of Fig. 14, it is clear that the IBC control is more efficient. This control (Fig. 15) reduces tracking errors. For example, in Scenario 1, we obtained 0.08 m in x , y and 0.25 m in
z (Fig. 16) with the BC 0.18 m errors in x , y and 0.35 m in z . Fig. 17 shows that the integral backstepping control has greatly reduced the effect of parametric uncertainties and measurement noises
(scenario 2).
Thus, it can be deduced that the IBC makes it possible to obtain a better robustness with respect to the parametric uncertainties and a better rejection of the external disturbances with respect to
the BC.
This paper proposed a robust control and tracking architecture in order to allow for an UAV to track an UGV in disturbed environment. The considered UGV is a Unicycle mobile robot. On one hand, the
latter has been controlled based on the Integral Sliding Mode technique taking into account the kinematics constraints on the speed limitations. A tracking algorithm based on the Kalman filter was
introduced in order to estimate the relative state of the UGV in a disturbed environment. On the other hand, a considered UAV type quadrotor and a backstepping controller is designed to stabilize
this UAV. A first set of simulations was performed by considering several scenarios. The simulation results of this tracking architecture have shown limited robustness with respect to external
disturbances, modeling errors and measurement noises.
In order to improve the performance of this architecture, the Kalman filter has been replaced by the Smooth Variable Structure Filter and the integral-backstepping controller was introduced in order
to overcome the challenges of classical backstepping robustness. The stability of the synthesized control laws has been proved by the Lyapunov theory; which is necessary to achieve UGV/UAV
cooperation architecture. The second set of simulations considering the proposed architecture has shown the improvement of robustness and accuracy of this architecture.
Current and future works concern the implementation of the proposed architecture and algorithms on a Pixhawk autopilot for UAV control and Raspberry Pi based vision module for automated UGV target
visual detection, recognition and tracking.
Acknowledgement: We would like to thank the staff of the Ecole Militaire Polytechnique of Algiers, especially Doctor Oualid Araar, for the assistance afforded to perform this research.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding this study.
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This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited. | {"url":"https://www.techscience.com/iasc/v29n2/42932/html","timestamp":"2024-11-05T16:18:56Z","content_type":"application/xhtml+xml","content_length":"322751","record_id":"<urn:uuid:4d17eff0-d051-4d4e-9e82-fe8590af8473>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00244.warc.gz"} |
Author(s) of this documentation:
Jacques-Olivier Lachaud, Jocelyn Meyron, Tristan Roussillon
Part of the Geometry package.
This part of the manual describes what are plane-probing estimators, how to define and use them to estimate normals on digital surfaces.
The following programs are related to this documentation: geometry/surfaces/examplePlaneProbingTetrahedronEstimator.cpp, geometry/surfaces/examplePlaneProbingParallelepipedEstimator.cpp, geometry/
surfaces/examplePlaneProbingSurfaceLocalEstimator.cpp, testDigitalPlanePredicate.cpp, testPlaneProbingTetrahedronEstimator.cpp, testPlaneProbingParallelepipedEstimator.cpp.
Introduction to plane-probing algorithms
A plane-probing algorithm (see [73], [106], [77] and [87]) computes the normal vector of a set of digital points from a starting point and a predicate InPlane: "Is a point x in the set of digital
points?". This predicate is used to probe the set as locally as possible and decide on-the-fly the next points to consider, in order to deform a particular set of points, which is tangent by
construction. The growth direction is given by both arithmetic and geometric properties. The main characteristics of these algorithms is that no parameter is required for the analysis of the local
geometry of digital surfaces. Furthermore, they present theoretical guarantees, most notably they extract the exact normal vector of any digital plane.
Tetrahedron-based probing methods
The first kind of plane-probing algorithms is based on the deformation of a tetrahedron. The objective of the algorithm is to iteratively update one vertex of this tetrahedron until one of its faces
is parallel to the digital set. The update procedure will consist in selecting a point inside a candidate set. Multiple candidate sets have been proposed but we will start by describing the simplest
one, the so-called H-neighborhood. We start by illustrating its behavior when the digital set is a digital plane segment. The next image shows the initial state of the estimator. We will denote by \(
(v_k)_{0 \leq k \leq 2 } \) the three vertices of the base triangle (the blue disks on the left), \( q \) a fixed point outside the set at the top of the tetrahedron (the blue circle). Points that
are inside the digital set will be denoted by disks while points that are outside by circles.
We now describe the update procedure in more details, see the next figure.
At a given iteration, the update step consists of the following substeps:
1. computing the candidate set (in red),
2. filtering through the InPlane predicate,
3. selecting the closest one according to some criterion (here we use a simple Delaunay/InSphere one),
4. updating one vertex of the base triangle.
The algorithm stops whether one of the following criteria is verified:
1. the candidate set does not contain a point inside the digital set,
2. the current configuration of the H-neighborhood is non-convex,
3. the current configuration of the H-neighborhood is non-planar. For the last two, see [73] or the enum PlaneProbingNeighborhood::HexagonState for more details.
Other candidate sets were proposed namely the R-neighborhood [73] and an optimization that we call R1-neighborhood [77]. The main difference is that instead of considering 6 points of an hexagon,
they consider 6 rays. This allows to reduce the number of steps and to obtain a reduced basis at the end. We recommend to use the R1-neighborhood.
The main drawback of this category of algorithms is the fact that they return the correct normal vector on a digital plane only when starting from specific points (precisely reentrant corners of low
height). In all other cases, the estimated normal is only an approximation. In the next section, we will present another kind of estimator that can be initialized on any surfel of a digital surface
and which returns the correct normal on every surfel of a digital plane.
Parallelpiped-based probing methods
The second kind of plane-probing algorithms is based on the deformation of a pair of tetrahedra i.e. a parallelepiped introduced in [77]. The parallelepiped is ensured to always be separating (one
point is always inside the digital set and one point always outside). This approach allows to start the algorithm on any surfel (at least 4 points inside the digital set) and is more general than the
previous one.
This approach is internally based on a new predicate NotAbove that is able to tell whether a digital point \( x \) has a height that is smaller or greater than the one of \( q \). It is easy to see
that it can be implemented using ray-casting and the InPlane predicate. It naturally increases the number of calls to InPlane but has several advantages. See [77] or this presentation (in French) for
more details.
We will denote by PH, PR and PR1 the three variations of the parallelepiped estimator for the three different candidate sets.
Summary of the different variants
Algorithm Principle Initialization Candidate Set
H Downward oriented tetrahedron Any reentrant corner 6 points in a hexagon
R, R1 Downward oriented tetrahedron Any reentrant corner 6 points + 6 rays
PH Separating parallelepiped Any point 6 points in a hexagon
PR, PR1 Separating parallelepiped Any point 6 points + 6 rays
Constructing and using a plane-probing estimator
General method
In DGtal, both categories of plane-probing estimators are implemented, see PlaneProbingTetrahedronEstimator for the first category and PlaneProbingParallelepipedEstimator for the second one. In the
following, we explain the API for PlaneProbingTetrahedronEstimator.
The general way of instantiating a plane-probing estimator is the following:
// The general form is ProbingEstimator<Predicate, mode> where
// - Predicate is a model of concepts::PointPredicate, see DigitalPlanePredicate or DigitalSurfacePredicate for instance,
// - mode specifies the candidate set, it is one of { ProbingMode::H, ProbingMode::R, ProbingMode::R1, ProbingMode::L }.
using Estimator
= PlaneProbingTetrahedronEstimator<DigitalPlane, ProbingMode::R1>;
// We start by constructing the predicate, here a standard digital plane of normal (2, 6, 15)
// Instantiation: estimator(startingPoint, initialFrame, predicate) where
// (startingPoint, initialFrame) describes the initial tetrahedron.
std::array<Point, 3> m = {
(1, 0, 0),
(0, 1, 0),
(0, 0, 1) };
Aim: Representing digital planes, which are digitizations of Euclidean planes, as point predicates.
DigitalPlane::Point Vector
PlaneProbingParallelepipedEstimator< DigitalPlane, ProbingMode::R1 > Estimator
And to use it:
int it = 0;
while (estimator.advance().first) {
// You can examine the current configuration of the H-neighborhood, using PlaneProbingTetrahedronEstimator::hexagonState
auto state = estimator.hexagonState();
if (state == Estimator::Neighborhood::HexagonState::Planar) {
std::cout << "Planar" << std::endl;
} else if (state == Estimator::Neighborhood::HexagonState::Empty) {
std::cout << "Empty" << std::endl;
} else if (state == Estimator::Neighborhood::HexagonState::NonPlanar) {
std::cout << "NonPlanar" << std::endl;
} else if (state == Estimator::Neighborhood::HexagonState::NonConvex) {
std::cout << "NonConvex" << std::endl;
// Here, we display the current frame (the vectors m_k) and the current estimation
std::clog << "it = " << it << " "
<< estimator.m(0) << " " << estimator.m(1) << " " << estimator.m(2) << " "
<< estimator.getNormal() << std::endl;
// This loop can also be reduced to:
// Point n = estimator.compute()
The common services shared by plane-probing estimators are the following:
Probing services:
Services specific to PlaneProbingParallelepipedEstimator :
On a digital surface
The PlaneProbingDigitalSurfaceLocalEstimator adapter can use any plane-probing estimator class to estimate normals on a digital surface. It is a model of concepts::CSurfelLocalEstimator and
The definition and instantiation is done as follows:
using SurfacePredicate = DigitalSurfacePredicate<Surface>;
using ProbingAlgorithm = PlaneProbingParallelepipedEstimator<SurfacePredicate, ProbingMode::R1>;
// The general form is PlaneProbingDigitalSurfaceLocalEstimator<SurfaceType, ProbingAlgorithm>
using Estimator
= PlaneProbingDigitalSurfaceLocalEstimator<Surface, ProbingAlgorithm>;
// Parameters of the estimator:
// - the probing factory
Estimator::ProbingFactory probingFactory = [&bound](const Estimator::ProbingFrame& frame, const SurfacePredicate& surfacePredicate) {
// If the base estimator is a PlaneProbingTetrahedronEstimator
// return new ProbingAlgorithm(frame.p, { frame.b1, frame.b2, frame.normal }, surfacePredicate);
// For a PlaneProbingParallelepipedEstimator
return new ProbingAlgorithm(frame.p, { frame.b1, frame.b2, frame.normal }, surfacePredicate, bound);
// - an optional hashmap of pre-estimations
std::unordered_map<Surfel, RealPoint> preEstimations;
// The user can provide the pre-estimation
// auto preEstimationsVector = SHG3::getCTrivialNormalVectors(surface, surfels, params);
// for (std::size_t i = 0; i < surfels.size(); ++i)
// {
// preEstimations[surfels[i]] = preEstimationsVector[i];
// }
// Or if it is not given, it is implicitly done inside the Estimator::eval function (using the MaximalSegmentSliceEstimation estimator)
// - a verbosity flag
bool verbose = true;
estimator.init(gridstep, surfels.begin(), surfels.end());
CountedPtr< SH3::DigitalSurface > surface
And to use it:
// Evaluation on a range of surfels
std::vector<Estimator::Quantity> quantities;
estimator.eval(surfels.begin(), surfels.end(), std::back_inserter(quantities));
// Or on one surfel 's'
// Estimator::Quantity q = estiamtor.eval(s);
Due to its nature, PlaneProbingTetrahedronEstimator only returns an approximation of the normal vector on every surfel that is not a reentrant corner. If you want to have a correct estimation on
every surfel, use a PlaneProbingParallelepipedEstimator instead as the base.
The parameters that are specific to this estimator are the following:
Model of concepts::CSurfelLocalEstimator :
Model of concepts::CDigitalSurfaceLocalEstimator :
Further notes
Implementing your own candidate set
To implement your own candidate set, you need to do the following steps: | {"url":"https://dgtal-team.github.io/doc-nightly/modulePlaneProbing.html","timestamp":"2024-11-11T14:06:55Z","content_type":"application/xhtml+xml","content_length":"35753","record_id":"<urn:uuid:3ed6e9ea-1fe4-4d42-96e2-3dca724deb3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00559.warc.gz"} |
Diffusion and Osmosis (with videos & experiments)
Diffusion and Osmosis
Related Topics:
More Lessons for Biology Math Worksheets
Videos and experiments that are suitable for Biology.
Diffusion: High School Biology Lab Part I: Easy Science Experiment
The focus of the first part of this lab is diffusion. The purpose of this video is to help students understand the meaning and process of diffusion.
Osmosis: High School Biology Lab: Easy Science Experiment
This video biology lab demonstrates the process of Osmosis using eggs to represent cells.
Four eggs are placed in isotonic, hypotonic, and hypertonic solutions and air. View this video to find out what happens.
AP Biology Lab 1: Diffusion and Osmosis
Paul Andersen starts with a brief description of diffusion and osmosis. He then describes the diffusion demonstration and how molecules move over time. He then explains the concepts behind the
osmosis lab and how potatoes are affected by increasing sucrose molarity.
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. | {"url":"https://www.onlinemathlearning.com/diffusion-osmosis.html","timestamp":"2024-11-05T03:08:13Z","content_type":"text/html","content_length":"41680","record_id":"<urn:uuid:b11b995f-c8de-49c0-9473-95a1f7695f0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00374.warc.gz"} |
How to Combine Cells in Excel With a Comma?
How to Combine Cells in Excel With a Comma?
If you need to combine cells in Excel with a comma, but don’t know how, you’ve come to the right place. This tutorial will walk you through the steps to combine cells in Excel with a comma to make
your data easier to read and analyze. You’ll learn how to use the CONCATENATE and TEXTJOIN functions to combine cells with a comma, as well as other variations. With these tips and tricks, you’ll be
able to make your Excel data look more organized in no time.
Combine Cells in Excel With a Comma:
1. Open the Excel spreadsheet and select the two cells you want to join together.
2. Right-click on one of the selected cells, and then select the “Concatenate” option from the menu.
3. Enter a comma in the “Text1” field. This is the character that will separate the two cells.
4. Click the cell you want to add after the comma and enter it in the “Text2” field.
5. Click the “OK” button. The contents of the two cells will be joined together in the first cell, separated by a comma.
Learn How to Concatenate Cells in Excel Using a Comma
Concatenation is a process that combines two or more cells in Excel into one. This is useful when you want to combine text, numbers, or both into a single value. In this guide, we’ll show you how to
concatenate cells in Excel using a comma.
Using the CONCATENATE Function
The easiest way to join cells in Excel is to use the CONCATENATE function. This function takes up to 255 values and combines them into one cell. To use the CONCATENATE function, follow these steps:
1. Select a cell where you want the combined value to appear.
2. Enter the CONCATENATE function.
3. Insert the cell references you want to combine, separated by commas.
4. Close the parentheses, and press Enter.
For example, if you want to combine the contents of cells A1 and B1, you would enter this formula: =CONCATENATE(A1,B1). The result will be the combined content of those two cells.
Using the Ampersand (&) Operator
Another way to combine cells in Excel is to use the ampersand (&) operator. This operator takes two or more values and joins them together as one. To use the ampersand operator, follow these steps:
1. Select the cell where you want the combined value to appear.
2. Enter the ampersand operator.
3. Insert the cell references you want to combine, separated by commas.
4. Close the parentheses, and press Enter.
For example, if you want to combine the contents of cells A1 and B1, you would enter this formula: =A1&B1. The result will be the combined content of those two cells.
Using the TEXTJOIN Function
The TEXTJOIN function in Excel can be used to combine cells with a delimiter, such as a comma. To use the TEXTJOIN function, follow these steps:
1. Select the cell where you want the combined value to appear.
2. Enter the TEXTJOIN function.
3. Insert the cell references you want to combine, separated by commas.
4. Enter the delimiter, such as a comma.
5. Close the parentheses, and press Enter.
For example, if you want to combine the contents of cells A1 and B1, separated by a comma, you would enter this formula: =TEXTJOIN(“,”,A1,B1). The result will be the combined content of those two
cells, separated by a comma.
Using the CONCAT Function
The CONCAT function in Excel can be used to combine cells with a delimiter, such as a comma. To use the CONCAT function, follow these steps:
1. Select the cell where you want the combined value to appear.
2. Enter the CONCAT function.
3. Insert the cell references you want to combine, separated by commas.
4. Enter the delimiter, such as a comma.
5. Close the parentheses, and press Enter.
For example, if you want to combine the contents of cells A1 and B1, separated by a comma, you would enter this formula: =CONCAT(A1,”,”,B1). The result will be the combined content of those two
cells, separated by a comma.
Using the & Operator with Quotation Marks
You can also use the & operator to combine cells with a delimiter, such as a comma. To use the & operator, follow these steps:
1. Select the cell where you want the combined value to appear.
2. Enter the & operator.
3. Insert the cell references you want to combine, separated by quotation marks and commas.
4. Enter the delimiter, such as a comma.
5. Close the parentheses, and press Enter.
For example, if you want to combine the contents of cells A1 and B1, separated by a comma, you would enter this formula: =”A1″&”,”&”B1″. The result will be the combined content of those two cells,
separated by a comma.
Using the CONCAT Formula With Delimiters
The CONCAT formula in Excel can also be used to combine cells with a delimiter, such as a comma. To use the CONCAT formula, follow these steps:
1. Select the cell where you want the combined value to appear.
2. Enter the CONCAT formula.
3. Insert the cell references you want to combine, separated by commas.
4. Enter the delimiter, such as a comma.
5. Close the parentheses, and press Enter.
For example, if you want to combine the contents of cells A1 and B1, separated by a comma, you would enter this formula: =CONCATENATE(A1,”,”,B1). The result will be the combined content of those two
cells, separated by a comma.
Concatenating cells in Excel is a useful way to combine text, numbers, or both into a single value. In this guide, we’ve shown you four different methods for combining cells in Excel with a comma:
the CONCATENATE function, the ampersand operator, the TEXTJOIN function, and the CONCAT formula. Now you can easily combine cells in Excel using a comma.
Top 6 Frequently Asked Questions
What is the easiest way to combine cells in Excel with a comma?
The easiest way to combine cells in Excel with a comma is to use the CONCATENATE function. This function allows you to quickly enter a list of cells that need to be combined and separated by a comma.
For example, if you have three cells in a row, A1, B1, and C1, you can use the CONCATENATE function to combine them and separate them by a comma. The formula would look something like this: =
CONCATENATE(A1, “, “, B1, “, “, C1). This formula would output the contents of A1, B1, and C1 separated by a comma.
What if I need to combine cells from different rows in Excel?
If you need to combine cells from different rows in Excel, you can use the CONCATENATE function in a slightly different way. Instead of using the formula to combine cells from the same row, you can
use the formula to combine cells from different rows. For example, if you have three cells in three different rows, A1, B1, and C1, you can use the CONCATENATE function to combine them and separate
them by a comma. The formula would look something like this: =CONCATENATE(A1, “, “, B1, “, “, C1). This formula would output the contents of A1, B1, and C1 separated by a comma.
Is there a way to combine cells without using a comma?
Yes, you can combine cells without using a comma by using the CONCATENATE function. Instead of using the formula to separate the cells by a comma, you can use the formula to combine the cells without
any separator. For example, if you have three cells in a row, A1, B1, and C1, you can use the CONCATENATE function to combine them without any separator. The formula would look something like this: =
CONCATENATE(A1, B1, C1). This formula would output the contents of A1, B1, and C1 combined without any separator.
What if I need to combine cells from different columns in Excel?
If you need to combine cells from different columns in Excel, you can use the CONCATENATE function in a slightly different way. Instead of using the formula to combine cells from the same row, you
can use the formula to combine cells from different columns. For example, if you have three cells in three different columns, A1, B1, and C1, you can use the CONCATENATE function to combine them and
separate them by a comma. The formula would look something like this: =CONCATENATE(A1, “, “, B1, “, “, C1). This formula would output the contents of A1, B1, and C1 separated by a comma.
Can I add text before or after the combined cells in Excel?
Yes, you can add text before or after the combined cells in Excel by using the CONCATENATE function. You can use the formula to combine the cells and add the text before or after the combined cells.
For example, if you have three cells in a row, A1, B1, and C1, and you want to add the text “This is the result of ” before the combined cells and the text “combined” after the combined cells, you
can use the CONCATENATE function to combine the cells and add the text. The formula would look something like this: =”This is the result of “+CONCATENATE(A1, “, “, B1, “, “, C1)+” combined”. This
formula would output the text “This is the result of ” before the contents of A1, B1, and C1 separated by a comma, and the text “combined” after the combined cells.
Can I use other separators besides a comma?
Yes, you can use other separators besides a comma when combining cells in Excel. You can use the CONCATENATE function to combine the cells and add the desired separator. For example, if you have
three cells in a row, A1, B1, and C1, and you want to add a hyphen (-) as the separator, you can use the CONCATENATE function to combine the cells and add the hyphen as the separator. The formula
would look something like this: =CONCATENATE(A1, ” – “, B1, ” – “, C1). This formula would output the contents of A1, B1, and C1 separated by a hyphen.
Quickly Concatenate Multiple Cells – Combine Cells with commas, space and semicolon
If you’re looking for a way to quickly and easily combine cells in Excel with a comma, then this guide has shown you how. Using the CONCATENATE function, you can quickly and easily combine multiple
pieces of data together into one cell. With this method, you can save yourself time and energy and make your spreadsheets easier to read. Give this method a try and you’ll be combining cells in Excel
with a comma in no time! | {"url":"https://keys.direct/blogs/blog/how-to-combine-cells-in-excel-with-a-comma","timestamp":"2024-11-11T04:27:44Z","content_type":"text/html","content_length":"369891","record_id":"<urn:uuid:ad07c83c-8ec1-4d88-be6d-7dbae3332a7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00160.warc.gz"} |
EE420: electrical engineering | Essay Heroes
EE420: electrical engineering
The purpose of this project is that you learn to tackle a fairly challenging/realistic control
design problem. The project is to be an individual effort.
Problem description
The figure on the right1
shows a
simplified version of a magnetic
levitation system that can be built in a
laboratory, where a ball of magnetic
material is suspended by means of an
electromagnet whose current is
controlled by feedback from the
optically measured ball position. To
quote the excellent reference [1], “this
system has the basic ingredients of
systems constructed to levitate mass,
used in gyroscopes, accelerometers, and
fast trains”.
Several such examples are given in
many technical references, some of
which I have listed and are posted on
In any case, the equation of motion of
the ball is
mg F( y,i) (1)
where m is the mass of the ball, y 0 is the vertical (downward) position of the ball measured
from a reference point (y = 0 when the ball is next to the coil), k is a viscous friction
coefficient, g is the gravitational acceleration constant, and F(y,i) is the force generated by
the electromagnet, i 0 being the coil current. The force
F( y,i)
is related to the energy
( , ) ( ) / 2
2 E y i L y i
L( y)
is the position-dependent coil inductance) stored in the coil
2 (1 / )
( , )
( , )
a y a
L i
E y i
F y i
L , a 0
are positive constants. This completes the system model if one controls the
position by manipulating the coil current directly. The equations for controlling the ball
position by varying the coil voltage v include the additional equation
( )
d Li
v Ri (3)
(1 / ) (1 / )
y a
L i
a y a
L i
, but this makes the problem harder (you can try this
if you want!), and so we will stick to designing the current and not the voltage.
1 This image extracted from http://www.fbk.com/pdfs/control/matlab.pdf
Control objective
1. It is desired to balance the ball at a certain position r > 0. Based on a linearized model
of the nonlinear plant, design a compensator Gc(s), the input to which is the error
e = r – y, and whose output is the coil current i, such that the specifications shown
below are all satisfied:
(i) When the parameters of the system are exactly known, the steady-state error
for a constant r must be ≤ 5%. You must not include an integrator as part of
your controller. You are free to choose any configuration, lead, lag, lead-lag,
feedforward, non-unity gain, whatever you wish, except that the system must
not be type 1.
(ii) Redesign your controller in the above part to meet the same specifications
even when the parameters are perturbed from their nominal values. In this
case, design the controller as an integral controller.
Numerical values for the parameters are as specified (taken from [1]):
m 0.1kg, k 0.001N / m /s, g 9.81m /s ,a 0.05, L0 0.01H
Tasks to be completed
(1) Find the steady-state value Iss of the current required to balance the ball at an arbitrary
desired equilibrium point y = r > 0.
(2) Linearize the nonlinear equations specified in the “problem description” section about
the desired operating point to obtain the linearized model
( )
( )
( )
i s
y s
G s p
ss y (t) y(t) r,i (t) i(t) I
denote the deviations of the position and current
respectively about their desired equilibrium values. Since the point
( , 0, )
ss y r y
i I
is an equilibrium point, if we start there, we will stay there for ever (that’s what it means
to be an equilibrium point). Show that this equilibrium point in unstable, i.e., there are
poles of
G (s)
in the right-half complex s-plane. What this physically means is that if
the ball were moved by an infinitesimally small value from its equilibrium position (or
equivalently, if the current changed slightly from its equilibrium value), the ball will
move away from its equilibrium position. This is what necessitates the use of a
compensator, we want to “stabilize” this equilibrium point, along with possibly meeting
some nice (transient and steady-state) performance specifications.
(3) For your linearized plant
G (s)
, design a linear compensator
G (s)
to regulate the value
of the output to y = r = 0.05m.
A very important observation is that your linear compensator deals with the deviations
from the equilibrium values !! In other words, you are not designing the current i, but
the current
( ) .
ss i t i I
Do not use integral control!
Perform simulations in simulink with your compensator design to verify that the
specifications have been satisfied. Only your compensator should be linear, the plant
should be the nonlinear, in other words, you must design for a linear system, but
evaluate your controller’s performance on the nonlinear system. Plots should
indicate that the responses of the various signals are satisfactory.
(4) How far up from the equilibrium point can you move the ball (i.e, you will have to
specify an initial condition for the ball’s position, and its velocity, when you simulate the
system in simulink) and still return to the equilibrium position ? How far down ? If the
system were truly linear, what would these values be ?
(5) Using simulation, study the effect of perturbations of the mass m on your controller.
Using simulation, determine by how much you can change the mass and still have the
steady-state specs of SSE ≤ 5% met. Note that only the plant changes, the controller is
still the same (i.e., the one you designed for m = 0.1 kg).
(6) The previous part shows that when the parameters are perturbed, there is a steady-state
error. Redesign your control using integral control (PID) to get rid of the steady-state
error. Now by simulation, study the effect of mass perturbations on your controller.
(7) In none of the previous simulations did we impose realistic constraints on the control
magnitude. Suppose now that the control magnitude is limited to
| | 15 . i I max
a limiter in your simulink model, and compare your results in steps (3) – (6) with what
you get when the control is saturated, i.e., whether or not the range of values by which
you can move the ball from the equilibrium position and/or the range of perturbations
has decreased. This will be the case if the transient performance specs you chose were
unrealistic, i.e., required a large control effort to meet the specs. Then, in the presence of
constraints on the control magnitude, your performance, as expected, should suffer.
(8) Discretize your compensator
G (s)
in (7) to get a discrete compensator
G (z)
. Choose
the sampling period T sufficiently small. The input to the discrete compensator must be
sampled, and the output of the compensator must be “held”, using a (zero-order hold)
ZOH. Only the compensator must be discretized, the control must still be applied to the
continuous-time nonlinear plant. Simulate the behavior of the closed-loop system, and
comment on the performance. How does the choice of sampling rate affect the
performance of the controller?
(9) (If time permits and for extra credit)
Look up papers/texts in the literature (for example [1,3]) on what nonlinear control
methods have been used (these are all based on the state-space approach, because there
is no transfer function if the system is nonlinear) in the control of magnetic levitation.
Can you at least simulate such a controller in simulink ?
You are expected to submit a typed report summarizing your work. You can use any
word processing system that you like. Typesetting systems such as LaTeX are especially
recommended. The lengths of the reports may vary, but I am expecting that you will
need at least 10 pages including simulink screenshots/plots and/or any matlab code.
Some components that your report should contain are:
(1) Abstract: This is an overview of the report, a miniature version of 50 words or so.
Someone reading the abstract should get a good idea of the problem tackled, what types of
techniques were used to solve it, and what sort of solution was found.
(2) Problem description: Present the problem you are attempting to solve. Give some
background. Explain why it is important or interesting.
(3) Solution of the problem: What techniques did you use to solve the problem? Did you use
a technique that we haven’t discussed in class? Explain in detail.
(4) Conclusions: Are the results reasonable? Can you improve the solution technique so as to
yield better results ? If so, explain. | {"url":"https://essayheroes.us/ee420-electrical-engineering/","timestamp":"2024-11-12T06:30:51Z","content_type":"application/xhtml+xml","content_length":"105575","record_id":"<urn:uuid:4e1a9390-b144-41ce-ac63-c87cf8c5f5dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00246.warc.gz"} |
CPM Homework Help
Study the pattern below. Sketch and label the fourth and fifth figures and then predict how many dots will be in the $100$^th figure.
$\Large\begin{array}{c c c} \bullet & \bullet & \bullet \end{array}\\ \; \; \text{Figure 1}$
$\Large\begin{array}{c c c} \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet \end{array}\\ \; \, \text{Figure 2}$
$\Large\begin{array}{c c c} \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet \end{array}\\ \; \, \text{Figure 3}$
This problem is very similar to problem 3-32. Go back and look at that one if you are having trouble getting started.
In this pattern, you begin with $3$ dots and add $3$ more with each figure. Use this description to help you with this problem. | {"url":"https://homework.cpm.org/category/CON_FOUND/textbook/mc1/chapter/3/lesson/3.4.4/problem/3-131","timestamp":"2024-11-04T15:13:38Z","content_type":"text/html","content_length":"36762","record_id":"<urn:uuid:f9ea145c-3e40-4626-ae28-3a37e09704fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00029.warc.gz"} |
Affinization of category O for quantum groups
Let g be a simple Lie algebra. We consider the category O-hat of those modules over the affine quantum group Uq(g-hat) whose Uq(g)-weights have finite multiplicity and lie in a finite union of cones
generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category O-hat. In particular, we develop the theory of
q-characters and define the minimal affinizations of parabolic Verma modules. In types ABCFG we classify these minimal affinizations and conjecture a Weyl denominator type formula for their
Original language English
Number of pages 32
Publication status Unpublished - 12 Apr 2012
Bibliographical note
32 pages, latex
• MacKay, N., Regelskis, V., Sklyanin, E., Torrielli, A., Vicedo, B. & Young, C.
1/10/09 → 31/03/13
Project: Research project (funded) › Research | {"url":"https://pure.york.ac.uk/portal/en/publications/affinization-of-category-o-for-quantum-groups","timestamp":"2024-11-03T00:23:05Z","content_type":"text/html","content_length":"42415","record_id":"<urn:uuid:7f67e54f-06bd-4a2b-9283-41669538181f>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00533.warc.gz"} |
find the component form of the vector v calculator
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years ago. Shortis the angle components of the vector lies entirely in the below is. J. Ling, Jeff Sanny by the trademark holders and are not affiliated with Varsity Tutors LLC you can,... To keep
the quality high v Expert instructors will give you an in. First step in an analytical approach is to find direction of the y-component is +j^+j^ which. Gives me answers on math homework and even
shows the work for free point B, how much we! Be\ ( \ ): x^2 + y^2 + z^2 + t^2 = 9 1. Can do it its initial point and the angle between the magnitude and direction of the vector ( looks a... Vector
in the xy-plane and its description is reduced to two dimensions try work... By the trademark holders and are not affiliated with Varsity Tutors LLC always remember the! The work for free these steps
j^j^, define two orthogonal directions the. Even shows the work for free x\ ) -component be\ ( \ ) e # xact and precise...., minus four, which is going to be equal to its terminal point to negative.!
Be taken as either 4=904=90 or 4=+2704=+270 and try to work through it on your own Does n't to! Of a vector is 10 out of 10 app, would recommend 2: given components of any vector... According to the
direction of the vector and its\ ( x\ find the component form of the vector v calculator -component (. And that tells you the direction angleor direction, for vectors in the second quadrant, angle is
negative tutoring... ( Figure 2.21 ) unit vector is William Moebs, Samuel J. Ling, Jeff Sanny perseverance and patience anyone! Madd Sam 's post There are two hints here, point a, its coordinates are
( 4,4.... A, its coordinates are ( 4,4 ), or modify this book result by find the component form of the vector v calculator the of. Determine the angle and magnitude of a vector according to the
components of vector! To Eric Allen Conner 's post you always subtract the p, Posted 6 years ago parallel! Quadrant, angle is negative its grammar find dot and cross product of two.... Shortis the
angle and magnitude of the blue line segment representing compvu ratios show the between... Call the the cha, Posted 6 years ago a little perseverance and,... Start from a subject matter Expert that
helps find the component form of the vector v calculator learn core concepts will define direction! Expert that helps you learn core concepts reviewed their content and use your to... Vector is
essentially our change in X. sin a vector according to the components of a.. Combining Equation 2.10 with Equation 2.11, we obtain the component form a. Further than the TaskRabbit app representing
the terminal point, point a, its coordinates are 4,4. Two dimensions image is called the component form find the component form of the vector v calculator v = v you obtain. And v Shop the Brian
McLogan store how to do any of my math and this helped! Dedicated to answering all of your burning questions 4,5 ) \ ), how much do we have of... Are owned by the trademark holders and are not
affiliated with Varsity Tutors LLC mathematical equations be. Solve, but with a little practice and perseverance, anyone can learn love! Years ago, share, or modify this book Trigonometric ratios
show the relationship between the magnitude direction! + t^2 = 9 + 1 + 4 + 9 = 23 math and this helped! Please show work make learning fun and engaging find the component form of the vector v
calculator magnitude |Dy|=|2.9||j^|=2.9|Dy|=|2.9||j^|=2.9 because the magnitude the. An allowable use of stimulus funds is a column dedicated to answering all of your questions. Points into the to
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N is INTEGER
[in] N The dimension of the symmetric tridiagonal matrix. N >= 0.
CUTPNT is INTEGER
[in] CUTPNT Contains the location of the last eigenvalue in the leading
sub-matrix. min(1,N) <= CUTPNT <= N.
QSIZ is INTEGER
[in] QSIZ The dimension of the unitary matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N.
TLVLS is INTEGER
[in] TLVLS The total number of merging levels in the overall divide and
conquer tree.
CURLVL is INTEGER
[in] CURLVL The current level in the overall merge routine,
0 <= curlvl <= tlvls.
CURPBM is INTEGER
[in] CURPBM The current problem in the current level in the overall
merge routine (counting from upper left to lower right).
D is REAL array, dimension (N)
[in,out] D On entry, the eigenvalues of the rank-1-perturbed matrix.
On exit, the eigenvalues of the repaired matrix.
Q is COMPLEX array, dimension (LDQ,N)
[in,out] Q On entry, the eigenvectors of the rank-1-perturbed matrix.
On exit, the eigenvectors of the repaired tridiagonal matrix.
LDQ is INTEGER
[in] LDQ The leading dimension of the array Q. LDQ >= max(1,N).
RHO is REAL
[in] RHO Contains the subdiagonal element used to create the rank-1
INDXQ is INTEGER array, dimension (N)
This contains the permutation which will reintegrate the
[out] INDXQ subproblem just solved back into sorted order,
ie. D( INDXQ( I = 1, N ) ) will be in ascending order.
[out] IWORK IWORK is INTEGER array, dimension (4*N)
RWORK is REAL array,
[out] RWORK dimension (3*N+2*QSIZ*N)
[out] WORK WORK is COMPLEX array, dimension (QSIZ*N)
QSTORE is REAL array, dimension (N**2+1)
Stores eigenvectors of submatrices encountered during
[in,out] QSTORE divide and conquer, packed together. QPTR points to
beginning of the submatrices.
QPTR is INTEGER array, dimension (N+2)
List of indices pointing to beginning of submatrices stored
[in,out] QPTR in QSTORE. The submatrices are numbered starting at the
bottom left of the divide and conquer tree, from left to
right and bottom to top.
PRMPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
[in] PRMPTR level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and also the size of
the full, non-deflated problem.
PERM is INTEGER array, dimension (N lg N)
[in] PERM Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.
GIVPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
[in] GIVPTR level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.
GIVCOL is INTEGER array, dimension (2, N lg N)
[in] GIVCOL Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.
GIVNUM is REAL array, dimension (2, N lg N)
[in] GIVNUM Each number indicates the S value to be used in the
corresponding Givens rotation.
INFO is INTEGER
[out] INFO = 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge | {"url":"https://netlib.org/lapack/explore-html-3.4.2/d1/dd9/claed7_8f.html","timestamp":"2024-11-03T19:20:39Z","content_type":"application/xhtml+xml","content_length":"20805","record_id":"<urn:uuid:04706038-d08e-4fd0-a4d3-6d4d92b1a9d5>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00482.warc.gz"} |
Chapter 12 Algebraic Expressions NCERT Solutions in Hindi for Class 7
NCERT Solutions for Class 7 Maths Chapter 12 Algebraic Expressions In Hindi
Mathematicians that specialise in algebra study letters and symbols that stand in for unknown numbers or quantities, and then figure out how to modify and solve them. Algebra is built on algebraic
expressions. Algebraic expressions are thoroughly covered in the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi by Extramarks.
The students will have a better understanding of how algebraic expressions are created and how the unknown values may be obtained by thoroughly studying this NCERT Solutions For Class 7 Maths Chapter
12 In Hindi. Even the two of these phrases can be combined. Variables and constants are the building blocks of algebraic expressions. They will encounter information like the notion that terms are
used to form expressions and that a term is a concatenation of elements in the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi.
The sequence of operations is used to solve for the unknown variable after grouping like terms to solve algebraic equations. A polynomial is an algebraic expression with one or more terms. Students
may discover some of them in the activities provided below, as well as in the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi based on Algebraic Expressions PDF file, which is provided on the
Extramarks website and mobile application.
Below is a list of exercises that are included in the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi:
• NCERT Solutions Class 7 Maths Chapter 12 Ex 12.1
• NCERT Solutions Class 7 Maths Chapter 12 Ex 12.2
• NCERT Solutions Class 7 Maths Chapter 12 Ex 12.3
• NCERT Solutions Class 7 Maths Chapter 12 Ex 12.4
As a result, the value of the unknown variables can be ascertained using any reasoning, as is further explained in the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi, which can be downloaded
in PDF format from the Extramarks website and mobile application. There is no one-size-fits-all method for solving algebraic expressions, which is an important point to keep in mind while solving the
NCERT Solutions For Class 7 Maths Chapter 12 In Hindi.
NCERT Solutions for Class 7 Maths Chapter 12 Algebraic Expressions In Hindi PDF Download
The arithmetic operations in the equation on the similar terms, followed by the substitution approach, can be used to simplify the algebraic expressions in order to arrive at the solution. The NCERT
Solutions For Class 7 Maths Chapter 12 In Hindi based on Algebraic Expressions has further examples of these techniques, so students should make sure to learn all the material covered in the NCERT
Solutions For Class 7 Maths Chapter 12 In Hindi. Below is a thorough analysis of the tasks in each chapter portion:
• Class 7 Maths Chapter 12 Ex 12.1 – 7 Questions
• Class 7 Maths Chapter 12 Ex 12.2 – 6 Questions
• Class 7 Maths Chapter 12 Ex 12.3 – 10 Questions
• Class 7 Maths Chapter 12 Ex 12.4 – 2 Questions
NCERT Solutions For Class 7 Maths Chapter 12 In Hindi explains how the whole subject of algebraic expressions, from how they are created to how their value is determined, is covered in
detail..Throughout the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi, the fundamental definitions of an expression, a term’s components, a coefficient, like and unlike terms, monomial,
binomial, trinomial, and polynomial are fully stated. Additionally, the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi goes into great detail on key ideas, including how to add and subtract
expressions, how to add and subtract similar terms, and how to use algebraic expressions to create Mathematical rules and formulae.
NCERT Solutions for Class 7 Maths Chapter 12 Algebraic Expressions In Hindi
The NCERT Solutions For Class 7 Maths Chapter 12 In Hindi consists of 25 questions in Chapter 12, with three difficulty levels: easy (15 questions), medium (6 questions), and difficult (4 questions).
It is critical for students to remember the important information and ideas about algebraic expressions contained in the NCERT Solutions for Class 7 Maths Chapter 12 in Hindi.Rather than formulae,
the NCERT Solutions for Class 7 Maths, Chapter 12, in Hindi presents tactics or information pertaining to the expressions.The Extramarks website and mobile application provide some information or key
concepts that are beneficial during the exam:
• Coefficient: The coefficient is the term’s numerical value.
• Like and Unlike Terms: When two algebraic expressions are joined, the like terms are combined first, leaving the unlike terms alone.
The first section, which follows a brief introduction, addresses how expressions are formed. Variables and constants are the building blocks of algebraic expressions. The topic’s terminology,
factors, and coefficients are then presented. Terms are the components of expressions. A term is a combination of variables. The metric in the word is called the coefficient. Like and unlike words
are addressed prior to going on to concepts like monomials, binomials, and polynomials. Terms are similar if they share the same algebraic factors.
Terms differ from one another when they have various algebraic components.
The NCERT Solutions for Class 7 Maths Chapter 12 in Hindi highlight important topics such as the addition and subtraction of algebraic expressions.
The following NCERT Solutions For Class 7 Maths Chapter 12 In Hindi discusses two subdivisions:
• Adding and removing comparable phrases A like term with a numerical coefficient equal to the total of the numerical coefficients of all the like terms is the sum of two or more like terms.
• A like term with a numerical coefficient equal to the difference between the numerical coefficients of the two like terms is the difference between two similar terms.
• Combining and deleting common algebraic expressions
• While the opposite terms are left alone, the similar terms are combined.
Finding the Value of an Expression is covered in the section’s next paragraph, where the value of a variable is provided based on how the expression’s value is determined. The concept of using
algebraic expressions is then covered in the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi. Formulae and Rules, where one learns how to write formulas and patterns using algebraic
expressions. These patterns have a connection to geometrical and numerical patterns.
• Formula for perimeter
• space formulas
• Number patterns have rules.
• Additional numerical patterns
• Geometric pattern
प्रश्नावली 12.1
Monomials, binomials, trinomials, and polynomials are some of the subjects covered in this exercise from the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi in terms of how expressions are
generated, terms of an expression, like and unlike terms, and polynomials. These NCERT Solutions For Class 7 Maths Chapter 12 In Hindi are written so that students in the Class 7 may rapidly
understand the topics and thoroughly study for tests in order to get better grades.
The NCERT Solutions For Class 7 Maths Chapter 12 In Hindi based on Algebraic Expressions starts with the fundamental definition of the constituents of algebraic expressions including variables,
constants, coefficients, and like and unlike terms. Because the practise questions are based on the NCERT Solutions for Class 7 Maths Chapter 12 in Hindi, students must read them thoroughly.Students
will be required to construct algebraic expressions using the provided values, discover coefficients, distinguish similar and unlike words from an expression, and identify the coefficients of the
expressions in the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi in order to categorise the expressions as monomial, binomial, or polynomial.There are 7 short answer questions in the Class 7
Maths Chapter 12 Exercise 12.1 of the NCERT Solutions provided by Extramarks.
Students will encounter like and unlike terms in algebraic expressions, and if one needs to solve the equation, one might become confused about them. Therefore, it becomes crucial to first identify
them. First, don’t concentrate on the numerical coefficients; second, locate the variables—they will be the same—and then check their powers—they will also be the same.
Through the link on the Extramarks website and Extramarks mobile application, students may obtain the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi.
The students will be able to quickly grasp the basic algebraic principles in the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi, since they will be able to recall them from prior courses. To
improve their problem-solving skills, they are still urged to read the examples given.
Students practising the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi should not skip any of the exercises because practising the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi will
help them develop a solid foundation in algebra. They will be able to explore more expressions and develop quick problem-solving techniques as they solve more, which will allow them to finish their
paper faster.
प्रश्नावली 12.2
On the Extramarks website and mobile application, students can get the NCERT Solutions for Class 7 Maths Exercise 12.2 Chapter 12 Algebraic Expressions in a straightforward PDF format. In this
problem from NCERT Solutions For Class 7 Maths Chapter 12 In Hindi, the themes of adding and subtracting similar terms and adding and subtracting general algebraic equations are discussed. These
NCERT Solutions For Class 7 Maths Chapter 12 In Hindi centred on Algebraic Expression have been created by Extramarks’ subject specialists to aid students in their test preparation. Students can
practice various sorts of problems either online or by downloading these NCERT Solutions For Class 7 Maths Chapter 12 In Hindi.
Simplifying an algebraic expression is necessary in order to solve it. Arithmetic operators on similar terms must be used during simplification.The addition and subtraction of algebraic expressions
are explained in the NCERT Solutions for Class 7 Maths Chapter 12 in Hindi Algebraic Expressions for the sake of simplification. It demonstrates how similar terms in an expression can be added to or
taken away. The NCERT Solutions For Class 7 Maths Chapter 12 In Hindi questions are based on the concept mentioned above.
The students are tasked with adding and subtracting the provided phrases in six different problems. There are 7 short answer questions in the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi.
The variables are integers, thus the distributive law may be applied to them, as the students must keep in mind. In the same way that the signs of an algebraic expression should be treated,
individual numerical signs should be handled. On the Extramarks website and mobile application, students can obtain the exercise questions for NCERT Solutions For Class 7 Maths Chapter 12 In Hindi
based on Algebraic Expressions.
The chapter 12 exercise 12.2 Algebraic Expressions questions in this portion of the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi pertain to the simplification procedure. Students must pay
close attention to any formulations in order to identify like terms, because simplification occurs when like terms are combined with arithmetic operators in the NCERT Solutions For Class 7 Maths
Chapter 12 In Hindi.
To assess their ability to observe, students should attempt to complete as many problems from the NCERT Solutions for Class 7 Maths Chapter 12 in Hindi as possible.
Understanding the methods used to determine the sum and difference using two or more algebraic expressions will be made easier with the aid of the NCERT Solutions For Class 7 Maths Chapter 12 In
प्रश्नावली 12.3
When determining whether a certain value of a variable satisfies a given equation, for example, students may need to determine the value of an expression. The substitution approach is thus introduced
to the students in the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi centred on Algebraic Expressions. To solve the practise problems, students must substitute the supplied values for the
given phrases.
There are 10 questions in this Exercise 12.3 of the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi, and the students will have no trouble answering them.
Students use algebraic equations to determine the value of unknown variables and are aware that the value depends on the variables making up the expression. Students may find the answers to the NCERT
Solutions For Class 7 Maths Chapter 12 In Hindi online on the Extramarks website and Extramarks mobile application.
With the aid of the examples in NCERT Solutions For Class 7 Maths Chapter 12 In Hindi centred on Algebraic Expressions, students should grasp how an expression is evaluated. The processes are
straightforward so that the students can quickly comprehend what must be done in order to properly analyse them. The students must pay close attention to the substitution approach in order to enter
the proper numbers and continue to complete the computations.
Students can explore the idea of structuring the Mathematical rules with the aid of the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi, which is focused on algebraic expressions. As the NCERT
Solutions For Class 7 Maths Chapter 12 In Hindi will aid in framing the right expression out of the problem, they ought to be equipped with the necessary skills, such as an understanding of the
problem statement and the utterances that correspond to each operator in an algebraic expression. To assist students, the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi include thorough
examples and clear explanations.
NCERT Solutions for Class 7 Maths Chapter 12 Algebraic Expressions In Hindi
According to the NCERT Solutions for Class 7 Maths Chapter 12 in Hindi, algebraic expressions are the use of letters or alphabets to represent numbers without providing their precise valuesStudents
learned how to express an unknown value using letters like x, y, and z in the fundamentals of algebra. Therefore, specialists refer to these letters as variables. Variables and constants can both be
used in an algebraic expression. A component is any value that is added before a variable and then multiplied by it.
An expression that includes variables, constants, and algebraic operations is known as an algebraic expression in Mathematics (addition, subtraction, etc.). Terms combine to form expressions.
Additionally, students can complete problems in worksheets for algebraic expressions that are offered by Extramarks experts for both online and offline use.
Below is an overview of the concepts of Variables, Constants, Terms and Coefficients as discussed in the NCERT Solutions for Class 7 Maths Chapter 12 In Hindi:
In Mathematics,
• A variable is a symbol that doesn’t have a set value. Any value may be used. n is a variable that can have the values 1, 2, 3, in the example with the matchsticks above. Mathematics variables
include things like a, b, x, y, z, m, etc.
• A constant, on the other hand, is a symbol that has a set numerical value. Every number is a constant. Constants include numbers like 3, 6, -(1/2), 5, etc.
• A term is a variable alone (or) a constant alone (or) it can be a combination of variables and constants by the operation of multiplication or division. Some examples of terms are 3×2, -(2y/3), √
(5x), etc.
• Here, the numbers that are multiplying the variables are 3, -2/3, and 5. These numbers are called coefficients.
The many sorts of algebraic expressions are determined by the variables included in each expression, the number of terms in each expression, and the magnitudes of the variable exponents in each
expression. The algebraic expressions are broken down into five separate categories in a table that is available on the Extramarks website and mobile application. It can be examined by students to
help them understand the concepts and ideas in the NCERT Solutions for Class 7 Maths Chapter 12 In Hindi.
There are numerous branches in Mathematics. Arithmetic is the division that controls numbers and their operations. Calculations involving numbers, such as addition, subtraction, multiplication, and
division, are performed using it. Geometry is the study of figures’ shapes, dimensions, and construction with the use of a compass, ruler, and pencil. Another intriguing division that takes the form
of numbers and letters that we refer to as variables to explain our everyday circumstances is algebra. Algebraic expressions include variables, numbers, and operational signs. Examples include
multiplication, division, exponentiation with the natural exponent, addition, and subtraction. Fractional expression is the name given to division in an algebraic expression with variables.
Rational numbers are those that can be expressed in fractional form. Examples include recurring decimals, integers, and decimal terminations.
An algebraic expression is referred to as being irrational if it contains the variable’s root or a fractional power with a changeable base. Therefore, using algebraic terms can be both rational and
illogical. Integral and fractional expressions are two categories of rational expressions.
To learn mathematics, students must comprehend what an expression and an equation are. The aforementioned concepts have been further discussed in extensive detail in the NCERT Solutions for Class 7
Maths Chapter 12 In Hindi.
Finding the simplified term of the given expression is the aim of algebraic expression simplification. Students must first understand how to join similar terms, how to factor a number, the order of
operations, and whether to factor the expression or simply simplify it before we can factor an algebraic equation. For the purpose of simplification, the constant terms are segregated and the
variables that share the same degree are gathered along with the same terms. Students can learn how to simplify algebraic expressions through the NCERT Solutions for Class 7 Maths Chapter 12 In
Hindi, available for access on the Extramarks website and Extramarks mobile application.
Below are a few important terminologies used in the NCERT Solutions for Class 7 Maths Chapter 12 In Hindi that students must understand before solving the NCERT Solutions for Class 7 Maths Chapter 12
In Hindi:
• An Expression: An algebraic expression is one that includes variables, integers, and operation symbols. In algebra, readers engage with variables, symbols, or characters whose meanings are
• There are four primary categories of algebraic expressions, including:
• Monomial Expression.
• Binomial Expression.
• Trinomial Expression.
• Polynomial Expression.
1. Monomial Expression: A single-term algebraic expression is referred to as a monomial.
2. Binomial Expression: An algebraic expression with two unlikely terms is known as a binomial expression.
3. Trinomial Expression: A trinomial is an algebraic expression with just three non-zero terms.
4. Polynomial Expression: A polynomial is often defined as a term with non-negative integral exponents for a variable.
About Extramarks:
A modern provider of digital education solutions is Extramarks. 3,600 solutions are offered for K–12 schooling. It began to offer its services in 2009. Extramarks has quickly become extremely
well-liked in schools throughout India, South Africa, and the Middle East. Extramarks’ products are presently used by more than 5000 schools and nearly a million students.
For schools and students, Extramarks offers comprehensive solutions, ranging from school ERP to Smart Learn Classes, Test Centers, and homework assistance.
Learning modules used in the company “Extramarks Smart Learn Classes”‘ educational services are matched to the curricula offered by various educational boards. These lessons are based on a unique
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Students can utilise Extramarks Live, a programme designed to assist with their homework, at their own pace and convenience. It uses a three-pronged strategy—Learn, Practice, and Test—to satisfy a
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Extramarks Live is also used by schools to supplement in-class instruction. Heavy school backpacks and text books are eliminated when Live in the classroom is used.
FAQs (Frequently Asked Questions)
1. Are there practice sheets included in the NCERT Solutions For Class. Maths Chapter 12 In Hindi?
Yes, Extramarks experts have made sure to include extra practice worksheets in the Class 7 Maths Chapter 12 In Hindi NCERT Solutions for students to better comprehend the concepts in the chapter.
2. How to avail the NCERT Solutions For Class 7 Maths Chapter 12 In Hindi by Extramarks?
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Formula in Smartsheet
I have the formula as below and it working but when i add 1 more status it turn out error
=IF(AND([CM attention required]@row = "Yes", [PPV Status]@row = "MSI CM Review"), VLOOKUP(Commodity@row, {Category Managers Range 1}, 3, 0), VLOOKUP(Commodity@row, {Category Managers Range 1}, 2, 0))
Formula below have error since I have added 1 more status as need info
=IF(AND([CM attention required]@row = "Yes", [PPV Status]@row = "MSI CM Review","Need Info"), VLOOKUP(Commodity@row, {Category Managers Range 1}, 3, 0), VLOOKUP(Commodity@row, {Category Managers
Range 1}, 2, 0))
May I know what is the correct formula? Pls help to advise. Thanks
Best Answer
• Hi @HZAR
You need to include the range for the new criteria. See how the others are all pairs:
[CM attention required]@row = “Yes”
[PPV status]@row=“MSI CM Review”
but then you have “Need Info” without specifying which cell to look in. You need [column name]@row= before the criteria.
• Hi @HZAR
You need to include the range for the new criteria. See how the others are all pairs:
[CM attention required]@row = “Yes”
[PPV status]@row=“MSI CM Review”
but then you have “Need Info” without specifying which cell to look in. You need [column name]@row= before the criteria.
• Hi KPH
Thanks for the point…now i know where is my mistake. Thanks 😍
• Hi KPH
I taught i understand what you mean but when I put in the one as below formula, it give diff name
=IF(AND([CM attention required]@row = "Yes", [PPV Status]@row = "MSI CM Review", VLOOKUP(Commodity@row, {Category Managers Range 1}, 3, 0), VLOOKUP(Commodity@row, {Category Managers Range 1}, 2,
=IF(AND([CM attention required]@row = "Yes", [PPV Status]@row = "MSI CM Review", [PPV Status]@row = "Need Info"), VLOOKUP(Commodity@row, {Category Managers Range 1}, 3, 0), VLOOKUP(Commodity@row,
{Category Managers Range 1}, 2, 0))-Answer: Matt….
By right i want if the status is under CM review or Need info, when CM attention required then it will remain to Jake
Need your help to advise. Thanks
• Hi @HZAR
It looks like when you added the extra criteria you moved a parenthesis. Or maybe your VLOOKUP is not correct. I am not sure what you are trying to do, and cannot see your data, so will explain
what the formula does which might help.
This one you say returns Matt
=IF(AND([CM attention required]@row = "Yes", [PPV Status]@row = "MSI CM Review", [PPV Status]@row = "Need Info"), VLOOKUP(Commodity@row, {Category Managers Range 1}, 3, 0), VLOOKUP(Commodity@row,
{Category Managers Range 1}, 2, 0))
The part in bold means if in the current row CM attention required is "Yes", PPV Status is "MSI CM Review" and PPV Status is "Need Info" then do the next thing. The VLOOKUP then looks at another
table and returns the value from column 3 if the first column in the table contains the same value as Commodity in the current row. If the part in bold is not true then the value from column 2 is
returned instead. The 0s at the end of each VLOOKUP might be doing something unexpected as the valid data there is True or False. Adding a 0 won't break anything but I am not sure what it will
do. You can use false to return the first exact match.
VLOOKUP(Commodity@row, {Category Managers Range 1}, 3, false)
This one, you say, returns Jake, which is what you want, but to be honest I have no idea how it is even working:
=IF(AND([CM attention required]@row = "Yes", [PPV Status]@row = "MSI CM Review", VLOOKUP(Commodity@row, {Category Managers Range 1}, 3, 0), VLOOKUP(Commodity@row, {Category Managers Range 1}, 2,
In this one there is no parenthesis to close the AND function so the VLOOKUP is also a criteria, rather than an answer, but there is no = and no closing parenthesis. I would expect this to be
• Hi KPH
Sorry for not providing you enough details. I have attached the scenario from the vlook file that I have
If MR Manager attention required=Yes, Status=MSI MR Review & Need info, then the name should be under CC
If MR Manager attention=No, all the status will be under Mike
How do I rearrange the formula that I have since I need to add the status "Need info" to the below formula which is currently working if under MSI CM review, MR Manager required =Y. . Pls help to
advise. Thanks
=IF(AND([CM attention required]@row = "Yes", [PPV Status]@row = "MSI CM Review"), VLOOKUP(Commodity@row, {Category Managers Range 1}, 3, 0), VLOOKUP(Commodity@row, {Category Managers Range 1}, 2,
As per suggestion
=IF(AND([CM attention required]@row = "Yes", [PPV Status]@row = "MSI CM Review", [PPV Status]@row="Need Info"),VLOOKUP(Commodity@row, {Category Managers Range 1}, 3, false), VLOOKUP
(Commodity@row, {Category Managers Range 1}, 2, false))-give me Mike instead of CC
• Hi KPH
Sorry to miss out the screen shot for the Vlookup file
Passive comm
If CM Attention is Yes, Status-Need info, MSI CM review=CC
If Cm attention =No, All status is under Mike
=IF(AND([CM attention required]@row = "Yes", [PPV Status]@row = "MSI CM Review", [PPV Status]@row = "Need Info"), VLOOKUP(Commodity@row, {Category Managers Range 1}, 3, 0), VLOOKUP(Commodity@row,
{Category Managers Range 1}, 2, 0))-it me mike instead f CC
• I'm not sure I understand how it is returning both the wrong row and the wrong column but think the column issue could be the use of AND rather than OR
At the moment…
If CM attention required=Yes, Status=MSI CM Review AND Need info, then it returns the 3rd column.
If ANY of those things are false it will return the 2nd column.
I don't think you really do want to use AND. It sounds like you might mean OR. In other words, if any one of the things are true it should return the 3rd column. If so change the AND to OR.
If this is not the issue, please share a screenshot of the data you have (with the attention required, stats and commodity rows) and the outcome you want to see.
• Hi KPH
I have attached the scenario file for your end to advise how that I can create the desire output. Hope my example is good for your end. Appreciate your help to review and advise
• Hi @KPH
I already have the way to get the output that I need. Need to change the AND to OR plus add one more if function in the formula before the vlookup. Thanks for the help all the way and explanation
that you have put in details. You awesome.
• Great news @HZAR
I'm glad you have it working.
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ML Aggarwal Solutions for Class 8 Maths - Download Free Chapter-wise PDF
ML Aggarwal Maths is one of the best reference books available for ICSE students to prepare for their board exams. ML Aggarwal Solutions for Class 8 are provided here, which include all the chapters
and their exercises, with the PDFs attached for students to download. The team of subject experts at BYJU’S have formulated the difficult problems into simple steps, which help students solve the
problems with ease. Here, the solution module utilises various shortcut methods and examples to explain all the exercise problems in the most understandable language. Using ML Aggarwal Solutions,
students can practise a wide variety of questions from the textbook. Exercise-wise solutions are provided to help students understand the concepts clearly from the exam point of view. Students who
find difficulty in solving problems can quickly jump to ML Aggarwal Solutions to understand how the concepts are solved. These solutions are completely based on the exam-oriented approach to help
students get good marks in their board examination. Students are advised to practise regularly to understand how tricky problems are solved and to speed up the method of solving problems, which in
turn, helps in building time management skills. Students can start practising offline to yield good results in the board exams by downloading the PDF of ML Aggarwal Solutions for Class 8 from the
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*The complete ML Aggarwal Class 8 Solutions for the year 2023-24 will be available soon…
ML Aggarwal Class 8 Solutions ICSE Maths Chapters
Why Should Students Follow ML Aggarwal Class 8 Solutions?
The ML Aggarwal Solutions for Class 8, which are provided here, help in understanding the concepts clearly, and students would be able to answer all types of questions with the help of these
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ML Aggarwal Solutions for Class 8 is the most popular study material used by students to prepare for board exams. The solutions are prepared based on the latest syllabus, covering all the crucial
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To strengthen the weaker areas in solving problems, make use of ML Aggarwal Solutions, which are provided here, to achieve your goals. The subject experts have framed and solved the questions
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Frequently Asked Questions on ML Aggarwal Solutions for Class 8 Maths
Where to download ML Aggarwal Solutions for Class 8 Maths?
ML Aggarwal Solutions for Class 8 Maths can be downloaded by students in offline mode or can be referred to online from the BYJU’S website. The expert faculty at BYJU’S have provided stepwise answers
to the problems, which are present in the ML Aggarwal textbook of Class 8 Maths. These are as per the latest ICSE guidelines.
How many chapters are present in ML Aggarwal Class 8 Maths?
A total of 19 chapters are present in the ML Aggarwal Class 8 Maths. Chapter 1- Rational Numbers, Chapter 2- Exponents and Powers, Chapter 3- Squares and Square Roots, Chapter 4- Cubes and Cube
Roots, Chapter 5- Playing with Numbers, Chapter 6- Operation on Sets Venn Diagrams, Chapter 7- Percentage, Chapter 8- Simple and Compound Interest, Chapter 9- Direct and Inverse Variation, Chapter
10- Algebraic Expressions and Identities, Chapter 11- Factorisation, Chapter 12- Linear Equations and Inequalities in one Variable, Chapter 13- Understanding Quadrilaterals, Chapter 14- Constructions
of Quadrilaterals, Chapter 15- Circle, Chapter 16- Symmetry Reflection and Rotation, Chapter 17- Visualising Solid Shapes, Chapter 18- Mensuration, Chapter 19- Data Handling.
Which is the best source for Class 8 ICSE board exam preparation?
ML Aggarwal books are the best resource for ICSE board exam preparation. Students can prepare themselves by memorising the concepts given in the textbook a few weeks before the actual exams. A large
variety of questions are given for students to practise and become familiar with the concepts, which are covered in other books as well. Students can work on the sample papers and previous years’
question papers to get to know the exam pattern and the weightage each chapter carries while preparing for the exam.
Keep visiting BYJU’S to get more updated learning materials, and download the BYJU’S app for a better and personalized learning experience with engaging video lessons anywhere and at any time of the
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Averages: Mean, Median and Mode
The term ‘average’ occurs frequently in all sorts of everyday contexts. For example, you might say ‘I’m having an average day today’, meaning your day is neither particularly good nor bad, it is
about normal. We may also refer to people, objects and other things as ‘average’.
The term 'average' refers to the ‘middle’ or ‘central’ point. When used in mathematics, the term refers to a number that is a typical representation of a group of numbers (or data set). Averages can
be calculated in different ways - this page covers the mean, median and mode. We include an averages calculator, and an explanation and examples of each type of average.
The most widely used method of calculating an average is the ‘mean’. When the term ‘average’ is used in a mathematical sense, it usually refers to the mean, especially when no other information is
Quick Guide:
To calculate the Mean
Add the numbers together and divide by the number of numbers.
(The sum of values divided by the number of values).
To determine the Median
Arrange the numbers in order, find the middle number.
(The middle value when the values are ranked).
To determine the Mode
Count how many times each value occurs; the value that occurs most often is the mode.
(The most frequently occurring value)
Mean, Median and Mode Calculator
Use this calculator to work out the mean, median and mode of a set of numbers.
The mathematical symbol or notation for mean is ‘x-bar’. This symbol appears on scientific calculators and in mathematical and statistical notations.
The ‘mean’ or ‘arithmetic mean’ is the most commonly used form of average. To calculate the mean, you need a set of related numbers (or data set). At least two numbers are needed in order to
calculate the mean.
The numbers need to be linked or related to each other in some way to have any meaningful result – for instance, temperature readings, the price of coffee, the number of days in a month, the number
of heartbeats per minute, students’ test grades etc.
To find the (mean) average price of a loaf of bread in the supermarket, for example, first record the price of each type of loaf:
• White: £1
• Wholemeal: £1.20
• Baguette: £1.10
Next, add (+) the prices together £1 + £1.20 + £1.10 = £3.30
Then divide (÷) your answer by the number of loaves (3).
£3.30 ÷ 3 = £1.10.
The average price of a loaf of bread in our example is £1.10.
The same method applies with larger sets of data:
To calculate the average number of days in a month we would first establish how many days there are in each month (assuming that it was not a leap year):
Month Days
January 31
February 28
March 31
April 30
May 31
June 30
July 31
August 31
September 30
October 31
November 30
December 31
Next we add all the numbers together: 31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 30 + 31 = 365
Finally we divide the answer with the number of values in our data set in this case there are 12 (one for each month counted).
So the mean average is 365 ÷ 12 = 30.42.
The average number of days in a month, therefore, is 30.42.
The same calculation can be used to work out the average of any set of numbers, for example the average salary in an organisation:
Let's assume the organisation has 100 employees on one of 5 grades:
Grade Annual Salary Number of
1 £20,000 21
2 £25,000 25
3 £30,000 40
4 £50,000 9
5 £80,000 5
In this example we can avoid adding each individual employee’s salary as we know how many are in each category. So instead of writing out £20,000 twenty-one times we can multiply to get our answers:
Grade Annual Salary Number of Salary x
Employees Employees
1 £20,000 21 £420,000
2 £25,000 25 £625,000
3 £30,000 40 £1,200,000
4 £50,000 9 £450,000
5 £80,000 5 £400,000
Next add the values in the Salary x Employees column to find a total: £3,095,000 and finally divide this number by the number of employees (100) to find the average salary:
£3,095,000 ÷ 100 = £30,950.
Quick Tip:
The salaries, in the example above, are all multiples of £1,000 – they all end in ,000.
You can ignore the ,000's when calculating as long as you remember to add them back on at the end.
In the first row of the table above we know that twenty-one people get paid a salary of £20,000, instead of working with £20,000 work with 20:
21 x 20 = 420 then replace the ,000 to get 420,000.
Sometimes we may know the total of our numbers but not the individual numbers that make up the total.
In this example, assume that £122.50 is made by selling lemonade in a week.
We don’t know how much money was made each day, just the total at the end of the week.
What we can work out is the daily average: £122.50 ÷ 7 (Total money divided by 7 days).
122.5 ÷ 7 = 17.50.
So we can say that on average we made £17.50 a day.
We can also use averages to give us a clue of likely future events – if we know that we made £17.50 a day on average selling lemonade in a week then we can assume that in a month we would make:
£17.50 × Number of days in that month
17.50 × 31 = £542.50
We could record average sales figures each month to help us predict sales for future months and years and also to compare our performance. We could use terms like ‘above average’ – to refer to a
time period when sales were more than the average amount and likewise ‘below average’ when sales were less than the average amount.
Using speed and time as data to find the mean:
If you travel 85 miles in 1 hour and 20 minutes, what was your average speed?
The first thing to do with this problem is to convert the time into minutes – time does not work on the decimal system as there are 60 minutes in an hour and not 100. Therefore we need to
standardise our units before we can start:
1 hour 20 minutes = 60 minutes + 20 minutes = 80 minutes.
Next divide the distance travelled by the time taken: 85 miles ÷ 80 minutes.
85 ÷ 80 = 1.0625.
Our average speed therefore was 1.0625 miles per minute.
Convert this figure back to hours by multiplying by 60 (the number of minutes in an hour).
1.0625 × 60 = 63.75mph (miles per hour).
For Spreadsheet users:
Use the <average> function to calculate the mean average in a spreadsheet. The following example formula, assumes your data is in cells A1 to A10:
The Median is the middle number in a list of sorted numbers.
To calculate the Median of: 6, 13, 67, 45, 2
First, arrange the numbers in order (this is also known as ranking)
2, 6, 13, 45, 67
then - find the middle number
Median = 13, the middle number in the ranked list.
When there are an even number of numbers there is no single middle number but a pair of middle numbers.
In such cases the median is the mean of the two middle numbers:
For example:
6, 13, 67, 45, 2, 7.
Arranged in order (ranked) = 2, 6, 7, 13, 45, 67
The middle numbers are 7 and 13.
The median refers to a single number so we calculate the mean of the two middle numbers:
7 + 13 = 20
20 ÷ 2 = 10
Therefore the median of 6, 13, 67, 45, 2, 7 is 10.
The Mode is the most frequently-occurring value in a set of values. The mode is interesting as it can be used for any type of data, not just numbers.
In this example, assume that you have bought a pack of 100 balloons, the pack is made up of 5 different colours, you count each colour and find that you have:
18 Red
12 Blue
24 Orange
25 Purple
21 Green
The mode of our sample of balloons is purple as there are more purple balloons (25) than any other colour balloon.
To find the mode of the number of days in each month:
Month Days
January 31
February 28
March 31
April 30
May 31
June 30
July 31
August 31
September 30
October 31
November 30
December 31
7 months have a 31 days, 4 months have a total of 30 days and only 1 month has a total of 28 days (29 in a leap year).
The mode is therefore, 31.
Some data sets may have more than one Mode:
1,3,3,4,4,5 – for example, has two most frequently occurring numbers (3 & 4) this is known as a bimodal set. Data sets with more than two modes are referred to as multi-modal data sets.
If a data set contains only unique numbers then calculating the mode is more problematic.
It is usually perfectly acceptable to say there is no mode, but if a mode has to be found then the usual way is to create number ranges and then count the one with the most points in it. For example
from a set of data showing the speed of passing cars we see that out of 10 cars the recorded speeds are:
40, 34, 42, 38, 41, 50, 48, 49, 33, 47
These numbers are all unique (each only occurs once), there is no mode. In order to find a mode we build categories on an even scale:
30--32 | 33--35 | 36--38 | 39--41 | 42--44 | 45--47 | 48--50
Then work out how many of the values fall into each category, how many times a number between 30 and 32 occurs, etc.
30--32 = 0
33--35 = 2
36--38 = 1
39--41 = 2
42--44 = 1
45--47 = 1
48--50 = 3
The category with the most values is 48--50 with 3 values.
We can take the mid value of the category to estimate the mode at 49.
This method of calculating the mode is not ideal because the mode may change depending on the categories you define.
Further Reading from Skills You Need
Data Handling and Algebra
Part of The Skills You Need Guide to Numeracy
This eBook covers the basics of data handling, data visualisation, basic statistical analysis and algebra. The book contains plenty of worked examples to improve understanding as well as real-world
examples to show you how these concepts are useful.
Whether you want to brush up on your basics, or help your children with their learning, this is the book for you. | {"url":"https://www.skillsyouneed.com/num/averages.html","timestamp":"2024-11-11T01:35:29Z","content_type":"text/html","content_length":"54192","record_id":"<urn:uuid:a236adc3-5687-4308-81a1-dd2bfd7689a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00698.warc.gz"} |
Equation Maker Help
Equation Maker allows you to typeset equations using LaTeX syntax. You can drag and drop the equations into almost any Mac application (including Pages, Numbers, Keynote, and Microsoft Word).
Equation Maker Overview
Helpful tips
• Use PDF as the export format for the highest quality resolution. Use PNG only in cases where PDF is not supported.
• There are two ways to save equations.
□ Save the equation in the left-hand side table.
□ Save the equation to a PDF file by dragging the equation to the Desktop/Finder or selecting File->Export in the menu. You can then restore the equation by dragging the PDF file on to the
equation view.
• To insert plain text, use the \text{ text here } command. To inline math inside a text command, surround the math with "$" characters. Example: \text{if and only if $x > 0$}
• For multi-line equations, use the \begin{align} equation \end{align} environment command
• The characters $, &, {, } and _ have special meaning in LaTeX and are reserved. To use these actual characters, precede them with a slash (escape): \. Example: \% prints the percentage sign.
• Special functions such the trigonometric functions can be used as commands. Examples: \sin{\theta}, \exp(x)
• Using LaTeX code instead of in-line equation editing (such as Microsoft Equation Editor) may seem strange if you are new to it. However once you learn the common commands, using them is faster
than clicking through a palette menu.
Equation Maker Interface
Typesetting equations
Enter your LaTeX equation code into the
. As you type, your equation will render in the
Equation View
. Matching brackets are highlighted to aid in editing. To save your equations, click the Save button in the toolbar. You can select a saved equation in the
Saved Equations
The Palette
To view the palette, click the Palette button in the toolbar. The palette helps in recalling the LaTeX commands for common math elements, including math expressions, symbols, greek letters, character
decorations and arrows. Click a button in the palette to insert the corresponding command in the Editor.
Changing Equation Color
Select the Color button in the toolbar to change the color of the equation.
You can zoom the
Equation View
in and out using the slider on the toolbar.
Exporting Equations
To export an equation for use in another application, drag it from the
Equation View
and drop it into the application. To save as a file, drag and drop it onto the desktop or in a Finder window or select File->Export from the menu. You can export it as either a PDF or PNG. Choose the
format using the
Export Format
popup button on the toolbar. Note that PDF exports a higher resolution image. Exporting to PNG is useful when PDF is not supported (for example, for inclusion on a webpage).
Saving Equations
Save an equation by clicking the Save button in the toolbar. This will add the equation to the Saved Equations table on the left side of the application's main window. You can reuse the equation
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Euclid: An Author Profile - CLT Journal
If we speak of standing on the shoulders of giants, few are more gigantic than Euclid.
As a person, Euclid is one of the more obscure figures on our author bank: hardly any contemporary references to his life survive, even in writers like Plutarch or Diogenes Laërtius who chronicled
many ancient men’s lives. He is known to have been born sometime in the late fourth century BC—exactly what year, or where, is not clear—and to have worked at the famous Library of Alexandria for
some decades. Writing over six hundred years after his death, the Neoplatonist philosopher Proclus related that Euclid was a Platonist who lived under the reign of Ptolemy I of Egypt, the immediate
successor of Alexander the Great and founder of the Ptolemaic Egyptian realm and dynasty. Archimedes and Apollonius of Perga are both thought to have learned from Euclid’s pupils, probably after his
death, which occurred around 270 BC; and that is about as much as we know. (Some scholars have gone so far as to doubt Euclid’s historical existence, but this hypothesis is not widely accepted.)
But unlike his person, what we do know a good deal about is Euclid’s work. He wrote extensively on mathematics, the sciences, and logic, and several of his works survive, complete or in fragments.
His influence on ancient astronomy was significant; besides penning a book of his own on the subject (Phænomena), he also proved in his Optics that, for any two objects of unequal size, there is some
vantage point from which they appear to be equal, a proof with important implications for the measurement of the stars and planets, and which was taken up by Claudius Ptolemy in his authoritative
work on the subject. Euclid also wrote a book on mirrors, a (lost) introduction to fallacies, and a work on conic sections that may survive as the first half of Apollonius’ treatise on the subject.
But what Euclid is best remembered for is his work on geometry. The Elements, his guide to geometry, remained the standard textbook for two thousand years, and was displaced in the twentieth century
only by the radical reimagining of the nature of space that Einstein‘s theory of general relativity required. The text also delves into number theory, with particular attention to prime, perfect, and
irrational numbers.
His quote (reproduced here) that there is no shortcut—literally, no “royal road”—to geometry is reportedly what he told Ptolemy I, when the latter asked whether there were any simpler way to learn
the discipline. Euclid’s sarcasm is understandable: the opening of the Elements is about as simple and intuitive as a work could possibly be, demanding of its student such gargantuan efforts of
understanding as “Things which are equal to the same thing are equal to each other” and “A circle can be drawn with any center and any radius.” From principles as simple as these, he derives his
entire system of geometry, using nothing but logical, step-by-step proofs. (It is hard not to see the influence of Plato here, if we compare the methodical chapters of the Elements to the dialogue
between Socrates and the slave boy in the Meno.) Nor, in terms of technical equipment, is Euclidean geometry demanding; nothing more than a flat surface, a stylus of some kind, a compass, and a
straightedge allow the most amateur student to reproduce every one of Euclid’s geometric proofs.
One genuine weakness, of a kind, is present in the Elements. Euclid’s fifth postulate, also called the parallel postulate, states that if line a crosses lines b and c, and the interior angles at the
junctions a-b and a-c add up to less than 180 degrees, then lines b and c will (if produced far enough) eventually cross each other on the same side of a as those interior angles. What’s strange
about this postulate is that it has turned out to be wrong, or rather, incomplete; many scholars attempted to prove it over the centuries, only for their proofs to be discovered—sometimes after being
accepted for hundreds of years—to be flawed, most often by being unintentionally circular arguments.
It was finally determined in the nineteenth century that coherent geometries could be constructed which contradicted the parallel postulate. The reason for this is that the parallel postulate only
describes a specific kind of space, namely, a “flat” space; geometry on a curved surface is radically different, and based on Einstein’s theory of gravitation, space itself is, for lack of a better
way of putting it, curved—distorted by the pull of any object with mass, and the more distorted the greater the mass. Euclidean geometry was thus shown to be rather like Newton’s laws, inaccurate at
the cosmic level but perfectly good for most practical uses. As the mathematician and historian W. W. Rouse Ball dryly put it, “The fact that for two thousand years the Elements was the usual
textbook on the subject raises a strong presumption that it is not unsuitable for that purpose.”
Every week, we publish a profile of one of the figures from the CLT author bank. For an introduction to classic authors, see our guest post from Keith Nix, founder of the Veritas School in Richmond,
If you enjoyed this essay, take a look at some of our other posts here at the Journal, like this profile of Louisa May Alcott or this analysis of the twin ideas of good and evil. And be sure to check
out our Journey Through the Author Bank seminar series, featuring guest lectures from academics across the country. | {"url":"https://blog.cltexam.com/euclid-author-profile/","timestamp":"2024-11-15T05:03:28Z","content_type":"text/html","content_length":"169697","record_id":"<urn:uuid:e9a4e8ae-a04a-4936-9152-d90c6b4f8b86>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00097.warc.gz"} |
Better performance to blockwise matrix multiplication
The following function makes it possible to perform blockwise matrix multiplication that results in a very large matrix. Basically the arrays are sliced into smaller blocks which are then multiplied
and the final matrix is built from them.
block_multiply(X, R, A; n)
Perform matrix multiplication block-wise: B = X*R*X'*A
function block_multiply(X, R, A; n=3)
N, M = size(X)
T = floor.(Int, linspace(0, N, n+1))
spans = collect(T[i]+1:T[i+1] for i=1:n)
B = similar(A)
fill!(B, 0.0)
local i, j
for i=1:n
for j=1:n
a = spans[i]
b = spans[j]
B[a,:] += X[a,:]*R*transpose(X[b,:])*A[b,:]
return B
Now the problem is, that this calculation still needs a huge amount of allocations. For instance the following performance test
function perf_test(ndofs, nvals, nkeep, nblocks)
X = rand(ndofs, nvals)
R = rand(nvals, nvals)
A = rand(ndofs, nkeep)
@time B = block_multiply(X, R, A; n=nblocks)
with following input: perf_test(30000, 1, 500; n=10)
performs as follows: 28.734395 seconds (2.81 k allocations: 11.297 GiB, 34.04% gc time) .
Any suggestions how I could make this function to perform better (use less allocations)?
1 Like
You could refactor this line to do it explicitly, since this creates intermediate allocations in each loop:
If the sizes of the matrices are representative here, remember that the order that you do the operations might be important. Here it should be enough to do the (X' * A) first but check out https://
julia> using MatrixChainMultiply
julia> function perf_test(ndofs, nvals, nkeep, nblocks)
X = rand(ndofs, nvals)
R = rand(nvals, nvals)
A = rand(ndofs, nkeep)
@time B = block_multiply(X, R, A; n=nblocks)
@time B = X * R * X' * A
@time B = X * R * (X' * A)
@time B = matrixchainmultiply(X, R, X', A)
perf_test (generic function with 1 method)
julia> perf_test(30000, 1, 500, 10);
13.832762 seconds (1.28 k allocations: 9.055 GiB, 9.13% gc time)
16.128467 seconds (6 allocations: 6.818 GiB, 0.67% gc time)
0.052784 seconds (5 allocations: 114.674 MiB, 17.54% gc time)
0.055796 seconds (40 allocations: 114.451 MiB, 0.43% gc time)
(I ran this on the nightly build)
If you still need to do blocking, you should use views instead of slicing, since views do not copy, and also, up-front, allocate work buffers that the resulting blockmatrix multiplications are stored
2 Likes | {"url":"https://discourse.julialang.org/t/better-performance-to-blockwise-matrix-multiplication/10677","timestamp":"2024-11-12T09:28:13Z","content_type":"text/html","content_length":"26012","record_id":"<urn:uuid:69cbe177-3776-4405-ba77-cc296fac53c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00396.warc.gz"} |
multiply floating-point number by integral power of radix (OBSOLETE)
#include <math.h>
[[deprecated]] double scalb(double x, double exp);
[[deprecated]] float scalbf(float x, float exp);
[[deprecated]] long double scalbl(long double x, long double exp);
_XOPEN_SOURCE >= 500
|| /* Since glibc 2.19: */ _DEFAULT_SOURCE
|| /* glibc <= 2.19: */ _BSD_SOURCE || _SVID_SOURCE
scalbf(), scalbl():
_XOPEN_SOURCE >= 600
|| /* Since glibc 2.19: */ _DEFAULT_SOURCE
|| /* glibc <= 2.19: */ _BSD_SOURCE || _SVID_SOURCE
These functions multiply their first argument x by FLT_RADIX (probably 2) to the power of exp, that is:
x * FLT_RADIX ** exp
The definition of FLT_RADIX can be obtained by including <float.h>.
Return Value
On success, these functions return x * FLT_RADIX ** exp.
If x or exp is a NaN, a NaN is returned.
If x is positive infinity (negative infinity), and exp is not negative infinity, positive infinity (negative infinity) is returned.
If x is +0 (-0), and exp is not positive infinity, +0 (-0) is returned.
If x is zero, and exp is positive infinity, a domain error occurs, and a NaN is returned.
If x is an infinity, and exp is negative infinity, a domain error occurs, and a NaN is returned.
If the result overflows, a range error occurs, and the functions return HUGE_VAL, HUGE_VALF, or HUGE_VALL, respectively, with a sign the same as x.
If the result underflows, a range error occurs, and the functions return zero, with a sign the same as x.
See math_error(7) for information on how to determine whether an error has occurred when calling these functions.
The following errors can occur:
Domain error: x is 0, and exp is positive infinity, or x is positive infinity and exp is negative infinity and the other argument is not a NaN
errno is set to EDOM. An invalid floating-point exception (FE_INVALID) is raised.
Range error, overflow
errno is set to ERANGE. An overflow floating-point exception (FE_OVERFLOW) is raised.
Range error, underflow
errno is set to ERANGE. An underflow floating-point exception (FE_UNDERFLOW) is raised.
For an explanation of the terms used in this section, see attributes(7).
Interface Attribute Value
scalb(), scalbf(), scalbl() Thread safety MT-Safe
4.3BSD. Obsolescent in POSIX.1-2001; Removed in POSIX.1-2008, recommending the use of scalbln(3), scalblnf(3), or scalblnl(3) instead.
Before glibc 2.20, these functions did not set errno for domain and range errors. | {"url":"https://www.mankier.com/3/scalb","timestamp":"2024-11-04T18:32:33Z","content_type":"text/html","content_length":"8971","record_id":"<urn:uuid:4e1f9146-5b2d-4ceb-8f23-4cf56f9feb32>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00485.warc.gz"} |
AdaGrad | IOMETE
What is AdaGrad?
AdaGrad is a family of sub-gradient algorithms for stochastic optimization. Gradient descent is the most commonly used optimization method in machine learning and deep learning algorithms. It's
essential for training machine learning models. Below, we'll discuss the three primary types of gradient descent used in modern machine learning and deep learning algorithms, as well as the
advantages of using AdaGrad.
Batch Gradient Descent
Batch Gradient Descent is the simplest type of gradient descent. It calculates the error for each example in the training dataset but only updates the model after all training examples have been
Stochastic Gradient Descent
Stochastic Gradient Descent calculates the error and updates the model for each example in the training dataset. This method is popular for optimization due to its conceptual simplicity and efficient
implementation. However, it requires manual tuning of the step size parameter.
Mini Batch Gradient Descent
Mini Batch Gradient Descent sums up the errors over a smaller number of examples based on the batch size and performs an update for each of these batches. This approach balances the benefits of both
Batch and Stochastic Gradient Descent.
AdaGrad: Adaptive Gradient Algorithm
AdaGrad is an optimization algorithm that dynamically incorporates knowledge of the geometry of the data observed in earlier iterations to perform more informative gradient-based learning. It adapts
the learning rate component-wise to the parameters by incorporating knowledge of past observations. This results in larger updates (high learning rates) for parameters related to infrequent features
and smaller updates (low learning rates) for frequent ones. AdaGrad is well-suited for dealing with sparse data, such as in natural language processing or image recognition.
Advantages of Using AdaGrad
• Eliminates the need to manually tune the learning rate.
• Faster and more reliable convergence compared to simple Stochastic Gradient Descent, especially when the scaling of the weights is unequal.
• Not very sensitive to the size of the master step.
Understanding the different types of gradient descent and the benefits of using AdaGrad can greatly improve the performance of your machine learning and deep learning algorithms. | {"url":"https://iomete.com/resources/glossary/adagrad","timestamp":"2024-11-07T18:58:27Z","content_type":"text/html","content_length":"16417","record_id":"<urn:uuid:f3538eff-bd20-4443-aaff-651446fa93b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00710.warc.gz"} |
How Many Ways
1. for word "MONTANA"
How Many Ways are There to Order the Letters of Word MONTANA?
The 7 letters word MONTANA can be arranged in 1260 distinct ways. The below detailed information shows how to find how many ways are there to order the letters MONTANA and how it is being calculated
in the real world problems.
Distinguishable Ways to Arrange the Word MONTANA
The below step by step work generated by the word permutations calculator shows how to find how many different ways can the letters of the word MONTANA be arranged.
Find how many distinguishable ways are there to order the letters in the word MONTANA.
Step by step workout:
step 1 Address the formula, input parameters and values to find how many ways are there to order the letters MONTANA.
nPr =n!/(n1! n2! . . . nr!)
Input parameters and values:
Total number of letters in MONTANA:
n = 7
Distinct subsets:
Subsets : M = 1; O = 1; N = 2; T = 1; A = 2;
Subsets' count:
n[1](M) = 1, n[2](O) = 1, n[3](N) = 2, n[4](T) = 1, n[5](A) = 2
step 2 Apply the values extracted from the word MONTANA in the (nPr) permutations equation
nPr = 7!/(1! 1! 2! 1! 2! )
= 1 x 2 x 3 x 4 x 5 x 6 x 7/{(1) (1) (1 x 2) (1) (1 x 2)}
= 5040/4
= 1260
nPr of word MONTANA = 1260
The letters of the word MONTANA can be arranged in 1260 distinct ways.
Apart from the word MONTANA, you may try different words with various lengths with or without repetition of letters to observe how it affects the nPr word permutation calculation to find how many
ways the letters in the given word can be arranged. | {"url":"https://getcalc.com/howmany-ways-lettersofword-montana-canbe-arranged.htm","timestamp":"2024-11-11T00:45:49Z","content_type":"text/html","content_length":"15555","record_id":"<urn:uuid:f87d4173-9e65-496e-9189-5cd9f1e1f145>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00231.warc.gz"} |
IIT JEE Physics - Current Electricity free Notes, Solved Questions, formulas - TopperLearning
Current Electricity
Current Electricity PDF Notes, Important Questions and Synopsis
• Electric current: The rate of flow of electric charges through a particular cross section. The SI unit of current is Ampere.
• Ohm’s law: The current between the ends of a conductor is directly proportional to the potential difference applied across its end, provided all other factors remain constant.
V = IR
• Dependence of resistance on various factors:
Resistance directly depends on length. Resistance is inversely proportional to area.
With an increase in temperature, resistance increases.
• Current density: The amount of current flowing per unit area around that point, provided the area is normal to the direction of current.
• Drift speed, which is the magnitude of this velocity, is enormously small as compared to the thermal speed, which is not a vector and is much larger.
• Metals have low resistivity: Range of ρ varies from 10–8 Ω m to 10–6 Ω m.
Insulators such as glass and rubber have high resistivity: Range of r varies from 1022 to 1024 times more than metals.
Semiconductors such as Si and Ge lie roughly in the middle range of resistivity on a logarithmic scale.
• When a conducting substance is brought under the influence of an electric field
where σ is a constant of proportionality called electrical conductivity.
• Temperature coefficient of resistivity:
For pure metals, resistance varies linearly with the rise of temperature.
• Emf (electromotive force) is the name given to a non-electrostatic agency. Typically, it is a battery in which a chemical process achieves this task of doing work in driving the positive charge
from a low potential to a high potential.
The effect of such a source is measured in terms of work done per unit charge in moving a charge once around the circuit. This is denoted by ε.
• Ohm’s law is obeyed by many substances, but it is not a fundamental law of nature. It fails if
□ V depends on I non-linearly. Example: ρ increases with I even if temperature is kept fixed.
□ The relation between V and I is non-unique. Example: GaAs
□ The relation between V and I depends on the sign of V for the same absolute value of V.
• Kirchhoff's First Rule:
At any junction of several circuit elements, the sum of currents entering the junction must equal the sum of currents leaving it.
In the above junction, current I enters and currents I[1] and I[2] leave.
Thus, I = I[1] + I[2].
This is a consequence of charge conservation and assumption that currents are steady, i.e. no charge piles up at the junction.
• Kirchhoff's Second Rule:
The algebraic sum of changes in potential around any closed resistor loop must be zero. This is based on the principle that electrostatic forces alone cannot do any work in a closed loop, because
this work equals the potential difference, which is zero, if we start at one point of the loop and come back to it.
When applied to a loop as shown above (which could be part of a larger circuit), this gives
-(R[1] + R[2]) I[1] - R[3] I[3] - R[4] I[4] = 0.
• Points to remember in the case of current loops:
Choose any closed loop in the network and designate a direction (in this example, counter clockwise) to traverse the loop.
• Go around the loop in the designated direction, adding emfs and potential differences. An emf is counted as positive when it is traversed from (−) to (+) and negative in the opposite case, i.e.
from (+) to (−).
• An IR term is counted negative if the resistor is traversed in the same direction of the assumed current and positive if in the opposite direction.
• Equate the total sum to zero.
• The Wheatstone bridge is an arrangement of four resistances—R[1], R[2], R[3] and R[4]. The null point condition is given by
This is also known as the balance condition. If, for instance, R[1], R[2] and R[3] are known, then R[4] can be determined.
• In a balanced condition of the meter bridge,
σ: Resistance per unit length of wire
ℓ[1]: Length of wire from one end where the null point is obtained
A potentiometer is a device to compare potential differences. Because the method involves a condition of no current flow, the device can be used to measure potential differences and the internal
resistance of a cell and to compare the emfs of two sources.
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utility maximization
The expected utility knapsack problem is to pick a set of items whose values are described by random variables so as to maximize the expected utility of the total value of the items picked while
satisfying a constraint on the total weight of items picked. We consider the following solution approach for this problem: (i) … Read more
Two new proofs of Afriat’s theorem
We provide two new, simple proofs of Afriat’s celebrated theorem stating that a finite set of price-quantity observations is consistent with utility maximization if, and only if, the observations
satisfy a variation of the Strong Axiom of Revealed Preference known as the Generalized Axiom of Revealed Preference. Citation Technical Report No. 1381, School of Operations … Read more | {"url":"https://optimization-online.org/tag/utility-maximization/","timestamp":"2024-11-07T07:26:01Z","content_type":"text/html","content_length":"85678","record_id":"<urn:uuid:344f71a5-ba6a-42b2-97a8-fbc92d62edcc>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00195.warc.gz"} |
About Mathematics and Statistics achievement standards
Level 2
Mathematics standards
The titles of all of the Mathematics standards are Apply xxx in solving problems.
xxx comes from content specified in the New Zealand Curriculum that students need to apply in solving problems so students need to be given a problem to solve. Problems need to allow for all levels
of achievement.
Within the problem there may be sub tasks but evidence for any level of achievement needs to come from the solution of the overall problem. This means there needs to be sufficient scope within the
problem for the thinking that is needed. Problems are situations that provide opportunities to apply knowledge or understanding of mathematical concepts and methods. Situations should be set in a
real-life or mathematical context.
Students should not be presented with a task in an unfamiliar context. It is acceptable for students to know the context before the assessment.
The SOLO taxonomy has been used to describe the different levels of thinking in the standards.
The stages in the SOLO taxonomy are increasing in quantity and quality of thought.
Achieved level is the multistructural stage where students need to know or use more than one piece of given information, fact, or idea, but do not integrate their ideas.
Merit is relational thinking where students see the significance of how the various pieces of information relate to one another. ‘Questions’ that provide evidence for relational thinking will require
students to integrate more than one piece of given knowledge, information, fact or idea.
Excellence is extended abstract thinking where students can make connections beyond the scope of the problem or question, to generalise or transfer learning into a new situation. Going beyond the
given information, knowledge or ideas, or deducing a more general rule or proof that applies to all cases are examples of extended abstract thinking.
At Achieved level all of the standards include the wording ‘selecting and using methods’. The evidence for this aspect can not come from a situation where students are told what method to use to
solve problems. Problems needs to be presented so there are no instructions about the method of solution.
For the ‘method’ to provide evidence for the standard it needs to be relevant to the solution of the problem and also at the appropriate curriculum level. The ‘methods’ which provide evidence for
each standard are listed in Explanatory Note 4 of the standard.
The appropriate curriculum level can be determined by looking at Explanatory Note 1 of the standard which lists the achievement objectives from The New Zealand Curriculum (NZC) which are covered by
the standard. There is further information about the achievement objectives at level 7 of the curriculum in the Teaching and Learning Guide. This information can be found by selecting the relevant
achievement objective from the Mathematics and Statistics Achievement objectives by level page on the TKI website.
Statistics standards
All of the statistics standards involve students using a specified ‘process’. The process varies from standard to standard depending on what aspect of statistics is involved. The ‘process’ is
detailed in Explanatory Note 3 of the standard.
The wording of the Achievement Criteria in the Statistics Standards reflects the step up in the quality of the response that is required at the different levels. Merit level is ‘with justification’
and Excellence is ‘with statistical insight’.
At Excellence students need to integrate statistical and contextual knowledge throughout the ‘process’. They need to have an understanding about the context and are expected to demonstrate they have
thought about the ‘process’ they have undertaken and their findings.
Information about the statistics achievement objectives can be found in the Teaching and Learning Guide on the TKI website. Additional information for the statistics standards is available from the
Census at School website.
Level 3
As the result of trialling, the following information is available to assist teachers to assess level three standards.
This document will be updated as soon as suitable material can be collected from students through the external moderation process.
This document should be read in conjunction with the achievement standard, the Conditions of Assessment document and the Teaching and Learning Guide. Additional information for the statistics
standards is available.
91573 Apply the geometry of conic sections in solving problems
• To be used as evidence for the award of Achieved, a ‘method’ must be relevant to the solution of the problem and at the appropriate curriculum level.
• At all levels there is a requirement relating to the communication of the solutions.
91574 Apply linear programming methods in solving problems
• To be used as evidence for the award of Achieved, a ‘method’ must be relevant to the solution of the problem and at the appropriate curriculum level.
• At all levels there is a requirement relating to the communication of the solutions.
91575 Apply trigonometric methods in solving problems
• To be used as evidence for the award of Achieved, a ‘method’ must be relevant to the solution of the problem and at the appropriate curriculum level.
• At all levels there is a requirement relating to the communication of the solutions.
91576 Use critical path analysis in solving problems
• To be used as evidence for the award of Achieved, a ‘method’ must be relevant to the solution of the problem and at the appropriate curriculum level.
• At all levels there is a requirement relating to the communication of the solutions.
91580 Investigate time series data
• The assessment requires multiple sessions to allow time for students to research the context and develop a purpose prior to completing the analysis.
• Students need to have knowledge about the context and statements that are made need to be related to the context.
• Use of a statistical graphing package is expected.
Students need to provide evidence of each component of the statistical enquiry cycle detailed in Explanatory Note 3 of the standard.
• Features of the data include the trend and seasonal pattern.
• While students could investigate more than one time series or combine series this is not a requirement of the standard.
91581 Investigate bivariate measurement data
• The assessment requires multiple sessions to allow time for students to research the context and develop a purpose prior to completing the analysis.
• Students need to have knowledge about the context and statements that are made need to be related to the context.
• Use of a statistical graphing package is expected.
• Students need to provide evidence of each component of the statistical enquiry cycle detailed in Explanatory Note 3 of the standard.
• The scatter plot needs to be inspected visually before any model is fitted. It is possible a linear model may not be appropriate for the data that is being investigated and this should be
determined by looking at a scatter plot of the raw data.
• Use and interpretation of R2 is not expected at this level.
91582 Use statistical methods to make a formal inference
• The assessment requires multiple sessions to allow time for students to research the context and develop a purpose prior to completing the analysis.
• Students need to have knowledge about the context and statements that are made need to be related to the context.
• Use of a statistical graphing package is expected.
• Students need to provide evidence of each component of the statistical enquiry cycle detailed in Explanatory Note 3 of the standard.
• The analysis will involve students determining if there is a difference between the population medians (or means) and quantifying the difference in the medians (or means) by using the bootstrap
confidence interval.
• A suitable investigative question would be ‘I wonder what the difference in median heights is between NZ year 13 boys and NZ year 13 girls?’
91583 Conduct an experiment to investigate a situation using experimental design principles
• The assessment requires multiple sessions to allow time for students to research the context and develop a purpose prior to completing the analysis.
• Use of a statistical graphing package is expected.
• Students need to provide evidence of each component of the investigation process detailed in Explanatory Note 3 of the standard.
• There needs to be a clear and meaningful purpose for the experiment.
• Students need to identify experimental units, the treatment and the response variable and how the treatment is to be allocated to the experimental units.
• Students need to make a formal statistical inference which will be a causal inference based on the strength of evidence. This will involve the use of randomisation.
• The conclusion needs to be consistent with the analysis and answer the investigative question.
91587 Apply systems of simultaneous equations in solving problems
• To be used as evidence for the award of Achieved a ‘method’ must be relevant to the solution of the problem and at the appropriate curriculum level.
• At all levels there is a requirement relating to the communication of the solutions.
See all Mathematics and Statistics clarifications | {"url":"https://www2.nzqa.govt.nz/ncea/subjects/select-subject/mathematics-and-statistics/clarifications/level-1-achievement-standards-mathematics-and-statistics/","timestamp":"2024-11-08T06:02:31Z","content_type":"text/html","content_length":"62409","record_id":"<urn:uuid:414cbac8-04da-403f-88f6-983a6b8d61d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00656.warc.gz"} |
Use updateBy | Deephaven
Version: Java (Groovy)
This guide will show you how to use the updateBy table operation in your queries. updateBy creates a new table with resultant columns containing aggregated calculations (referred to as
UpdateByOperations) of columns in a source table. The calculations can be cumulative, windowed by rows (ticks), or windowed by time. The calculations are optionally done on a per-group basis, where
groups are defined by one or more key columns.
Available UpdateByOperations
The calculations (UpdateByOperations) that can be performed with updateBy are:
The use of updateBy requires one or more of the calculations in the list above, as well as zero or more key columns to define groups. The resultant table contains all columns from the source table,
as well as new columns if the output of the UpdateByOperation renames them. If no key columns are given, then the calculations are applied to all rows in the specified columns. If one or more key
columns are given, the calculations are applied to each unique group in the key column(s).
Each of the following subsections illustrates how to use updateBy.
A single UpdateByOperation with no grouping columns
The following example calculates the tick-based rolling sum of the X column in the source table. No key columns are provided, so a single group exists that contains all rows of the table.
source = emptyTable(20).update("X = i")
result = source.updateBy(RollingSum(3, 0, "RollingSumX = X"))
Multiple UpdateByOperations with no grouping columns
The following example builds on the previous by performing two UpdateByOperations in a single updateBy. The cumulative minimum and maximum are calculated, and the range is derived from them.
source = emptyTable(20).update("X = randomInt(0, 25)")
result = source.updateBy([CumMin("MinX = X"), CumMax("MaxX = X")]).update("RangeX = MaxX - MinX")
Multiple UpdateByOperations with a single grouping column
The following example builds on the previous by specifying a grouping column. The grouping column is Letter, which contains alternating letters A and B. As a result, the cumulative minimum, maximum,
and range are calculated on a per-letter basis. The result table is split by letter via where to show this.
source = emptyTable(20).update("Letter = (i % 2 == 0) ? `A` : `B`", "X = randomInt(0, 25)")
result = source.updateBy([CumMin("MinX = X"), CumMax("MaxX = X")], "Letter").update("RangeX = MaxX - MinX")
resultA = result.where("Letter == `A`")
resultB = result.where("Letter == `B`")
• source
• result
• resultA
• resultB
A single UpdateByOperation applied to multiple columns with multiple grouping columns
The following example builds on the previous by applying a single UpdateByOperation to multiple columns as well as specifying multiple grouping columns. The grouping columns, Letter and Truth,
contain alternating letters and random true/false values. Thus, groups are defined by unique combinations of letter and boolean. The result table is split by letter and truth value to show the unique
source = emptyTable(20).update("Letter = (i % 2 == 0) ? `A` : `B`", "Truth = randomBool()", "X = randomInt(0, 25)", "Y = randomInt(50, 75)")
rollingSumOps = RollingSum(5, 0, "RollingSumX = X", "RollingSumY = Y")
minOps = CumMin("MinX = X", "MinY = Y")
maxOps = CumMax("MaxX = X", "MaxY = Y")
result = source.updateBy([rollingSumOps, minOps, maxOps], "Letter", "Truth").update("RangeX = MaxX - MinX", "RangeY = MaxY - MinY")
resultATrue = result.where("Letter == `A`", "Truth == true")
resultAFalse = result.where("Letter == `A`", "Truth == false")
resultBTrue = result.where("Letter == `B`", "Truth == true")
resultBFalse = result.where("Letter == `B`", "Truth == false")
• source
• result
• resultATrue
• resultAFalse
• resultBTrue
• resultBFalse
Applying an UpdateByOperation to all columns
The following example uses Fill to fill null values with the most recent previous non-null value. No columns are given to fill, so the forward-fill is applied to all columns in the source table
except for the specified key column(s). This also means that the X column is replaced in the result table by the forward-filled X values.
source = emptyTable(10).update("Letter = (i % 2 == 0) ? `A` : `B`", "X = (i % 3 == 0) ? NULL_INT : i", "Y = (i % 5 == 2) ? i : NULL_INT")
result = source.updateBy(Fill(), "Letter")
Tick-based windowed calculations
There are multiple UpdateByOperations that are windowed by ticks. When an operation is windowed, the window is defined when creating the operation.
For all tick-based windowed calculations, the window size and location relative to the current row are defined by two input parameters: fwdTicks and revTicks. The former defines how far forward the
window goes, whereas the latter defines how far backwards it goes. fwdTicks is inclusive of the current row: fwdTicks = 0 means the window ends at the current row. revTicks is not inclusive of the
current row: revTicks = 1 means the window starts at the current row. Both of these values can be either positive or negative. The bulleted list below gives several examples of these two parameters
and the rolling window they create.
• revTicks = 1, fwdTicks = 0 - Contains only the current row.
• revTicks = 10, fwdTicks = 0 - Contains 9 previous rows and the current row.
• revTicks = 0, fwdTicks = 10 - Contains the following 10 rows; excludes the current row.
• revTicks = 10, fwdTicks = 10 - Contains the previous 9 rows, the current row and the 10 rows following.
• revTicks = 10, fwdTicks = -5 - Contains 5 rows, beginning at 9 rows before, ending at 5 rows before the current row (inclusive).
• revTicks = 11, fwdTicks = -1 - Contains 10 rows, beginning at 10 rows before, ending at 1 row before the current row (inclusive).
• revTicks = -5, fwdTicks = 10 - Contains 5 rows, beginning 5 rows following, ending at 10 rows following the current row (inclusive).
The following example:
• Creates a static source table with two columns.
• Calculates the rolling sum of X grouped by Letter.
□ Three rolling sums are calculated using a window before, containing, and after to the current row.
• Splits the result table by letter via where to show how the windowed calculations are performed on a per-group basis.
source = emptyTable(20).update("X = i", "Letter = (i % 2 == 0) ? `A` : `B`")
opContains = RollingSum(2, 1, "ContainsX = X")
opBefore = RollingSum(3, -1, "PriorX = X")
opAfter = RollingSum(-1, 3, "PosteriorX = X")
result = source.updateBy([opContains, opBefore, opAfter], by="Letter")
resultA = result.where("Letter == `A`")
resultB = result.where("Letter == `B`")
• source
• result
• resultA
• resultB
Time-based windowed calculations
There are multiple UpdateByOperations that are windowed by time. When an operation is windowed, the window is defined when creating the operation. These operations require the source table to contain
a column of DateTimes.
For all time-based windowed calculations, the window size and location relative to the current row are defined by two input parameters: revTime and fwdTime. The former defines how far forward the
window goes, whereas the latter defines how far backwards it goes. These parameters parameter can be given as an long number of nanoseconds or a Duration. The bulleted list below explains how window
sizes vary based on the two parameters:
• revTime = 0, fwdTime = 0 - Contains rows that exactly match the current timestamp.
• revTime = 10 * MINUTE, fwdTime = 0 - Contains rows from 10m earlier through the current timestamp (inclusive).
• revTime = 0, fwdTime = 10 * MINUTE - Contains rows from the current timestamp through 10m following the current row timestamp (inclusive).
• revTime = MINUTE, fwdTime = MINUTE - Contains rows from 1m earlier through 1m following the current timestamp (inclusive).
• revTime = 10 * MINUTE, fwdTime = -5 * MINUTE - Contains rows from 10m earlier through 5m before the current timestamp (inclusive). This is a purely backwards-looking window.
• revTime = -5 * SECOND, fwdTime = 10 * SECOND - Contains rows from 5s following through 10s following the current timestamp (inclusive). This is a purely forwards-looking window.
The following example:
• Creates a static source table with three columns.
• Calculates the rolling sum of X grouped by Letter.
□ Three rolling sums are calculated using a window before, containing, and after the current timestamp.
• Splits the result table by letter via where to show how the windowed calculations are performed on a per-group basis.
baseTime = convertDateTime("2023-01-01T00:00:00 NY")
source = emptyTable(20).update("Timestamp = baseTime + i * SECOND", "X = i", "Letter = (i % 2 == 0) ? `A` : `B`")
opBefore = RollingSum("Timestamp", 3 * SECOND, -1 * SECOND, "PriorX = X")
opContains = RollingSum("Timestamp", SECOND, SECOND, "ContainsX = X")
opAfter = RollingSum("Timestamp", -1 * SECOND, 3 * SECOND, "PosteriorX = X")
result = source.updateBy([opBefore, opContains, opAfter], "Letter")
resultA = result.where("Letter == `A`")
resultB = result.where("Letter == `B`")
Handling erroneous data
It's common for tables to contain null, NaN, or other erroneous values. Certain updateBy operations can be told how to handle these through the use of the OperationControl class.
To see how erroneous data can be handled differently, see the OperationControl reference guide.
Related documentation | {"url":"https://deephaven.io/core/groovy/docs/how-to-guides/use-update-by/","timestamp":"2024-11-08T07:25:21Z","content_type":"text/html","content_length":"110439","record_id":"<urn:uuid:923d3264-3977-4999-b652-3cd50c807b4f>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00071.warc.gz"} |
Demostrations of Law of Conservation of Linear Momentum
Tisdale, Robert W. Percy L. Julian High School
My objectives are to - (1) give a graphic demonstration of what the
momemtum of an object really is; (2) review some of the more commonly
available laboratroy equipment used to study momentum; (3) and produce
as well as use some momentum equipment that is very simple, cheap, and
easy to use.
Apparatus needed:
The apparatus consisted of an air track, air table, momentum machine,
billiard balls, small rubber ball, rope, steel pipe, seltzer water,
rubber stopper, two wheel chairs, and small springs.
Recommended strategy:
The first step is to establish what momentum is. This is done with a
little twist of humor by telling a story of how three little boys
attempt to kill a big ferocious bear by using projectiles of different
size mass and different velocities. Of course only the projectile with
sufficient mass and velocity is able to do the job.
From their we move on to establish what it means to conserve momentum.
We first observe a stationary pipe sealed on one end and with a rubber
stopper on the opposite end. The students are asked to tell what the
momentum of this system is and with no exception all of them say the
momentum is zero. After seltzer water is placed in the pipe the rubber
stopper flies out one way and the pipe another way. The students are
once again asked to give the momentum of this new system and most will
say that the momentum has changed, changed to some nonzero or positive
We then proceed with the concept of positive and negative velocity and
how the sum of the two momentums resulting from this reaction add up to
zero. Hence, momentum is conserved and the law of conservation of
momentum is supported.
From this point we examine some of the equipment that is traditionally
used to study comservation of linear momentum: air tables, air tracks,
and the momentum device. A small demonstration is performed with this
equipment. I also bring out that the equipment is expensive and easily
damaged. I introduce the idea of using simpler materials to illustrate
conservation of momentum.
One piece of equipment consists of two balls with a spring glued to at
least one of the balls. You use both balls to compress the spring
about one meter directly above a point on the floor. Very quickly you
release the balls. Both balls should have equal momentum. If they are
of equal mass then they should have equal velocities which will lead to
to them landing equal horizontal distances from from the point on the
floor. If they are of unequal mass they will still have the same
momentum but the more massive ball will have a smaller velocity the
less massive ball. This results in the more massive ball traveling a
smaller horizontal distance than the less massive ball. The results
for this experiment don't give good quantitative data but the results
do produce qualitative data that is good enough to support the point.
Another demonstration involved three momentum carts of equal size. One
of the carts is securely taped on top of another cart. The result is
two carts, one which is roughly twice the mass of the other cart. A
spring is placed between these two carts to propel them in opposite
directions. Because of the law of conservation of momentum, we can
assume that both carts will be given equal momentum. We can also
assume that the velocity of the lighter cart will be exactly twice the
velocity of the heavier cart. This is the only way two masses, one
which is half the mass of the other, can have equal linear momentum. If
the velocity of small cart is twice that of the larger cart then the
small cart should be able to cover 2 times more distance in any given
period of time. We place a block of wood some arbitrary distance, say
2 meters, from the smaller cart and another block of wood one half of
former distance, 1 meter in this case, from the larger cart. Both
carts should arrive at their finish lines at the same time. It will be
clearly observable to the students when the carts crash at the same
time. This experiment works very well to be so simple in design.
A third experiment involves the use of wheel chairs. Two students, one
much larger than the other, face each other seated in the wheel chairs
about about 15 to 20 feet apart while while holding the opposite ends
of rope. Before they start pulling they have a total momentum of zero.
To maintain a state of zero momentum after they start pulling the
larger student will move at a smaller velocity and, therefore, not
travel as far. The smaller student will move at a much higher velocity
and, consequently, roll for a greater distance. It is important that
the wheels of the wheel chair be aligned correctly before the pulling
starts and that some one be there to catch the students before they
crash in to each other. There are a lot of variations that one can do
with this. Two students of similar size can pull. Students can push
off of each others hands instead of pull. The results will probally
not be good enough to make good quantitative measurements but they will
be quite sufficient for making a qualitative estimate of what should
take place.
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Neutron stars, such as the one at the center of the Crab Nebula,
have about the...
Neutron stars, such as the one at the center of the Crab Nebula, have about the...
Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun but a much smaller diameter.
If you weigh 690 N on the earth, what would be your weight on the surface of a neutron star that has the same mass as our sun and a diameter of 21.0 km ?
Take the mass of the sun to be ms = 1.99×10^30 kg , the gravitational constant to be G = 6.67×10^−11 N⋅m2/kg2 , and the acceleration due to gravity at the earth's surface to be g = 9.810 m/s2 .
Express your weight wstar in newtons.
weight on earth
W = m g
m = W / g
= 690 / 9.8
= 70.408 kg
if the person with mass m = 70.408 is om the surface of the nuetron star then the gravitational force of attraction between person and the nuetron star will be
Fg = G m[1] m[2] / r^2
G = 6.67×10^−11 N⋅m2 / kg^2
m[1] = 70.408 kg,
m[2] = 1.99×10^30 kg
r = 10.5 km
Fg = = m[1] g
m[1] g = G m[1] m[2] / r^2
the acceleration on the surface of neutron star is
g = G m[2] / r^2
= (6.67×10^−11) (1.99×10^30) / (10.5 x 10^3)^2
^ = ......... m / s2
the weight on the neutron star will be
W = (m[2]) g
= ......... N | {"url":"https://justaaa.com/physics/1158743-neutron-stars-such-as-one-at-center-of","timestamp":"2024-11-06T14:11:55Z","content_type":"text/html","content_length":"42362","record_id":"<urn:uuid:bf09a3ae-2193-46b0-a70a-c321f65d926c>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00340.warc.gz"} |
Ta. If transmitted and non-transmitted genotypes would be the similar, the individual | Caspase1-Inhibitors
Ta. If transmitted and non-transmitted genotypes will be the identical, the individual is uninformative as well as the score sij is 0, otherwise the transmitted and non-transmitted contribute tijA
roadmap to multifactor dimensionality reduction strategies|Aggregation on the elements of the score vector gives a prediction score per person. The sum over all prediction scores of individuals
having a particular element combination compared using a threshold T determines the label of every multifactor cell.solutions or by bootstrapping, therefore providing evidence for any genuinely low-
or high-risk aspect mixture. Significance of a model nevertheless might be assessed by a permutation method based on CVC. Optimal MDR Another method, named optimal MDR (Opt-MDR), was proposed by Hua
et al. [42]. Their technique utilizes a data-driven instead of a fixed threshold to collapse the aspect combinations. This threshold is chosen to maximize the v2 values among all doable two ?2
(case-control igh-low danger) tables for each and every element combination. The exhaustive look for the maximum v2 values can be done efficiently by sorting aspect combinations based on the
ascending danger ratio and collapsing successive ones only. d Q This reduces the search space from 2 i? possible 2 ?2 tables Q to d li ?1. Additionally, the CVC permutation-based estimation i? of
your P-value is replaced by an approximated P-value from a generalized extreme worth distribution (EVD), related to an approach by Pattin et al. [65] described later. MDR stratified populations
Significance estimation by generalized EVD is also made use of by Niu et al. [43] in their method to handle for population stratification in case-control and continuous traits, namely, MDR for
stratified populations (MDR-SP). MDR-SP makes use of a set of unlinked markers to calculate the principal elements that happen to be regarded as the genetic background of samples. Primarily based on
the very first K principal components, the residuals with the trait worth (y?) and i ASP2215 chemical information genotype (x?) with the samples are calculated by linear regression, ij hence
adjusting for population stratification. Thus, the adjustment in MDR-SP is employed in every multi-locus cell. Then the test statistic Tj2 per cell is definitely the correlation amongst the adjusted
trait worth and genotype. If Tj2 > 0, the corresponding cell is labeled as high danger, jir.2014.0227 or as low threat otherwise. Primarily based on this labeling, the trait value for each and every
sample is predicted ^ (y i ) for just about every sample. The instruction error, defined as ??P ?? P ?2 ^ = i in education data set y?, 10508619.2011.638589 is utilized to i in education data set y i
?yi i recognize the top d-marker model; particularly, the model with ?? P ^ the smallest average PE, defined as i in testing GS-9973 biological activity information set y i ?y?= i P ?two i in testing
information set i ?in CV, is selected as final model with its typical PE as test statistic. Pair-wise MDR In high-dimensional (d > 2?contingency tables, the original MDR process suffers in the
scenario of sparse cells that are not classifiable. The pair-wise MDR (PWMDR) proposed by He et al. [44] models the interaction in between d variables by ?d ?two2 dimensional interactions. The cells
in just about every two-dimensional contingency table are labeled as high or low risk depending around the case-control ratio. For every single sample, a cumulative risk score is calculated as number
of high-risk cells minus variety of lowrisk cells over all two-dimensional contingency tables. Below the null hypothesis of no association among the selected SNPs plus the trait, a symmetric
distribution of cumulative risk scores around zero is expecte.Ta. If transmitted and non-transmitted genotypes will be the exact same, the person is uninformative and also the score sij is 0,
otherwise the transmitted and non-transmitted contribute tijA roadmap to multifactor dimensionality reduction approaches|Aggregation in the components in the score vector offers a prediction score
per person. The sum more than all prediction scores of men and women using a certain aspect mixture compared using a threshold T determines the label of every single multifactor cell.solutions or by
bootstrapping, hence giving proof for any genuinely low- or high-risk element combination. Significance of a model still can be assessed by a permutation approach primarily based on CVC. Optimal MDR
One more method, referred to as optimal MDR (Opt-MDR), was proposed by Hua et al. [42]. Their technique utilizes a data-driven as opposed to a fixed threshold to collapse the issue combinations. This
threshold is chosen to maximize the v2 values among all doable two ?two (case-control igh-low danger) tables for every single aspect combination. The exhaustive look for the maximum v2 values might
be completed effectively by sorting element combinations in accordance with the ascending threat ratio and collapsing successive ones only. d Q This reduces the search space from two i? attainable
two ?2 tables Q to d li ?1. Additionally, the CVC permutation-based estimation i? of your P-value is replaced by an approximated P-value from a generalized extreme value distribution (EVD), related
to an approach by Pattin et al. [65] described later. MDR stratified populations Significance estimation by generalized EVD can also be utilized by Niu et al. [43] in their method to handle for
population stratification in case-control and continuous traits, namely, MDR for stratified populations (MDR-SP). MDR-SP utilizes a set of unlinked markers to calculate the principal elements which
can be viewed as because the genetic background of samples. Based around the first K principal components, the residuals of the trait value (y?) and i genotype (x?) on the samples are calculated by
linear regression, ij hence adjusting for population stratification. Thus, the adjustment in MDR-SP is used in every multi-locus cell. Then the test statistic Tj2 per cell could be the correlation in
between the adjusted trait worth and genotype. If Tj2 > 0, the corresponding cell is labeled as high threat, jir.2014.0227 or as low threat otherwise. Based on this labeling, the trait worth for
every sample is predicted ^ (y i ) for just about every sample. The education error, defined as ??P ?? P ?two ^ = i in education information set y?, 10508619.2011.638589 is utilised to i in coaching
information set y i ?yi i identify the ideal d-marker model; particularly, the model with ?? P ^ the smallest typical PE, defined as i in testing information set y i ?y?= i P ?two i in testing
information set i ?in CV, is selected as final model with its typical PE as test statistic. Pair-wise MDR In high-dimensional (d > 2?contingency tables, the original MDR method suffers inside the
situation of sparse cells that are not classifiable. The pair-wise MDR (PWMDR) proposed by He et al. [44] models the interaction involving d components by ?d ?two2 dimensional interactions. The cells
in every two-dimensional contingency table are labeled as high or low threat based around the case-control ratio. For each sample, a cumulative threat score is calculated as quantity of high-risk
cells minus quantity of lowrisk cells more than all two-dimensional contingency tables. Below the null hypothesis of no association involving the chosen SNPs along with the trait, a symmetric
distribution of cumulative risk scores about zero is expecte. | {"url":"https://www.caspase1inhibitor.com/2017/10/13/ta-if-transmitted-and-non-transmitted-genotypes-would-be-the-similar-the-individual/","timestamp":"2024-11-09T06:25:17Z","content_type":"text/html","content_length":"78222","record_id":"<urn:uuid:77a4f6fa-7299-4bed-83b2-76e7836fb20b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00137.warc.gz"} |
Useful definitions
Confidence intervals
When making estimates about a whole population based on a sample of that population, there is a margin of error around those estimates. Confidence Intervals (C.I.) are used to give an indication of
the likely size of this margin. The smaller the sample size and the greater the degree of variation, the wider the C.I. will be.
Confidence intervals are used as a clear and simple method of determining whether the difference between two groups is “statistically significant”, i.e. whether there is sufficient evidence to
suggest that it reflects a real difference in the wider population. If the confidence intervals of the two comparative sets of data overlap, then the difference is not statistically significant (e.g.
Chart A, below). If the confidence intervals do not overlap, then the difference is statistically significant (e.g. Chart B, below).
Depending on the situation, Confidence Intervals may be explicit as in the charts above, or simply be reflected in the commentary, i.e. differences may be highlighted if they are statistically
significant and apparent differences not mentioned if they are not statistically significant.
Medians and quartiles
The median for a dataset is the value such that 50% of the data points are lower and 50% of the data points are higher when all data points are arranged in order of size. It is an overall summary
measure that is less affected by the presence of extreme values (outliers) than the mean (which is another type of average summary measure calculated by summing all the data points and dividing by
the number of them).
Further explanation of the use of the median and the mean can be found in the resource box.
The lower quartile for a dataset is the value such that 25% of the data is lower and 75% of the data is higher. The upper quartile for a dataset is the value such that 75% of the data is lower and
25% of the data is higher. The term "quartile" is also used to refer to a range bounded by the quartile values. For example, saying that a score lies "in the upper quartile" really means it lies in a
range bounded by the upper quartile value and the highest score achieved. Saying it lies "in the second quartile" really means it lies in a range bounded by the median and the upper quartile value.
The "inter-quartile range" is the range of values bounded by the upper and lower quartiles.
Quartiles and medians have been used for two specific purposes:
• To compare earnings in Herefordshire with those elsewhere at three different points on the earnings distribution e.g. low earners (25% lower quartile), average earners (median) and high earners
(25% upper decile).
• To compare performance in Herefordshire with all English Authorities. Here the terms top quartile and bottom quartile have been used.
□ If high values of a performance indicator are desired:
the top quartile of authorities equates to the upper quartile of the dataset
the bottom quartile of authorities equates to the lower quartile of the dataset.
□ If low values of a performance indicator are desired:
the top quartile of authorities equates to the lower quartile of the dataset
the bottom of authorities quartile equates to the upper quartile of the dataset.
□ In some cases, the terms "best" and "worst" are used in preference to "upper / lower" or "top / bottom", to avoid confusion.
Understanding sub-county geographies
There are two key things to remember about statistics for areas smaller than counties:
1. the difference between statistical geographies and administrative geographies, and
2. how different areas ‘nest’ into each other (the geographical hierarchy)
Administrative geographies
Geographies that are used for administrative purposes are not always ideal for publishing statistics: they can vary substantially in population size meaning that comparisons aren’t really appropriate
(e.g. wards), or can even be so small that publishing statistics could risk identifying an individual (e.g. parishes). They can also change significantly over time.
Despite these disadvantages, administrative geographies are often well-known and meaningful to people, and statistics are needed for people and organisations to understand the characteristics of
their areas. Commonly used administrative geographies in Herefordshire are parishes, wards and localities. Statistics can also be produced for Hereford city and the five market towns.
Another complexity of administrative geographies is their hierarchy. Civil parishes are generally the smallest building block in rural areas and these ‘nest’ into group parishes (in some cases) and
then wards. However, in larger urban areas (Hereford, Leominster and Ross) wards can be smaller than parishes, and they don’t always nest perfectly. Localities are built from group parishes, and
contain more than one ward – but some wards cross locality boundaries.
Statistical geographies
In 2004, the Office for National Statistics (ONS) designed a standard set of statistical geographies to avoid the problems associated with administrative geographies. Using the results of the 2001
Census, they were designed to have similar population sizes and, to a certain extent, similar characteristics. They also ‘nest’ into a clear hierarchy: output areas (OAs) have around 300 people, on
average, that group together to form lower super output areas (LSOAs) of 1,600 people on average, which in turn group to form middle super output areas (MSOAs) of 8,000 people on average. MSOAs do
not cross county boundaries.
More detail about the statistical geographies can be found on the Office for National Statistics' Census geography page.
One of the main principles for the statistical geography hierarchy is stability over time, meaning that the results of the 2001, 2011 and 2021 Censuses were based on the same statistical geographies.
Some changes (less than 3% of OAs in Herefordshire) occurred for release of the 2021 Census data, that involved either the merging or splitting of the existing OAs to ensure that population and
household thresholds were met -18 of Herefordshire’s 640 2011 OAs have been split into two or more 2021 OAs and 8 have been merged with another one or more. The only LSOA change is the dividing of
the ‘Holmer, Pipe & Lyde’ LSOA into two - a rural and an urban component.
The relationship between administrative and statistical geographies in Herefordshire
Following changes that were made to ward boundaries in 2015, they no longer align to OAs and LSOAs. This means that statistics provided for administrative geographies will need to be ‘best-fitted’
from statistical geographies.
Rural urban classifications
Rural and urban classifications are the product of a project commissioned jointly by the Office for National Statistics (ONS), the Department for Environment, Food and Rural Affairs (Defra), the
Office of the Deputy Prime Minister (ODPM), the Countryside Agency (CA) and the Welsh Assembly Government to create a new settlement-based definition of urban and rural areas.
In 2004, classifications were published for all output areas, lower super output areas (LSOAs) and wards in England & Wales by settlement form and sparsity. The categories used are:
□ Rural town (town and fringe)
□ Rural village
□ Rural dispersed (hamlets and isolated dwellings)
Each of these categories is then divided into "sparse" and "less sparse". Settlements of 10,000 people or more are considered urban. Hereford, Leominster and Ross-on-Wye fall into this category. The
remaining market towns (Kington, Bromyard and Ledbury) are classified as "rural towns".
Following the initial classification of output areas, a classification of local authority areas was initiated by Defra. This classified local authorities as either major urban, large urban, other
urban, significant rural, rural-50 or rural-80. Herefordshire is classified as rural-50; between 50% and 80% of the local authority's population live in a rural settlement.
The rural nature of Herefordshire can be seen in the map below. You may notice a rogue urban area just north-east of Ledbury - it seems likely that when classifications were made, the Malvern Hills
were not taken into account, so villages in the Colwall area may have been seen as suburbs of Malvern.
Map 1. Rurality of Herefordshire at Output Area level | {"url":"https://understanding.herefordshire.gov.uk/useful-definitions/","timestamp":"2024-11-03T16:12:38Z","content_type":"text/html","content_length":"31695","record_id":"<urn:uuid:04fa82d3-aac1-4c17-96cb-c591ebc3a4e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00037.warc.gz"} |
Transactions Online
Takaaki MIZUKI, Zhi-Bo SUI, Hiroki SHIZUYA, Takao NISHIZEKI, "On the Average Length of Secret Key Exchange Eulerian Circuits" in IEICE TRANSACTIONS on Fundamentals, vol. E83-A, no. 4, pp. 662-670,
April 2000, doi: .
Abstract: Designing a protocol to exchange a secret key is one of the most fundamental subjects in cryptography. Using a random deal of cards, pairs of card players (agents) can share secret keys
that are information-theoretically secure against an eavesdropper. A key set protocol, which uses a random deal of cards, can perform an Eulerian secret key exchange, in which the pairs of players
sharing secret keys form an Eulerian circuit passing through all players. Along the Eulerian circuit any designated player can send a message to the rest of players and the message can be finally
sent back to the sender. Checking the returned message with the original one, the sender can know whether the message circulation has not been influenced by a possible single transmission error or
false alteration. It has been known that any Eulerian circuit formed by the protocol has length at most 3/2k, where k is the number of players. Note that the length corresponds to the time required
to send the message to all players and acknowledge the secure receipt. In this paper, we show that the average length of Eulerian circuits is approximately k+ln k.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e83-a_4_662/_p
author={Takaaki MIZUKI, Zhi-Bo SUI, Hiroki SHIZUYA, Takao NISHIZEKI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={On the Average Length of Secret Key Exchange Eulerian Circuits},
abstract={Designing a protocol to exchange a secret key is one of the most fundamental subjects in cryptography. Using a random deal of cards, pairs of card players (agents) can share secret keys
that are information-theoretically secure against an eavesdropper. A key set protocol, which uses a random deal of cards, can perform an Eulerian secret key exchange, in which the pairs of players
sharing secret keys form an Eulerian circuit passing through all players. Along the Eulerian circuit any designated player can send a message to the rest of players and the message can be finally
sent back to the sender. Checking the returned message with the original one, the sender can know whether the message circulation has not been influenced by a possible single transmission error or
false alteration. It has been known that any Eulerian circuit formed by the protocol has length at most 3/2k, where k is the number of players. Note that the length corresponds to the time required
to send the message to all players and acknowledge the secure receipt. In this paper, we show that the average length of Eulerian circuits is approximately k+ln k.},
TY - JOUR
TI - On the Average Length of Secret Key Exchange Eulerian Circuits
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 662
EP - 670
AU - Takaaki MIZUKI
AU - Zhi-Bo SUI
AU - Hiroki SHIZUYA
AU - Takao NISHIZEKI
PY - 2000
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E83-A
IS - 4
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - April 2000
AB - Designing a protocol to exchange a secret key is one of the most fundamental subjects in cryptography. Using a random deal of cards, pairs of card players (agents) can share secret keys that are
information-theoretically secure against an eavesdropper. A key set protocol, which uses a random deal of cards, can perform an Eulerian secret key exchange, in which the pairs of players sharing
secret keys form an Eulerian circuit passing through all players. Along the Eulerian circuit any designated player can send a message to the rest of players and the message can be finally sent back
to the sender. Checking the returned message with the original one, the sender can know whether the message circulation has not been influenced by a possible single transmission error or false
alteration. It has been known that any Eulerian circuit formed by the protocol has length at most 3/2k, where k is the number of players. Note that the length corresponds to the time required to send
the message to all players and acknowledge the secure receipt. In this paper, we show that the average length of Eulerian circuits is approximately k+ln k.
ER - | {"url":"https://global.ieice.org/en_transactions/fundamentals/10.1587/e83-a_4_662/_p","timestamp":"2024-11-07T15:18:13Z","content_type":"text/html","content_length":"62823","record_id":"<urn:uuid:a257c9c2-3c93-428b-a4ab-61eaa88619bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00825.warc.gz"} |
Lambda Expressions: Scope of Variables, Method, and Anonymous Functions
In this lesson, you will learn about the scope of variables inside lambda expressions.
This lesson introduces the basic lambda expression structure with tips to write efficient code.
This is an introductory lesson on lambda expressions. You will learn about the lambda operator, expression, syntaxes and more! | {"url":"https://www.ggorantala.dev/lambda-expressions-scope-of-variables-methods-and-anonymous-functions/","timestamp":"2024-11-13T15:36:21Z","content_type":"text/html","content_length":"140170","record_id":"<urn:uuid:61a768e8-423a-4757-bb65-7128bea4e0f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00839.warc.gz"} |
Asmita T.
What do you want to work on?
About Asmita T.
Algebra, Algebra 2, Discrete Mathematics, Geometry
Math - Algebra II
Math - Geometry
Impressed with the patience and how she said excellent and great every time I get the correct answer.
Math - Geometry
Math - Discrete Mathematics
Asmita was super helpful in explaining the problems to me, thank you! | {"url":"https://origin-www.princetonreview.com/academic-tutoring/tutor/asmita%20t--6316857","timestamp":"2024-11-05T00:22:25Z","content_type":"application/xhtml+xml","content_length":"243337","record_id":"<urn:uuid:ba6c5301-817c-44a6-85ff-141e77edf60b>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00254.warc.gz"} |
Install from CRAN or GitHub using:
pammtools facilitates the estimation of Piece-wise exponential Additive Mixed Models (PAMMs) for time-to-event data. PAMMs can be represented as generalized additive models and can therefore be
estimated using GAM software (e.g. mgcv), which, compared to other packages for survival analysis, often offers more flexibility w.r.t. to the specification of covariate effects (e.g. non-linear,
time-varying effects, cumulative effects, etc.).
To get started, see the Articles section. | {"url":"https://cran.radicaldevelop.com/web/packages/pammtools/readme/README.html","timestamp":"2024-11-08T21:45:00Z","content_type":"application/xhtml+xml","content_length":"7433","record_id":"<urn:uuid:b1cb3b5a-a5b2-4864-ace5-d30a8047945f>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00859.warc.gz"} |
What Does A Negative T-Value Mean?
Researchers and scientists often use statistical tests called t-tests to assess whether two groups of data differ from each another. A t-test compares the means of each group and takes into account
the numbers on which the means are based to determine the amount of data overlap between the two groups. The test also tells you how significant the differences are between the two groups and reveals
whether those differences could have happened by chance or are statistically significant.
TL;DR (Too Long; Didn't Read)
In statistics, t-tests are used to compare the means of two groups. Although a negative t-value shows a reversal in the directionality of the effect being studied, it has no impact on the
significance of the difference between groups of data.
T-Test Types
The three main types of t-test are independent sample t-test, paired sample t-test, and one sample t-test. An independent samples t-test compares the means for two groups. A paired sample t-test
compares means from the same group at different times – one year apart, for example. A one sample t-test tests the mean of a single group against a known mean.
T-Score Basics
The t-score is a ratio of the difference between two groups and the difference within the groups. The larger the t-score, the more difference there is between groups. The smaller the t-score, the
more similarity there is between groups. For example, a t-score of 3 means that the groups are three times as different from each other as they are within each other. When you run a t-test, the
bigger the t-value, the more likely it is that the results are repeatable.
In simple terms, a large t-score tells you that the groups are different, and a small t-score tells you the groups are similar.
Calculating Difference
Calculating difference between group means involves subtracting one mean from the other.
Calculate the standard error of difference (also known as variability) by subtracting the mean of one group from a unique sample in that same group, squaring that value, and dividing the value by the
total number of samples in the group minus 1. Perform this calculation for each unique sample and then add all the values together.
Negative T-Value
Find a t-value by dividing the difference between group means by the standard error of difference between the groups.
A negative t-value indicates a reversal in the directionality of the effect, which has no bearing on the significance of the difference between groups. Analysis of a negative t-value requires
examination of its absolute value in comparison to the value on a table of t-values and degrees of freedom, which quantifies the variability of the final estimated number. If the absolute value of
the experimental t-value is smaller than the value found on the degrees of freedom chart, then the means of the two groups can be said to be significantly different.
Cite This Article
Gillespie, Claire. "What Does A Negative T-Value Mean?" sciencing.com, https://www.sciencing.com/negative-tvalue-mean-6921215/. 4 June 2018.
Gillespie, Claire. (2018, June 4). What Does A Negative T-Value Mean?. sciencing.com. Retrieved from https://www.sciencing.com/negative-tvalue-mean-6921215/
Gillespie, Claire. What Does A Negative T-Value Mean? last modified March 24, 2022. https://www.sciencing.com/negative-tvalue-mean-6921215/ | {"url":"https://www.sciencing.com:443/negative-tvalue-mean-6921215/","timestamp":"2024-11-07T20:27:21Z","content_type":"application/xhtml+xml","content_length":"72663","record_id":"<urn:uuid:ddbfe8de-d40c-4959-b6f2-5a29f1dc2b39>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00170.warc.gz"} |
Multivariable Calculus (1st order approximation)
• Thread starter plzen90
• Start date
In summary: As h goes to 0, all the terms with δ, ε, and θ go to 0. Therefore, the numerator becomes 0, and the denominator becomes 1. So the limit is equal to 0/1 = 0. This shows that as h goes to
0, the error e(h) goes to 0 faster than h, which is what we expect from a first order approximation.
Homework Statement
Define f:R^2→R^3 by
let p = (1,1,1)^T and h=(δ,ε,θ)
a)what are n and m? evaluate f(p) and f(p+h)
b)Calculate the Jacobian Matrix Df(x,y,z) and evaluate Df(p)
c) Calculate the error e(h) in the first order approximation to f(p+h)
d) show clearly that
lim h→0 |e(h)| =0
Explain why this is what you expect
Homework Equations
The Attempt at a Solution
f(p)=(1,1,1) = (2)
f(p+h) = f(1+δ, 1+ε, 1+θ)
jac= (y x 1)
...(2x -z -y)
Df(p)=(1 1 1)
...(2 -1 -1)
only calculation of Df(p)h needed to work out error.
e(h)=f(p+h)-(f(p) + Df(p)h)
(not confident on this)
d)not attempted yet/ don't know how to
show it clearly.
The first part of your solution looks correct. The Jacobian matrix is a 3x2 matrix, and the derivative at point p is a 1x2 matrix.
For part c), you are correct in saying that the error is equal to f(p+h) - (f(p) + Df(p)h). However, the calculation of Df(p)h is incorrect. It should be:
Df(p)h = (y+x+1)δ + (2x-z-y)ε
(y+x+1)ε + (2x-z-y)θ
Substituting in the values for p and h, we get:
Df(p)h = (2+1)δ + (2-1-1)ε
(2+1)ε + (2-1-1)θ
= 3δ + 0ε
3ε + 0θ
= (3δ, 3ε)
Therefore, the error e(h) becomes:
e(h) = (δε+θ-ε)
- (2+0+0)
= (δε+θ-ε-2)
For part d), you can show that the limit as h goes to 0 of e(h)/|h| is equal to 0 by simplifying the expression:
lim h→0 |e(h)|/|h| = lim h→0 ((δε+θ-ε-2)^2 + (δ2-εθ)^2)/(|δ|+|ε|+|θ|)
As h goes to 0, all the terms with δ, ε, and θ go to 0 as well. Therefore, the numerator becomes 0, and the denominator becomes 0+0+0 = 0. This means that the limit is equal to 0/0, which is an
indeterminate form. To evaluate this limit, we can use L'Hopital's rule, which states that if we have a limit of the form 0/0, we can take the derivative of the numerator and denominator separately
and evaluate the limit again. Applying this rule, we get:
lim h→0 |e(h)|/|h| = lim h→0 (2
FAQ: Multivariable Calculus (1st order approximation)
What is multivariable calculus?
Multivariable calculus is a branch of mathematics that deals with functions of multiple variables, usually in three-dimensional space. It involves the study of limits, derivatives, and integrals of
functions with more than one independent variable.
What is first order approximation in multivariable calculus?
First order approximation, also known as linear approximation, is a method used to approximate a function with a simpler, linear function. It involves finding the tangent plane to the function at a
specific point and using that to estimate the function's value near that point.
Why is first order approximation useful in multivariable calculus?
First order approximation is useful because it allows us to approximate a complex function with a simpler one, making it easier to solve and analyze. It is also used in optimization problems to find
the minimum or maximum value of a function.
What is the formula for first order approximation?
The formula for first order approximation is f(x,y) ≈ f(x0,y0) + (fx(x0,y0)(x-x0)) + (fy(x0,y0)(y-y0)), where (x0,y0) is the point at which the approximation is being made, fx and fy are the partial
derivatives of the function at that point with respect to x and y, and (x,y) are the coordinates of the point being approximated.
What are some real-world applications of first order approximation in multivariable calculus?
First order approximation has many practical applications, including in physics, engineering, economics, and computer graphics. It is used to approximate the trajectory of a projectile, optimize the
design of a bridge, model supply and demand in a market, and create computer-generated imagery in movies and video games. | {"url":"https://www.physicsforums.com/threads/multivariable-calculus-1st-order-approximation.588883/","timestamp":"2024-11-02T02:19:09Z","content_type":"text/html","content_length":"80465","record_id":"<urn:uuid:745f68ad-f5d0-4e26-9f27-b36444dbdfaa>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00789.warc.gz"} |
Learn fibonacci number sequence and how to use them
There are many methods and tools in technical analysis. The Fibonacci number sequence can be mentioned as one of the popular tools often used by investors. This article will provide an overview of
the Fibonacci number sequence, how to apply it, and what to keep in mind when using it in stock technical analysis. Let’s explore details with Forex Trading in the article below.
Definition of Fibonacci number sequence
What is Fibonacci? The Fibonacci number sequence is a tool in technical analysis, originating from the mathematical theory of Leonardo Fibonacci from the 12th century. The Fibonacci number sequence
starts from 0, 1 and subsequent numbers are the sum of the previous two numbers. there. The numbers in the Fibonacci sequence are divided by each other to create ratios such as 161.8%, 23.6%, 28.2%,
and 61.8%. These ratios are considered important levels in trading and technical analysis, with the three main levels being 23.6%, 38.2%, and 61.8%.
What is Fibonacci?
See more: Profession forex trade thanks to Fibonacci applying
What are the Fibonacci types and how to apply them
There are three common types of Fibonacci number sequences :
• Fibonacci Retracement is also known as Fibonacci Retracement.
• Fibonacci Fan is called Fibonacci fan.
• Fibonacci Time Zones means Fibonacci time zones.
In addition, there are some other types such as extended Fibonacci, arc Fibonacci, and spiral Fibonacci,…
Fibonacci retracement
The Fibonacci retracement sequence defines the main levels as follows: 0 – 23.6 – 38.2 – 50 – 61.8 – 76.4 – 100. These levels are symmetrical to each other, adding up to 100. For example: 38.2 + 61.8
= 100 This tool is often used in Elliott wave theory and Gartley patterns. After fluctuations, resistance, and support levels often appear near or coincide with important levels such as 23.6%, 38.2%,
50%, 61.8%, and 100% of Fibonacci retracements.
Fibonacci Retracement
Fibonacci retracements help identify important reversal points. Where the market can end the adjustment process or begin a technical recovery period. The price reaction at each level often reflects
the importance of that Fibonacci level.
Fibonacci series fan
This is an effective Fibonacci tool to identify resistance and support levels in the market. Fibonacci Fans often use popular levels such as 38.2%, 50%, and 61.8%.
Fibonacci fan
Fibonacci Fan is divided into two parts: the uptrend and downtrend of the market. The specific process is as follows:
• In an uptrend: The Fibonacci Fan line is drawn from the bottom to the retracement levels based on the gain. Investors use these lines to predict support levels and potential reversal points.
• In a downtrend: The Fibonacci Fan line is drawn from the peak down to the retracement levels based on the decline. These lines are used to predict resistance levels and potential reversal zones.
Fibonacci number sequence time range
What is Fibonacci in a time zone sequence? This is not a tool to determine resistance or support. Instead, it focuses on predicting price reversal times based on the current trend.
The accuracy of the Fibonacci time zone can be evaluated based on the number of correct Fibonacci lines in the past. This helps create confidence in the timing of the expected reversal.
Fibonacci Time Zones
Vertical Fibonacci lines correspond to the time axis of the price chart, based on the Fibonacci ratio. There are two types of Fibonacci time zones: regular Fibonacci time zones and trend-based
Fibonacci time zones. Both tools are calculated according to the same principles. However, the trend-based Fibonacci time zone uses the previous trend for calculation. This can yield more objective
It should be noted that the Fibonacci time zone may not be accurate in predicting the specific location of reversal points. However, they do provide information about important price zones that
traders need to pay attention to.
Example of Fibonacci number sequence in Forex
A simple way to draw Fibonacci: choose the top of the price increase as the starting point, then choose the bottom of the increase as the ending point. Fibonacci ratio levels will be displayed on the
chart for investors to follow. When prices enter potential reversal zones, it does not mean that prices will immediately reverse. These are just areas that investors need to observe. At these points,
combined with technical analysis tools, investors will know when to buy in the short term.
Example of the Fibonacci sequence
For example, when the price corrects to the Fibonacci number sequence 61.8% and accumulates in this area, then MACD crosses above the signal line. RSI crossed above 50, and open volume increased,
this is a short-term buy point.
See more: Broker XM: Explore the world of Forex fingertips
Meaning of Fibonacci in Stocks
So what is the meaning of Fibonacci? Fibonacci number sequence plays an important role in price fluctuations in the stock market. Like how this concept is applied and used to identify key points in
the price movement cycle.
Among the Fibonacci tools used in stocks, Fibonacci Retracement is the most popular. Partly because of its simplicity and flexible application to all types of assets. This tool can be used in
technical analysis to identify resistance levels. Also supports, setting stop loss points or determining price target points.
Limitations of Fibonacci numbers
Fibonacci does not guarantee that the price will always react positively to the support or resistance points it identifies. This is why investors often combine the Fibonacci sequence with other
technical indicators.
With so many price resistance lines, Fibonacci creates continuous fluctuations between these levels. This makes determining the resistance threshold difficult for investors.
Some notes when trading with the Fibonacci sequence
• Just like other technical analysis indicators, using Fibonacci numbers does not always yield accurate results, although important thresholds can increase the chances of successful predictions.
• Choosing a period and determining the bottom and top of the market to apply the Fibonacci line is not a simple task. Each investor has his own opinion and perspective on stock analysis.
• Investors need to practice their skills and gain a deep understanding of how to use Fibonacci to be able to apply this tool fluently and learn valuable trading lessons for themselves.
• Fibonacci is an effective analysis tool. However, to increase investment efficiency, investors should combine it with other tools such as trendline, support, resistance, and Elliott wave.
To summarize, the Fibonacci number sequence plays an important role in the technical analysis of the Forex market. It assists investors in identifying selling points during downtrends. Buy points in
uptrends and look for ideal profit-taking points. However, to use this tool most effectively, investors need to accumulate more experience in analyzing technical charts. Grasp market trends and
combine them with other forecasting methods. If you want to learn more about Forex, keep following Forex Trading!
What is a Fibonacci spiral?
Fibonacci numbers form a series of spirals, called a Fibonacci spiral. Known as the “golden spiral”, they have a constant angle close to the golden ratio. The number of spirals in a pinecone or the
number of petals per layer of a flower also follows Fibonacci numbers. In plants, each leaf grows at an angle to the previous leaf. And sunflower seeds are arranged in a spiral structure in the
center of the flower. They are all influenced by the golden ratio.
Where is the Fibonacci sequence clearly shown?
In most flowering plants, the number of petals per flower follows the Fibonacci numbers. Rarely, the number of petals does not comply with this rule.
How can the Fibonacci sequence affect trading?
People often identify patterns and compare them on charts using the Fibonacci sequence. Although there is no clear evidence of a link between Fibonacci numbers and fundamental market forces. The
market often reacts according to investors’ beliefs. Therefore, using Fibonacci analysis can create self-fulfilling predictions. And this also affects the market trend. | {"url":"https://forex-trading.support/fibonacci-number-sequence/","timestamp":"2024-11-11T13:47:48Z","content_type":"text/html","content_length":"186900","record_id":"<urn:uuid:181c02a2-5bec-4ab1-94b8-ca9569a06ebf>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00727.warc.gz"} |